BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 CHAPTER 21 SEISMIC DESIGN OF CONCRETE BRIDGES TABLE OF CONTENTS 21.1 INTRODUCTION .......................................................................................................... 21-1 21.2 DESIGN CONSIDERATIONS ...................................................................................... 21-1 21.3 21.2.1 Preliminary Member and Reinforcement Sizes ........................................... 21-1 21.2.2 Minimum Local Displacement Ductility Capacity ...................................... 21-7 21.2.3 Displacement Ductility Demand Requirements .......................................... 21-9 21.2.4 Displacement Capacity Evaluation............................................................ 21-12 21.2.5 P- Effects ............................................................................................... 21-15 21.2.6 Minimum Lateral Strength ........................................................................ 21-15 21.2.7 Column Shear Design ................................................................................ 21-16 21.2.8 Bent Cap Flexural and Shear Capacity ...................................................... 21-18 21.2.9 Seismic Strength of Concrete Bridge Superstructures .............................. 21-18 21.2.10 Joint Shear Design ..................................................................................... 21-26 21.2.11 Torsional Capacity .................................................................................... 21-34 21.2.12 Abutment Seat Width Requirements ......................................................... 21-34 21.2.13 Hinge Seat Width Requirements ............................................................... 21-35 21.2.14 Abutment Shear Key Design ..................................................................... 21-36 21.2.15 No-Splice Zone Requirements .................................................................. 21-38 21.2.16 Seismic Design Procedure Flowchart ........................................................ 21-39 DESIGN EXAMPLE - THREE-SPAN CONTINUOUS CAST-IN-PLACE CONCRETE BOX GIRDER BRIDGE ...................................................................... 21-43 21.3.1 Bridge Data................................................................................................ 21-43 21.3.2 Design Requirements................................................................................. 21-43 21.3.3 Step 1- Select Column Size, Column Reinforcement, and Bent Cap Width ......................................................................................... 21-46 21.3.4 Step 2 - Perform Cross-section Analysis ................................................... 21-47 21.3.5 Step 3 - Check Span Configuration/Balanced Stiffness ............................ 21-48 21.3.6 Step 4 - Check Frame Geometry ............................................................... 21-49 Chapter 21 – Seismic Design of Concrete Bridges 21-i BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 21.3.7 Step 5 – Calculate Minimum Local Displacement Ductility Capacity and Demand .............................................................................................. 21-49 21.3.8 Step 6 – Perform Transverse Pushover Analysis....................................... 21-51 21.3.9 Step 7- Perform Longitudinal Pushover Analysis ..................................... 21-56 21.3.10 Step 8- Check P - Δ Effects ..................................................................... 21-60 21.3.11 Step 9- Check Bent Minimum Lateral Strength ........................................ 21-60 21.3.12 Step 10- Perform Column Shear Design ................................................... 21-61 21.3.13 Step 11- Design Column Shear Key .......................................................... 21-64 21.3.14 Step 12 - Check Bent Cap Flexural and Shear Capacity ........................... 21-65 21.3.15 Step 13- Calculate Column Seismic Load Moments ............................... 21-67 21.3.16 col @ soffit into the Superstructure ............................ 21-74 Step 14 - Distribute M eq 21.3.17 Step 15- Calculate Superstructure Seismic Moment Demands at Location of Interest ................................................................................... 21-74 21.3.18 Step 16- Calculate Superstructure Seismic Shear Demands at Location of Interest ................................................................................... 21-78 21.3.19 Step 17- Perform Vertical Acceleration Analysis ..................................... 21-82 21.3.20 Step 18- Calculate Superstructure Flexural and Shear Capacity ............... 21-82 21.3.21 Step 19- Design Joint Shear Reinforcement .............................................. 21-86 21.3.22 Step 20- Determine Minimum Hinge Seat Width ..................................... 21-95 21.3.23 Step 21- Determine Minimum Abutment Seat Width .............................. 21-95 21.3.24 Step 22 - Design Abutment Shear Key Reinforcement ............................. 21-96 21.3.25 Step 23 - Check Requirements for No-splice Zone ................................... 21-97 APPENDICES .......................................................................................................................... 21-98 NOTATION ........................................................................................................................... 21-137 REFERENCES ....................................................................................................................... 21-144 Chapter 21 – Seismic Design of Concrete Bridges 21-ii BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 CHAPTER 21 SEISMIC DESIGN OF CONCRETE BRIDGES 21.1 INTRODUCTION This chapter is intended primarily to provide guidance on the seismic design of Ordinary Standard Concrete Bridges as defined in Caltrans Seismic Design Criteria (SDC), Version 1.7 (Caltrans 2013). Information presented herein is based on SDC (Caltrans 2013), AASHTO LRFD Bridge Design Specifications (AASHTO 2012) with California Amendments (Caltrans 2014), and other Caltrans Structure Design documents such as Bridge Memo to Designers (MTD). Criteria on the seismic design of nonstandard bridge features are developed on a project-by-project basis and are beyond the scope of this chapter. The first part of the chapter, i.e., Section 21.2, describes general seismic design considerations including pertinent formulae, interpretation of Caltrans SDC provisions, and a procedural flowchart for seismic design of concrete bridges. In the second part, i.e., Section 21.3, a seismic design example of a three-span continuous cast-in-place, prestressed (CIP/PS) concrete box girder bridge is presented to illustrate various design applications following the seismic design procedure flowchart. The example is intended to serve as a model seismic design of an ordinary standard bridge using the current SDC Version 1.7 provisions. 21.2 DESIGN CONSIDERATIONS 21.2.1 Preliminary Member and Reinforcement Sizes Bridge design is inherently an iterative process. It is common practice to design bridges for the Strength and Service Limit States and then, if necessary, to refine the design of various components to satisfy Extreme Events Limit States such as seismic performance requirements. In practice, however, engineers should keep certain seismic requirements in mind even during the Strength and Service Limit States design. This is especially true while selecting the span configuration, column size, column reinforcement requirements, and bent cap width. 21.2.1.1 Sizing the Column and Bent Cap (1) Column size SDC Section 7.6.1 specifies that the column size should satisfy the following equations: Chapter 21 – Seismic Design of Concrete Bridges 21-1 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 Dc  1.00 Ds D ftg 0.70  0.7  (SDC 7.6.1-1) (SDC 7.6.1-2) Dc where: Dc Ds Dftg = = = column cross sectional dimension in the direction of interest (in.) depth of superstructure at the bent cap (in.) depth of footing (in.) If Dc > Ds, it may be difficult to meet the joint shear, superstructure capacity, and ductility requirements. (2) Bent Cap Width SDC Section 7.4.2.1 specifies the minimum cap width required for adequate joint shear transfer as follows: Bcap  Dc  2 21.2.1.2 (ft) (SDC 7.4.2.1-1) Column Reinforcement Requirements (1) Longitudinal Reinforcement Maximum Longitudinal Reinforcement Area, Ast ,max  0.04  Ag (SDC 3.7.1-1) Minimum Longitudinal Reinforcement Area: Ast ,min  0.01( Ag ) for columns (SDC 3.7.2-1) Ast ,min  0.005( Ag ) for Pier walls (SDC 3.7.2-2) where: Ag = the gross cross sectional area (in.2) Normally, choosing column Ast  0.015( Ag ) is a good starting point. (2) Transverse Reinforcement Either spirals or hoops can be used as transverse reinforcement in the column. However, hoops are preferred (see MTD 20-9) because of their discrete nature in the case of local failure. Chapter 21 – Seismic Design of Concrete Bridges 21-2 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015  Inside the Plastic Hinge Region The amount of transverse reinforcement inside the analytical plastic hinge region (see SDC Section 7.6.2 for analytic plastic hinge length formulas), expressed as volumetric ratio,s, shall be sufficient to ensure that the column meets the performance requirements as specified in SDC Section 4.1. 4( A )  s  ' b for columns with circular or interlocking cores (SDC 3.8.1-1) D (s) For rectangular columns with ties and cross ties, the corresponding equation for  s , is: s  where: Av = Dc' = s = Av Dc' s (SDC 3.8.1-2) sum of area of the ties and cross ties running in the direction perpendicular to the axis of bending (in.2) confined column cross-section dimension, measured out to out of ties, in the direction parallel to the axis of bending (in.) transverse reinforcement spacing (in.) In addition, the transverse reinforcement should meet the column shear requirements as specified in SDC Section 3.6.3.  Outside the Plastic Hinge Region As specified in SDC Section 3.8.3, the volume of lateral reinforcement outside the plastic hinge region shall not be less than 50 % of the minimum amount required inside the plastic hinge region and meet the shear requirements. (3) Spacing Requirements The selected bar layout should satisfy the following spacing requirements for effectiveness and constructability:  Longitudinal Reinforcement Maximum and minimum spacing requirements are given in AASHTO Article 5.10 (2012).  Transverse Reinforcement According to SDC Section 8.2.5, the maximum spacing in the plastic hinge region shall not exceed the smallest of: Chapter 21 – Seismic Design of Concrete Bridges 21-3 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015  1 of the least column cross-section dimension for columns and ½ of the least cross-section dimension for piers  6 times the nominal diameter of the longitudinal bars  8 in. 5 Outside this region, the hoop spacing can be and should be increased to economize the design. 21.2.1.3 Balanced Stiffness (1) Stiffness Requirements For an acceptable seismic response, a structure with well-balanced mass and stiffness across various frames is highly desirable. Such a structure is likely to respond to a seismic activity in a simple mode of vibration and any structural damage will be well distributed among all the columns. The best way to increase the likelihood that the structure responds in its fundamental mode of vibration is to balance its stiffness and mass distribution. To this end, the SDC recommends that the ratio of effective stiffness between any two bents within a frame or between any two columns within a bent satisfy the following: kie k ej  0.5  ke / m i 2   ie k /m j  j For constant width frame    0.5   For variable width frame (SDC 7.1.1-1) (SDC 7.1.1-2) SDC further recommends that the ratio of effective stiffness between adjacent bents within a frame or between adjacent columns within a bent satisfies the following: kie k ej  0.75 For constant width frame  ke / m i 1.33   ie k /m j  j    0.75 For variable width frame   (SDC 7.1.1-3) (SDC 7.1.1-4) where: kie mi = = smaller effective bent or column stiffness (kip/in.) tributary mass of column or bent i (kip-sec2/ft) k ej = larger effective bent or column stiffness (kip/in.) mj = tributary mass of column or bent j (kip-sec2/ft) Chapter 21 – Seismic Design of Concrete Bridges 21-4 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 Bent stiffness shall be based on effective material properties and also include the effects of foundation flexibility if it is determined to be significant by the Geotechnical Engineer. If these requirements of balanced effective stiffness are not met, some of the undesired consequences include:  The stiffer bent or column will attract more force and hence will be susceptible to increased damage  The inelastic response will be distributed non-uniformly across the structure  Increased column torsion demands will be generated by rigid body rotation of the superstructure (2) Material and Effective Column Section Properties To estimate member ductility, column effective section properties as well as the moment-curvature ( M   ) relationship are determined by using a computer program such as xSECTION (Mahan 2006) or similar tool. Effort should be made to keep the dead load axial forces in columns to about 10% of their ultimate compressive capacity (Pu = Ag f c' ) to ensure that the column does not experience brittle compression failure and also to ensure that any potential P- effects remain within acceptable limits. When the column axial load ratio starts approaching 15%, increasing the column size or adding an extra column should be considered. Material Properties  Concrete Concrete compressive stress f c = 4,000 psi is commonly used for superstructure, columns, piers, and pile shafts. For other components like abutments, wingwalls, and footings, f c =3,600 psi is typically specified. SDC Section 3.2 requires that expected material properties shall be used to calculate section capacities for all ductile members. To be consistent between the demand and capacity, expected material properties will also be used to calculate member stiffness. For concrete, the expected compressive strength, f ce' , is taken as: f ce'  1.3( f c' )     Greater of  and   5,000 psi    (SDC 3.2.6-3) Other concrete properties are listed in SDC Section 3.2.6. Chapter 21 – Seismic Design of Concrete Bridges 21-5 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015  Steel Grade A706/A706M is typically used for reinforcing steel bar. Material properties for Grade A706/A706M steel are given in SDC Section 3.2.3. Effective Moment of Inertia It is well known that concrete cover spalls off at very low ductility levels. Therefore, the effective (cracked) moment of inertia values are used to assess the seismic response of all ductile members. This is obtained from a moment-curvature analysis of the member cross-section. 21.2.1.4 Balanced Frame Geometry SDC Section 7.1.2 requires that the ratio of fundamental periods of vibration for adjacent frames in the longitudinal and transverse directions satisfy: Ti (SDC 7.1.2-1)  0.7 Tj where: = = Ti Tj natural period of the less flexible frame (sec.) natural period of the more flexible frame (sec.) The consequences of not meeting the fundamental period requirements of SDC Equation 7.1.2-1 include a greater likelihood of out-of-phase response between adjacent frames leading to large relative displacements that increase the probability of longitudinal unseating and collision between frames at the expansion joints. For bents/frames that do not meet the SDC requirements for fundamental period of vibration and/or balanced stiffness, one or more of the following techniques (see SDC Section 7.1.3) may be employed to adjust the dynamic characteristics:         Use of oversized shafts Adjust the effective column length. This may be achieved by lowering footings, using isolation casings, etc. Modify end fixities Redistribute superstructure mass Vary column cross section and longitudinal reinforcement ratios Add or relocate columns Modify the hinge/expansion joint layout, if applicable Use isolation bearings or dampers Chapter 21 – Seismic Design of Concrete Bridges 21-6 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 If the column reinforcement exceeds the preferred maximum, the following additional revisions as outlined in MTD 6-1 (Caltrans 2009) may help:  Pin columns in multi-column bents and selected single columns adjacent to abutments at their bases Use higher strength concrete Shorten spans and add bents Use pile shafts in lieu of footings Add more columns per bent     21.2.2 Minimum Local Displacement Ductility Capacity Before undertaking a comprehensive analysis to consider the effects of changes in column axial forces (for multi-column bents) due to seismic overturning moments and the effects of soil overburden on column footings, it is good practice to ensure that basic SDC ductility requirements are met. SDC Section 3.1 requires that each ductile member shall have a minimum local displacement ductility capacity c of 3 to ensure dependable rotational capacity in the plastic hinge regions regardless of the displacement demand imparted to the member. Δc  ΔYcol  Δp L2 (Y ) 3 Lp     p   p  L  2   ΔYcol  (SDC 3.1.3-1) (SDC 3.1.3-2) (SDC 3.1.3-3)  p  Lp p (SDC 3.1.3-4)  p  u  Y (SDC 3.1.3-5) col  c1  col Y 1   p1 ;  c 2   Y 2   p 2 (SDC 3.1.3-6) L12 L22 (Y 1 ); col  (Y 2 ) Y2 3 3 L p1  L    ;  p 2   p 2  L2  p 2   p1   p1  L1    2  2     p1  L p1 p1;  p 2  L p 2 p 2 ΔYcol 1   p1  u1  Y1;  p 2  u 2  Y 2 (SDC 3.1.3-7) (SDC 3.1.3-8) (SDC 3.1.3-9) (SDC 3.1.3-10) where: L = distance from the point of maximum moment to the point of contra-flexure (in.) Chapter 21 – Seismic Design of Concrete Bridges 21-7 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 LP = Y = p = equivalent analytical plastic hinge length as defined in SDC Section 7.6.2 (in.) idealized plastic displacement capacity due to rotation of the plastic hinge (in.) idealized yield displacement of the column at the formation of the plastic hinge (in.) idealized yield curvature defined by an elastic-perfectly-plastic representation of the cross section’s M- curve, see SDC Figures 3.3.1-1 and 3.3.1-2 (rad/in.) idealized plastic curvature capacity (assumed constant over Lp) (rad/in.) p = p = = plastic rotation capacity (radian) curvature capacity at the Failure Limit State, defined as the concrete strain Ycol = u reaching cu or the longitudinal reinforcing steel reaching the reduced ultimate strain suR (rad/in.) It is Caltrans’ practice to use an idealized bilinear M- curve to estimate the idealized yield displacement and deformation capacity of ductile members. c Ycol C.L. Column p C.G. L Force Idealized Yield Curvature Capacity EquivalentCurvature Lp p Actual Curvature P u p   Y Y c Displacement Figure SDC 3.1.3-1 Local Displacement Capacity – Cantilever Column with Fixed Base Chapter 21 – Seismic Design of Concrete Bridges 21-8 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 C.L. Column Y1 u1 p1 P1 Lp1 P2 Idealized L1 c2 Yield Curvature P1 Actual Curvature colY1 colY2 c1 Idealized Equivalent Curvature L2 Lp2 P2 u2 p2 Y2 Figure SDC 3.1.3-2 Local Displacement Capacity – Framed Column, Assumed as Fixed-Fixed 21.2.3 Displacement Ductility Demand Requirements The displacement ductility demand is mathematically defined as  D  D Y (i ) (SDC 2.2.3-1) where: D Y(i) = = the estimated global/frame displacement demand the yield displacement of the subsystem from its initial position to the formation of plastic hinge (i) To reduce the required strength of ductile members and minimize the demand imparted to adjacent capacity protected components, SDC Section 2.2.4 specifies target upper limits of displacement ductility demand values,  D , for various bridge components. Chapter 21 – Seismic Design of Concrete Bridges 21-9 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 D  4 D  5 D  5 D  1 Single Column Bents supported on fixed foundation Multi-Column Bents supported on fixed or pinned footings Pier Walls (weak direction) supported on fixed or pinned footings Pier Walls (strong direction) supported on fixed or pinned footings In addition, SDC Section 4.1 requires each bridge or frame to satisfy the following equation: D  C (SDC 4.1.1-1) where: = C the bridge or frame displacement capacity when the first ultimate capacity is reached by any plastic hinge (in.) = the displacement generated from the global analysis, stand-alone analysis, or the larger of the two if both types of analyses are necessary (in.) D The seismic demand can be estimated using Equivalent Static Analysis (ESA). As described in SDC Section 5.2.1, this method is most suitable for structures with well-balanced spans and uniformly distributed stiffness where the response can be captured by a simple predominantly translational mode of vibration. Effective properties shall be used to obtain realistic values for the structure’s period and demand. The displacement demand,  D , can be calculated from Equation 21.2-1. D  ma ke (21.2-1) where: m a ke = tributary superstructure mass on the bent/frame = demand spectral acceleration = effective frame stiffness For ordinary bridges that do not meet the criteria for ESA or where ESA does not provide an adequate level of sophistication to estimate the dynamic behavior, Elastic Dynamic Analysis (EDA) may be used. Refer to SDC Section 5.2.2 for more details regarding EDA. Chapter 21 – Seismic Design of Concrete Bridges 21-10 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 Figure SDC 3.1.4.1-1 Local Ductility Assessment Chapter 21 – Seismic Design of Concrete Bridges 21-11 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 21.2.4 Displacement Capacity Evaluation SDC Section 5.2.3 specifies the use of Inelastic Static Analysis (ISA), or “pushover” analysis, to determine reliable displacement capacities of a structure or frame. ISA captures non-linear bridge response such as yielding of ductile components and effects of surrounding soil as well as the effects of foundation flexibility and flexibility of capacity protected components such as bent caps. The effect of soil-structure interaction can be significant in the case where footings are buried deep in the ground. Pushover analysis shall be performed using expected material properties of modeled members to provide a more realistic estimate of design strength. As required by SDC Section 3.4, capacity protected concrete components such as bent caps, superstructures and footings shall be designed to remain essentially elastic when the column reaches its overstrength capacity. This is required in order to ensure that no plastic hinge forms in these components. Caltrans’ in-house computer program wFRAME (Mahan 1995) or similar tool may be used to perform pushover analysis. If wFRAME program is used, the following conventions are applicable to both the transverse and longitudinal analyses:     The model is two-dimensional with beam elements along the c.g. of the superstructure/bent cap and columns. The dead load of superstructure/bent cap, and of columns, if desired, is applied as a uniformly distributed load along the length of the superstructure/bent cap. The element connecting the superstructure c.g. to the column end point at the soffit level is modeled as a super stiff element with stiffness much greater than the regular column section. The moment capacity for such element is also specified much higher than the plastic moment capacity of the column. This is done to ensure that for a column-to-superstructure fixed connection, the plastic hinge forms at the top of the column below the superstructure soffit. The soil effect can be included as p-y, t-z, and q-z springs. Though “pushover” is mainly a capacity estimating procedure, it can also be used to estimate demand for structures having characteristics outlined previously in Section 21.2.3. 21.2.4.1 Foundation Soil Springs The p-y curves are used in the lateral modeling of soil as it interacts with the bent/column foundations. The Geotechnical Engineer generally produces these curves, the values of which are converted to proper soil springs within the push analysis. The spacing of the nodes selected on the pile members would naturally Chapter 21 – Seismic Design of Concrete Bridges 21-12 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 change the values of spring stiffness, however, a minimum of 10 elements per pile is advised (recommended optimum is 20 elements or 2 ft to 5 ft pile segments). The t-z curves are used in the modeling of skin friction along the length of piles. Vertical springs are attached to the nodes to support the dead load of the bridge system and to resist overturning effects caused by lateral bridge movement. The bearing reaction at tip of the pile is usually modeled as a q-z spring. This spring may be idealized as a bi-linear spring placed in the boundary condition section of the push analysis input file. 21.2.4.2 Transverse Pushover Analysis During the transverse movement of a multi-column frame, a strong cap beam provides a framing action. As a result of this framing action, the column axial force can vary significantly from the dead load state. If the seismic overturning forces are large, the top of the column might even go into tension. The effect of change in the axial force in a column section due to overturning moments can be summarized as follows:    Mp changes The tension column(s) will become more ductile while the compression column(s) will become less ductile. The required flexural capacity of cap beam that is needed to make sure that the hinge forms at column top will obviously become larger. With the changes in column axial loads, the section properties (Mp and Ie) should be updated and a second iteration of the wFRAME program performed if using wFRAME for the analysis. The effective bent cap width to be used for the pushover analysis is calculated as follows: Beff  Bcap  (12t ) (SDC 7.3.1.1-1) where: = thickness of the top or bottom slab (in.) = bent cap width (in.) t Bcap 21.2.4.3 Longitudinal Pushover Analysis Although the process of calculating the section capacity and estimating the seismic demand is similar for the transverse and longitudinal directions, there are some significant differences. For longitudinal push analysis:  If wFRAME program is used, columns are lumped together Chapter 21 – Seismic Design of Concrete Bridges 21-13 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015  For prestressed superstructures, gross moment of inertia is used for the superstructure Bent overturning is ignored The abutment is modeled as a linear spring whose stiffness is calculated as described in this Section.   If the column or pier cross-section is rectangular, section properties along the longitudinal direction of the bridge as shown in Figure 21.2-1 must be calculated and used. If using xSECTION , this can be achieved by specifying in the xSECTION input file, the angle between the column section coordinate system and the longitudinal direction of the bridge as shown in the sketch below. Both left and right longitudinal pushover analyses of the bridge should be performed. Bridge longitudinal direction Bent Line Figure 21.2-1 Bridge Longitudinal Direction It is Caltrans’ practice to design the abutment backwall so that it breaks off in shear during a seismic event. SDC Section 7.8.1 requires that the linear elastic demand shall include an effective abutment stiffness that accounts for expansion gaps and incorporates a realistic value for the embankment fill response. The abutment embankment fill stiffness is non-linear and is highly dependent upon the properties of the backfill. The initial embankment fill stiffness, Ki, is estimated at 50 kip/in./ft for embankment fill material meeting the requirements of Caltrans Standard Specifications and 25 kip/in./ft, if otherwise. The initial stiffness, K i shall be adjusted proportional to the backwall/diaphragm height as follows:  h  K abut  K i w   5.5  (SDC 7.8.1-2) where: w h = projected width of the backwall or diaphragm for seat and diaphragm abutments, respectively (ft) = height of the backwall or diaphragm for seat and diaphragm abutments, respectively (ft) Chapter 21 – Seismic Design of Concrete Bridges 21-14 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 The passive pressure resisting movement at the abutment, Pw, is given as:  h or hdia  Pw  Ae (5 ) bw  5.5   kip  ft (SDC 7.8.1-3) where: hbw wbw Ae   hdia wdia For seat abutments For diaphragm abutments (SDC 7.8.1-4) The terms hbw, hdia, wbw, and wdia, are defined in SDC Figure 7.8.1-2. SDC Section 7.8.1 specifies that the effectiveness of the abutment shall be assessed by the coefficient: RA   D eff (SDC 7.8.1-5) where: RA D eff = = = abutment displacement coefficient the longitudinal displacement demand at the abutment from elastic analysis the effective longitudinal abutment displacement at idealized yield Details on the interpretation and use of the coefficient RA value are given in SDC Section 7.8.1. 21.2.5 P- Effects In lieu of a rigorous analysis to determine P- effects, SDC recommends that such effects can be ignored if the following equation is satisfied: Pdl  r  0.20M col p where: M col p = idealized plastic moment capacity of a column calculated from M- = = analysis dead load axial force relative lateral offset between the base of the plastic hinge and the point of contra-flexure Pdl r 21.2.6 (SDC 4.2-1) Minimum Lateral Strength SDC Section 3.5 specifies that each bent shall have a minimum lateral flexural capacity (based on expected material properties) to resist a lateral force of 0.1Pdl, Chapter 21 – Seismic Design of Concrete Bridges 21-15 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 where Pdl is the tributary dead load applied at the center of gravity of the superstructure. 21.2.7 Column Shear Design The seismic shear demand shall be based upon the overstrength shear Vo , associated with the column overstrength moment M 0col . Since shear failure tends to be brittle, shear capacity for ductile members shall be conservatively determined using nominal material properties, as follows: Vn  V0 (SDC 3.6.1-1) Vn = Vc + Vs (SDC 3.6.1-2) where:   0.90 21.2.7.1 Shear Demand Vo Shear demand associated with overstrength moment may be calculated from: V0  M 0col L (21.2-2) where: M 0col  1.2M col p L (SDC 4.3.1-1) = clear length of column Alternately, the maximum shear demand may be determined from wFRAME pushover analysis results. The maximum column shear demand obtained from wFRAME analysis is multiplied by a factor of 1.2 to obtain the shear demand associated with the overstrength moment. 21.2.7.2 Concrete Shear Capacity Vc   c Ae (SDC 3.6.2-1) Ae  (0.8) Ag (SDC 3.6.2-2) where:  c  f1 f 2 f c'  4 fc' = 3 f 2 f c'  4 f c' (Inside the plastic hinge region) (SDC 3.6.2-3) (Outside the plastic hinge region) (SDC 3.6.2-4) Chapter 21 – Seismic Design of Concrete Bridges 21-16 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 0.3  f1   s f yh 0.150  3.67  d  3 ( f yh in ksi)  s f yh  0.35 ksi f2  1  21.2.7.3 (SDC 3.6.2-5) (21.2-3) Pc  1.5 2,000Ag ( Pc is in lb, Ag is in in.2) (SDC 3.6.2-6) Transverse Reinforcement Shear Capacity Vs  Av f yh D'   Vs   s   (SDC 3.6.3-1) where:   Av  n  Ab (SDC 3.6.3-2) 2 n = number of individual interlocking spiral or hoop core sections 21.2.7.4 Maximum Shear Reinforcement Strength, Vs,max Vs, max  8 f c' Ae 21.2.7.5 (SDC 3.6.5.1-1) Minimum Shear Reinforcement Av, min  0.025 21.2.7.6 (psi) D's f yh in.  2 (SDC 3.6.5.2-1) Column Shear Key The area of interface shear key reinforcement, Ask in hinged column bases shall be calculated as shown in the following equations: 1.2( Fsk  0.25P) if P is compressive (SDC 7.6.7-1) Ask  fy Ask  1.2( Fsk  P) fy if P is tensile (SDC 7.6.7-2) where: Ask  4 in.2 Fsk = (21.2-4) shear force associated with the column overstrength moment, including overturning effects (kip) Chapter 21 – Seismic Design of Concrete Bridges 21-17 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 P = absolute value of the net axial force normal to the shear plane (kip) = lowest column axial load if net P is compressive considering overturning effects = largest column axial load if net P is tensile, considering overturning effects The hinge shall be proportioned such that the area of concrete engaged in interface shear transfer, Acv satisfies the following equations: 4.0 Fsk f c' Acv  0.67Fsk Acv  (SDC 7.6.7-3) (SDC 7.6.7-4) In addition, the area of concrete section used in the hinge must be enough to meet the axial resistance requirements as provided in AASHTO Article 5.7.4.4 (AASHTO 2012), based on the column with the largest axial load. 21.2.8 Bent Cap Flexural and Shear Capacity According to SDC Section 3.4, a bent cap is considered a capacity protected member and shall be designed flexurally to remain essentially elastic when the column reaches its overstrength capacity. The expected nominal moment capacity Mne for capacity protected members may be determined either by a traditional strength method or by a more complete M-ϕ analysis. The expected nominal moment capacity shall be based on expected concrete and steel strength values when either R concrete strain reaches 0.003 or the steel strain reaches  SU as derived from the applicable stress-strain relationship. The shear capacity of the bent cap is calculated according to AASHTO Article 5.8 (AASHTO 2012). The seismic flexural and shear demands in the bent cap are calculated corresponding to the column overstrength moment. These demands are obtained from a pushover analysis with column moment capacity as M0 and then compared with the available flexural and shear capacity of the bent cap. The effective bent cap width to be used is calculated as follows: Beff  B cap 12t  t 21.2.9 (SDC 7.3.1.1-1) = thickness of the top or bottom slab Seismic Strength of Concrete Bridge Superstructures When moment-resisting superstructure-to-column details are used, seismic forces of significant magnitude are induced into the superstructure. If the superstructure does not have adequate capacity to resist such forces, unexpected and unintentional Chapter 21 – Seismic Design of Concrete Bridges 21-18 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 hinge formation may occur in the superstructure leading to potential failure of the superstructure. According to SDC Sections 3.4 and 4.3.2, a capacity design approach is adopted to ensure that the superstructure has an appropriate strength reserve above demands generated from probable column plastic hinging. MTD 20-6 (Caltrans 2001a) describes the philosophy, design criteria, and a procedure for determining the seismic demands in the superstructure, and also recommends a method for determining the flexural capacity of the superstructure at all critical locations. 21.2.9.1 General Assumptions As discussed in MTD 20-6, some of the assumptions made to simplify the process of calculating seismic demands in the superstructure include:  The superstructure demands are based upon complete plastic hinge formation in all columns or piers within the frame. Effective section properties shall be used for modeling columns or piers while gross section properties may be used for superstructure elements. Additional column axial force due to overturning effects shall be considered when calculating effective section properties and the idealized plastic moment capacity of columns and piers. Superstructure dead load and secondary prestress demands are assumed to be uniformly distributed to each girder, except in the case of highly curved or highly skewed structures. While assessing the superstructure member demands and available section capacities, an effective width, Beff as defined in SDC Section 7.2.1.1 will be used.      D  2 Ds Beff   c  Dc  Ds Box girders and slab superstructures Open soffit superstructures (SDC 7.2.1.1-1) where: Dc Ds 21.2.9.2 = = cross sectional dimension of the column (in.) depth of the superstructure (in.) Superstructure Seismic Demand The force demand in the superstructure corresponds to its Collapse Limit State. The Collapse Limit State is defined as the condition when all the potential plastic hinges in the columns and/or piers have been formed. When a bridge reaches such a state during a seismic event, the following loads are present: Dead Loads, Secondary Forces from Post-tensioning (i.e., prestress secondary effects), and Seismic Loads. Since the prestress tendon is treated as an internal component of the superstructure and is included in the member strength calculation, only the secondary effects which are caused by the support constraints in a statically indeterminate prestressed frame are considered to contribute to the member demand. Chapter 21 – Seismic Design of Concrete Bridges 21-19 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 The procedure for determining extreme seismic demands in the superstructure considers each of these load cases separately and the final member demands are obtained by superposition of the individual load cases. Since different tools may be used to calculate these demands, it is very important to use a consistent sign convention while interpreting the results. The following sign convention (see Figure 21.2-2a) for positive moments, shears and axial forces, is recommended. The sign convention used in wFRAME program is shown in Figure 21.2-2b. It should be noted that although the wFRAME element level sign convention is different from the standard sign convention adopted here, the resulting member force conditions (for example, member in positive or negative bending, tension or compression, etc.) are the same as furnished by the standard convention. In particular, note that the inputs Mp and Mn for the beam element in the wFRAME program correspond to tension at the beam bottom (i.e., positive bending) and tension at the beam top (i.e., negative bending), respectively. The engineer should also ensure that results obtained while using the computer program CTBridge (Caltrans 2007) are consistent with the above sign conventions when comparing outputs or employing the results of one program as inputs into another program. (i) Beam (j) (j) (i) (j) Beam (i) Column Column (i) (a) Standard (j) (b) wFRAME Figure 21.2-2 Sign Convention for Positive Moment, Shear and Axial force (Element Level) Prior to the application of seismic loading, the columns are “pre-loaded” with moments and shears due to dead loads and secondary prestress effects. At the Collapse Limit State, the “earthquake moment” applied to the superstructure may be greater or less than the overstrength moment capacity of the column or pier depending on the direction of these “pre-load” moments and the direction of the seismic loading under consideration. Figure 21.2-3 shows schematically this approach of calculating columns seismic forces. Chapter 21 – Seismic Design of Concrete Bridges 21-20 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 As recommended in MTD 20-6, due to the uncertainty in the magnitude and distribution of secondary prestress moments and shears at the extreme seismic limit state, it is conservative to consider such effects only when their inclusion results in increased demands in the superstructure. Once the column moment, Meq, is known at each potential plastic hinge location below the joint regions, the seismic moment demand in the superstructure can be determined using currently available Caltrans’ analysis tools. One such method entails application of Meq at the column-superstructure joints and then using computer program SAP2000 (CSI 2007) to compute the moment demand in the superstructure members. Another method involves using the wFRAME program to perform a longitudinal pushover analysis by specifying the required seismic moments in the columns as the plastic hinge capacities of the column ends. The pushover is continued until all the plastic hinges have formed. + +  +    M 0   M dl    M ps   M eq M eq M ps M dl  When earthquake forces add to dead load and secondary prestress forces. When earthquake forces add to dead load and secondary prestress forces. = Collapse Limit Collapse Limit State + State M ps M dl + -      M 0   M dl    M ps   M eq M eq =  When earthquake forces counteract dead load and secondary prestress When earthquake forces counteract dead forces. load and secondary prestress forces. Figure 21.2-3 Column Forces Corresponding to Two Seismic Loading Cases Note that CTBridge is a three-dimensional analysis program where force results are oriented in the direction of each member’s local axis. If wFRAME (a twodimensional frame analysis program) is used to determine the distribution of seismic forces to the superstructure, it must be ensured that the dead load and secondary prestress moments lie in the same plane prior to using them in any calculations. This must be done especially when horizontal curves or skews are involved. Chapter 21 – Seismic Design of Concrete Bridges 21-21 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 (1) Dead Load Moments, Additional Dead Load Moments, and Prestress Secondary Moments These moments are readily available from CTBridge output and are assumed to be uniformly distributed along each girder. (2) Earthquake Moments in the Superstructure (Reference MTD 20-6, SDC 4.3.2) The aim here is to determine the amount of seismic loading needed to ensure that potential plastic hinges have formed in all the columns of the framing system. To form a plastic hinge in the column, the seismic load needs to produce a moment at the potential plastic hinge location of such a magnitude that, when combined with the “pre-loaded” dead load and prestress moments, the column will reach its overstrength plastic moment capacity, M 0col . @ soffit col @ soffit M 0col @ soffit  M dlcol @ soffit  M col  M eq ps (21.2-5) It should be kept in mind that dead load moments will have positive or negative values depending on the location along the span length. Also, the direction of seismic loading will determine the nature of the seismic moments. Two cases of longitudinal earthquake loading shall be considered, namely, (a) bridge movement to the right, and (b) bridge movement to the left. col , are calculated from Equation 21.2-5 The column seismic load moments, M eq based upon the principle of superposition as follows:  col @ soffit @ soffit M eq  M 0col @ soffit  M dlcol @ soffit  M col ps  In the above equation, the overstrength column moment M M ocol  1.2M col p (21.2-6) col o is given as: (SDC 4.3.1-1) (3) Earthquake Shear Forces in the Superstructure A procedure similar to that used for moments can be followed to calculate the seismic shear force demand in the superstructure. As in the case of moments, the shear forces in the superstructure member due to dead load, additional dead load, and secondary prestress are readily available from CTBridge output. Chapter 21 – Seismic Design of Concrete Bridges 21-22 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 The superstructure seismic shear forces due to seismic moments can be obtained directly from the wFRAME output or calculated by using the previously computed values of the superstructure seismic moments, M eqL and M eqR , for each span. (4) Moment and Shear Demand at Location of Interest The extreme seismic moment demand in the superstructure is calculated as the summation of all the moments obtained from the above sections, taking into account the proper direction of bending in each case as well as the effective section width. The superstructure demand moments at the adjacent left and right superstructure span are given by: L M DL  M dlL  M ps  M eqL (21.2-7) R M DR  M dlR  M ps  M eqR (21.2-8) Similarly, the extreme seismic shear force demand in the superstructure is calculated as the summation of shear forces due to dead load, secondary prestress effects and the seismic loading, taking into account the proper direction of bending in each case and the effective section width. The superstructure demand shear forces at the adjacent left and right superstructure spans are defined as: VDL  VdlL  VpsL  VeqL (21.2-9) VDR  VdlR  VpsR  VeqR (21.2-10) As stated previously in this section, the secondary effect due to the prestress will be considered only when it results in an increased seismic demand. Dead load and secondary prestress moment and shear demands in the superstructure are proportioned on the basis of the number of girders falling within the effective section width. The earthquake moment and shear imparted by column is also assumed to act within the same effective section width. (5) Vertical Acceleration In addition to the superstructure demands discussed above, SDC Sections 2.1.3 and 7.2.2 require an equivalent static vertical load to be applied to the superstructure to estimate the effects of vertical acceleration in the case of sites with Peak Ground Acceleration (PGA) greater than or equal to 0.6g. For such sites, the effects of vertical acceleration may be accounted for by designing the superstructure to resist an additional uniformly applied vertical force equal to 25% of the dead load applied upward and downward. Chapter 21 – Seismic Design of Concrete Bridges 21-23 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 21.2.9.3 Superstructure Section Capacity (1) General To ensure that the superstructure has sufficient capacity to resist the extreme seismic demands determined in Section 21.2.9.2, SDC Section 4.3.2 requires the superstructure capacity in the longitudinal direction to be greater than the demand distributed to it (the superstructure) on each side of the column by the largest combination of dead load moment, secondary prestress moment, and column earthquake moment, i.e., sup(R ) M ne   M dlR  M pR/ s  M eqR (SDC 4.3.2-1) sup(L ) M ne   M dlL  M pL / s  M eqL (SDC 4.3.2-2) where: sup R , L = M ne expected nominal moment capacity of the adjacent right (R) or left (L) superstructure span (2) Superstructure Flexural Capacity MTD 20-6 (Caltrans 2001a) describes the philosophy behind the flexural section capacity calculations. Expected material properties are used to calculate the flexural capacity of the superstructure. The member strength and curvature capacities are assessed using a stress-strain compatibility analysis. Failure is reached when either the ultimate concrete, mild steel or prestressing ultimate strain is reached. The internal resistance force couple is shown in Figure 21.2-4. p/s se se sa sa Ap/s As s Ts Tp/s MLn d’c ds Stress Strain c Cc Cs Cp/s d’s s ’s c c MR n T’ ’s A’s Strain Note: Axial forces not shown V M Stress col o col o Figure 21.2-4 Superstructure Capacity Provided by Internal Couple Chapter 21 – Seismic Design of Concrete Bridges dc dp/s N/A N/A C ’  s s dp/s Cc p/s 21-24 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 Caltrans in-house computer program PSSECx or similar program, may be used to calculate the section flexural capacity. The material properties for 270 ksi and 250 ksi prestressing strands are given in SDC Section 3.2.4. According to MTD 20-6, at locations where additional longitudinal mild steel is not required by analysis, a minimum of #8 bars spaced 12 in. (maximum spacing) should be placed in the top and bottom slabs at the bent cap. The mild steel reinforcement should extend beyond the inflection points of the seismic moment demand envelope. As specified in SDC Section 3.4, the expected nominal moment capacity, Mne, for capacity protected concrete components shall be determined by either M- analysis or strength design. Also, SDC Section 3.4 specifies that expected material properties shall be used in determining flexural capacity. Expected nominal moment capacity for capacity-protected concrete members shall be based on the expected concrete and steel strengths when either the concrete strain reaches its ultimate value based on the stress-strain model or the reduced ultimate prestress steel strain,  suR = 0.03 is reached. In addition to these material properties, the following information is required for the capacity analysis:    Eccentricity of prestressing steel - obtained from CTBridge output file. This value is referenced from the CG of the section. Prestressing force - obtained from CTBridge output file under the “P/S Response After Long Term Losses” Tables. Prestressing steel area, Aps - calculated for 270ksi steel as A ps    P jack (21.2-11) (0.75)(270) Reinforcement in top and bottom slab, per design including #8 @12. Location of top and bottom reinforcement, referenced from center of gravity of section, slab steel section depth and assumed cover, etc. Both negative (tension at the top) and positive (tension at the bottom) capacities are calculated at various sections along the length of the bridge by the PSSECx computer program. The resistance factor for flexure, flexure = 1.0, as we are dealing with extreme conditions corresponding to column overstrength. (3) Superstructure Shear Capacity MTD 20-6 specifies that the superstructure shear capacity is calculated according to AASHTO Article 5.8. As shear failure is brittle, nominal rather than expected material properties are used to calculate the shear capacity of the superstructure. Chapter 21 – Seismic Design of Concrete Bridges 21-25 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 21.2.10 Joint Shear Design 21.2.10.1 General (1) Principal Stresses In a ductility-based design approach for concrete structures, connections are key elements that must have adequate strength to maintain structural integrity under seismic loading. In moment resisting connections, the force transfer across the joint typically results in sudden changes in the magnitude and nature of moments, resulting in significant shear forces in the joint. Such shear forces inside the joint can be many times greater than the shear forces in individual components meeting at the joint. SDC Section 7.4 requires that moment resisting connections between the superstructure and the column shall be designed to transfer the maximum forces produced when the column has reached its overstrength capacity, M 0col , including the effects of overstrength shear V0col . Accordingly, SDC Section 7.4.2 requires all superstructure/column moment-resisting joints to be proportioned so that the principal stresses satisfy the following equations: For principal compression, pc: pc  0.25 fc For principal tension, pt: pt  12 fc (psi) (psi) (SDC 7.4.2-1) (SDC 7.4.2-2) 2 pt  ( f h  fv )  f  fv  2   h   v jv 2 2   (SDC 7.4.4.1-1) 2 ( f  fv )  f  fv  2   h pc  h   v jv 2  2  T v jv  c A jv Ajv  lacBcap fv  Pc A jh (SDC 7.4.4.1-2) (SDC 7.4.4.1-3) (SDC 7.4.4.1-4) (SDC 7.4.4.1-5) Ajh  Dc  Ds Bcap (SDC 7.4.4.1-6) Pb Bcap Ds (SDC 7.4.4.1-7) fh  where: fh = average normal stress in the horizontal direction (ksi) Chapter 21 – Seismic Design of Concrete Bridges 21-26 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 f Bcap Dc Ds lac Pc Pb = = = = = = = Tc = = h average normal stress in the vertical direction (ksi) bent cap width (in.) cross sectional dimension of column in the direction of bending (in.) depth of superstructure at the bent cap for integral joints (in.) length of column reinforcement embedded into the bent cap (in.) column axial force including the effects of overturning (kip) beam axial force at the center of the joint, including the effects of prestressing (kip) column tensile force (defined as M 0col h ) associated with the column overstrength plastic hinging moment, M 0col . Alternatively, Tc may be obtained from the moment-curvature analysis of the cross section (kip) distance from the center of gravity of the tensile force to the center of gravity of the compressive force of the column section (in.) In the above equations, the value of f h may be taken as zero unless prestressing is specifically designed to provide horizontal joint compression. (2) Minimum Bent Cap Width – See Section 21.2.1.1 (3) Minimum Joint Shear Reinforcement SDC 7.4.4.2 specifies that, if the principal tensile stress, pt is less than or equal to 3.5 f c ' (psi) , no additional joint reinforcement is required. However, a minimum area of joint shear reinforcement in the form of column transverse steel continued into the bent cap shall be provided. The volumetric ratio of the transverse column reinforcement (  s , min ) continued into the cap shall not be less than:  s, min  3.5 f c f yh (psi) (SDC 7.4.4.2-1) If pt is greater than 3.5 f c ' , joint shear reinforcement shall be provided. The amount and type of joint shear reinforcement depend on whether the joint is classified as a “T” joint or a Knee Joint. 21.2.10.2 Joint Description The following types of joints are considered as “T” joints for joint shear analysis (SDC Section 7.4.3):   Integral interior joints of multi-column bents in the transverse direction All integral column-to-superstructure joints in the longitudinal direction Chapter 21 – Seismic Design of Concrete Bridges 21-27 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015  Exterior column joints for box girder superstructures if the cap beam extends beyond the joint (i.e. column face) far enough to develop the longitudinal cap reinforcement Any exterior column joint that satisfies the following equation shall be designed as a Knee joint. For Knee joints, it is also required that the main bent cap top and bottom bars be fully developed from the inside face of the column and extend as closely as possible to the outside face of the cap (see SDC Figure 7.4.3-1). S  Dc where: S = Dc = (SDC 7.4.3-1) cap beam short stub length, defined as the distance from the exterior girder edge at soffit to the face of the column measured along the bent centerline (see Figure SDC 7.4.3-1), column dimension measured along the centerline of bent Bent Cap Top and Bottom Reinforcement S Dc Figure SDC 7.4.3-1 Knee Joint Parameters 21.2.10.3 T Joint Shear Reinforcement (1) Vertical Stirrups in Joint Region Vertical stirrups or ties shall be placed transversely within a distance Dc extending from either side of the column centerline. The required vertical stirrup area Asjv is given as Asjv  0.2  Ast (SDC 7.4.4.3-1) where Ast = Total area of column main reinforcement anchored in the joint. Refer to SDC Section 7.4.4.3 for placement of the vertical stirrups. Chapter 21 – Seismic Design of Concrete Bridges 21-28 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 (2) Horizontal Stirrups Horizontal stirrups or ties, Asjh , shall be placed transversely around the vertical stirrups or ties in two or more intermediate layers spaced vertically at not more than 18 inches. Asjh  0.1  Ast (SDC 7.4.4.3-2) This horizontal reinforcement shall be placed within a distance Dc extending from either side of the column centerline. (3) Horizontal Side Reinforcement The total longitudinal side face reinforcement in the bent cap shall at least be equal to the greater of the area specified in SDC Equation 7.4.4.3-3. top 0.1 Acap  Assf  max  0.1 Abot cap  (SDC 7.4.4.3-3) where: Acap = area of bent cap top or bottom flexural steel (in.2). The side reinforcement shall be placed near the side faces of the bent cap with a maximum spacing of 12 inches. Any side reinforcement placed to meet other requirements shall count towards meeting this requirement. (4) “J” Dowels For bents skewed more than 20o, “J” bars (dowels) hooked around the longitudinal top deck steel extending alternately 24 in. and 30 in. into the bent cap are required. The J-dowel reinforcement shall be equal to or greater than the area specified as: Asj bar  0.08Ast (SDC 7.4.4.3-4) This reinforcement helps to prevent any potential delamination of concrete around deck top reinforcement. The J-dowels shall be placed within a rectangular region defined by the width of the bent cap and the distance Dc on either side of the centerline of the column. (5) Transverse Reinforcement Transverse reinforcement in the joint region shall consist of hoops with a minimum reinforcement ratio specified as:    2  lac , provided    s  0.4 Ast Chapter 21 – Seismic Design of Concrete Bridges (SDC 7.4.4.3-5) 21-29 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 where: area of longitudinal column reinforcement (in.2) actual length of column longitudinal reinforcement embedded into the bent cap (in.) For interlocking cores,  s shall be based on area of reinforcement Ast of each core. All vertical column bars shall be extended as close as possible to the top bent cap reinforcement. Ast lac = = (6) Anchorage for Main Column Reinforcement The main column reinforcement shall extend into the cap as deep as possible to fully develop the compression strut mechanism in the joint. If the minimum joint shear reinforcement prescribed in SDC Equation 7.4.4.2-1 is met, and the column longitudinal reinforcement extension into the cap beam is confined by transverse hoops or spirals with the same volumetric ratio as that required at the top of the column, the anchorage for longitudinal column bars developed into the cap beam for seismic loads shall not be less than: lac, required  24dbl (SDC 8.2.1-1) With the exception of slab bridges where the provisions of MTD 20-7 shall govern, the development length specified above shall not be reduced by use of hooks or mechanical anchorage devices. 21.2.10.4 Knee Joint Shear Reinforcement Knee joints may fail in either “opening” or “closing” modes (see Figure SDC 7.4.5-1). Therefore, both loading conditions must be evaluated. Refer to SDC Section 7.4.5 for the description of Knee joint failure modes. (a) (b) Figure SDC 7.4.5-1 Knee Joint Failure Modes Chapter 21 – Seismic Design of Concrete Bridges 21-30 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 Two cases of Knee joints are identified as follows: Case 1: S Dc 2 Dc  S  Dc 2 The following reinforcement is required for Knee joints. Case 2: (SDC 7.4.5.1-1) (SDC 7.4.5.1-2) (1) Bent Cap Top and Bottom Flexural Reinforcement - Use for both Cases 1 and 2 The top and bottom reinforcement within the bent cap width used to meet this provision shall be in the form of continuous U-bars with minimum area: Asu barmin  0.33Ast (SDC 7.4.5.1-3) where: Ast = total area of column longitudinal reinforcement anchored in the joint (in.2) The “U” bars may be combined with bent cap main top and bottom reinforcement using mechanical couplers. Splices in the “U” bars shall not be located within a distance, ld, from the interior face of the column. (2) Vertical Stirrups in Joint Region - Use for both Cases 1 and 2 Vertical stirrups or ties, Asjv as specified in SDC Equation 7.4.5.1-4, shall be placed transversely within each of regions 1, 2, and 3 of Figure SDC 7.4.5.1-1. Asjv  0.2  Ast (SDC 7.4.5.1-4) The stirrups provided in the overlapping areas shown in Figure SDC 7.4.5.1-1 shall count towards meeting the requirements of both areas creating the overlap. These stirrups can be used to meet other requirements documented elsewhere including shear in the bent cap. (3) Horizontal Stirrups - Use for both Cases 1 and 2 Horizontal stirrups or ties, Asjh , as specified in SDC Equation 7.4.5.1-5, shall be placed transversely around the vertical stirrups or ties in two or more intermediate layers spaced vertically at not more than 18 inches (see Figures SDC 7.4.4.3-2, 7.4.4.3-4, and 7.4.5.1-5 for rebar placement). Asjh  0.1 Ast (SDC 7.4.5.1-5) The horizontal reinforcement shall be placed within the limits shown in Figures SDC 7.4.5.1-2 and SDC 7.4.5.1-3. Chapter 21 – Seismic Design of Concrete Bridges 21-31 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 jv in each of As 1 2 Bent cap stirrups 3 2 CL Bent 3 Bcap Dc 1 CL Girder S Dc/2 S < Dc Figure SDC 7.4.5.1-1 Location of Knee Joint Vertical Shear Reinforcement (Plan View) (4) Horizontal Side Reinforcement- Use for both Cases 1 and 2 The total longitudinal side face reinforcement in the bent cap shall be at least equal to the greater of the area specified as: Assf top 0.1  Acap    or 0.1  Abot cap  (SDC 7.4.5.1-6) where: top Acap = Area of bent cap top flexural steel (in.2) bot Acap = Area of bent cap bottom flexural steel (in.2) This side reinforcement shall be in the form of “U” bars and shall be continuous over the exterior face of the Knee Joint. Splices in the U bars shall be located at least a distance ld from the interior face of the column. Any side reinforcement placed to Chapter 21 – Seismic Design of Concrete Bridges 21-32 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 meet other requirements shall count towards meeting this requirement. Refer to SDC Figures 7.4.5.1-4 and 7.4.5.1-5 for placement details. (5) Horizontal Cap End Ties for Case 1 Only The total area of horizontal ties placed at the end of the bent cap is specified as: Asjhcmin  0.33Asu bar (SDC 7.4.5.1-7) This reinforcement shall be placed around the intersection of the bent cap horizontal side reinforcement and the continuous bent cap U-bar reinforcement, and spaced at not more than 12 inches vertically and horizontally. The horizontal reinforcement shall extend through the column cage to the interior face of the column. (6) J-Dowels - Use for both Cases 1 and 2 Same as in Section 21.2.10.3 for T joints, except that placement limits shall be as shown in SDC Figure 7.4.5.1-3. (7) Transverse Reinforcement Transverse reinforcement in the joint region shall consist of hoops with a minimum reinforcement ratio as specified in SDC Equations 7.4.5.1-9 to 7.4.5.1-11. s  0.76 Ast Dclac, provided  (For Case 1 Knee joint)  Ast  2  lac, provided   s  0.4 (SDC 7.4.5.1-9) (For Case 2 Knee joint, Integral bent cap) (SDC 7.4.5.1-10)   Ast  2  lac, provided   s  0.6 (For Case 2 Knee joint, Non-integral bent cap) (SDC 7.4.5.1-11) where: lac,provided = Ast = Dc = actual length of column longitudinal reinforcement embedded into the bent cap (in.) total area of column longitudinal reinforcement anchored in the joint (in.2) diameter or depth of column in the direction of loading (in.) The column transverse reinforcement extended into the bent cap may be used to satisfy this requirement. For interlocking cores, ρs shall be calculated on the basis of Chapter 21 – Seismic Design of Concrete Bridges 21-33 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 Ast and Dc of each core (for Case 1 Knee joints) and on area of reinforcement, Ast of each core (for Case 2 Knee joints). All vertical column bars shall be extended as close as possible to the top bent cap reinforcement. 21.2.11 Torsional Capacity There is no history of damage to bent caps of Ordinary Standard Bridges from previous earthquakes attributable to torsional forces. Therefore, these bridges are not usually analyzed for torsional effects. However, non-standard bridge features (for example, superstructures supported on relatively long outrigger bents) may experience substantial torsional deformation and warping and should be designed to resist torsional forces. 21.2.12 Abutment Seat Width Requirements Sufficient seat width shall be provided to prevent the superstructure from unseating when the Design Seismic Hazards occur. Per SDC Section 7.8.3, the abutment seat width measured normal to the centerline of the bent, N A , as shown in Figure SDC 7.8.3-1 shall be calculated as follows: CL Brg. NA p/s+cr+sh+temp eq 4 Minimum Seat Width, NA = 30 in. Figure SDC 7.8.3-1 Abutment Seat Width Requirements N A   p / s  cr  sh  temp  eq  4 (in.) (SDC 7.8.3-1) where: NA = abutment seat width normal to the centerline of bearing. Note that for abutments skewed at an angle sk, the minimum seat width measured along the longitudinal axis of the bridge is NA/cos sk (in.) Chapter 21 – Seismic Design of Concrete Bridges 21-34 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 p/s = displacement attributed to pre-stress shortening (in.)  cr  sh =  temp = displacement attributed to creep and shrinkage (in.) displacement attributed to thermal expansion and contraction (in.) Δeq displacement demand, ΔD for the adjacent frame. Displacement of the abutment is assumed to be zero (in.) = The minimum seat width normal to the centerline of bearing as calculated above shall be not less than 30 in. 21.2.13 Hinge Seat Width Requirements For adjacent frames with ratio of fundamental periods of vibration of the less flexible and more flexible frames greater than or equal to 0.7, SDC Section 7.2.5.4 requires that enough hinge seat width be provided to accommodate the anticipated thermal movement (Δtemp), prestress shortening (Δp/s), creep and shrinkage (Δcr+sh), and the relative longitudinal earthquake displacement demand between the two frames (Δeq) - see Figure SDC 7.2.5.4-1. The minimum hinge seat width measured normal to the centerline of bent, N H is given by:  NH where:  eq  eq (iD) = =    p / s  cr  sh  temp  eq  4 in.  or  the larger of  24 (in.)       1 2 D 2 2 D (SDC 7.2.5.4-1) (SDC 7.2.5.4-2) relative earthquake displacement demand at an expansion joint (in.) the larger earthquake displacement demand for each frame calculated by the global or stand-alone analysis (in.) NH p/s+cr+sh + (Time-dependent terms) eq temp NH 4  24 Figure SDC 7.2.5.4-1 Minimum Hinge Seat Width Chapter 21 – Seismic Design of Concrete Bridges 21-35 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 21.2.14 Abutment Shear Key Design 21.2.14.1 General According to SDC Section 7.8.4, abutment shear key force capacity, Fsk shall be determined as follows: Fsk   (0.75V piles  Vww ) Fsk   Pdl 0.5    1 For Abutment on piles (SDC 7.8.4-1) For Abutment on Spread footing (SDC 7.8.4-2) (SDC 7.8.4-3) where: V piles = Sum of lateral capacity of the piles (kip) Vww = Pdl = Shear capacity of one wingwall (kip) Superstructure dead load reaction at the abutment plus the weight of the abutment and its footing (kip) factor that defines the range over which Fsk is allowed to vary α = For abutments supported by a large number of piles, it is permitted to calculate the shear key capacity using the following equation, provided the value of Fsk is less than that furnished by SDC Equation 7.8.4-1: Fsk  Pdlsup (SDC 7.8.4-4) where: Pdlsup = 21.2.14.2 superstructure dead load reaction at the abutment (kip) Abutment Shear Key Reinforcement The SDC provides two methods for designing abutment shear key reinforcement, namely, Isolated and Non-isolated methods. (1) Vertical Shear Key Reinforcement, Ask Ask  Fsk 1.8 f ye Ask  1 Fsk  0.4  Acv  1.4 f ye  0.25 f ce' Acv   0.4 Acv  Fsk  min  A 1 . 5 cv   Chapter 21 – Seismic Design of Concrete Bridges Isolated shear key Non-isolated shear key (SDC 7.8.4.1A-1) (SDC 7.8.4.1A-2) (SDC 7.8.4.1A-3) 21-36 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 Ask  0.05Acv f ye (SDC 7.8.4.1A-4) where: Acv = area of concrete engaged in interface shear transfer (in.2) In the above equations, f ye and f ce' have units of ksi, Acv and Ask are in in2, and Fsk is in kip. See SDC Figure 7.8.4.1-1 for placement of shear key reinforcement for both methods. (2) Horizontal Reinforcement in the Stem Wall (Hanger Bars), Ash Ash  (2.0) AskIso( provided)  iso (2.0) AskNon ( provided)  Ash  max Fsk f  ye Isolated shear key (SDC 7.8.4.1B-1) Non-isolated shear key (SDC 7.8.4.1B-2) where: AskIso( provided) =  iso AskNon ( provided) = area of interface shear reinforcement provided in SDC Equation 7.8.4.1A-1(in.2) area of interface shear reinforcement provided in SDC Equation 7.8.4.1A-2 (in.2) For the isolated key design method, the vertical shear key reinforcement, Ask should be positioned relative to the horizontal reinforcement, Ash to maintain a minimum length Lmin given by (see Figure SDC 7.8.4.1-1A): Lmin,hooked  0.6(a  b)  ldh (SDC 7.8.4.1B-3) Lmin,headed  0.6(a  b)  3 in. (SDC 7.8.4.1B-4) where: a = b = ldh = vertical distance from the location of the applied force on the shear key to the top surface of the stem wall, taken as one-half the vertical length of the expansion joint filler plus the pad thickness (see Figure SDC 7.8.4.1-1(A)) vertical distance from the top surface of the stem wall to the centroid of the lowest layer of shear key horizontal reinforcement development length in tension of standard hooked bars as specified in AASHTO (2012) Chapter 21 – Seismic Design of Concrete Bridges 21-37 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 * Smooth construction joint is required at the shear key interfaces with the stemwall and backwall to effectively isolate the key except for specifically designed reinforcement. These interfaces should be trowel-finished smooth before application of a bond breaker such as construction paper. Form oil shall not be used as a bond breaker for this purpose. (A) Isolatedy Shear Key (B) Non-Isolated Shear Key NOTES: (a) Not all shear key bars shown (b) On high skews, use 2-inch expanded polystyrene with 1 inch expanded polystyrene over the 1-inch expansion joint filler to prevent binding on post-tensioned bridges. Figure SDC 7.8.4.1-1 Abutment Shear Key Reinforcement Details 21.2.15 No-Splice Zone Requirements No splices in longitudinal column reinforcement are allowed in the plastic hinge regions (see SDC Section 7.6.3) of ductile members. These plastic hinge regions are called “No-Splice Zones,” and shall be detailed with enhanced lateral confinement and shown on the plans. Chapter 21 – Seismic Design of Concrete Bridges 21-38 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 In general, for seismic critical elements, no splices in longitudinal rebars are allowed if the rebar cage is less than 60 ft. long. Refer to SDC Section 8.1.1 for more provisions for “No-Splice Zones” in ductile members. 21.2.16 Seismic Design Procedure Flowchart 1. Select column size, column reinforcement, and bent cap width (SDC Sections 3.7.1, 3.8.1, 7.6.1, 8.2.5) 2. Perform cross-section analysis to determine column effective moment of inertia (Material properties - SDC Sections 3.2.6, 3.2.3) Ensure that dead load on column ~ 10 % column ultimate ' compressive capacity Ag f c 3. Check span configuration/balanced stiffness (SDC Section 7.1.1) No Multi-frame bridge ? Yes o 4. Check frame geometry (SDC Eq. 7.1.2-1) 5. Calculate minimum local displacement ductility capacity and demand (SDC Sections 3.1.3, 3.1.4, 3.1.4.1) Check that local displacement ductility capacity,  c  3 Chapter 21 – Seismic Design of Concrete Bridges 21-39 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 6. Perform transverse pushover analysis (SDC Section 5.2.3, SDC Eq. 7.3.1.1-1) Check that: (a) displacement demand,  D < displacement capacity,  C (SDC Eq. 4.1.1-1) (b)  D  target value from SDC Section 2.2.4 7. Perform longitudinal pushover analysis (SDC Sections 5.2.3, 7.8.1) Check that: (a)  D <  C (b)  D  target value from SDC Section 2.2.4 8. Check P-  effects in transverse and longitudinal directions (SDC Eq. 4.2-1) 9. Check bent minimum lateral strength in transverse and longitudinal directions (SDC Section 3.5) 10. Perform column shear design in transverse and longitudinal directions (SDC Sections 3.6.1, 3.6.2, 3.6.3, 3.6.5, 4.3.1) No Are column bases pinned? Yes 11. Design column shear key (SDC Section 7.6.7) Chapter 21 – Seismic Design of Concrete Bridges 21-40 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 12. Check bent cap flexural and shear capacity (AASHTO Article 5.8, SDC Sect. 3.4, SDC Eq. 7.3.1.1-1) 13. Calculate column seismic load moments (SDC Sect. 4.3.2, MTD 20-6, SDC Eq. 4.3.1-1) col @ soffit @ soffit M eq  M 0col @ soffit  M dlcol @ soffit  M col ps   14. Distribute column seismic moments into the superstructure (S/S) to obtain S/S seismic demands (Perform right and left Pushover analyses) 15. Calculate S/S moment demands at location of interest (SDC Eq. 7.2.1.1-1, MTD 20-6) L L R M D  M dlL  M ps  M eqL ; M DR  M dlR  M ps  M eqR 16. Calculate S/S shear demands at location of interest (SDC Eq. 7.2.1.1-1, MTD 20-6) L L VD  VdlL  V ps  VeqL ; VDR  VdlR  VpsR  VeqR Is Site PGA  0.6g ? No Yes o 17. Perform vertical acceleration analysis (SDC Sections 7.2.2 and 2.1.3) Chapter 21 – Seismic Design of Concrete Bridges 21-41 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 18. Calculate superstructure flexural and shear Capacity (MTD 20-6, AASHTO Article 5.8) 19. Design joint shear reinforcement (SDC Sections 7.4.2, 7.4.4, 7.4.4.3, 7.4.5.1) .) Multi-frame bridge ? Yes No Yes 20. Determine minimum hinge seat width (SDC Section 7.2.5.4) No Seat type abutments ? Yes o 21. Determine minimum abutment seat width (SDC Section 7.8.3) 22. Design abutment shear key reinforcement (SDC Section 7.8.4) 23. Check requirements for No-splice Zone (SDC Section 8.1.1, MTD 20-9) END Chapter 21 – Seismic Design of Concrete Bridges 21-42 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 21.3 DESIGN EXAMPLE - THREE-SPAN CONTINUOUS CASTIN-PLACE CONCRETE BOX GIRDER BRIDGE 21.3.1 Bridge Data The three-span Prestress Reinforced Concrete Box Girder Bridge shown in Figure 21.3-1 will be used to illustrate the principles of seismic bridge design. The span lengths are 126 ft, 168 ft and 118 ft. The column height varies from 44 ft at Bent 2 to 47 ft at Bent 3. Both bents have a skew angle of 20 degrees. The columns are pinned at the bottom. The bridge ends are supported on seat-type abutments. Material Properties: Concrete: Reinforcing steel: fc  4 ksi A706, f y  60 ksi ; Es  29,000 ksi ; f ye  68 ksi ; fue  95 ksi Bridge Site Conditions: This example bridge crosses a roadway and railroad tracks. Because of poor soil conditions, the footing is supported on piles. The ground motion at the bridge site is assumed to be: Soil Profile: Type C Magnitude: 8.0  0.25 Peak Ground Acceleration: 0.5g Figure 21.3-2 shows the assumed design spectrum. For more information on Design Spectrum development, refer to SDC Section 2.1.1 and Appendix B. 21.3.2 Design Requirements Perform seismic analysis and design in accordance with Caltrans SDC Version 1.7 (Caltrans 2013). Chapter 21 – Seismic Design of Concrete Bridges 21-43 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 Figure 21.3-1 General Plan (Bridge Design Academy Prototype Bridge) Chapter 21 – Seismic Design of Concrete Bridges 21-44 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 Figure 21.3-2 Design Spectrum for Soil Profile C (M = 8.00.25) Chapter 21 – Seismic Design of Concrete Bridges 21-45 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 21.3.3 Step 1- Select Column Size, Column Reinforcement, and Bent Cap Width 21.3.3.1 Column size Given Ds  6.75 ft from the strength limit state design, we select a column width OK. (SDC 7.6.1-1) Dc  6.00 ft so that 0.70  Dc / Ds  0.89  1.00 . 21.3.3.2 Bent Cap Width Bcap  Dc  2 = 6 + 2 = 8 ft 21.3.3.3 (SDC 7.4.2.1-1) Column Longitudinal and Transverse Reinforcement   As  0.015Ag  0.015 6.00122  0.0154071.5  61.07 in.2 4 Use: #14 bars for longitudinal reinforcement #8 hoops @ 5 in c/c for the plastic hinge region Maximum spacing of hoops = 5 in. < 8 in. < 6×1.693 = 10.2 in. < 72/5 = 14.4 in. OK. (SDC Section 8.2.5) 61.07  27.1 Number of #14 bars = 2.25 Let us use 26-#14 longitudinal bars (i.e., 1.44% of Ag ) 1.0  1.44  4.0 OK. (SDC 3.7.1-1/3.7.2-1) Assuming a concrete cover of 2 in. as specified in CA Amendment Table 5.12.3-1 for minimum concrete cover (Caltrans 2014). Diameter of longitudinal reinforcement loop (from centerline to centerline of longitudinal bars):  1.88  d M = 72  2(2)  2(1.13)  2   63.86 in.  2   dM  7.7 in. > 1.5(1.693) in. > 1.5 in.  Spacing of longitudinal bars = 26 OK. (AASHTO 5.10.3.1) Note: If the provided spacing turns out to be more that the maximum spacing allowed, then a smaller bar size can be used. Chapter 21 – Seismic Design of Concrete Bridges 21-46 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 21.3.4 Step 2 - Perform Cross-section Analysis 21.3.4.1 Calculate Dead Load Axial Force As a first step toward calculating effective section properties of the column, the dead load axial force at column top (location of potential plastic hinge) is calculated. These column axial forces are obtained from CTBridge output. It should also be noted that these loads do not include the weight of the integral bent cap. The CTBridge model has the regular superstructure cross-section with flared bottom slab instead of solid cap section. In this example, weight of the whole solid cap was added to the CTBridge results (conservative). As read from the CTBridge output results, the column dead load axial forces are: Column 1 1,489 1,425 Bent 2 (Pc) (kip) Bent 3 (Pc) (kip) Average Bent Cap Length = Column 2 1,494 1,453  Deck Width  Soffit Width  1   2  cos(Skew Angle)  =  49.83  43.08  1    49.44ft 0   2  cos(20 )  Bent Cap Weight = 8(6.75)(49.44)(0.150)  400 kips Adding this bent cap weight, the total axial force in each column becomes: Bent 2 (Pc) (kip) Bent 3 (Pc) (kip) 21.3.4.2 Column 1 1,689 1,625 Column 2 1,694 1,653 Check Column Dead Load Axial Force Ratio Using Column 2 of Bent 2 (worst case): Chapter 21 – Seismic Design of Concrete Bridges 1694100%  10.4 % ~ 10% 4071.5 4  OK. 21-47 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 21.3.4.3 Material and Section Properties for Section Analysis Using xSECTION Program Expected compressive strength of concrete f ce'  1.3(4,000)  5,200 psi * > 5,000 psi OK. (SDC 3.2.6-3) * The xSECTION input file was originally created with the value of f ce'  5.28 ksi. The resulting values of ductility parameters are not significantly different from the corresponding values obtained using f ce'  5.20 ksi. Therefore, the results with f ce'  5.28 ksi are retained. Other concrete properties used are listed in SDC Section 3.2.6. The following values are used as input to xSECTION program:     Column Diameter = 72.0 in. Concrete cover = 2 in. Main Reinforcement: #14 bars, total 26. Lateral Reinforcement: #8 hoops @ 5 in c/c, f ce'  5,200 psi * The program calculates the modulus of elasticity of concrete internally. For Grade A706 bar reinforcing steel, 0.09 0.06 T ransverse steel Longitudinal steel  R   su  Select Output for Bent 2 Column xSECTION run is shown in Appendix 21.3-1. Moment-Curvature ( M   ) diagram for Bent 2 Column is shown in Appendix 21.3-2. Bent 2 Column Axial Force, Pc = 1,694 kips. Bent 3 Column Axial Force, Pc = 1,653 kips.    From M-ϕ analysis results, cracked moment of inertia, Ie = 23.717 ft4 for Bent 2 columns (See Appendices 21.3-1 and 21.3-2). For Bent 3, Ie = 23.612 ft4. 21.3.5 Step 3 - Check Span Configuration/Balanced Stiffness 21.3.5.1 Bent 2 Stiffness  331501.5 5,200  Ec  33wc 1.5 f c' ( psi)     4,372 ksi 1,000   k2e  (2) (SDC 3.2.6-1)  (3)(4,372)(23.717)(124 )  3EIe  ( 2 )   87.64 kip/in.  (44  (12))3 L3   Chapter 21 – Seismic Design of Concrete Bridges 21-48 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 21.3.5.2 Bent 3 Stiffness  (3)(4,372)(23.612)(124 )  k3e  (2)    71.59 kip/in. (47(12))3   (2)(1,694) m2 = Total tributary mass at Bent 2 =  8.77 kip  s 2 /in. (32.2)(12) (2)(1,653) m3 = Total tributary mass at Bent 3 =  8.56 kip  s 2 /in. (32.2)(12) kie 71.59   0.82  0.75  0.5 k ej 87.64 OK. (SDC 7.1.1-1 and 7.1.1-3) It is seen that the balanced stiffness criteria and span layout configuration are satisfied. Note that since this is a constant width bridge with only two bents and two columns in each bent, we only need to satisfy the more onerous of SDC Equations (7.1.1-1) and (7.1.1-3). 21.3.6 Step 4 - Check Frame Geometry Since this is a single-frame bridge, this step does not apply. 21.3.7 Step 5 – Calculate Minimum Local Displacement Ductility Capacity and Demand 21.3.7.1 Displacement Ductility Capacity (1) Bent 2 Columns L = 44 ft Y  0.000078rad/in. as read from the M   data listed in Appendix 21.3-1. Lp  0.08L  0.15 f ye dbl  0.3 f ye dbl  0.08(528)  0.15(68)(1.693)  59.51in.  0.3(68)(1.693)  34.54in. OK. (SDC 7.6.2.1-1) 2 L  1 (SDC 3.1.3-2) Y   Y  (528) 2 (0.000078)  7.25 in.  3  3   Plastic curvature,  p  0.000747 rad/in. (See M   data shown in Appendices 21.3-1 and 21.3-2). Plastic rotation,  p  L p p  59.51 0.000747 0.044454 rad. Chapter 21 – Seismic Design of Concrete Bridges (SDC 3.1.3-4) 21-49 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 Lp   59.51    0.044454 528  Plastic displacement,  p   p  L    22.15 in. 2  2    (SDC 3.1.3-8) Total Displacement Capacity, c  Y   p  7.25  22.15  29.40 in. (SDC 3.1.3-1)  c  29.40  = Local displacement ductility capacity, c    4.1  3 Y  7.25  OK. (SDC Section 3.1.4.1) (2) Bent 3 Columns Similarly,  p = 24.93 in., Y = 8.27 in. c  21.3.7.2  c  33.20  =   4.0  3 Y  8.27  OK. Displacement Ductility Demand (1) Bent 2 The period of fundamental mode of vibration is as: 8.77 m2 T2  2  2  1.99 sec. e 87.64 k2 From the Design spectrum shown in Figure 21.3-2, the value of spectral acceleration for T = 1.99 sec is read as: a2  0.36g ma 8.77(0.36)(32.2)(12)  13.92 in. 87.64 ke 13.92  1.9  5 OK. (SDC Section 2.2.4) Displacement Demand ductility,  D  7.25 Displacement demand,  D   (2) Bent 3 Similarly, for Bent 3, T3  2.17 sec. The longer period is expected because Bent 3 columns are longer. The corresponding value of spectral acceleration, a3  0.33g (Figure 21.3-2) Displacement demand,  D  8.56(0.33)(32.2)(12)  15.25 in. 71.59 Displacement Demand ductility,  D  Chapter 21 – Seismic Design of Concrete Bridges 15.25  1.8  5 8.27 OK. 21-50 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 21.3.8 Step 6 – Perform Transverse Pushover Analysis 21.3.8.1 Modeling Figure 21.3-3 shows a schematic model of the frame in the transverse direction. Data used for the soil springs are shown in Appendix 21.3-3. The following values of column effective section properties for Bent 2 and idealized plastic moment capacity (under dead loads only) obtained from xSECTION output (see Appendix 21.3-1) are used as input in wFRAME program for pushover analysis. Pc (kip) Mp (kip-ft) Ie (ft4) ϕy ( rad/in.) ϕp ( rad/in.) 1,694 13,838 23.717 0.000078 0.000747 Appendices 21.3-4 and 21.3-5 show select portions of xSECTION output for the cap section for positive and negative bending, respectively. The following section properties are used for the wFRAME run: ve ve  55.57 ft 4 , I eff  48.94 ft 4 A  62.62 ft 2 *, I eff *Note that per SDC Equation 7.3.1.1-1, the value of A (effective bent cap cross sectional area) would be 66.62 ft2. The value of 62.62 ft4 used is based on effective bent cap overhang width of 34 in. required by California Amendment Article 4.6.2.6.1 (Caltrans 2014). However, any errors introduced by using A = 62.62 ft4 instead of A = 66.62 ft4 would result in a conservative design. As the frame is pushed toward the right, the resulting overturning moment causes redistribution of the axial forces in the columns. This overturning causes an additional axial force on the front side column, which will experience additional compression. The column on the backside experiences the same value in tension, reducing the net axial load. Based on their behavior, these columns are usually known as compression and tension columns, respectively. At the instant the first plastic hinge forms (in this case at the top of the compression column), the following superstructure displacement and corresponding lateral force values are obtained from the wFRAME output (see Appendix 21.3-6):  y  8.49 in. Corresponding lateral force = 0.171(3,382) = 578 kips, where, 3,382 kips is the total tributary weight on the bent. At this stage, the axial forces in tension and compression columns as read from the wFRAME analysis output are 907 kips and 2,474 kips, respectively. Chapter 21 – Seismic Design of Concrete Bridges 21-51 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 3.38 Rigid Links 35.80 7.72* 7.72* 34* L OOSE Sand SAND: Loose N=10, , o, N = 10,ϕ=30 = O30 KK==2525PCI pci 3.28 8.20 Medium Dense Sand MEDIUM D ENSE SAND N = 20, ϕ = 33o, N=20, =33 degrees K = 150 pci K=150 pci * Dimensions along the skewed bent line Figure 21.3-3 Transverse Pushover Analysis Model These values can be quickly checked using simple hand calculations as described below: M overturning  57844  25,432 kip - ft Axial compression corresponding to M overturning , P   Chapter 21 – Seismic Design of Concrete Bridges 25,432  748 kips 34 21-52 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 The axial force in the compression column will increase to 1,694 + 748 = 2,242 kips. The tension column will see its axial compression drop to 1,694-748 = 946 kips. These values compare very well with the wFRAME results. The small differences are probably due to the presence of soil in the more realistic wFRAME model. Column section properties corresponding to the updated axial forces (i.e. with overturning) are obtained from new xSECTION runs and summarized in the table below (see Appendices 21.3-7 and 21.3-8 for select portions of the output for the compression and tension columns, respectively). Column Type Pc (kip) Mp (kip-ft) Ie (ft4) ϕy ( rad/in.) ϕp ( rad/in.) Tension 907 12,636 21.496 0.000079 0.000836 Compression 2,474 14,964 25.572 0.000079 0.000682 Note that higher compression produces a higher value of Mp but a reduction in p. This trend occurs in all columns and is a reminder that Mp is not the only indicator of column performance. With updated values of Mp and Ie, we run a second iteration of the wFRAME program. As the frame is pushed laterally, the compression column yields at the top at a displacement y(1) = 8.79 inches. The tension column has not reached its capacity yet. See Appendix 21.3-9 for these results. At this stage, the column axial forces are read to be 880 kips and 2,502 kips for tension and compression columns, respectively. Since, the change in column axial load is now less than 5%, there is no need for further iteration. As the frame is pushed further, the already yielded compression column is able to undergo additional displacement because of its plastic hinge rotational capacity. As the bent is pushed further, the top of the tension column yields at a displacement, y(2) = 10.52 in (see Appendix 21.3-9). At this point the effective bent stiffness approaches zero and will not attract any additional force if pushed further. The bent, however, will be able to undergo additional displacement until the rotational capacity of one of the hinges is reached. The force-displacement relationship is shown in Appendix 21.3-10. The idealized yield y, which was calculated previously based upon the assumption that cap beam is infinitely rigid, is updated to 8.79 inches. The corresponding lateral force = 0.176(3,382) = 595 kips. Chapter 21 – Seismic Design of Concrete Bridges 21-53 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 21.3.8.2 Displacement Ductility Capacity The main purpose of the preliminary calculation for C was to size up the members and ensure that they meet the minimum local displacement ductility capacity of 3 before proceeding with the more realistic and comprehensive pushover analysis that includes the effects of bent cap flexibility. The displacement capacities for both columns are calculated as before (see Step 5) using updated values of ϕp, and summarized below: Tension Column Compression Column L = 44 ft, Lp = 59.51 in. L = 44 ft, L p  59.51 in. ϕp = 0.000836 rad/in. ϕp = 0.000682 rad/in. Δp = 24.79 in. Δp = 20.22 in. Δc = 10.52 + 24.79 = 35.31 in. Δc = 8.79 + 20.22 = 29.01 in. For bents having a large number of columns or more locations for potential hinging, tabulation of these results provides a quick way to determine the critical hinge. Hinge Location Compression Column Top Tension Column Top Yield Displacement (in.) Plastic Deformation (in.) Total Displacement Capacity (in.) 8.79 20.22 29.01* 10.52 24.79 35.31 * Critical bent displacement capacity, C. 21.3.8.3 Displacement Ductility Demand (1) Bent 2 k2e  Fy y T  2  595 k  67.69 8.79 in 8.77  2.26 sec 67.69 From the Design Spectrum (DS) curve, the spectral acceleration a2 is read as 0.32g. The maximum seismic displacement demand is estimated as: 8.77  (0.32  32.2  12)  16.02 in. 67.69 16.02 D   1.82 < 5 8.79 D  Chapter 21 – Seismic Design of Concrete Bridges OK. (SDC Section 2.2.4) 21-54 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 Also, D  16.02 in.  C  29.01 in. OK. (SDC 4.1.1-1) Note that the bent is forced well beyond its yield displacement but that collapse is prevented because of ductile capacity. This is what we expect of Caltrans “No Collapse” Performance Criteria. Based upon these checks one might conclude that the column is over designed for the anticipated seismic demand. However, as shown later, the P- effect controls the column flexural design. The above calculation was made assuming the bent stiffness equals the stiffness at first yield. This assumption is valid because the two hinges occurred close to each other (i.e., 8.79 in and 10.52 in). If this assumption is not valid, a more exact calculation may be carried out using idealized stiffness as follows (see Figure 21.34): Fy 640   66.67 kip/in. k 2e  y 9.6 T  2 8.77  2.28 sec . 66.67 a2 = 0.3g; D  D  8.77(0.32)(32.2)(12)  16.27 in. 66.67 16.27  1.69  5 9.6 OK.  D  16.27 in.  C  29.01in. OK. 640 595 640 Force (kip) 8.79 9.60 10.52 Displacement (in.) Figure 21.3-4 Force – Displacement Relations Chapter 21 – Seismic Design of Concrete Bridges 21-55 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 (2) Bent 3 The same procedure is repeated to perform transverse pushover analysis for Bent 3. The results are summarized below: Tension Column L  47 ft, L p  62.39 in. Compression Column L  47 ft, L p  62.39 in.  p  0.000842rad/in.  p  0.000685rad/in.  p  27.99 in.  p  22.77 in.  c  11.48  27.99  39.47 in.  c  9.71  22.77  32.48 in.* * Critical bent displacement capacity, C. Seismic Demand ke3  Fy y  0.180(3,278)  60.77 kip/in. 9.71 8.56  2.36 sec 60.77 From Design Spectrum, the spectral acceleration a3 is read as 0.31g. The period of vibration, T  2 8.56(0.31)(32.2)(12)  16.87 in. 60.77 16.87 D   1.74  5 9.71 D  Also, D  16.87 in.  C  32.48 in. 21.3.9 Step 7- Perform Longitudinal Pushover Analysis 21.3.9.1 Abutment Soil Springs OK. OK. This bridge is supported on seat type abutments (see Figure 21.3-5 for effective abutment dimensions). The effective area is calculated as: Ae  hbwwbw  6.75(46.46)  313.6ft 2 ( wbw  (49.83  43.08) / 2  46.46ft ) h   6.75  Pw  Ae (5 ) bw   (313.6)(5)   1,924 kips  5.5   5.5  Chapter 21 – Seismic Design of Concrete Bridges (SDC 7.8.1-4) (SDC 7.8.1-3) 21-56 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 wbw Figure 21.3-5 hbw Effective Area of Seat Type Abutment Using initial embankment fill stiffness,  kips/in.  Ki  50    ft  Initial abutment stiffness (SDC 7.8.1-1)  h   6.75  K abut  Ki w   50(46.46)   2,851 kip/in.  5.5   5.5   (SDC 7.8.1-2) F 1,924   0.67 in. (See Figure 21.3-6) K 2,851 effective     gap  0.67  2.60  3.27 in.  0.272ft See Appendix 21.3-11 for calculations for gap, the combined effect of thermal movement and anticipated shortening. Average contributory length is used in the calculation for gap. Abut Kinitial  1,924  588 kip/in. 7,061 kip/ft 3.27 1,924 kip 588 kip/in. 2.6 in. 0.67 in. Figure 21.3-6 Initial Abutment Stiffness Iteration Chapter 21 – Seismic Design of Concrete Bridges 21-57 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 This value is used as the starting abutment stiffness for the longitudinal pushover analysis. When the structure has reached its plastic limit state (i.e., when both bents 2 and 3 columns have yielded), the longitudinal bridge stiffness is calculated as follows: 0.38(8,430) klong   351 kip/in. 9.13 (See Appendix 21.3-12 for the force-deflection curve for Right Push). Mass, m  T  2 W 8,430   21.82 kip  s 2 /in. g 32.2  12 m klong  2 21.82  1.57 sec 351 Sa  0.48g F ma 21.82(0.48)(32.2)(12) D     11.53 in. K K 351 RA  D  effective  11.53  3.53 3.27 Abut Abut  Kinitial  1.0  0.45( RA  2) Since 2 < R A < 4, K final (SDC Section 7.8.1) Abut K final  588(0.312)  183 kip/in.  2196kip/ft The following stiffness values as shown in Figure 21.3-7 shall be used for all subsequent wFRAME longitudinal pushover analyses: K1 = 2,196 kip/ft and 1 = 0.272 ft K2 = 0 kip/ft and 2 = 1.0 ft K2 K1 0.272 ft 1.0 ft Figure 21.3-7 Final Abutment Stiffness Chapter 21 – Seismic Design of Concrete Bridges 21-58 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 21.3.9.2 Displacement Ductility Capacity and Demand From the wFRAME results (see Appendix 21.3-13 for the force-displacement relationship for the right push), the yield displacements of Bent 2 and Bent 3 are: Location Bent 2 Bent 3 Yield Displacement (Right Push) (in.) 8.86 9.11 Yield Displacement (Left Push) (in.) 8.36 9.84 The plastic deformation capacities for both Bent 2 and Bent 3 are exactly the same as calculated for the transverse bending for the case of gravity loading. This is because the longitudinal case has very little overturning to change the column axial loads.   p = 22.15 in. for Bent 2 and p = 24.93 in. for Bent 3. (1) Bent 2 Min c  c  8.86  22.15  =   3.5  3 Y  8.86  OK. (SDC Section 3.1.4) (2) Bent 3 Min c  c  9.84  24.93     3.5  3 y  9.84  OK. (SDC Section 3.1.4) From wFRAME force-displacement relationship of Appendix 21.3-13, the bridge longitudinal stiffness is calculated when the first bent has yielded. 0.22(8,430) klong   209 kip/in. 8.86 T = 2.03 sec for which Sa = 0.35g    D = 14.12 in. This demand is the same at Bents 2 and 3 because the superstructure constrains the bents to move together. This might not be the case when the bridge has significant foundation flexibility that can result from rotational and/or translational foundation movements. 14.12 Max  D   1.7  5 (Bent 2) OK. (SDC Section 2.2.4) 8.36 14.12 Max  D   1.5  5 (Bent 3) OK. (SDC Section 2.2.4) 9.11 Chapter 21 – Seismic Design of Concrete Bridges 21-59 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 21.3.10 Step 8 - Check P - Δ Effects 21.3.10.1 Transverse direction We have relatively heavily loaded tall columns. P-effects could be significant for this type of situation. (1) Bent 2 Columns Pdl = 1,694 kips, M p = 13,838 kip-ft, Maximum Seismic Displacement Δr = 16.02 in. Pdl  r 1,694(16.02)   0.16  0.20 13,838(12) M col p OK. (SDC 4.2-1) (2) Bent 3 Columns Pdl = 1,653 kips, M p = 13,777 kip-ft, Maximum Seismic Displacement Δr = 16.87 in. Pdl  r 1,653(16.87)   0.17  0.20 13,777(12) M col p OK. (SDC 4.2-1) Now we can see that although the selected column section has more than enough ductility capacity, the column sections meet the P- requirements only by a small margin. 21.3.10.2 Longitudinal Direction (1) Bent 2 Columns Pdl  r 1,694(14.12)   0.14  0.20 13,838(12) M col p OK. (SDC 4.2-1) OK. (SDC 4.2-1) (2) Bent 3 Columns Pdl  r 1,653(14.12)   0.14  0.20 13,777(12) M col p 21.3.11 Step 9 - Check Bent Minimum Lateral Strength 21.3.11.1 Transverse direction From the force deflection data shown in Appendix 21.3-10, Chapter 21 – Seismic Design of Concrete Bridges 21-60 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 Minimum lateral strength per bent = 0.193,383  643kips  0.1(3,383)  338 kips 2.3.11.2 OK. (SDC Section 3.5) Longitudinal Direction From the force deflection data shown in Appendix 21.3-13, Minimum lateral strength per column =  8,430   8,430  0.22 OK.   927 kips  0.1   422 kips  2   2  21.3.12 Step 10 - Perform Column Shear Design 21.3.12.1 Transverse Bending (SDC Section 3.5) (1) Bent 2 M 0  1.2M p  1.2(14,964)  17,957 kip  ft (includes overturning effects). Shear demand associated with overstrength moment is as: M 17,957 V0  0   408 kips L 44 Alternatively, from wFRAME output (see Appendix 21.3-9), the maximum column shear demand = 1.2(349)  419 kips. The presence of soil around the footing results in a slightly shorter effective column length, which in turn causes slightly higher column shear demand in the wFRAME output. Concrete Shear Capacity, Vc For #8 hoops @ 5 in o.c., Ab  0.79 in.2 , D'  72  2  2  1.13 1.13   66.87 in. , s  5 in. 2 2 4 Ab  0.009451 D's f yh = 60 ksi s  (SDC 3.8.1-1) s f yh  0.009451(60)  0.57  0.35  Use  s f yh  0.35 ksi (SDC Section 3.6.2) Using the maximum value of the displacement ductility demand, d = 1.82 (see calculation for Bent 2 – Transverse pushover analysis), the shear capacity factor f1 is calculated as: Chapter 21 – Seismic Design of Concrete Bridges 21-61 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 f1   s f yh 0.150  3.67   d  0.35  3.67  1.82  4.18  3 0.150  Use f1 = 3 f2  1  Pc 1 2,000Ag  Use f 2  1.11 (SDC 3.6.2-5) 88010  1.11  1.5   2 2,000 6  (12)  4 3 OK. It is seen from the equations for concrete shear capacity, that the plastic hinge region is more critical as the capacity will be lower in this region. Furthermore, the shear capacity is reduced when the axial load is decreased. The controlling shear capacity will be found in the tension column. vc  f1 f 2 f c'  3(1.11) 4,000  211psi  4 4,000  253 psi OK.   Ae  0.8 6(12) 2  3,257in.2 4 Vc  vc Ae  211(3,257)  687,227 lb  687 kips Transverse Reinforcement Shear Capacity, Vs  nAv f yh D'    0.79(60)(66.87)    Vs     996 kips  2 2 5 s    Maximum shear strength is as: Vs, max  8 f c' Ae  8 4,0003,257 / 1,000  1,648 kips  996 kips Minimum shear reinforcement is as:  D's   Av, min  0.025  f yh     66.87(5)  2 2  0.025   0.14 in.  0.79 in.  60  Shear capacity OK. (SDC 3.6.5.1-1) OK. (SDC 3.6.5.2-1) Vn  0.9Vc  Vs   0.9(687  996)  1,515 kips  V0  419 kips Chapter 21 – Seismic Design of Concrete Bridges OK. 21-62 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 (2) Bent 3 V0  M 0 1.2(14,893)   380 kips L 47 From the wFRAME analysis results, the maximum column shear demand = 1.2 × 340 = 408 kips. Going through a similar calculation as was done for Bent 2 columns, we determine that Vn  0.9Vc  Vs   0.9(681 996)  1,509 kips  V0  408 kips 21.3.12.2 OK. Longitudinal bending (1) Bent 2 V0 = 1.2Vp = 1.2(645/2) = 377 kips This corresponds to the maximum shear value of Vp = 323 kips/column obtained from the wFRAME pushover analysis. For  D = 1.7, f 1 = 4.3 > 3. Use f 1 = 3. For dead load axial force, factor f 2 = 1.21 vc = 230 psi which gives Vc = 749 kips Vs = 996 kips as calculated before. Vn  0.9(749  996)  1,571 kips  V0  307 kips OK. (2) Bent 3 V0 = 1.2Vp = 1.2(629/2) = 378 kips This corresponds to the maximum shear value of Vp = 315 kips/column obtained from the wFRAME pushover analysis. For  D = 1.5, factor 1 = 4.5 > 3. Use f 1 = 3 For dead load axial force, factor f 2 = 1.20  vc = 228 psi which gives Vc = 743 kips Vs = 996 kips as calculated earlier. Vn  0.9(743  996)  1,565 kips  V0  378 kips Chapter 21 – Seismic Design of Concrete Bridges OK. 21-63 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 21.3.13 Step 11- Design Column Shear Key 21.3.13.1 Determine Shear Key Reinforcement Since the net axial force on both columns of Bent 2 is compressive, the area of interface shear key, required Ask is given by Ask  1.2( Fsk  0.25P) fy (SDC 7.6.7-1) P = 815 kips (for column with the lowest axial load) – see Appendix 21.3-9 Shear force associated with column overstrength moment is as: Fsk = shear force associated with column overstrength moment 1.2349  419 kips For Bent 2 Fsk   1.2340  408 kips For Bent 3 See Step 10 – Perform Column Shear Design and Appendix 21.3-9. Therefore, Fsk = 419 kips 1.2419  0.25(815)  4.3 in.2  4 in.2 OK. 60 Provide 6#8 dowels in column key ( Ask , provided = 4.74 in.2 > 4.3 in.2 OK.) Ask  Dowel Cage diameter: Preferred spacing of #8 bars = 4.25 in. (see BDD 13-20) Diameter of dowel cage = (6)(4.25)/ = 8.1 in. say 9 in. cage 21.3.13.2 Determine Concrete Area Engaged in Shear Transfer, Acv Acv  4.0(419)  419 in.2 4 4.0 Fsk = f c' (SDC 7.6.7-3) Acv  0.67Fsk  281 in.2 (SDC 7.6.7-4) Per SDC Section 7.6.7, Acv must not be less than that required to meet the axial resistance requirements specified in AASHTO Article 5.7.4.4 (AASHTO 2012).  Pn   0.85 0.85 f c' ( Ag  Ast )  f y Ast  (AASHTO 5.7.4.4-2) Using the largest axial load with overturning effects P = 2,567 kips (see Appendix 21.3-9) and  = 1 (seismic), we have:   Pn  1.00.85 0.854( Ag  4.74)  604.74  2,567 kips Ag = 809 in.2 > 419 in.2 Chapter 21 – Seismic Design of Concrete Bridges 21-64 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 Therefore, Acv,reqd  809 in.2 Diameter of Acv  8094  32  in. Use Acv diameter = 32 in. (see Figure 21.3-8) 9 in. Cage 6-#8 Dowel s Acv Diameter = 32 in. Figure 21.3-8 Column Shear Key 21.3.14 Step 12 - Check Bent Cap Flexural and Shear Capacity 21.3.14.1 Check Bent Cap Flexural Capacity The design for strength limit states had resulted in the following main reinforcement for the bent cap: Top Reinforcement 22 - #11 rebars Bottom Reinforcement 24 - #11 rebars Ignoring the side face reinforcement, the positive and negative flexural capacity of the bent cap is estimated to be M+ve = 21,189 kip-ft and M-ve = 19,436 kip-ft. Appendices 21.3-4 and 21.3-5 show these values, which are based on when either the R as required for capacity concrete strain reaches 0.003 or the steel strain reaches  SU protected members (See SDC Section 3.4). The seismic flexural and shear demands in the bent cap are calculated corresponding to the column overstrength moment. These demands are obtained from Chapter 21 – Seismic Design of Concrete Bridges 21-65 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 a new wFRAME pushover analysis of Bent 2 with column moment capacity taken as Mo. As shown in Appendix 21.3-14 (right pushover), bent cap moment demands are: M Dve  14,350 kip  ft  M ve  21,189 kip  ft OK. M Dve  15,072 kip  ft  M ve  19,436 kip  ft OK. The associated shear demand obtained from the above pushover analysis, Vo = 2,009 kips. 21.3.14.2 Check Bent Cap Shear Capacity Nominal shear resistance of the bent cap, Vn is the lesser of: Vn  Vc  Vs + Vp (AASHTO 5.8.3.3-1) Vn  0.25 f c'bv dv + Vp (AASHTO 5.8.3.3-2) Vc  0.0316 (AASHTO 5.8.3.3-3) and where: Vs  bv dv f c' bv d v Av f y d v cot (AASHTO C5.8.3.3-1) s Vp = 0 (bent cap is not prestressed) = effective web width = 8 ft = 96 in. = effective shear depth = distance between the resultants of the tensile and compressive forces due to flexure, not to be taken less than the greater of 0.9d e or 0.72h (see AASHTO Article 5.8.2.9). 0.72 h = 0.72 (81) = 58.3 in. Assuming clear distance from cap bottom to main bottom bars = 5 in. d e = cap effective depth = 81-5-1.63/2 = 75.2 in. 0.9d e = 0.9(75.2) = 67.7 in. > 58.3 in. Therefore, d v, min = 67.7 in Method 1 of AASHTO Article 5.8.3.4 (AASHTO 2012) is used to determine the values of β and  (the bent cap section is non-prestressed and the effect of any axial tension is assumed to be negligible).  Use β = 2.0 and  = 45o per AASHTO Article 5.8.3.4.1. Vc  0.0316 f c' bv dv  0.0316(2)( 4 )(96)(67.7)  821 kips Assuming 6-legged, #6 stirrups @ 7 in. o.c. transverse reinforcement (see Figure 21.3-19). Chapter 21 – Seismic Design of Concrete Bridges 21-66 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 Av f y d v cot 6(0.44)(60)(67.7)(cot 45)  1,532 kips 7 s Vn  Vc  Vs  821 1,532  2,353kips Vs   Vn  0.25 f c'bv dv  0.25(4)(96)(67.7)  6,499kips  2,353kips  Vn  2,353kips   Vn  0.9(2,353)  2,118kips  V0  2,009kips OK. 21.3.15 Step 13 - Calculate Column Seismic Load Moments 21.3.15.1 Determine Dead Load, Additional Dead Load, and Prestress Secondary Moments at Column Tops/Deck Soffit For this bridge, the top of bent support results from CTBridge (Table 21.3-1) will need to be transformed to the consistent planar coordinate system (i.e., the plane formed by the centerline of the bridge and the vertical axis) to ensure consistency with wFRAME results and to account for the bridge skew. To do so, the following coordinate transformation (see Figure 21.3-9) will be applied to the top of column moments from CTBridge. Table 21.3-1 Top of Bent Column Moments (kip-ft) from CTBridge Bent 2 3 Skew Mz (Degree) 20 -1,189 20 1,305 DL My Mlong Mz ADL My Mlong Mz 91 -1 -1,148 1,227 -213 234 17 -1 -207 220 82 -127 Sec. PS My Mlong -371 287 204 -218 It is noted that the above values are for both columns in each bent. (1) Moment at Column top - Bent 2  Dead load and additional dead load moments (Figure 21.3-10) col,bottom  0 kip - ft Column moment at base, M dl Column moment at deck soffit, (CTBridge Output) M dlcol,top @ jo int   1,148   207  1,355 kip - ft  Secondary Prestress Moments (Figure 21.3-11) ,bottom Column moment at base, M col  0 kip - ft ps (CTBridge Output) ,top @ jo int Column moment at deck soffit, M col  204 kip - ft ps Chapter 21 – Seismic Design of Concrete Bridges 21-67 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 CL BRIDGE  (positive rotation) My My Mz Mz  (positive skew) Tx = Tx (out of page) CL BENT Column Mz = Mz cos - My sin My = Mz sin + My cos Tx = Tx (Longitudinal Moment) (Transverse Moment) (Torsional Moment) Figure 21.3-9 Coordinate Transformation from Skewed to Unskewed Configuration Deck Soffit 30.8 kips 1,355 kip-ft Column 30.8 kips Figure 21.3-10 Free Body Diagram Showing Equilibrium of Dead Loading at Bent 2 Chapter 21 – Seismic Design of Concrete Bridges 21-68 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 204 kip-ft Deck Soffit 4.6 kips Column 4.6 kips Figure 21.3-11 Free Body Diagram Showing Equilibrium of Secondary Prestress Forces at Bent 2 (2) Moment at Column Top - Bent 3  Dead load and additional dead load moments col,top @ jo int  0 kip - ft Column moment at base, M dl (CTBridge Output) Column moment at deck soffit, M dlcol,top @ jo int   1,227   220  1,447 kip - ft  Secondary Prestress Moments ,bottom  0 kip  ft Column moment at base, M col ps (CTBridge Output) ,top @ jo int  218 kip - ft Column Moment at deck soffit, M col ps 21.3.15.2 Determine Earthquake Moments in the Superstructure (1) Dead Load and Additional Dead Load Moments CTBridge output lists these moments at every 1/10th point of the span length and at the face of supports (see Table 21.3-2). (2) Secondary Prestress Moments CTBridge output lists these moments at every 1/10th point of the span length and at the face of supports (see Table 21.3-2). Chapter 21 – Seismic Design of Concrete Bridges 21-69 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 Table 21.3-2 Dead Load and Secondary Prestress Moments from CTBridge Output Location Span 3 Span 2 Span 1 x/L Support 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Support Support 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Support Support 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Support x (ft) 1.5 12.6 25.2 37.8 50.4 63 75.6 88.2 100.8 113.4 123 129 142.8 159.6 176.4 193.2 210 226.8 243.6 260.4 277.2 291 297 305.8 317.6 329.4 341.2 353 364.8 376.6 388.4 400.2 410.5 Whole Superstructure Width M DL M ADL M PS (kip-ft) (kip-ft) (kip-ft) 619 114 647 7110 1275 1462 12158 2178 2272 14741 2640 3096 14857 2661 3956 12508 2240 4705 7693 1377 5617 412 74 6400 -9334 -1671 7911 -21553 -3857 8498 -32599 -5819 8672 -33654 -6009 8468 -17502 -3136 9516 -1955 -354 9005 9208 1645 8318 15989 2859 8281 18388 3289 8027 16406 2935 8072 10043 1795 7905 -699 -128 8355 -15820 -2835 8645 -31614 -5646 7554 -30429 -5434 7482 -20789 -3723 7275 -9854 -1766 6861 -1093 -197 5559 5506 986 4870 9943 1781 4085 12219 2189 3417 12333 2210 2669 10286 1844 1945 6077 1091 1230 637 117 529 Chapter 21 – Seismic Design of Concrete Bridges Per Girder M DL M ADL M PS (kip-ft) (kip-ft) (kip-ft) 124 23 129 1422 255 292 2432 436 454 2948 528 619 2971 532 791 2502 448 941 1539 275 1123 82 15 1280 -1867 -334 1582 -4311 -771 1700 -6520 -1164 1734 -6731 -1202 1694 -3500 -627 1903 -391 -71 1801 1842 329 1664 3198 572 1656 3678 658 1605 3281 587 1614 2009 359 1581 -140 -26 1671 -3164 -567 1729 -6323 -1129 1511 -6086 -1087 1496 -4158 -745 1455 -1971 -353 1372 -219 -39 1112 1101 197 974 1989 356 817 2444 438 683 2467 442 534 2057 369 389 1215 218 246 127 23 106 21-70 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 (3) Case 1 Earthquake Loading: Bridge moves from Abutment 1 towards Abutment 4 As shown in Figure 21.3-12, such loading results in positive moments in the columns according to the sign convention used here. Abut 1 Abut 4 Bent 2 Bent 3 Figure 21.3-12 Seismic Loading Case “1” Producing Positive Moments in Columns As calculated previously, the columns have already been “pre-loaded” by: col @ soffit @ soffit M dl  M col  (1,355)  (204)  1,151kip - ft (Bent 2) ps col @ soffit @ soffit M dl  M col  (1,447)  (218)  1229 kip - ft (Bent 3) ps Column moment generated by seismic loading at column soffit is: col @ soffit @ soffit @ soffit M eq  1.2M col (M dlcol  M col ) p ps  1.2(2)(13,838)  (1,355)  0)  34,566 kip - ft (Bent 2) It should be noted that the secondary prestress moment is neglected because doing so results in increased seismic demand on the column and hence in the superstructure. Figure 21.3-13 schematically explains this superposition approach. col @ soffit M eq  1.2(2)(13,777)  (1,447  218)  31,835 kip - ft (Bent 3) It should be noted that for Bent 3, the effect of secondary prestress moments is included because doing so results in increased seismic moment in the columns and hence in the superstructure. Chapter 21 – Seismic Design of Concrete Bridges 21-71 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 30.8 kips 4.6 kips Deck Soffit col Veq 204 kip-ft 1,355 kip-ft col M eq Column n + + 4.6 kips 30.8 kips Dead Load and Additional Dead Load and Dead Load col Veq Secondary Prestress State Secondary Prestress Seismic State 755 kips M 0col  1.2M Pcol  33,211kip - ft 755 kips Collapse Limit State Figure 21.3-13 Superposition of Column Forces at Bent 2 for Loading Case “1” (4) Case 2 Earthquake Loading: Bridge moves from Abutment 4 towards Abutment 1 As shown in Figure 21.3-14, such loading results in negative moments in the columns according to our sign convention. Abut 4 Abut 1 Bent 2 Bent 3 Figure 21.3-14 Seismic Loading Case “2” Producing Negative Moments in Columns Chapter 21 – Seismic Design of Concrete Bridges 21-72 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 Bent 2 col @ soffit col col  1.2M col M eq p  ( M dl  M ps )  1.2(2)(13,838)  (1,355  204)  32,060 kip  ft Bent 3 col @ soffit col col M eq  1.2M col p  ( M dl  M ps )  1.2(2)(13,777)  (1,447  0)  34,512 kip - ft Figure 21.3-15 schematically shows the Free Body Diagram at Bent 2 for this seismic loading case. 30.8 kips 1,355 kip-ft 4.6 kips Deck Soffit Veqcol 204 kip-ft col M eq Column + + + 4.6 kips 30.8 Secondary Prestress State Secondary Prestress Dead Load Additional Deadand Load and Dead Load col Veq Seismic State 755 kips M 0col  1.2M Pcol  33,211 kip - ft = 755 kips Limit State Figure 21.3-15 Superposition of Column Forces at Bent 2 for Loading Case “2” Chapter 21 – Seismic Design of Concrete Bridges 21-73 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 21.3.16 Step 14 - Distribute M eqcol @ soffit into the Superstructure The static non-linear “push-over” frame analysis program wFRAME is used to col @ soffit into the superstructure. distribute the column earthquake moments M eq Note the difference in sign convention between the wFRAME model and the one adopted here. Therefore, for the input file, the positive column earthquake moments corresponding to “Case 1” loading are used as negative column moment capacities for pushover analysis while the negative column earthquake moments corresponding to “Case 2” are modeled as positive column moment capacities. Also, the superstructure dead load is removed from the wFRAME model. Appendix 21.3-15 shows portions of the output file for Case 1 (i.e., right push). Table 21.3-3 lists the distribution of earthquake moments in the superstructure as obtained from these pushover analyses. 21.3.17 Step 15 - Calculate Superstructure Seismic Moment Demand at Location of Interest Let us calculate superstructure moment demand at the face of the cap on each side of the column. (1) Example Calculation - Bent 2: Left and Right Faces of Bent Cap The effective section width is: beff = Dc + 2Ds = 6.00 + 2(6.75) = 19.50 ft. (SDC 7.2.1.1-1) Based on the column location and the girder spacing, it can easily be concluded that the girder aligned along the centerline of the bridge lies outside the effective width. Therefore, at the face of bent cap, four girders are within the effective section. All five girders fall within the effective width for all the other tenth point locations (see Table 21.3-4). Note that the per-girder values used below have previously been listed in Table 21.3-2. Case 1 M dlL   6,520   1,164(4 )  30,736 kip - ft M dlR   6,731   1,202(4 )  31,732 kip - ft L M ps   1,7344   6,936 kip - ft R M ps   1,6944   6,776 kip - ft L M eq  15,015 kip - ft (see Table 21.3-3) R M eq  21,135 kip - ft (see Table 21.3-3) Chapter 21 – Seismic Design of Concrete Bridges 21-74 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 The superstructure moment demand is then calculated as: L M DL  M dlL  M ps  M eqL = (-30,736) + (6,936*) + (-15,015) = -45,751 kip - ft R M DR  M dlR  M ps  M eqR = (-31,732) + (6,776) + (21,135) = -3,821 kip - ft Table 21.3-4 lists these superstructure seismic moment demands. Case 2 L M eq  13,201kip - ft; M eqR = -20,299 kip - ft M DL = (-30,736) + (6,936) + (13,201) = -10,599 kip - ft M DR = (-31,732) + (6,776*) + (-20,295) = -52,027 kip - ft * The prestressing secondary effect is ignored as doing so results in a conservatively higher seismic demand in the superstructure. (2) Bent 3 Similarly, we obtain the following:  49,702 kip - ft Case 1 M DL     3,001kip - ft Case 2   9,434 kip - ft Case 1 M DR    43,915 kip - ft Case 2 Seismic moment demands along the superstructure length have been summarized in the form of moment envelope values (see Table 21.3-4). M positive  M EQ,max  M DL  M ADL  M *ps M negative  M EQ,min  M DL  M ADL  M *ps* * Only ** include Mps when it maximizes Mpositive Only include Mps when it minimizes Mnegative Chapter 21 – Seismic Design of Concrete Bridges 21-75 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 Table 21.3-3 Earthquake Moments from wFRAME Output Location Span 3 Span 2 Span 1 0.0 Support 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Support 1.0 0.0 Support 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Support 1.0 0.0 Support 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Support 1.0 M EQ (kip-ft) Standard Convention wFRAME Convention Case 1 Case 2 Case 1 Case 2 0 0 0 0 -183 161 -1538 1352 -3076 2705 -4614 4057 -6152 5409 -7691 6761 -9229 8114 -10767 9466 -12305 10818 -13843 12170 -15015 13201 -15381 13523 -15381 13523 -21895 21055 21895 -21055 21135 -20295 17640 -16798 13385 -12540 9131 -8282 4876 -4024 621 234 -3634 4492 -7889 8750 -12144 13008 -16399 17266 -19894 20763 -20653 21524 -20653 21524 -13620 15621 13620 -15621 13274 -15223 12258 -14059 10896 -12496 9534 -10934 8172 -9372 6810 -7810 5448 -6248 4086 -4686 2724 -3124 1362 -1562 173 -199 0 0 0 0 Start Node End Node wFRAME Positive Convention Chapter 21 – Seismic Design of Concrete Bridges Start Node End Node Standard Positive Convention 21-76 Span 3 Span 2 Span 1 Location 21-77 x/L Support 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Support Support 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Support Support 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Support x (ft) 1.5 12.6 25.2 37.8 50.4 63.0 75.6 88.2 100.8 113.4 123.0 129.0 142.8 159.6 176.4 193.2 210.0 226.8 243.6 260.4 277.2 291.0 297.0 305.8 317.6 329.4 341.2 353.0 364.8 376.6 388.4 400.2 410.5 No. of Girders in Effective Section 4 4 5 5 5 5 5 5 5 5 5 4 4 5 5 5 5 5 5 5 5 5 4 4 5 5 5 5 5 5 5 5 5 4 Case 1 Case 2 Case 1 Case 2 Envelope M DL M ADL M PS M EQ M EQ M postive M negative M positive M negative M positive M negative (kip-ft) 496 7110 12158 14741 14857 12508 7693 412 -9334 -21553 -26079 -26923 -17502 -1955 9208 15989 18388 16406 10043 -699 -15820 -25291 -24344 -20789 -9854 -1093 5506 9943 12219 12333 10286 6077 509 (kip-ft) 91 1275 2178 2640 2661 2240 1377 74 -1671 -3857 -4656 -4807 -3136 -354 1645 2859 3289 2935 1795 -128 -2835 -4517 -4347 -3723 -1766 -197 986 1781 2189 2210 1844 1091 94 (kip-ft) 517 1462 2272 3096 3956 4705 5617 6400 7911 8498 6937 6774 9516 9005 8318 8281 8027 8072 7905 8355 8645 6043 5986 7275 6861 5559 4870 4085 3417 2669 1945 1230 423 (kip-ft) -183 -1538 -3076 -4614 -6152 -7691 -9229 -10767 -12305 -13843 -15015 21135 17640 13385 9131 4876 621 -3634 -7889 -12144 -16399 -19894 13274 12258 10896 9534 8172 6810 5448 4086 2724 1362 173 (kip-ft) 161 1352 2705 4057 5409 6761 8114 9466 10818 12170 13201 -20295 -16798 -12540 -8282 -4024 234 4492 8750 13008 17266 20763 -15223 -14059 -12496 -10934 -9372 -7810 -6248 -4686 -3124 -1562 -199 (kip-ft) 921 8309 13532 15862 15321 11762 5459 -3881 -15399 -30755 -38815 -3821 6518 20082 28301 32005 30324 23779 11854 -4616 -26409 -43658 -9434 -4979 6138 13804 19533 22619 23273 21298 16799 9760 1199 (kip-ft) 403 6847 11260 12766 11365 7057 -159 -10281 -23310 -39254 -45751 -10597 -2998 11077 19984 23724 22297 15707 3950 -12970 -35054 -49702 -15418 -12254 -724 8244 14663 18534 19856 18629 14854 8530 776 (kip-ft) 1265 11199 19313 24533 26883 26213 22801 16352 7724 -4742 -10599 -45251 -27920 -5843 10889 23105 29938 31905 28493 20536 7256 -3001 -37931 -31296 -17255 -6665 1988 7999 11576 12526 10950 6836 827 (kip-ft) 748 9737 17041 21438 22927 21509 17184 9952 -187 -13240 -17535 -52027 -37436 -14848 2571 14824 21911 23833 20588 12181 -1390 -9045 -43915 -38571 -24116 -12224 -2881 3913 8159 9857 9006 5606 404 (kip-ft) 1265 11199 19313 24533 26883 26213 22801 16352 7724 -4742 -10599 -3821 6518 20082 28301 32005 30324 31905 28493 20536 7256 -3001 -9434 -4979 6138 13804 19533 22619 23273 21298 16799 9760 1199 (kip-ft) 403 6847 11260 12766 11365 7057 -159 -10281 -23310 -39254 -45751 -52027 -37436 -14848 2571 14824 21911 15707 3950 -12970 -35054 -49702 -43915 -38571 -24116 -12224 -2881 3913 8159 9857 9006 5606 404 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 Chapter 21 – Seismic Design of Concrete Bridges Table 21.3-4 Moment Demand Envelope BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 21.3.18 Step 16 - Calculate Superstructure Seismic Shear Demand at Location of Interest Values of shear forces due to dead load, additional dead load, and secondary prestress, as read from CTBridge output, are listed in Table 21.3-5. Superstructure Seismic Shear Forces due to Seismic Moments, Veq Span 1, Case 1 (1) Seismic Moment at Abutment 1, M eq  0 kip - ft ( 2) Seismic Moment at Bent 2 M eq  15,381kip - ft Shear force in Span 1, Veq  M ( 2) eq (1)  M eq  Length of Span 1  15,381 0  122 kips = 126 Span 1, Case 2 (1) Seismic Moment at Abutment 1, M eq  0 kip - ft ( 2) Seismic Moment at Bent 2, M eq  13,523 kip - ft Shear force in Span 1, Veq  M ( 2) eq (1)  M eq  = 13,523 0  107kips Length of Span 1 126 Similarly, the seismic shear forces for the remaining spans are calculated to be:  253 kips Case 1 Span 2, Veq    253 kips Case 2  115 kips Case 1 Span 3, Veq    133 kips Case 2 Table 21.3-6 lists these values. Table 21.3-7 lists the maximum shear demands summarized as a shear envelope. * V positive  VEQ,max  VDL  VADL  V ps ** Vnegative  VEQ,min  VDL  VADL  V ps Vmax  Greater of Absolute (V posittive ) or Absolute(Vnegative ) * ** Only include VPS when it maximizes Vpositive Only include VPS when it minimizes Vnegative Chapter 21 – Seismic Design of Concrete Bridges 21-78 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 Table 21.3-5 Dead Load and Secondary Prestress Shears Forces from CTBridge Output Span 3 Span 2 Span 1 Location x/L Support 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Support Support 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Support Support 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Support x (ft) 1.5 12.6 25.2 37.8 50.4 63.0 75.6 88.2 100.8 113.4 123.0 129.0 142.8 159.6 176.4 193.2 210.0 226.8 243.6 260.4 277.2 291.0 297.0 305.8 317.6 329.4 341.2 353.0 364.8 376.6 388.4 400.2 410.5 Whole Superstructure Width V DL V ADL V PS (kip) (kip) (kip) 671 120 79 498 89 78 303 54 76 107 19 76 -89 -16 75 -284 -51 75 -480 -86 75 -675 -121 75 -871 -156 75 -1070 -191 30 -1232 -218 134 1287 227 -44 1056 189 -22 795 142 2 534 96 2 273 49 2 13 2 2 -248 -45 1 -509 -91 1 -770 -138 1 -1031 -185 -28 -1261 -223 37 1171 207 -118 1021 182 -69 834 149 -48 651 117 -48 468 84 -48 284 51 -49 101 18 -48 -82 -15 -48 -265 -47 -48 -448 -80 -48 -608 -109 -68 Chapter 21 – Seismic Design of Concrete Bridges V DL (kip) 134 100 61 21 -18 -57 -96 -135 -174 -214 -246 257 211 159 107 55 3 -50 -102 -154 -206 -252 234 204 167 130 94 57 20 -16 -53 -90 -122 Per Girder V ADL (kip) 24 18 11 4 -3 -10 -17 -24 -31 -38 -44 45 38 28 19 10 0 -9 -18 -28 -37 -45 41 36 30 23 17 10 4 -3 -9 -16 -22 V PS (kip) 16 16 15 15 15 15 15 15 15 6 27 -9 -4 0 0 0 0 0 0 0 -6 7 -24 -14 -10 -10 -10 -10 -10 -10 -10 -10 -14 21-79 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 Table 21.3-6 Earthquake Shear Forces from wFRAME Output Span 3 Span 2 Span 1 Location 0 Support 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Support 1 0 Support 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Support 1 0 Support 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Support 1 0.0 1.5 12.6 25.2 37.8 50.4 63.0 75.6 88.2 100.8 113.4 123.0 126.0 126.0 129.0 142.8 159.6 176.4 193.2 210.0 226.8 243.6 260.4 277.2 291.0 294.0 294.0 297.0 305.8 317.6 329.4 341.2 353.0 364.8 376.6 388.4 400.2 410.5 412.0 Start Node End Node wFRAME Positive Convention Chapter 21 – Seismic Design of Concrete Bridges V EQ (kip) Standard Convention wFRAME Convention Case 1 Case 2 Case 1 Case 2 -122 107 -122 107 0 0 -122 107 0 0 -122 107 0 0 -122 107 0 0 -122 107 0 0 -122 107 0 0 -122 107 0 0 -122 107 0 0 -122 107 0 0 -122 107 0 0 -122 107 0 0 -122 107 -122 107 -122 107 -253 253 -253 253 0 0 -253 253 0 0 -253 253 0 0 -253 253 0 0 -253 253 0 0 -253 253 0 0 -253 253 0 0 -253 253 0 0 -253 253 0 0 -253 253 0 0 -253 253 0 0 -253 253 -253 253 -253 253 -115 133 -115 133 0 0 -115 133 0 0 -115 133 0 0 -115 133 0 0 -115 133 0 0 -115 133 0 0 -115 133 0 0 -115 133 0 0 -115 133 0 0 -115 133 0 0 -115 133 0 0 -115 133 -115 133 -115 133 Start Node End Node Standard Positive Convention 21-80 No. of Girders in Effective Section Span 3 Span 2 Span 1 Location 21-81 x/L Support 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Support Support 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Support Support 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Support x (ft) 12.6 12.6 25.2 37.8 50.4 63.0 75.6 88.2 100.8 113.4 123.0 129.0 142.8 159.6 176.4 193.2 210.0 226.8 243.6 260.4 277.2 291.0 297.0 305.8 317.6 329.4 341.2 353.0 364.8 376.6 388.4 400.2 410.5 5 5 5 5 5 5 5 5 5 5 4 4 5 5 5 5 5 5 5 5 5 4 4 5 5 5 5 5 5 5 5 5 4 Case 1 Case 2 Case 1 Case 2 Envelope V DL V ADL V PS V EQ V EQ V positive V negative V positive V negative V positive V negative V max (kip) 498 498 303 107 -89 -284 -480 -675 -871 -1070 -986 1029 1056 795 534 273 13 -248 -509 -770 -1031 -1009 937 1021 834 651 468 284 101 -82 -265 -448 -486 (kip) 89 89 54 19 -16 -51 -86 -121 -156 -191 -174 182 189 142 96 49 2 -45 -91 -138 -185 -178 165 182 149 117 84 51 18 -15 -47 -80 -87 (kip) 78 78 76 76 75 75 75 75 75 30 107 -35 -22 2 2 2 2 1 1 1 -28 30 -94 -69 -48 -48 -48 -49 -48 -48 -48 -48 -54 (kip) -122 -122 -122 -122 -122 -122 -122 -122 -122 -122 -122 -253 -253 -253 -253 -253 -253 -253 -253 -253 -253 -253 -115 -115 -115 -115 -115 -115 -115 -115 -115 -115 -115 (kip) 107 107 107 107 107 107 107 107 107 107 107 253 253 253 253 253 253 253 253 253 253 253 133 133 133 133 133 133 133 133 133 133 133 (kip) 543 543 311 80 -151 -382 -613 -843 -1075 -1354 -1175 958 992 686 378 71 -237 -544 -852 -1160 -1469 -1411 987 1088 868 652 436 220 4 -212 -428 -644 -689 (kip) 465 465 235 4 -227 -457 -688 -918 -1149 -1383 -1282 923 971 684 377 69 -238 -546 -853 -1161 -1496 -1440 893 1020 820 604 388 172 -44 -259 -476 -692 -743 (kip) 772 772 540 309 78 -153 -384 -614 -846 -1125 -946 1464 1498 1192 884 577 269 -38 -346 -654 -963 -905 1234 1336 1116 900 684 468 252 36 -180 -396 -441 (kip) 694 694 464 233 3 -228 -459 -689 -920 -1154 -1053 1429 1477 1190 883 575 268 -40 -347 -655 -990 -934 1141 1267 1068 852 636 420 204 -11 -228 -444 -495 (kip) 772 772 540 309 78 -153 -384 -614 -846 -1125 -946 1464 1498 1192 884 577 269 -38 -346 -654 -963 -905 1234 1336 1116 900 684 468 252 36 -180 -396 -441 (kip) 465 465 235 4 -227 -457 -688 -918 -1149 -1383 -1282 923 971 684 377 69 -238 -546 -853 -1161 -1496 -1440 893 1020 820 604 388 172 -44 -259 -476 -692 -743 (kip) 772 772 540 309 227 457 688 918 1149 1383 1282 1464 1498 1192 884 577 269 546 853 1161 1496 1440 1234 1336 1116 900 684 468 252 259 476 692 743 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 Chapter 21 – Seismic Design of Concrete Bridges Table 21.3-7 Shear Demand Envelope BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 21.3.19 Step 17 - Perform Vertical Acceleration Analysis Since the site PRA = 0.5g < 0.6g, vertical acceleration analysis is not required. 21.3.20 Step 18 - Calculate Superstructure Flexural and Shear Capacity 21.3.20.1 Superstructure Flexural Capacity Table 21.3-8 lists the data that will be used to calculate the flexural section capacity using the computer program PSSECx. Symbols in Table 21.38 are shown in Figure 21.3-16. Appendix 21.3-16 lists the PSSECx input for the superstructure section that lies just to the left of Bent 2. The model is shown in Appendix 21.3-17. The results for negative capacity calculations are shown in Appendix 21.3-18. The limiting condition for flexural capacity in this case was the steel reaching its maximum allowable strain. Figure 21.3-16 Typical Superstructure Cross Section PSSECx was run repeatedly to calculate superstructure flexural capacities at various points along the span length. Table 21.3-9 lists these capacities and also compares them with the maximum moment demands. As can be seen from these results, the superstructure has sufficient flexural capacity to meet the anticipated seismic demands. The worst D/C ratio of 0.63 suggests overdesign. If such case is found across a broad spectrum of various Caltrans bridges, perhaps the requirement of #8 spaced 12 in. may be revised in the future. Chapter 21 – Seismic Design of Concrete Bridges 21-82 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 Table 21.3-8 Section Flexural Capacity Calculation Data Span 3 Span 2 Span 1 Location No. No. Eccentricity Girders Girders in Effective e ps Section (in.) 5 4 -2.6628 5 5 -14.9760 5 5 -25.1328 5 5 -31.2264 5 5 -33.2568 5 5 -31.4076 5 5 -25.8576 5 5 -16.6068 5 5 -3.6576 5 5 14.9160 5 4 25.4412 5 4 25.6116 5 5 12.0432 5 5 -8.2824 5 5 -22.1568 5 5 -30.4824 5 5 -33.2568 5 5 -30.4824 5 5 -22.1568 5 5 -8.2824 5 5 12.0432 5 4 25.6116 5 4 25.3668 5 5 15.1068 5 5 -3.6576 5 5 -16.6068 5 5 -25.8576 5 5 -31.4076 5 5 -33.2568 5 5 -31.2264 5 5 -25.1328 5 5 -14.9760 5 4 -2.7900 PS Force After All Losses x/L x (ft) (kip) Support 1.5 7439 0.1 12.6 7508 0.2 25.2 7582 0.3 37.8 7650 0.4 50.4 7712 0.5 63.0 7766 0.6 75.6 7814 0.7 88.2 7859 0.8 100.8 7839 0.9 113.4 7765 Support 123.0 7697 Support 129.0 7595 0.1 142.8 7413 0.2 159.6 7370 0.3 176.4 7327 0.4 193.2 7272 0.5 210.0 7212 0.6 226.8 7148 0.7 243.6 7079 0.8 260.4 6999 0.9 277.2 6922 Support 291.0 6844 Support 297.0 6742 0.1 305.8 6572 0.2 317.6 6545 0.3 329.4 6522 0.4 341.2 6484 0.5 353.0 6443 0.6 364.8 6398 0.7 376.6 6345 0.8 388.4 6287 0.9 400.2 6225 Support 410.5 6174 P jack = 9,689 kips * Area of mild steel based on minimum seismic requirement only For Effective Section Area of Distance to PS Force Area of Top Mild Top Mild After All PS Steel* Steel Losses A ps A st,top y st,top (kip) 5952 7508 7582 7650 7712 7766 7814 7859 7839 7765 6157 6076 7413 7370 7327 7272 7212 7148 7079 6999 6922 5475 5393 6572 6545 6522 6484 6443 6398 6345 6287 6225 4940 (in.2) 38.28 47.85 47.85 47.85 47.85 47.85 47.85 47.85 47.85 47.85 38.28 38.28 47.85 47.85 47.85 47.85 47.85 47.85 47.85 47.85 47.85 38.28 38.28 47.85 47.85 47.85 47.85 47.85 47.85 47.85 47.85 47.85 38.28 (in.2) 8.00 8.00 8.00 8.00 8.00 47.40 47.40 47.40 47.40 47.40 47.40 47.40 47.40 47.40 47.40 8.00 8.00 8.00 47.40 47.40 47.40 47.40 47.40 47.40 47.40 47.40 47.40 47.40 8.00 8.00 8.00 8.00 8.00 (in.) 31.80 31.80 31.80 31.80 31.80 31.80 31.80 31.80 31.80 31.80 31.80 31.80 31.80 31.80 31.80 31.80 31.80 31.80 31.80 31.80 31.80 31.80 31.80 31.80 31.80 31.80 31.80 31.80 31.80 31.80 31.80 31.80 31.80 Area of Bottom Mild Steel* A st,bot (in.2) 6.00 6.00 6.00 6.00 6.00 34.76 34.76 34.76 34.76 34.76 34.76 34.76 34.76 34.76 34.76 6.00 6.00 6.00 34.76 34.76 34.76 34.76 34.76 34.76 34.76 34.76 34.76 34.76 6.00 6.00 6.00 6.00 6.00 Distance to Bottom Mild Steel y st,bot (in.) -42.13 -42.13 -42.13 -42.13 -42.13 -42.13 -42.13 -42.13 -42.13 -42.13 -42.13 -42.13 -42.13 -42.13 -42.13 -42.13 -42.13 -42.13 -42.13 -42.13 -42.13 -42.13 -42.13 -42.13 -42.13 -42.13 -42.13 -42.13 -42.13 -42.13 -42.13 -42.13 -42.13 (Remaining limit state requirements need to be satisfied, A st,top = 56.6 in.2 at right face of Bent 2 Chapter 21 – Seismic Design of Concrete Bridges 21-83 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 Table 21.3-9 Section Flexural Capacity Calculation Data Moment Demand Span 3 Span 2 Span 1 Location x/L Support 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Support Support 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Support Support 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Support x (ft) 1.5 12.6 25.2 37.8 50.4 63.0 75.6 88.2 100.8 113.4 123.0 129.0 142.8 159.6 176.4 193.2 210.0 226.8 243.6 260.4 277.2 291.0 297.0 305.8 317.6 329.4 341.2 353.0 364.8 376.6 388.4 400.2 410.5 M positive (kip-ft) 1265 11199 19313 24533 26883 26213 22801 16352 7724 -4742 -10599 -3821 6518 20082 28301 32005 30324 31905 28493 20536 7256 -3001 -9434 -4979 6138 13804 19533 22619 23273 21298 16799 9760 1199 M negative (kip-ft) 403 6847 11260 12766 11365 7057 -159 -10281 -23310 -39254 -45751 -52027 -37436 -14848 2571 14824 21911 15707 3950 -12970 -35054 -49702 -43915 -38571 -24116 -12224 -2881 3913 8159 9857 9006 5606 404 Chapter 21 – Seismic Design of Concrete Bridges Moment Capacity M positive (kip-ft) 34530 54933 65503 72025 73892 86109 81016 71684 58705 38646 26587 26432 41672 63619 77024 71217 73881 71218 77020 63615 41623 26344 26540 38316 58628 71666 80998 86086 74193 72006 65484 54917 34611 M negative (kip-ft) -39444 -34040 -23133 -16558 -14352 -36067 -41879 -51573 -65648 -85898 -81787 -81933 -82802 -60763 -45653 -17311 -14256 -17293 -45591 -60698 -82794 -81924 -81708 -86086 -65419 -51881 -41529 -35585 -13993 -16314 -22996 -34070 -39346 D/C Ratio Postive Moment 0.04 0.20 0.29 0.34 0.36 0.30 0.28 0.23 0.13 0.00 0.00 0.00 0.16 0.32 0.37 0.45 0.41 0.45 0.37 0.32 0.17 0.00 0.00 0.00 0.10 0.19 0.24 0.26 0.31 0.30 0.26 0.18 0.03 Negative Moment 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.20 0.36 0.46 0.56 0.63 0.45 0.24 0.00 0.00 0.00 0.00 0.00 0.21 0.42 0.61 0.54 0.45 0.37 0.24 0.07 0.00 0.00 0.00 0.00 0.00 0.00 21-84 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 21.3.20.2 Superstructure Shear Capacity As shown in Table 21.3-10, seismic shear demands do not control as they are less than the demands from the controlling limit state (i.e. Strength I, Strength II, etc.) calculated using CTBridge. Therefore, the superstructure has sufficient shear capacity to resist seismic demands. Table 21.3-10 Shear Demand vs. Capacity Span 3 Span 2 Span 1 Location Support 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Support Support 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Support Support 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Support 1.5 12.6 25.2 37.8 50.4 63.0 75.6 88.2 100.8 113.4 123.0 129.0 142.8 159.6 176.4 193.2 210.0 226.8 243.6 260.4 277.2 291.0 297.0 305.8 317.6 329.4 341.2 353.0 364.8 376.6 388.4 400.2 410.5 Shear Demand V max 803 772 540 309 227 457 688 918 1149 1383 1282 1464 1498 1192 884 577 269 546 853 1161 1496 1440 1234 1336 1116 900 684 468 252 259 476 692 743 Chapter 21 – Seismic Design of Concrete Bridges Shear Capacity = Strength Shear Demand φV n = V u, strength 2851 2317 1687 1101 681 1207 1782 2341 2901 3596 3966 4378 3759 2961 2160 1399 686 1375 2139 2942 3792 4367 3760 3388 2817 2312 1774 1238 738 1000 1548 2138 2653 D/C Ratio D/C 0.28 0.33 0.32 0.28 0.33 0.38 0.39 0.39 0.40 0.38 0.32 0.33 0.40 0.40 0.41 0.41 0.39 0.40 0.40 0.39 0.39 0.33 0.33 0.39 0.40 0.39 0.39 0.38 0.34 0.26 0.31 0.32 0.28 21-85 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 21.3.21 Step 19 - Design Joint Shear Reinforcement Figure 21.3-17 shows the bent cap-to-column joint. Bent Cap Main Top Reinforcement: 22 #11 Bent Cap Main Bottom Reinforcement: 24 #11 3.06 5.32 Basic Development length, ldb = 58.5 8.38 S =2.92 Dc =6.00 14.3 Dimensions along skew direction Figure 21.3-17 Bent Cap-to-Column Joint Cap beam short stub length, S = 14.3 – 8.38 – 3 = 2.92 ft < Dc = 6 ft (SDC 7.4.3-1). Therefore the joint will be designed as a knee joint in the transverse direction and a T joint in the longitudinal direction. 21.3.21.1 Transverse Direction (Knee Joint Design) 6.00  3.0 ft 2 Therefore, the joint is classified as Case 1 Knee joint. S  2.92 ft  Chapter 21 – Seismic Design of Concrete Bridges (SDC 7.4.5.1-1) 21-86 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 (1) Closing Failure Mode - Bent 2 Knee Joint Given: Superstructure depth, Ds = 6.75 ft Column diameter, Dc = 6 ft, Concrete cover = 2 in. Column reinforcement:  Main reinforcement anchored into cap beam: #14 bars, total 26 giving Ast = 58.50 in.2  Transverse reinforcement: #8 hoops spaced at 5 in. c/c. Column main reinforcement embedment length into the bent cap, lac, provided  66 in. From the xSECTION analysis of Bent 2 with overturning effects (see Appendix 21.3-7): Column plastic moment, Mp = 14,964 kip-ft Column axial force (including the effect of overturning), Pc = 2,474 kips Cap Beam main reinforcement: top: #11 bars, total 2 and bottom: #11 bars, total 24. Calculate principal stresses, p t and p c Vertical Shear Stress,  jv Tc  1.2 (2862) kips = 3,434 kips (Using xSECTION results of Appendix 21.3-7) A jv  l ac Bcap  6696  6,336 in.2 (SDC 7.4.4.1-4) Tc 3,434   0.542 ksi A jv 6336 (SDC 7.4.4.1-3)  jv  Normal Stress (Vertical), f v fv  Pc Pc 2,474    0.168 ksi A jh Dc  Ds Bcap 6.00  6.75(8.00)(144) (SDC 7.4.4.1-5) Normal Stress (Horizontal) Assume Pb = 0 since no prestressing is specifically designed to provide horizontal joint compression. Therefore, horizontal normal stress fh = (Pb / BcapDs) = 0. Chapter 21 – Seismic Design of Concrete Bridges 21-87 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 pt  fh  fv   2 2  f  fv    v2   h jv   2   0.00  0.168  2  0.464 ksi 2  0.00  0.168  2    0.542 (+ for joint in tension) 2   (SDC 7.4.4.1-1) pc  0.00  0.168  2  0.632 ksi 2  0.00  0.168  2    0.542 (+ for joint in compression) 2   (SDC 7.4.4.1-2) Check Joint Size Adequacy Principal compression, pc = 0.632 ksi  [0.25 f c' = 0.25 (4.0) = 1 ksi] OK (SDC 7.4.2-1) Principal tension, pt = 0.464 ksi  [12 fc'  12 4000 / 1000= 0.76 ksi] OK (SDC 7.4.2-2) Check the Need for Additional Joint Reinforcement Since pt = 0.464 ksi > [3.5 f c'  3.5 4000 / 1000 = 0.221 ksi], additional joint reinforcement is required (see SDC Section 7.4.4.2). Similar calculations can be performed for Bent 3. (2) Opening Failure Mode - Bent 2 Knee Joint From wFRAME push-over analysis results (see Appendix 21.3-6), Column axial force (including the effect of overturning), Pc = 907 kips* Column plastic moment, Mp = 12,636 kip-ft* * These values were obtained from xSECTION analysis of Bent 2 with overturning effects (see Appendix 21.3-8) Tc  1.2 (3,148) kips = 3,778 kips using xSECTION results. Ajv = 66 (96) = 6,336 in.2  jv  3,778  0.596 ksi 6,336 Chapter 21 – Seismic Design of Concrete Bridges 21-88 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 fv  Pc Pc 907    0.062 ksi A jh Dc  Ds  Bcap 6.00  6.75 8.00 144 fh = 0 (since Pb = 0) pt  0.00  0.062  2  0.00  0.062  2   0.596  0.566 ksi  2   pc  0.00  0.062   0.00  0.062  2    0.596  0.628 ksi 2   2 2 2 Check Joint Size Adequacy Principal compression, pc = 0.628 ksi < [0.25  4.0 = 1 ksi] Principal tension, [ p t = 0.566 ksi] < [ 12 4000 / 1000= 0.760 ksi] OK OK Check the Need for Additional Joint Reinforcement Since pt = 0.566 ksi > [ 3.5 4000 / 1000 = 0.221 ksi], additional joint reinforcement is required. Based upon joint stress condition evaluation for both closing and opening modes of failure, the joint needs additional joint reinforcement. Refer to Figure 21.3-18 for regions of additional joint shear reinforcement. Figure 21.3-18 Regions of Additional Joint Shear Reinforcement Chapter 21 – Seismic Design of Concrete Bridges 21-89 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 Joint Shear Requirement  Bent Cap Top and Bottom Flexural Reinforcement, AsU  Bar (Refer to Figure 21.3-19) AsU  Bar required = 0.33 Ast = 0.33 (58.5) = 19.3 in.2 (SDC 7.4.5.1-3) The bent cap reinforcement based upon service and seismic loading consists of: Top Reinforcement #11, total 22 bars giving Ast = 34.32 in.2 Bottom Reinforcement #11, total 24 bars giving Ast = 37.44 in.2 AsU  Bar provided = 12 (1.56) = 18.72 in.2 (within 4 % of 19.3 in.2) Say OK See Figure 21.3-19 for the rebar layout. Figure 21.3-19 Location of Joint Shear Reinforcement (Elevation View)  Vertical Stirrups in Joint Region Asjv required = 0.2 Ast = 0.20 (58.5) = 11.7 in.2 (SDC 7.4.5.1-4) Provide 5 sets of 6-legged, #6 stirrups so that Asjv provided = (6 legs)(5 sets)(0.44) = 13.2 in.2 > 11.7 in.2 Chapter 21 – Seismic Design of Concrete Bridges OK 21-90 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 Place stirrups transversely within a distance Dc = 72 inches extending from either side of the column centerline. These vertical stirrups are shown in Figure 21.3-19 and also in Figure 21.3-20.  Horizontal Stirrups in Joint Region Asjh required = 0.1 Ast = 0.1 (58.5) = 5.85 in.2 (SDC 7.4.5.1-5) As shown in Figure 21.3-19, provide 3-legged #6 stirrups, total 14 sets Asjh provided = (3 legs)(14 sets)(0.44) = 18.48 in.2 > 5.85 in.2 Placed within a distance Dc = 72 in. extending from either side of the column centerline as shown in Figure 21.3-19. Figure 21.3-20 Location of Vertical Stirrups, Asjv  Horizontal Side Reinforcement top 0.1  Acap  Assf   or 0.1  Abot cap  (SDC 7.4.5.1-6) top Acap = 34.32 in.2 bot Acap = 37.44 in.2 Chapter 21 – Seismic Design of Concrete Bridges 21-91 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 Assf 0.1 (34.32)  3.43 in.2   or 0.1 (37.44)  3.74 in.2   Assf = 3.74 in.2 As shown in Figures 21.3-21 and 21.3-22, provide #6, total 5 (“U” shaped), giving: Assf provided = (2 legs)(5 bars) (0.44)= 4.4 in.2 > 3.74 in.2  Horizontal Cap End Ties Asjhc  0.33Asu bar  0.33(19.3)  6.37 in.2 (SDC 7.4.5.1-7) 2 2 Provide #8, total 10 ( Asjhc , provided  10(0.79)  7.9 in.  6.37 in. ) OK See SDC Figures 7.4.5.1-2, 7.4.5.1-3, and 7.4.5.1-5 for placement of Asjhc  J-Dowels Strictly following SDC guidelines, there is no need for J-Dowels for this bridge. Let us provide it anyway. Asj bar = 0.08 Ast = 0.08 (58.5) = 4.68 in.2 Use 16, #5 J-Dowels. (SDC 7.4.5.1-8) Asj bar provided = (16 bars)(0.31) = 4.96 in.2 > 4.68 in.2 These dowels are placed within the rectangle defined by Dc on either side of the column centerline and the cap width. They are shown in Figures 21.3-21 and 21.3-22.  Check Transverse Reinforcement Minimum reinforcement ratio of transverse reinforcement (hoops)   Ast   0.76 58.5   0.00936  72(66)   Dclac, provided      (SDC 7.4.5.1-9)  s, required  0.76  Column transverse reinforcement that extends into the joint region consists of #8 hoops at 5 in. spacing. Chapter 21 – Seismic Design of Concrete Bridges 21-92 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 s,  provided  4 Ab 4(0.79)   0.0095  0.00936 OK D' s  1.13   72  2(2)  2  5  2   Check Anchorage for Main Column Reinforcement lac, required = 24dbl = 24 (1.69) = 40.6 in. < [lac, provided = 66 in.] OK (SDC 8.2.1-1) Figure 21.3-21 Joint Reinforcement Within the Column Region Figure 21.3-22 Joint Reinforcement Outside the Column Region Chapter 21 – Seismic Design of Concrete Bridges 21-93 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 21.3.21.2 Longitudinal Direction (T-Joint) Let us calculate joint stresses for the tension column, which will provide higher value of principal tensile stress (generally more critical than principal compressive stress). Column plastic moment, Mp = 13,838 kip-ft* Column axial force, Pc = 1,694 kips* Cap beam main reinforcement: top: #11 bars, total 22 and bottom: #11 bars, total 24. * These values were obtained from the xSECTION analysis of Bent 2 columns without overturning effects (see Appendix 21.3-1). (1) Calculate Principal Stresses, pt and pc Tc  1.2(2,948) kips = 3,538 kips using xSECTION results Ajv = lac Bcap = 66 (96) = 6,336 in.2 Vertical Shear Stress  jv  Tc 3,538   0.558 ksi A jv 6,336 Normal Stress (Vertical) fv  Pc Pc 1,694    0.115 ksi A jh Dc  Ds Bcap 6.00  6.75(8.00)(144) Assume Pb = 0 since no prestressing is specifically designed to provide horizontal joint compression. Therefore, horizontal normal stress fh = (Pb/BcapDs) = 0. pt  0.00  0.115   0.00  0.115  2    0.558  0.503 ksi (i.e., tension) 2   pc  0.00  0.115   0.00  0.115  2    0.558  0.618 ksi (i.e., compression) 2   2 2 2 2 Check Joint Size Adequacy Principal compression, pc = 0.618 ksi  [0.25 (4.0) = 1 ksi] OK Principal tension, pt = 0.503 ksi  [ 12 4000 / 1000 = 0.760 ksi] OK Chapter 21 – Seismic Design of Concrete Bridges 21-94 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 Check the Need for Additional Joint Reinforcement pt = 0.503 ksi > [ 3.5 4000 / 1000 = 0.221 ksi], therefore additional joint reinforcement is required. The horizontal stirrups, cap beam u-bar requirements, continuous cap side face reinforcement, J-dowels, minimum transverse reinforcement, and column reinforcement anchorage provided for transverse bending will also satisfy the joint shear requirements for longitudinal bending. The only additional joint reinforcement requirement that needs to be satisfied for longitudinal bending is to provide vertical stirrups in Regions 1 and 2 of Figure 21.3-18.  Vertical Stirrups in Joint Region – Regions 1 and 2 of Figure 21.3-18 Asjv required = 0.2 Ast = 0.2 (58.5) = 11.7 in.2 Provide: total 14 sets of 2-legged #6 stirrups or ties on each side of the column. Asjv provided = (2 legs)(14 sets)(0.44) = 12.32 in.2 > 11.7 in.2 OK As shown in Figures 21.3-19 and 21.3-20, these vertical stirrups and ties are placed transversely within a distance Dc extending from either side of the column centerline. Note that in the overlapping portions of regions 1 and 2 with region 3, the outside two legs of the 6-legged vertical stirrups provided for transverse bending are also counted toward the two legs of the vertical stirrups required for the longitudinal bending. 21.3.22 Step 20 - Determine Minimum Hinge Seat Width This bridge is not a multi-frame bridge. Therefore this step does not apply. 21.3.23 Step 21 - Determine Minimum Abutment Seat Width Minimum required abutment seat width, NA = 30 in. NA  p/s + cr+sh +temp +eq + 4 (in.) (SDC Section 7.8.3) (SDC 7.8.3-1) The combined effect of p/s, cr+sh, and temp, is calculated as 2.6 inches (see Joint Movement Calculation form - Appendix 21.3-11). The maximum seismic demand along the longitudinal direction of the bridge is calculated in a conservative way assuming that maximum longitudinal and transverse (along the bent line) demand displacements occur simultaneously, i.e., Chapter 21 – Seismic Design of Concrete Bridges 21-95 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015    eq,longitudinal = 14.12 + 16.87 sin (20) = 19.89 in. NA, required (normal to centerline of bearing) = (19.89 + 2.6) cos (20) + 4 = 25.13 in. < 30 in. Provide abutment seat width NA = 36 in. > 30 in. 21.3.24 OK Step 22 - Design Abutment Shear Key Reinforcement Shear key force capacity, Fsk   (0.75 Vpiles  Vww ) (SDC 7.8.4-1) We shall assume the following information to be available from the abutment foundation and wingwall design:     14 piles for the abutment, and 40 k/pile as ultimate shear capacity of the pile (see MTD 5-1) f c = 3.6 ksi Wingwall thickness = 12 in. Wingwall height to top of abutment footing = 14 ft (i.e., 6.75 ft Superstructure depth + 7.25 ft abutment stem height) V piles = 14 (40) = 560 kips Using Method 1 of AASHTO Article 5.8.3.4, the shear capacity of one wingwall, Vww may conservatively be estimated as follows: Effective shear depth d v = 0.72 (12 in) = 8.64 in. Effective width bv = [6.75  1 (7.25)](12) in = 110 in. 3 Vww = 0.0316 fcbv dv = 0.0316(2) 3.6 (110)(8.64) = 114 kips Assuming   0.75, Fsk  0.75 0.75(560)  114  401 kips We shall use the Isolated Shear Key Method for this example. Vertical shear key reinforcement: F 401 (SDC 7.8.4.1A-1)  3.28 in.2 Ask  sk  1.8 f ye 1.8(68) Provide 8 #6 – bundle bars as shown in Figure SDC 7.8.4.1-1A ( Ask , provided = 3.52 in.2 > 3.28 in.2) Hanger bars, Ash  2.0 AskIso( provided) = 2 (3.52) = 7.04 in.2 Provide 5 #11 hooked bars ( Ash, provided = 7.8 in.2 > 7.04 in.2) Chapter 21 – Seismic Design of Concrete Bridges OK (SDC 7.8.4.1B-1) OK 21-96 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 Place the vertical shear key bars, Ask at least Lmin from the end of the lowest layer of the hanger bars, where Lmin,hooked  0.6(a  b)  ldh (SDC 7.8.4.1B-3) Assuming 5-inch thick bearing pads and 12 in. vertical height of expansion joint filler (see SDC Figure 7.8.4.1-1A), a = (Bearing pad thichness + 6 in.) = 11 in. Assuming 2 in cover and #4 distribution bars for the hanger bars, b = 2 + 0.56 + 0.5 (1.63) = 3.4 in. of dimension “b”) ldh  (see SDC Figure 7.8.4.1-1A for definition 38db 38(1.41)   28.2 in. fc 3.6 Lmin,hooked  0.6(a  b)  ldh  0.6(11 3.4)  28.2  37 in. Place vertical shear key bars Ask 40 in. from the hooked ends of the hanger bars Ash . 21.3.25 Step 23 - Check Requirements for No-splice Zone For this bridge, only columns have been designated as “seismic critical” elements. Maximum length of column rebar can be estimated as Lmax = 47.00 + 5.5 = 52.5 ft < 60.00 ft Therefore, we will specify on the plans that no splices will be permitted for column main reinforcement. The superstructure rebars, however, will need to be spliced with Service Splice per MTD 20-9 (Caltrans 2001b). Chapter 21 – Seismic Design of Concrete Bridges 21-97 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 APPENDIX 21.3-1 Output from xSECTION 04/17/2006, 11:45 ************************************************************ * * * xSECTION * * * * DUCTILITY and STRENGTH of * * Circular, Semi-Circular, full and partial Rings, * * Rectangular, T-, I-, Hammer head, Octagonal, Polygons * * or any combination of above shapes forming * * Concrete Sections using Fiber Models * * * * VER._2.40,_MAR-14-99 * * * * Copyright (C) 1994, 1995, 1999 By Mark Seyed Mahan. * * * * A proper license must be obtained to use this software. * * For GOVERNMENT work call 916-227-8404, otherwise leave a * * message at 530-756-2367. The author makes no expressed or* * implied warranty of any kind with regard to this program.* * In no event shall the author be held liable for * * incidental or consequential damages arising out of the * * use of this program. * * * ************************************************************ This output was generated by running: xSECTION VER._2.40,_MAR-14-99 LICENSE (choices: LIMITED/UNLIMITED) UNLIMITED ENTITY (choices: GOVERNMENT/CONSULTANT) Government NAME_OF_FIRM Caltrans BRIDGE_NAME EXAMPLE BRIDGE_NUMBER 99-9999 JOB_TITLE PROTYPE BRIDGE - BRIDGE DESIGN ACADEMY Concrete Type Information: ----------strains-------Type e0 e2 ecc eu 1 0.0020 0.0040 0.0055 0.0145 2 0.0020 0.0040 0.0020 0.0050 --------strength-------f0 f2 fcc fu E 5.28 6.98 7.15 6.11 4313 5.28 3.61 5.28 2.64 4313 W 148 148 Steel Type Information: -----strains------ --strengthType ey eh eu fy fu E 1 0.0023 0.0150 0.0900 68.00 95.00 29000 2 0.0023 0.0075 0.0600 68.00 95.00 29000 Steel Fiber Information: Fiber No. type 1 2 2 2 3 2 4 2 5 2 6 2 xc in 31.93 31.00 28.27 23.90 18.14 11.32 yc in 0.00 7.64 14.84 21.17 26.28 29.86 area in^2 2.25 2.25 2.25 2.25 2.25 2.25 Chapter 21 – Seismic Design of Concrete Bridges 21-98 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 7 ... ... 2 3.85 31.70 2.25 25 26 2 2 28.27 31.00 -14.84 -7.64 2.25 2.25 Force Equilibrium Condition of the x-section: Max. Conc. Strain step epscmax 0 0.00000 1 0.00029 2 0.00032 ... ... 25 0.00322 26 0.00356 27 0.00394 28 0.00435 29 0.00481 30 0.00532 31 0.00588 32 0.00650 33 0.00718 34 0.00794 35 0.00878 36 0.00971 37 0.01073 38 0.01186 39 0.01312 40 0.01450 Max. Neutral Steel Axis Strain Conc. in. Tens. Comp. 0.00 0.0000 0 -12.30 -0.0001 1570 -9.09 -0.0002 1585 16.99 17.39 17.67 17.91 18.07 18.11 18.15 18.21 18.27 18.30 18.33 18.34 18.34 18.34 18.38 18.41 -0.0083 -0.0094 -0.0106 -0.0119 -0.0134 -0.0148 -0.0164 -0.0183 -0.0203 -0.0225 -0.0249 -0.0275 -0.0304 -0.0336 -0.0373 -0.0414 3210 3249 3309 3361 3388 3413 3461 3515 3570 3630 3686 3743 3792 3834 3847 3857 Steel force Comp. Tens. 0 0 174 -49 182 -73 889 929 952 978 1008 1037 1048 1060 1072 1087 1103 1122 1148 1181 1217 1256 -2406 -2483 -2568 -2646 -2703 -2756 -2816 -2881 -2948 -3021 -3096 -3171 -3246 -3321 -3371 -3420 P/S force 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Net Curvature Moment force rad/in (K-ft) 0.00 0.000000 0 1.52 0.000006 2588 0.95 0.000007 2843 -0.96 0.66 -1.69 -1.26 -0.57 0.59 -0.56 -0.42 -0.93 1.38 -1.20 -0.61 0.07 -0.67 -0.48 -1.66 0.000170 0.000192 0.000215 0.000241 0.000269 0.000298 0.000330 0.000366 0.000406 0.000449 0.000497 0.000550 0.000608 0.000672 0.000745 0.000825 12508 12718 12926 13129 13267 13362 13495 13660 13834 14017 14194 14368 14536 14695 14841 14976 First Yield of Rebar Information (not Idealized): Rebar Number 20 Coordinates X and Y (global in.) Yield strain = 0.00230 Curvature (rad/in)= 0.000054 Moment (ft-k) = 9537 -3.85, -31.70 Cross Section Information: Axial Load on Section (kips) = 1694 Percentage of Main steel in Cross Section = 1.44 Concrete modulus used in Idealization (ksi) = 4313 Cracked Moment of Inertia (ft^4) = 23.717 Idealization of Moment-Curvature Curve by Various Methods: Points on Curve =============== Method ID Conc. | Strain Curv. | in/in rad/in Strain @ 0.003 0.000155 Strain @ 0.004 0.000219 Strain @ 0.005 0.000279 CALTRANS 0.00720 0.000407 UCSD@5phy0.00483 0.000270 Moment (K-ft) 12388 12957 13302 13838 13271 Idealized Values ============================= Yield symbol Plastic | Curv. Moment for Curv. | rad/in (K-ft) moment rad/in 0.000070 12388 Mn 0.000755 0.000073 12957 Mn 0.000752 0.000075 13302 Mn 0.000750 0.000078 13838 Mp 0.000747 0.000075 13271 Mn 0.000750 Chapter 21 – Seismic Design of Concrete Bridges 21-99 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 APPENDIX 21.3-2 Moment – Curvature Relationship Chapter 21 – Seismic Design of Concrete Bridges 21-100 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 APPENDIX 21.3-3 Soil Spring Data p-y Data - Bent 2 (Location 1) 250 Stiffness (lbs/in) 200 150 100 50 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 3.0 3.5 4.0 4.5 Displacement (in) p-y Data - Bent 2 (Location 2) 1400 Stiffness (lbs/in) 1200 1000 800 600 400 200 0 0.0 0.5 1.0 1.5 2.0 2.5 Displacement (in) Chapter 21 – Seismic Design of Concrete Bridges 21-101 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 p-y Data - Bent 2 (Location 3) 1600 Stiffness (lbs/in) 1400 1200 1000 800 600 400 200 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 3.0 3.5 4.0 4.5 Displacement (in) p-y Data - Bent 2 (Location 4) 5000 4500 Stiffness (lbs/in) 4000 3500 3000 2500 2000 1500 1000 500 0 0.0 0.5 1.0 1.5 2.0 2.5 Displacement (in) Chapter 21 – Seismic Design of Concrete Bridges 21-102 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 APPENDIX 21.3-4 Bent Cap – Positive Bending Section Capacities – Select Output 05/16/2006, 10:17 ************************************************************ * xSECTION * * * * DUCTILITY and STRENGTH of * * Circular, Semi-Circular, full and partial Rings, * * Rectangular, T-, I-, Hammer head, Octagonal, Polygons * * or any combination of above shapes forming * * Concrete Sections using Fiber Models * * * * VER._2.40,_MAR-14-99 * * * * Copyright (C) 1994, 1995, 1999 By Mark Seyed Mahan. * * * * A proper license must be obtained to use this software. * * For GOVERNMENT work call 916-227-8404, otherwise leave a * * message at 530-756-2367. The author makes no expressed or* * implied warranty of any kind with regard to this program.* * In no event shall the author be held liable for * * incidental or consequential damages arising out of the * * use of this program. * * * ************************************************************ This output was generated by running: xSECTION VER._2.40,_MAR-14-99 …………………………………………………………………………………………………….. …………………………………………………………………………………………………….. Concrete Type Information: ----------strains-------- --------strength-------Type e0 e2 ecc eu f0 f2 fcc fu E 1 0.0020 0.0040 0.0027 0.0115 5.00 5.01 5.35 2.63 4200 2 0.0020 0.0040 0.0020 0.0050 5.00 3.52 5.00 2.50 4200 W 148 148 Steel Type Information: -----strains------ --strengthType ey eh eu fy fu E 1 0.0023 0.0150 0.0900 68.00 95.00 29000 2 0.0023 0.0075 0.0600 68.00 95.00 29000 ………………………………………………………………………………………………………….. First Yield of Rebar Information (not Idealized): Rebar Number 1 Coordinates X and Y (global in.) -44.80, -35.49 Yield strain = 0.00230 Curvature (rad/in)= 0.000037 Moment (ft-k) = 14873 Cross Section Information: Axial Load on Section (kips) = 1 Percentage of Main steel in Cross Section = 0.80 Concrete modulus used in Idealization (ksi) = 4200 Cracked Moment of Inertia (ft^4) = 55.568 Idealization of Moment-Curvature Curve by Various Methods: Points on Curve Idealized Values =============== ============================= Method Conc. Yield symbol Plastic ID | Strain Curv. Moment | Curv. Moment for Curv. | in/in rad/in (K-ft) | rad/in (K-ft) moment rad/in Strain @ 0.003 0.000520 21189 0.000053 21189 Mn 0.000665 Strain @ 0.004 0.000684 21635 0.000054 21635 Mn 0.000664 Strain @ 0.005 0.000000 0 0.000000 0 Mn 0.000718 CALTRANS 0.00187 0.000306 19484 0.000048 19484 Mp 0.000669 UCSD@5phy0.00126 0.000184 17426 0.000043 17426 Mn 0.000674 Chapter 21 – Seismic Design of Concrete Bridges 21-103 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 APPENDIX 21.3-5 Bent Cap – Negative Bending Section Capacities – Select Output 05/15/2006, 08:26 ************************************************************ * xSECTION * * * * DUCTILITY and STRENGTH of * * Circular, Semi-Circular, full and partial Rings, * * Rectangular, T-, I-, Hammer head, Octagonal, Polygons * * or any combination of above shapes forming * * Concrete Sections using Fiber Models * * * * VER._2.40,_MAR-14-99 * * * * Copyright (C) 1994, 1995, 1999 By Mark Seyed Mahan. * * * * A proper license must be obtained to use this software. * * For GOVERNMENT work call 916-227-8404, otherwise leave a * * message at 530-756-2367. The author makes no expressed or* * implied warranty of any kind with regard to this program.* * In no event shall the author be held liable for * * incidental or consequential damages arising out of the * * use of this program. * * * ************************************************************ This output was generated by running: xSECTION VER._2.40,_MAR-14-99 ……………………………………………………………………………………………………………………………. ………………………………………………………………………………………………………………………….. Concrete Type Information: ----------strains-------Type e0 e2 ecc eu 1 0.0020 0.0040 0.0027 0.0115 2 0.0020 0.0040 0.0020 0.0050 --------strength-------f0 f2 fcc fu E 5.00 5.01 5.35 2.63 4200 5.00 3.52 5.00 2.50 4200 W 148 148 Steel Type Information: -----strains------ --strengthType ey eh eu fy fu E 1 0.0023 0.0150 0.0900 68.00 95.00 29000 2 0.0023 0.0075 0.0600 68.00 95.00 29000 First Yield of Rebar Information (not Idealized): Rebar Number 25 Coordinates X and Y (global in.) 44.80, -34.49 Yield strain = 0.00230 Curvature (rad/in)= 0.000037 Moment (ft-k) = 13030 Cross Section Information: Axial Load on Section (kips) = 1 Percentage of Main steel in Cross Section = 0.80 Concrete modulus used in Idealization (ksi) = 4200 Cracked Moment of Inertia (ft^4) = 48.938 Idealization of Moment-Curvature Curve by Various Methods: Points on Curve Idealized Values =============== ============================= Method Conc. Yield symbol Plastic ID | Strain Curv. Moment | Curv. Moment for Curv. | in/in rad/in (K-ft) | rad/in (K-ft) moment rad/in Strain @ 0.003 0.000593 19436 0.000055 19436 Mn 0.000563 Strain @ 0.004 0.000000 0 0.000000 0 Mn 0.000618 Strain @ 0.005 0.000000 0 0.000000 0 Mn 0.000618 CALTRANS 0.00159 0.000282 17307 0.000049 17307 Mp 0.000569 UCSD@5phy0.00117 0.000183 15735 0.000044 15735 Mn 0.000573 Chapter 21 – Seismic Design of Concrete Bridges 21-104 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 APPENDIX 21.3-6 wFRAME Output File 05/15/2006, 07:47 Design Academy Example No: 1 (Bent 2) ************************************************************ * * * wFRAME * * * * PUSH ANALYSIS of BRIDGE BENTS and FRAMES. * * * * Indicates formation of successive plastic hinges. * * * * VER._1.12,_JAN-14-95 * * * * Copyright (C) 1994 By Mark Seyed. * * * * This program should not be distributed under any * * condition. This release is for demo ONLY (beta testing * * is not complete). The author makes no expressed or * * implied warranty of any kind with regard to this program.* * In no event shall the author be held liable for * * incidental or consequential damages arising out of the * * use of this program. * * * ************************************************************ Node Point Information: Fixity condition definitions: s=spring and value r=complete release f=complete fixity with imposed displacement node # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 name S01.00 S01.01 S01.02 C01.01 C01.02 C01.03 C01.04 P01.01 P01.02 P01.03 P01.04 S02.01 S02.02 S02.03 S02.04 S02.05 S02.06 C02.01 C02.02 C02.03 C02.04 P02.01 P02.02 P02.03 P02.04 S03.01 S03.02 coordinates -----------fixity -------X Y X-dir. Y-dir. Rotation 0.00 0.00 r r r 4.72 0.00 r r r 7.72 0.00 r r r 7.72 -3.38 r r r 7.72 -15.31 r r r 7.72 -27.24 r r r 7.72 -39.17 r r r 7.72 -41.22 s 1.4e+002 r r 7.72 -43.27 s 4.1e+002 r r 7.72 -45.32 s 6.7e+002 r r 7.72 -47.37 f 0.0000 f 0.0000 r 10.72 0.00 r r r 17.72 0.00 r r r 24.72 0.00 r r r 31.72 0.00 r r r 38.72 0.00 r r r 41.72 0.00 r r r 41.72 -3.38 r r r 41.72 -15.31 r r r 41.72 -27.24 r r r 41.72 -39.17 r r r 41.72 -41.22 s 1.4e+002 r r 41.72 -43.27 s 4.1e+002 r r 41.72 -45.32 s 6.7e+002 r r 41.72 -47.37 f 0.0000 f 0.0000 r 44.72 0.00 r r r 49.44 0.00 r r r Spring Information at node points: k's = k/ft or ft-k/rad.; node spring k1 d's = ft or rad. d1 k2 d2 Chapter 21 – Seismic Design of Concrete Bridges 21-105 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 # 8 9 10 22 23 24 name P01X01 P01X02 P01X03 P02X01 P02X02 P02X03 136.37 414.83 665.70 136.37 414.83 665.70 0.149 0.105 0.106 0.149 0.105 0.106 0.00 0.00 0.00 0.00 0.00 0.00 1.000 1.000 1.000 1.000 1.000 1.000 0.00 0.00 0.00 0.00 0.00 0.00 1000.000 1000.000 1000.000 1000.000 1000.000 1000.000 Structural Setup: Spans= 3, Columns= 2, Piles= 2, Link Beams= 0 Element Information: element nodes depth # name fix i j L d area Ei Ef 1 S01-01 rn 1 2 4.72 6.8 62.6 629528 62953 2 S01-02 rn 2 3 3.00 6.8 62.6 629528 62953 3 C01-01 rn 3 4 3.38 6.0 28.3 629528 62953 4 C01-02 rn 4 5 11.93 6.0 28.3 629528 62953 5 C01-03 rn 5 6 11.93 6.0 28.3 629528 62953 6 C01-04 rn 6 7 11.93 6.0 28.3 629528 62953 7 P01-01 rn 7 8 2.05 6.0 28.3 629528 62953 8 P01-02 rn 8 9 2.05 6.0 28.3 629528 62953 9 P01-03 rn 9 10 2.05 6.0 28.3 629528 62953 10 P01-04 rn 10 11 2.05 6.0 28.3 629528 62953 11 S02-01 rn 3 12 3.00 6.8 62.6 629528 62953 12 S02-02 rn 12 13 7.00 6.8 62.6 629528 62953 13 S02-03 rn 13 14 7.00 6.8 62.6 629528 62953 14 S02-04 rn 14 15 7.00 6.8 62.6 629528 62953 15 S02-05 rn 15 16 7.00 6.8 62.6 629528 62953 16 S02-06 rn 16 17 3.00 6.8 62.6 629528 62953 17 C02-01 rn 17 18 3.38 6.0 28.3 629528 62953 18 C02-02 rn 18 19 11.93 6.0 28.3 629528 62953 19 C02-03 rn 19 20 11.93 6.0 28.3 629528 62953 20 C02-04 rn 20 21 11.93 6.0 28.3 629528 62953 21 P02-01 rn 21 22 2.05 6.0 28.3 629528 62953 22 P02-02 rn 22 23 2.05 6.0 28.3 629528 62953 23 P02-03 rn 23 24 2.05 6.0 28.3 629528 62953 24 P02-04 rn 24 25 2.05 6.0 28.3 629528 62953 25 S03-01 rn 17 26 3.00 6.8 62.6 629528 62953 26 S03-02 rn 26 27 4.72 6.8 62.6 629528 62953 bandwidth of the problem = 10 Number of rows and columns in strage = 81 x 30 Cumulative Results of analysis at end of stage Icr 52.25 52.25 47.44 23.72 23.72 23.72 23.72 23.72 23.72 23.72 52.25 52.25 52.25 52.25 52.25 52.25 47.44 23.72 23.72 23.72 23.72 23.72 23.72 23.72 52.25 52.25 q -68.40 -68.40 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 -68.40 -68.40 -68.40 -68.40 -68.40 -68.40 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 -68.40 -68.40 Mpp 27676 27676 27676 13838 13838 13838 13838 13838 13838 13838 27676 27676 27676 27676 27676 27676 27676 13838 13838 13838 13838 13838 13838 13838 27676 27676 Mpn 27676 27676 27676 13838 13838 13838 13838 13838 13838 13838 27676 27676 27676 27676 27676 27676 27676 13838 13838 13838 13838 13838 13838 13838 27676 27676 tol status 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0 Plastic Action at: Element/ Stage/ Code/ node# name 1 2 3 4 5 6 7 8 9 10 11 12 13 S01.00 S01.01 S01.02 C01.01 C01.02 C01.03 C01.04 P01.01 P01.02 P01.03 P01.04 S02.01 S02.02 Lat. Force *g (DL= 3381.7) ---------- GLOBAL Displ.x Displ.y 0.00001 0.00633 0.00001 -0.00014 0.00001 -0.00450 -0.00484 -0.00418 -0.01446 -0.00304 -0.01376 -0.00191 -0.00662 -0.00078 -0.00503 -0.00058 -0.00338 -0.00039 -0.00170 -0.00019 0.00000 0.00000 0.00001 -0.00941 0.00000 -0.02023 / Deflection / (in) --------Rotation -0.00136 -0.00140 -0.00152 -0.00135 -0.00032 0.00038 0.00076 0.00079 0.00081 0.00083 0.00083 -0.00170 -0.00121 Chapter 21 – Seismic Design of Concrete Bridges 21-106 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 14 15 16 17 18 19 20 21 22 23 24 25 26 27 S02.03 S02.04 S02.05 S02.06 C02.01 C02.02 C02.03 C02.04 P02.01 P02.02 P02.03 P02.04 S03.01 S03.02 -0.00001 -0.00001 -0.00002 -0.00002 0.00483 0.01445 0.01375 0.00662 0.00503 0.00338 0.00170 0.00000 -0.00002 -0.00002 element node # name fix 1 S01-01 rn 2 S01-02 rn 3 C01-01 rn 4 C01-02 rn 5 C01-03 rn 6 C01-04 rn 7 P01-01 rn 8 P01-02 rn 9 P01-03 rn 10 P01-04 rn 11 S02-01 rn 12 S02-02 rn 13 S02-03 rn 14 S02-04 rn 15 S02-05 rn 16 S02-06 rn 17 C02-01 rn 18 C02-02 rn 19 C02-03 rn 20 C02-04 rn 21 P02-01 rn 22 P02-02 rn 23 P02-03 rn 24 P02-04 rn 25 S03-01 rn -0.02467 -0.02023 -0.00941 -0.00450 -0.00417 -0.00304 -0.00191 -0.00078 -0.00058 -0.00039 -0.00019 0.00000 -0.00014 0.00633 0.00000 0.00121 0.00170 0.00152 0.00135 0.00032 -0.00038 -0.00076 -0.00079 -0.00081 -0.00083 -0.00083 0.00140 0.00136 -------- local ----------- ------ element ---------displ.x displ.y rotation axial shear moment 1 0.00001 0.00633 -0.00136 0.00 0.00 0.00 2 0.00001 -0.00014 -0.00140 0.00 322.85 -761.94 2 0.00001 -0.00014 -0.00140 0.00 -322.85 761.93 3 0.00001 -0.00450 -0.00152 0.00 528.05 -2038.28 3 0.00450 0.00001 -0.00152 1690.85 -34.15 -1605.41 4 0.00418 -0.00484 -0.00135 -1690.85 34.15 1489.98 4 0.00418 -0.00484 -0.00135 1690.85 -34.15 -1489.98 5 0.00304 -0.01446 -0.00032 -1690.85 34.15 1082.57 5 0.00304 -0.01446 -0.00032 1690.85 -34.15 -1082.57 6 0.00191 -0.01376 0.00038 -1690.85 34.15 675.15 6 0.00191 -0.01376 0.00038 1690.85 -34.15 -675.15 7 0.00078 -0.00662 0.00076 -1690.85 34.15 267.73 7 0.00078 -0.00662 0.00076 1690.85 -34.14 -267.71 8 0.00058 -0.00503 0.00079 -1690.85 34.14 197.70 8 0.00058 -0.00503 0.00079 1690.85 -33.46 -197.70 9 0.00039 -0.00338 0.00081 -1690.85 33.46 129.10 9 0.00039 -0.00338 0.00081 1690.85 -32.05 -129.10 10 0.00019 -0.00170 0.00083 -1690.85 32.05 63.39 10 0.00019 -0.00170 0.00083 1690.85 -30.92 -63.40 11 0.00000 0.00000 0.00083 -1690.85 30.92 0.00 3 0.00001 -0.00450 -0.00152 34.15 1162.80 3643.68 12 0.00001 -0.00941 -0.00170 -34.15 -957.60 -463.07 12 0.00001 -0.00941 -0.00170 34.15 957.61 463.06 13 0.00000 -0.02023 -0.00121 -34.15 -478.81 4564.38 13 0.00000 -0.02023 -0.00121 34.15 478.81 -4564.38 14 -0.00001 -0.02467 0.00000 -34.15 -0.01 6240.23 14 -0.00001 -0.02467 0.00000 34.15 0.01 -6240.23 15 -0.00001 -0.02023 0.00121 -34.15 478.79 4564.47 15 -0.00001 -0.02023 0.00121 34.15 -478.80 -4564.46 16 -0.00002 -0.00941 0.00170 -34.15 957.60 -462.91 16 -0.00002 -0.00941 0.00170 34.15 -957.59 462.89 17 -0.00002 -0.00450 0.00152 -34.15 1162.79 -3643.48 17 0.00450 -0.00002 0.00152 1690.83 34.15 1605.20 18 0.00417 0.00483 0.00135 -1690.83 -34.15 -1489.77 18 0.00417 0.00483 0.00135 1690.83 34.15 1489.77 19 0.00304 0.01445 0.00032 -1690.83 -34.15 -1082.42 19 0.00304 0.01445 0.00032 1690.83 34.15 1082.42 20 0.00191 0.01375 -0.00038 -1690.83 -34.15 -675.06 20 0.00191 0.01375 -0.00038 1690.83 34.15 675.06 21 0.00078 0.00662 -0.00076 -1690.83 -34.15 -267.70 21 0.00078 0.00662 -0.00076 1690.83 34.14 267.71 22 0.00058 0.00503 -0.00079 -1690.83 -34.14 -197.71 22 0.00058 0.00503 -0.00079 1690.83 33.46 197.71 23 0.00039 0.00338 -0.00081 -1690.83 -33.46 -129.11 23 0.00039 0.00338 -0.00081 1690.83 32.06 129.12 24 0.00019 0.00170 -0.00083 -1690.83 -32.06 -63.39 24 0.00019 0.00170 -0.00083 1690.83 30.93 63.40 25 0.00000 0.00000 -0.00083 -1690.83 -30.93 0.00 17 -0.00002 -0.00450 0.00152 0.00 528.05 2038.28 Chapter 21 – Seismic Design of Concrete Bridges 21-107 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 26 S03-02 rn 26 -0.00002 -0.00014 26 -0.00002 -0.00014 27 -0.00002 0.00633 0.00140 0.00140 0.00136 Cumulative Results of analysis at end of stage 0.00 0.00 0.00 -322.85 322.85 0.00 -761.92 761.93 0.00 1 Plastic Action at: Element/ Stage/ Code/ C02-02 1 rs node# name 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 S01.00 S01.01 S01.02 C01.01 C01.02 C01.03 C01.04 P01.01 P01.02 P01.03 P01.04 S02.01 S02.02 S02.03 S02.04 S02.05 S02.06 C02.01 C02.02 C02.03 C02.04 P02.01 P02.02 P02.03 P02.04 S03.01 S03.02 Lat. Force *g (DL= 3381.7) 0.1712 ---------- GLOBAL Displ.x Displ.y 0.70748 0.02708 0.70748 0.00919 0.70747 -0.00241 0.69197 -0.00224 0.58279 -0.00163 0.39898 -0.00103 0.16941 -0.00042 0.12746 -0.00031 0.08515 -0.00021 0.04263 -0.00010 0.00000 0.00000 0.70749 -0.01286 0.70750 -0.02604 0.70750 -0.02467 0.70749 -0.01442 0.70746 -0.00595 0.70744 -0.00658 0.70163 -0.00611 0.61168 -0.00445 0.42648 -0.00280 0.18265 -0.00114 0.13751 -0.00085 0.09192 -0.00057 0.04603 -0.00028 0.00000 0.00000 0.70745 -0.00948 0.70745 -0.01442 element node # name fix 1 S01-01 rn 2 S01-02 rn 3 C01-01 rn 4 C01-02 rn 5 C01-03 rn 6 C01-04 rn 7 P01-01 rn 8 P01-02 rn 9 P01-03 rn 10 P01-04 rn 11 S02-01 rn 12 S02-02 rn / Deflection / (in) 8.4898 --------Rotation -0.00378 -0.00382 -0.00394 -0.00522 -0.01268 -0.01773 -0.02035 -0.02056 -0.02070 -0.02078 -0.02080 -0.00301 -0.00077 0.00103 0.00165 0.00039 -0.00090 -0.00252 -0.01204 -0.01849 -0.02187 -0.02214 -0.02233 -0.02243 -0.02246 -0.00102 -0.00106 -------- local ----------- ------ element ---------displ.x displ.y rotation axial shear moment 1 0.70748 0.02708 -0.00378 27.69 -0.01 -0.01 2 0.70748 0.00919 -0.00382 -27.69 322.85 -761.97 2 0.70748 0.00919 -0.00382 71.78 -322.85 761.95 3 0.70747 -0.00241 -0.00394 -71.78 528.05 -2038.29 3 0.00241 0.70747 -0.00394 907.25 253.50 11715.99 4 0.00224 0.69197 -0.00522 -907.25 -253.50 -10859.41 4 0.00224 0.69197 -0.00522 907.18 253.90 10859.10 5 0.00163 0.58279 -0.01268 -907.18 -253.90 -7830.08 5 0.00163 0.58279 -0.01268 907.18 253.90 7830.12 6 0.00103 0.39898 -0.01773 -907.18 -253.90 -4801.02 6 0.00103 0.39898 -0.01773 907.18 253.90 4801.03 7 0.00042 0.16941 -0.02035 -907.18 -253.90 -1771.96 7 0.00042 0.16941 -0.02035 907.18 253.53 1771.40 8 0.00031 0.12746 -0.02056 -907.18 -253.53 -1250.91 8 0.00031 0.12746 -0.02056 907.18 236.71 1251.16 9 0.00021 0.08515 -0.02070 -907.18 -236.71 -765.72 9 0.00021 0.08515 -0.02070 907.18 201.31 766.32 10 0.00010 0.04263 -0.02078 -907.18 -201.31 -353.62 10 0.00010 0.04263 -0.02078 907.18 172.67 354.11 11 0.00000 0.00000 -0.02080 -907.18 -172.67 0.01 3 0.70747 -0.00241 -0.00394 -147.11 379.20 -9677.96 12 0.70749 -0.01286 -0.00301 147.11 -174.00 10507.76 12 0.70749 -0.01286 -0.00301 -88.03 173.93 -10507.76 Chapter 21 – Seismic Design of Concrete Bridges 21-108 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 13 S02-03 rn 14 S02-04 rn 15 S02-05 rn 16 S02-06 rn 17 C02-01 rn 18 C02-02 rs 19 C02-03 rn 20 C02-04 rn 21 P02-01 rn 22 P02-02 rn 23 P02-03 rn 24 P02-04 rn 25 S03-01 rn 26 S03-02 rn 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 17 26 26 27 0.70750 0.70750 0.70750 0.70750 0.70749 0.70749 0.70746 0.70746 0.70744 0.00658 0.00611 0.00611 0.00445 0.00445 0.00280 0.00280 0.00114 0.00114 0.00085 0.00085 0.00057 0.00057 0.00028 0.00028 0.00000 0.70744 0.70745 0.70745 0.70745 -0.02604 -0.02604 -0.02467 -0.02467 -0.01442 -0.01442 -0.00595 -0.00595 -0.00658 0.70744 0.70163 0.70163 0.61168 0.61168 0.42648 0.42648 0.18265 0.18265 0.13751 0.13751 0.09192 0.09192 0.04603 0.04603 0.00000 -0.00658 -0.00948 -0.00948 -0.01442 Chapter 21 – Seismic Design of Concrete Bridges -0.00077 -0.00077 0.00103 0.00103 0.00165 0.00165 0.00039 0.00039 -0.00090 -0.00090 -0.00252 -0.00252 -0.01204 -0.01204 -0.01849 -0.01849 -0.02187 -0.02187 -0.02214 -0.02214 -0.02233 -0.02233 -0.02243 -0.02243 -0.02246 -0.00090 -0.00102 -0.00102 -0.00106 88.03 -5.78 5.78 76.09 -76.09 158.08 -158.08 216.23 -216.23 2474.51 -2474.51 2474.46 -2474.46 2474.46 -2474.46 2474.46 -2474.46 2474.46 -2474.46 2474.46 -2474.46 2474.46 -2474.46 2474.46 -2474.46 -72.95 72.95 -28.11 28.11 304.87 -304.87 783.67 -783.67 1262.47 -1262.47 1741.27 -1741.27 1946.47 322.57 -322.57 322.17 -322.17 322.18 -322.18 322.18 -322.18 322.17 -322.17 303.41 -303.41 265.35 -265.35 234.74 -234.74 528.06 -322.86 322.85 0.00 10049.47 -10049.47 6239.57 -6239.57 -921.93 921.94 -11435.05 11435.04 -16966.67 14926.95 -13838.61 13838.00 -9994.56 9994.59 -6150.91 6150.92 -2307.32 2307.40 -1646.88 1647.21 -1024.71 1024.96 -481.13 481.23 0.07 2038.30 -761.92 761.91 0.00 21-109 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 APPENDIX 21.3-7 Select Output from xSECTION, Compression Column ************************************************************ * * * xSECTION * * * * DUCTILITY and STRENGTH of * * Circular, Semi-Circular, full and partial Rings, * * Rectangular, T-, I-, Hammer head, Octagonal, Polygons * * or any combination of above shapes forming * * Concrete Sections using Fiber Models * * * * VER._2.40,_MAR-14-99 * * * * Copyright (C) 1994, 1995, 1999 By Mark Seyed Mahan. * * * * A proper license must be obtained to use this software. * * For GOVERNMENT work call 916-227-8404, otherwise leave a * * message at 530-756-2367. The author makes no expressed or* * implied warranty of any kind with regard to this program.* * In no event shall the author be held liable for * * incidental or consequential damages arising out of the * * use of this program. * * * ************************************************************ This output was generated by running: xSECTION VER._2.40,_MAR-14-99 LICENSE (choices: LIMITED/UNLIMITED) UNLIMITED ENTITY (choices: GOVERNMENT/CONSULTANT) Government NAME_OF_FIRM Caltrans BRIDGE_NAME EXAMPLE BRIDGE_NUMBER 99-9999 JOB_TITLE PROTYPE BRIDGE - BRIDGE DESIGN ACADEMY Concrete Type Information: ----------strains-------Type e0 e2 ecc eu 1 0.0020 0.0040 0.0055 0.0145 2 0.0020 0.0040 0.0020 0.0050 --------strength-------f0 f2 fcc fu E 5.28 6.98 7.15 6.11 4313 5.28 3.61 5.28 2.64 4313 W 148 148 Steel Type Information: -----strains------ --strengthType ey eh eu fy fu E 1 0.0023 0.0150 0.0900 68.00 95.00 29000 2 0.0023 0.0075 0.0600 68.00 95.00 29000 Steel Fiber Information: Fiber xc yc No. type in in 1 2 31.93 0.00 2 2 31.00 7.64 ... 25 2 28.27 -14.84 26 2 31.00 -7.64 area in^2 2.25 2.25 2.25 2.25 Chapter 21 – Seismic Design of Concrete Bridges 21-110 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 Force Equilibrium Condition of the x-section: Max. Conc. Strain step epscmax 0 0.00000 1 0.00029 ... 24 0.00291 25 0.00322 26 0.00356 27 0.00394 28 0.00435 29 0.00481 30 0.00532 31 0.00588 32 0.00650 33 0.00718 34 0.00794 35 0.00878 36 0.00971 37 0.01073 38 0.01186 39 0.01312 40 0.01450 Neutral Axis in. 0.00 -29.19 14.11 14.74 15.28 15.73 16.07 16.24 16.23 16.38 16.52 16.66 16.77 16.86 16.91 16.97 16.96 16.95 16.91 Max. Steel Strain Conc. Tens. Comp. 0.0000 0 0.0000 2256 -0.0061 -0.0070 -0.0081 -0.0092 -0.0104 -0.0117 -0.0129 -0.0144 -0.0161 -0.0180 -0.0200 -0.0223 -0.0248 -0.0275 -0.0304 -0.0335 -0.0370 3742 3778 3813 3856 3904 3950 4008 4043 4089 4135 4180 4226 4271 4310 4366 4415 4458 Steel force Comp. Tens. 0 0 222 -2 926 963 991 1018 1049 1075 1092 1106 1121 1137 1156 1177 1201 1231 1242 1255 1269 -2194 -2268 -2330 -2399 -2478 -2552 -2623 -2675 -2734 -2797 -2862 -2928 -2997 -3069 -3132 -3195 -3255 P/S force 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Net Curvature Moment force rad/in (K-ft) 0.00 0.000000 0 1.88 0.000004 2346 0.52 -1.28 0.34 0.71 0.63 -0.48 1.90 -0.34 1.91 0.76 0.35 1.07 0.93 -2.02 1.47 0.47 -1.79 0.000133 0.000152 0.000172 0.000194 0.000219 0.000244 0.000269 0.000300 0.000334 0.000372 0.000414 0.000459 0.000509 0.000565 0.000624 0.000689 0.000761 13492 13683 13834 14012 14204 14332 14424 14544 14706 14879 15055 15231 15403 15573 15730 15869 15987 First Yield of Rebar Information (not Idealized): Rebar Number 20 Coordinates X and Y (global in.) Yield strain = 0.00230 Curvature (rad/in)= 0.000057 Moment (ft-k) = 10802 -3.85, -31.70 Cross Section Information: Axial Load on Section (kips) = 2474 Percentage of Main steel in Cross Section = 1.44 Concrete modulus used in Idealization (ksi) = 4313 Cracked Moment of Inertia (ft^4) = 25.572 Idealization of Moment-Curvature Curve by Various Methods: Points on Curve =============== Method ID Conc. | Strain Curv. | in/in rad/in Strain @ 0.003 0.000138 Strain @ 0.004 0.000198 Strain @ 0.005 0.000253 CALTRANS 0.00755 0.000392 UCSD@5phy0.00558 0.000283 Moment (K-ft) 13546 14042 14366 14964 14479 Idealized Values ============================= Yield symbol Plastic | Curv. Moment for Curv. | rad/in (K-ft) moment rad/in 0.000071 13546 Mn 0.000689 0.000074 14042 Mn 0.000687 0.000075 14366 Mn 0.000685 0.000079 14964 Mp 0.000682 0.000076 14479 Mn 0.000685 Chapter 21 – Seismic Design of Concrete Bridges 21-111 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 APPENDIX 21.3-8 Select Output from xSECTION, Tension Column 05/10/2006, 07:43 ************************************************************ * * * xSECTION * * * * DUCTILITY and STRENGTH of * * Circular, Semi-Circular, full and partial Rings, * * Rectangular, T-, I-, Hammer head, Octagonal, Polygons * * or any combination of above shapes forming * * Concrete Sections using Fiber Models * * * * VER._2.40,_MAR-14-99 * * * * Copyright (C) 1994, 1995, 1999 By Mark Seyed Mahan. * * * * A proper license must be obtained to use this software. * * For GOVERNMENT work call 916-227-8404, otherwise leave a * * message at 530-756-2367. The author makes no expressed or* * implied warranty of any kind with regard to this program.* * In no event shall the author be held liable for * * incidental or consequential damages arising out of the * * use of this program. * * * ************************************************************ This output was generated by running: xSECTION VER._2.40,_MAR-14-99 LICENSE (choices: LIMITED/UNLIMITED) UNLIMITED ENTITY (choices: GOVERNMENT/CONSULTANT) Government NAME_OF_FIRM Caltrans BRIDGE_NAME EXAMPLE BRIDGE_NUMBER 99-9999 JOB_TITLE PROTYPE BRIDGE - BRIDGE DESIGN ACADEMY Concrete Type Information: ----------strains-------Type e0 e2 ecc eu 1 0.0020 0.0040 0.0055 0.0145 2 0.0020 0.0040 0.0020 0.0050 --------strength-------f0 f2 fcc fu E 5.28 6.98 7.15 6.11 4313 5.28 3.61 5.28 2.64 4313 W 148 148 Steel Type Information: -----strains------ --strengthType ey eh eu fy fu E 1 0.0023 0.0150 0.0900 68.00 95.00 29000 2 0.0023 0.0075 0.0600 68.00 95.00 29000 Steel Fiber Information: Fiber xc yc area No. type in in in^2 1 2 31.93 0.00 2.25 2 2 31.00 7.64 2.25 ................................ ................................ 14 2 -31.93 0.00 2.25 15 2 -31.00 -7.64 2.25 16 2 -28.27 -14.84 2.25 17 2 -23.90 -21.17 2.25 18 2 -18.14 -26.28 2.25 19 2 -11.32 -29.86 2.25 Chapter 21 – Seismic Design of Concrete Bridges 21-112 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 20 21 22 23 24 25 26 2 2 2 2 2 2 2 -3.85 3.85 11.32 18.14 23.90 28.27 31.00 -31.70 -31.70 -29.85 -26.28 -21.17 -14.84 -7.64 2.25 2.25 2.25 2.25 2.25 2.25 2.25 Force Equilibrium Condition of the x-section: Max. Max. Conc. Neutral Steel Steel Strain Axis Strain Conc. force P/S Net Curvature Moment step epscmax in. Tens. Comp. Comp. Tens. force force rad/in (K-ft) 0 0.00000 0.00 0.0000 0 0 0 0 0.00 0.000000 0 1 0.00029 2.87 -0.0003 949 131 -173 0 -0.82 0.000009 2393 ............................................................................... ............................................................................... 27 0.00394 19.79 -0.0125 2770 862 -2726 0 -0.83 0.000243 11770 28 0.00435 19.97 -0.0140 2820 878 -2792 0 -0.67 0.000272 11964 29 0.00481 20.02 -0.0156 2862 903 -2859 0 -0.64 0.000301 12123 30 0.00532 19.99 -0.0172 2893 936 -2922 0 -0.21 0.000333 12243 31 0.00588 20.03 -0.0191 2927 973 -2993 0 0.00 0.000368 12404 32 0.00650 20.00 -0.0210 2993 980 -3067 0 -0.17 0.000407 12576 33 0.00718 19.97 -0.0232 3066 989 -3148 0 0.19 0.000449 12758 34 0.00794 20.02 -0.0257 3114 993 -3201 0 -0.80 0.000498 12933 35 0.00878 20.05 -0.0285 3160 999 -3252 0 -0.36 0.000551 13102 36 0.00971 20.08 -0.0316 3203 1005 -3302 0 -0.67 0.000611 13262 37 0.01073 20.10 -0.0350 3245 1016 -3355 0 -0.91 0.000676 13418 38 0.01186 20.10 -0.0387 3280 1032 -3405 0 0.08 0.000747 13563 39 0.01312 20.12 -0.0429 3308 1052 -3453 0 -0.24 0.000827 13700 40 0.01450 20.13 -0.0474 3326 1077 -3496 0 0.08 0.000915 13815 First Yield of Rebar Information (not Idealized): Rebar Number 20 Coordinates X and Y (global in.) Yield strain = 0.00230 Curvature (rad/in)= 0.000051 Moment (ft-k) = 8190 -3.85, -31.70 Cross Section Information: Axial Load on Section (kips) = 907 Percentage of Main steel in Cross Section = 1.44 Concrete modulus used in Idealization (ksi) = 4313 Cracked Moment of Inertia (ft^4) = 21.496 Idealization of Moment-Curvature Curve by Various Methods: Points on Curve =============== Method ID Conc. | Strain Curv. | in/in rad/in Strain @ 0.003 0.000176 Strain @ 0.004 0.000248 Strain @ 0.005 0.000313 CALTRANS 0.00673 0.000421 UCSD@5phy0.00412 0.000256 Moment (K-ft) 11159 11800 12168 12636 11855 Chapter 21 – Seismic Design of Concrete Bridges Idealized Values ============================= Yield symbol Plastic | Curv. Moment for Curv. | rad/in (K-ft) moment rad/in 0.000070 11159 Mn 0.000845 0.000074 11800 Mn 0.000841 0.000076 12168 Mn 0.000839 0.000079 12636 Mp 0.000836 0.000074 11855 Mn 0.000841 21-113 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 APPENDIX 21.3-9 wFRAME, Output File 05/15/2006, 08:02 Design Academy Example No: 1 (Bent 2) ************************************************************ * * * wFRAME * * * * PUSH ANALYSIS of BRIDGE BENTS and FRAMES. * * * * Indicates formation of successive plastic hinges. * * * * VER._1.12,_JAN-14-95 * * * * Copyright (C) 1994 By Mark Seyed. * * * * This program should not be distributed under any * * condition. This release is for demo ONLY (beta testing * * is not complete). The author makes no expressed or * * implied warranty of any kind with regard to this program.* * In no event shall the author be held liable for * * incidental or consequential damages arising out of the * * use of this program. * * * ************************************************************ Node Point Information: Fixity condition definitions: s=spring and value r=complete release f=complete fixity with imposed displacement node # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 name S01.00 S01.01 S01.02 C01.01 C01.02 C01.03 C01.04 P01.01 P01.02 P01.03 P01.04 S02.01 S02.02 S02.03 S02.04 S02.05 S02.06 C02.01 C02.02 C02.03 C02.04 P02.01 P02.02 P02.03 P02.04 S03.01 S03.02 coordinates -----------fixity -------X Y X-dir. Y-dir. Rotation 0.00 0.00 r r r 4.72 0.00 r r r 7.72 0.00 r r r 7.72 -3.38 r r r 7.72 -15.31 r r r 7.72 -27.24 r r r 7.72 -39.17 r r r 7.72 -41.22 s 1.4e+002 r r 7.72 -43.27 s 4.1e+002 r r 7.72 -45.32 s 6.7e+002 r r 7.72 -47.37 f 0.0000 f 0.0000 r 10.72 0.00 r r r 17.72 0.00 r r r 24.72 0.00 r r r 31.72 0.00 r r r 38.72 0.00 r r r 41.72 0.00 r r r 41.72 -3.38 r r r 41.72 -15.31 r r r 41.72 -27.24 r r r 41.72 -39.17 r r r 41.72 -41.22 s 1.4e+002 r r 41.72 -43.27 s 4.1e+002 r r 41.72 -45.32 s 6.7e+002 r r 41.72 -47.37 f 0.0000 f 0.0000 r 44.72 0.00 r r r 49.44 0.00 r r r Spring Information at node points: k's = k/ft or ft-k/rad.; d's = ft or rad. Chapter 21 – Seismic Design of Concrete Bridges 21-114 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 node # 8 9 10 22 23 24 spring name P01X01 P01X02 P01X03 P02X01 P02X02 P02X03 k1 d1 136.37 414.83 665.70 136.37 414.83 665.70 k2 d2 0.149 0.105 0.106 0.149 0.105 0.106 0.00 0.00 0.00 0.00 0.00 0.00 1.000 1.000 1.000 1.000 1.000 1.000 0.00 0.00 0.00 0.00 0.00 0.00 1000.000 1000.000 1000.000 1000.000 1000.000 1000.000 Structural Setup: Spans= 3, Columns= 2, Piles= 2, Link Beams= 0 Element Information: element nodes depth # name fix i j L d area Ei Ef 1 S01-01 rn 1 2 4.72 6.8 62.6 629528 62953 2 S01-02 rn 2 3 3.00 6.8 62.6 629528 62953 3 C01-01 rn 3 4 3.38 6.0 28.3 629528 62953 4 C01-02 rn 4 5 11.93 6.0 28.3 629528 62953 5 C01-03 rn 5 6 11.93 6.0 28.3 629528 62953 6 C01-04 rn 6 7 11.93 6.0 28.3 629528 62953 7 P01-01 rn 7 8 2.05 6.0 28.3 629528 62953 8 P01-02 rn 8 9 2.05 6.0 28.3 629528 62953 9 P01-03 rn 9 10 2.05 6.0 28.3 629528 62953 10 P01-04 rn 10 11 2.05 6.0 28.3 629528 62953 11 S02-01 rn 3 12 3.00 6.8 62.6 629528 62953 12 S02-02 rn 12 13 7.00 6.8 62.6 629528 62953 13 S02-03 rn 13 14 7.00 6.8 62.6 629528 62953 14 S02-04 rn 14 15 7.00 6.8 62.6 629528 62953 15 S02-05 rn 15 16 7.00 6.8 62.6 629528 62953 16 S02-06 rn 16 17 3.00 6.8 62.6 629528 62953 17 C02-01 rn 17 18 3.38 6.0 28.3 629528 62953 18 C02-02 rn 18 19 11.93 6.0 28.3 629528 62953 19 C02-03 rn 19 20 11.93 6.0 28.3 629528 62953 20 C02-04 rn 20 21 11.93 6.0 28.3 629528 62953 21 P02-01 rn 21 22 2.05 6.0 28.3 629528 62953 22 P02-02 rn 22 23 2.05 6.0 28.3 629528 62953 23 P02-03 rn 23 24 2.05 6.0 28.3 629528 62953 24 P02-04 rn 24 25 2.05 6.0 28.3 629528 62953 25 S03-01 rn 17 26 3.00 6.8 62.6 629528 62953 26 S03-02 rn 26 27 4.72 6.8 62.6 629528 62953 bandwidth of the problem = 10 Number of rows and columns in strage = 81 x 30 Cumulative Results of analysis at end of stage Icr 52.25 52.25 43.00 21.50 21.50 21.50 21.50 21.50 21.50 21.50 52.25 52.25 52.25 52.25 52.25 52.25 51.14 25.57 25.57 25.57 25.57 25.57 25.57 25.57 52.25 52.25 q -68.40 -68.40 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 -68.40 -68.40 -68.40 -68.40 -68.40 -68.40 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 -68.40 -68.40 Mpp 29928 29928 29928 12636 12636 12636 12636 12636 12636 12636 29928 29928 29928 29928 29928 29928 29928 14964 14964 14964 14964 14964 14964 14964 29928 29928 Mpn 29928 29928 29928 12636 12636 12636 12636 12636 12636 12636 29928 29928 29928 29928 29928 29928 29928 14964 14964 14964 14964 14964 14964 14964 29928 29928 tol status 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0.02 e 0 Plastic Action at: Element/ Stage/ Code/ Lat. Force *g (DL= 3381.7) / Deflection / (in) node# name ---------- GLOBAL --------Displ.x Displ.y Rotation 1 S01.00 -0.00603 0.00640 -0.00137 2 S01.01 -0.00603 -0.00012 -0.00141 ………………………………………………………………………………………………. …………………………………………………………………………………………….. 25 P02.04 0.00000 0.00000 -0.00064 26 S03.01 -0.00606 -0.00012 0.00141 27 S03.02 -0.00606 0.00639 0.00137 element node # name fix 1 S01-01 rn 2 S01-02 rn -------- local ----------- ------ element ---------displ.x displ.y rotation axial shear moment 1 -0.00603 0.00640 -0.00137 0.00 0.00 0.00 2 -0.00603 -0.00012 -0.00141 0.00 322.85 -761.94 2 -0.00603 -0.00012 -0.00141 0.01 -322.84 761.93 3 -0.00603 -0.00450 -0.00153 -0.01 528.04 -2038.28 Chapter 21 – Seismic Design of Concrete Bridges 21-115 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 25 S03-01 rn 26 S03-02 rn 17 26 26 27 -0.00606 -0.00450 -0.00606 -0.00012 -0.00606 -0.00012 -0.00606 0.00639 0.00153 0.00141 0.00141 0.00137 Cumulative Results of analysis at end of stage 0.00 0.00 0.00 0.00 528.05 -322.85 322.85 0.00 2038.27 -761.93 761.93 0.00 1 Plastic Action at: Element/ Stage/ Code/ rs 1 C02-02 node# name 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 S01.00 S01.01 S01.02 C01.01 C01.02 C01.03 C01.04 P01.01 P01.02 P01.03 P01.04 S02.01 S02.02 S02.03 S02.04 S02.05 S02.06 C02.01 C02.02 C02.03 C02.04 P02.01 P02.02 P02.03 P02.04 S03.01 S03.02 Lat. Force *g (DL= 3381.7) 0.1760 ---------- GLOBAL Displ.x Displ.y 0.73219 0.02482 0.73218 0.00836 0.73218 -0.00234 0.71753 -0.00217 0.60724 -0.00158 0.41672 -0.00099 0.17709 -0.00040 0.13324 -0.00030 0.08902 -0.00020 0.04456 -0.00010 0.00000 0.00000 0.73219 -0.01192 0.73220 -0.02352 0.73220 -0.02138 0.73218 -0.01148 0.73215 -0.00478 0.73213 -0.00665 0.72472 -0.00618 0.62890 -0.00450 0.43749 -0.00283 0.18720 -0.00115 0.14093 -0.00086 0.09420 -0.00058 0.04717 -0.00029 0.00000 0.00000 0.73214 -0.01097 0.73214 -0.01813 node element name fix # 1 S01-01 rn 2 S01-02 rn 3 C01-01 rn 4 C01-02 rn 5 C01-03 rn 6 C01-04 rn 7 P01-01 rn 8 P01-02 rn 9 P01-03 rn 10 P01-04 rn 11 S02-01 rn / Deflection (in) / 8.7862 --------Rotation -0.00348 -0.00351 -0.00364 -0.00501 -0.01304 -0.01846 -0.02127 -0.02149 -0.02164 -0.02172 -0.02175 -0.00274 -0.00059 0.00106 0.00151 0.00003 -0.00137 -0.00300 -0.01255 -0.01903 -0.02242 -0.02270 -0.02288 -0.02299 -0.02302 -0.00149 -0.00153 -------- local ----------- ------ element ---------moment shear displ.x displ.y rotation axial 0.01 -0.01 29.06 1 0.73219 0.02482 -0.00348 -761.97 322.86 -29.06 2 0.73218 0.00836 -0.00351 761.94 -322.85 74.05 2 0.73218 0.00836 -0.00351 528.05 -2038.29 -74.05 3 0.73218 -0.00234 -0.00364 248.18 11431.10 879.83 3 0.00234 0.73218 -0.00364 -248.18 -10591.82 -879.83 4 0.00217 0.71753 -0.00501 248.19 10591.19 879.86 4 0.00217 0.71753 -0.00501 -248.19 -7630.29 -879.86 5 0.00158 0.60724 -0.01304 7630.31 248.21 879.86 5 0.00158 0.60724 -0.01304 -248.21 -4669.24 -879.86 6 0.00099 0.41672 -0.01846 4669.30 248.20 879.86 6 0.00099 0.41672 -0.01846 -248.20 -1708.25 -879.86 7 0.00040 0.17709 -0.02127 1708.95 248.12 879.86 7 0.00040 0.17709 -0.02127 -248.12 -1200.10 -879.86 8 0.00030 0.13324 -0.02149 1200.16 230.01 879.86 8 0.00030 0.13324 -0.02149 -728.78 -230.01 -879.86 9 0.00020 0.08902 -0.02164 729.12 192.70 879.86 9 0.00020 0.08902 -0.02164 -334.12 -192.70 -879.86 10 0.00010 0.04456 -0.02172 334.16 163.00 879.86 10 0.00010 0.04456 -0.02172 -0.02 -163.00 -879.86 11 0.00000 0.00000 -0.02175 351.79 -9393.48 -137.94 3 0.73218 -0.00234 -0.00364 -146.59 10141.05 137.94 12 0.73219 -0.01192 -0.00274 Chapter 21 – Seismic Design of Concrete Bridges 21-116 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 12 S02-02 rn 13 S02-03 rn 14 S02-04 rn 15 S02-05 rn 16 S02-06 rn 17 C02-01 rn 18 C02-02 rs 19 C02-03 rn 20 C02-04 rn 21 P02-01 rn 22 P02-02 rn 23 P02-03 rn 24 P02-04 rn 25 S03-01 rn 26 S03-02 rn 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 17 26 26 27 0.73219 0.73220 0.73220 0.73220 0.73220 0.73218 0.73218 0.73215 0.73215 0.73213 0.00665 0.00618 0.00618 0.00450 0.00450 0.00283 0.00283 0.00115 0.00115 0.00086 0.00086 0.00058 0.00058 0.00029 0.00029 0.00000 0.73213 0.73214 0.73214 0.73214 -0.01192 -0.02352 -0.02352 -0.02138 -0.02138 -0.01148 -0.01148 -0.00478 -0.00478 -0.00665 0.73213 0.72472 0.72472 0.62890 0.62890 0.43749 0.43749 0.18720 0.18720 0.14093 0.14093 0.09420 0.09420 0.04717 0.04717 0.00000 -0.00665 -0.01097 -0.01097 -0.01813 -0.00274 -0.00059 -0.00059 0.00106 0.00106 0.00151 0.00151 0.00003 0.00003 -0.00137 -0.00137 -0.00300 -0.00300 -0.01255 -0.01255 -0.01903 -0.01903 -0.02242 -0.02242 -0.02270 -0.02270 -0.02288 -0.02288 -0.02299 -0.02299 -0.02302 -0.00137 -0.00149 -0.00149 -0.00153 -78.08 78.08 6.47 -6.47 91.08 -91.08 175.49 -175.49 236.81 -236.81 2501.90 -2501.90 2501.84 -2501.84 2501.84 -2501.84 2501.84 -2501.84 2501.84 -2501.84 2501.84 -2501.84 2501.84 -2501.84 2501.84 -2501.84 -74.72 74.72 -28.84 28.84 146.67 332.13 -332.13 810.93 -810.93 1289.73 -1289.73 1768.53 -1768.53 1973.73 348.38 -348.38 348.05 -348.05 348.04 -348.04 348.03 -348.03 347.72 -347.72 328.53 -328.53 289.56 -289.56 257.88 -257.88 528.17 -322.97 322.85 0.00 -10141.02 9491.90 -9491.90 5491.19 -5491.19 -1861.15 1861.14 -12565.08 12565.06 -18178.47 16141.41 -14963.38 14964.00 -10811.87 10811.93 -6659.75 6659.80 -2507.76 2507.71 -1795.38 1795.20 -1121.56 1122.16 -528.43 528.77 -0.07 2038.64 -761.94 761.91 0.01 …………………………………………………………………………………… …………………………………………………………………………………… …………………………………………………………………………………… …………………………………………………………………………………… Cumulative Results of analysis at end of stage 6 Plastic Action at: Element/ Stage/ Code/ C02-02 1 rs P02X01 2 2 P01X01 3 2 P02X02 4 2 P01X02 5 2 C01-02 6 rs node# name 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 S01.00 S01.01 S01.02 C01.01 C01.02 C01.03 C01.04 P01.01 P01.02 P01.03 P01.04 S02.01 S02.02 S02.03 S02.04 S02.05 Lat. Force *g (DL= 3381.7) 0.1760 0.1818 0.1847 0.1875 0.1891 0.1903 ---------- GLOBAL Displ.x Displ.y 0.87699 0.03093 0.87698 0.01084 0.87698 -0.00217 0.85928 -0.00201 0.72688 -0.00147 0.49873 -0.00092 0.21194 -0.00038 0.15947 -0.00028 0.10654 -0.00019 0.05333 -0.00009 0.00000 0.00000 0.87699 -0.01377 0.87701 -0.02802 0.87702 -0.02622 0.87700 -0.01502 0.87697 -0.00605 Chapter 21 – Seismic Design of Concrete Bridges / Deflection / (in) 8.7862 9.4798 9.8322 10.1774 10.3724 10.5239 --------Rotation -0.00425 -0.00428 -0.00441 -0.00605 -0.01563 -0.02210 -0.02546 -0.02572 -0.02590 -0.02600 -0.02603 -0.00332 -0.00079 0.00115 0.00178 0.00039 21-117 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 17 18 19 20 21 22 23 24 25 26 27 S02.06 C02.01 C02.02 C02.03 C02.04 P02.01 P02.02 P02.03 P02.04 S03.01 S03.02 0.87696 0.87079 0.73519 0.50407 0.21424 0.16121 0.10771 0.05392 0.00000 0.87696 0.87697 element node # name fix 1 S01-01 rn 2 S01-02 rn 3 C01-01 rn 4 C01-02 rs 5 C01-03 rn 6 C01-04 rn 7 P01-01 rn 8 P01-02 rn 9 P01-03 rn 10 P01-04 rn 11 S02-01 rn 12 S02-02 rn 13 S02-03 rn 14 S02-04 rn 15 S02-05 rn 16 S02-06 rn 17 C02-01 rn 18 C02-02 rs 19 C02-03 rn 20 C02-04 rn 21 P02-01 rn 22 P02-02 rn 23 P02-03 rn 24 P02-04 rn 25 S03-01 rn 26 S03-02 rn -0.00682 -0.00634 -0.00462 -0.00290 -0.00118 -0.00089 -0.00059 -0.00030 0.00000 -0.01003 -0.01545 -0.00100 -0.00263 -0.01588 -0.02235 -0.02573 -0.02600 -0.02618 -0.02628 -0.02631 -0.00112 -0.00116 -------- local ----------- ------ element ---------displ.x displ.y rotation axial shear moment 1 0.87699 0.03093 -0.00425 31.37 -0.01 0.00 2 0.87698 0.01084 -0.00428 -31.37 322.86 -761.98 2 0.87698 0.01084 -0.00428 79.93 -322.85 761.94 3 0.87698 -0.00217 -0.00441 -79.93 528.05 -2038.30 3 0.00217 0.87698 -0.00441 814.86 295.87 13637.29 4 0.00201 0.85928 -0.00605 -814.86 -295.87 -12636.74 4 0.00201 0.85928 -0.00605 814.88 295.88 12636.00 5 0.00147 0.72688 -0.01563 -814.88 -295.88 -9106.20 5 0.00147 0.72688 -0.01563 814.88 295.89 9106.22 6 0.00092 0.49873 -0.02210 -814.88 -295.89 -5576.25 6 0.00092 0.49873 -0.02210 814.88 295.88 5576.32 7 0.00038 0.21194 -0.02546 -814.88 -295.88 -2046.37 7 0.00038 0.21194 -0.02546 814.88 295.73 2047.11 8 0.00028 0.15947 -0.02572 -814.88 -295.73 -1440.69 8 0.00028 0.15947 -0.02572 814.88 275.51 1440.64 9 0.00019 0.10654 -0.02590 -814.88 -275.51 -876.00 9 0.00019 0.10654 -0.02590 814.88 231.57 876.37 10 0.00009 0.05333 -0.02600 -814.88 -231.57 -401.75 10 0.00009 0.05333 -0.02600 814.88 196.02 401.81 11 0.00000 0.00000 -0.02603 -814.88 -196.02 -0.02 3 0.87698 -0.00217 -0.00441 -176.75 286.81 -11599.76 12 0.87699 -0.01377 -0.00332 176.75 -81.61 12152.41 12 0.87699 -0.01377 -0.00332 -112.03 81.70 -12152.37 13 0.87701 -0.02802 -0.00079 112.03 397.10 11048.49 13 0.87701 -0.02802 -0.00079 -20.60 -397.10 -11048.48 14 0.87702 -0.02622 0.00115 20.60 875.90 6593.00 14 0.87702 -0.02622 0.00115 70.88 -875.90 -6593.00 15 0.87700 -0.01502 0.00178 -70.88 1354.70 -1214.09 15 0.87700 -0.01502 0.00178 162.24 -1354.70 1214.09 16 0.87697 -0.00605 0.00039 -162.24 1833.50 -12372.80 16 0.87697 -0.00605 0.00039 228.60 -1833.50 12372.78 17 0.87696 -0.00682 -0.00100 -228.60 2038.70 -18181.08 17 0.00682 0.87696 -0.00100 2566.87 349.18 16143.98 18 0.00634 0.87079 -0.00263 -2566.87 -349.18 -14963.33 18 0.00634 0.87079 -0.00634 2566.81 348.82 14964.00 19 0.00462 0.73519 -0.01588 -2566.81 -348.82 -10802.60 19 0.00462 0.73519 -0.01588 2566.81 348.82 10802.66 20 0.00290 0.50407 -0.02235 -2566.81 -348.82 -6641.19 20 0.00290 0.50407 -0.02235 2566.81 348.81 6641.24 21 0.00118 0.21424 -0.02573 -2566.81 -348.81 -2479.92 21 0.00118 0.21424 -0.02573 2566.81 348.49 2479.88 22 0.00089 0.16121 -0.02600 -2566.81 -348.49 -1765.94 22 0.00089 0.16121 -0.02600 2566.81 328.24 1765.77 23 0.00059 0.10771 -0.02618 -2566.81 -328.24 -1092.73 23 0.00059 0.10771 -0.02618 2566.81 284.77 1093.33 24 0.00030 0.05392 -0.02628 -2566.81 -284.77 -509.41 24 0.00030 0.05392 -0.02628 2566.81 248.60 509.76 25 0.00000 0.00000 -0.02631 -2566.81 -248.60 -0.07 17 0.87696 -0.00682 -0.00100 -80.75 528.17 2038.65 26 0.87696 -0.01003 -0.00112 80.75 -322.97 -761.94 26 0.87696 -0.01003 -0.00112 -31.13 322.85 761.91 27 0.87697 -0.01545 -0.00116 31.13 0.00 0.01 Chapter 21 – Seismic Design of Concrete Bridges 21-118 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 APPENDIX 21.3-10 Force – Displacement Relationship, Bent 2, Right Push with Overturning Chapter 21 – Seismic Design of Concrete Bridges 21-119 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 APPENDIX 21.3-11 Joint Movement Calculation STATE OF CALIFORNIA.DEPARTMENT OF TRANSPORTATION a JOINT MOVEMENTS CALCULATIONS Note: Specific instructions are included as footnotes. DS-D-0129(Rev.5/93) EA DISTRICT 910076 COUNTY 59 ROUTE PM (KP) 999 99 ES BRIDGE NAME AND NUMBER Prototype Bridge TYPE OF STRUCTURE TYPE ABUTMENT TYPE EXPANSION(2" elasto pads, etc.) CIP/PS BOX GIRDER Seat Elastomeric Bearing Pads (1) TEMPERATURE EXTREMES(from Preliminary Rerport) Type Of Structure ANTICIPATED SHORTENING (inches/100 feet) (inches/100 feet) (3)MOVEMENT FACTOR (inches/100 feet) o Steel Range( o F)(0.0000065X1200) = + 0 o Concrete (Conventional) Range( o F)(0.0000060X1200) = + 0.06 Concrete(Pretensioned) Range( o F)(0.0000060X1200) = + 0.12 g = Concrete(Post Tensioned) Range( + 0.63 g = MAXIMUM 110 F - MINIMUM 23 F o 87 F = Range (2)THERMAL MOVEMENT ITEM(1) DESIGNER o 87 F)(0.0000060X1200) = DATE 0.6264 = = 1.26 ITEM(2)CHECKED BY DESIGNER DATE CHECKER To be filled in by Office of Structures Design b To be filled in by SR c Date: Seal Width Limits Location d Groove (saw cut) Width or Installation Width Skew (4) Calculated M.R. Seal Type (degrees) Contributing Movement (inches) A,B, Catalog W1 (5) W2 Structure Do not Length (inches) (Round up (Others) Number (inches) (inches) Temperature use in (feet) (3)X(4)/100 to 1/2") or Maximum Min.@Max. ( F) calculation Open Joint Abut 1 0 202 2.53 2.50 Joint Seal Assembly(strip seal) Abut 4 0 210 2.64 3.00 Joint Seal Assembly(strip seal) Temperature o f e (6)Adjust from Width at Maximum Temp. Temp. Listed (inches) (inches)  /(1)X(2)X(4)/100 see XS-12-59 Anticipated Shortening = 1.26  202  210     2.60 in. 100  2  Chapter 21 – Seismic Design of Concrete Bridges 21-120 w=(5)+(6) BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 APPENDIX 21.3-12 wFRAME Longitudinal Push Over – Force/Displacement Relationship, Right Push Chapter 21 – Seismic Design of Concrete Bridges 21-121 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 APPENDIX 21.3-13 wFRAME Longitudinal Push Over – Force vs. Displacement Relationship Chapter 21 – Seismic Design of Concrete Bridges 21-122 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 APPENDIX 21.3-14 Cap Beam – Seismic Moment and Shear Demands 05/15/2006, 15:50 Design Academy Example No: 1 (Bent 2) ************************************************************ * * * wFRAME * * * * PUSH ANALYSIS of BRIDGE BENTS and FRAMES. * * * * Indicates formation of successive plastic hinges. * * * * VER._1.12,_JAN-14-95 * * * * Copyright (C) 1994 By Mark Seyed. * * * * This program should not be distributed under any * * condition. This release is for demo ONLY (beta testing * * is not complete). The author makes no expressed or * * implied warranty of any kind with regard to this program.* * In no event shall the author be held liable for * * incidental or consequential damages arising out of the * * use of this program. * * * ************************************************************ Node Point Information: Fixity condition definitions: s=spring and value r=complete release f=complete fixity with imposed displacement node # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 name S01.00 S01.01 S01.02 C01.01 C01.02 C01.03 C01.04 P01.01 P01.02 P01.03 P01.04 S02.01 S02.02 S02.03 S02.04 S02.05 S02.06 C02.01 C02.02 C02.03 C02.04 coordinates -----------fixity -------X Y X-dir. Y-dir. Rotation 0.00 0.00 r r r 4.72 0.00 r r r 7.72 0.00 r r r 7.72 -3.38 r r r 7.72 -15.31 r r r 7.72 -27.24 r r r 7.72 -39.17 r r r 7.72 -41.22 s 1.4e+002 r r 7.72 -43.27 s 4.1e+002 r r 7.72 -45.32 s 6.7e+002 r r 7.72 -47.37 f 0.0000 f 0.0000 r 10.72 0.00 r r r 17.72 0.00 r r r 24.72 0.00 r r r 31.72 0.00 r r r 38.72 0.00 r r r 41.72 0.00 r r r 41.72 -3.38 r r r 41.72 -15.31 r r r 41.72 -27.24 r r r 41.72 -39.17 r r r Cumulative Results of analysis at end of stage 6 Plastic Action at: Element/ Stage/ Code/ P02X01 1 2 P01X01 2 2 P02X02 3 2 P01X02 4 2 C02-02 5 rs Lat. Force *g (DL= 3381.7) 0.1863 0.1958 0.1966 0.2059 0.2147 Chapter 21 – Seismic Design of Concrete Bridges / Deflection / (in) 9.3076 9.7870 9.8292 10.3036 10.7599 21-123 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 C01-02 6 node# name 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 S01.00 S01.01 S01.02 C01.01 C01.02 C01.03 C01.04 P01.01 P01.02 P01.03 P01.04 S02.01 S02.02 S02.03 S02.04 S02.05 S02.06 C02.01 C02.02 C02.03 C02.04 P02.01 P02.02 P02.03 P02.04 S03.01 S03.02 rs ---------- GLOBAL Displ.x Displ.y 1.03149 0.03456 1.03149 0.01254 1.03148 -0.00170 1.01183 -0.00158 0.85856 -0.00115 0.59020 -0.00072 0.25113 -0.00029 0.18897 -0.00022 0.12627 -0.00015 0.06321 -0.00007 0.00000 0.00000 1.03150 -0.01419 1.03152 -0.02840 1.03153 -0.02512 1.03151 -0.01281 1.03148 -0.00492 1.03145 -0.00729 1.02247 -0.00677 0.86615 -0.00493 0.59507 -0.00310 0.25323 -0.00126 0.19057 -0.00095 0.12734 -0.00063 0.06376 -0.00032 0.00000 0.00000 1.03146 -0.01250 1.03147 -0.02108 element node # name fix 1 S01-01 rn 2 S01-02 rn 3 C01-01 rn 4 C01-02 rs 5 C01-03 rn 6 C01-04 rn 7 P01-01 rn 8 P01-02 rn 9 P01-03 rn 10 P01-04 rn 11 S02-01 rn 12 S02-02 rn 13 S02-03 rn 14 S02-04 rn 15 S02-05 rn 16 S02-06 rn 0.2275 12.3779 --------Rotation -0.00466 -0.00469 -0.00482 -0.00679 -0.01829 -0.02608 -0.03015 -0.03047 -0.03069 -0.03081 -0.03085 -0.00350 -0.00064 0.00137 0.00182 -0.00001 -0.00167 -0.00363 -0.01853 -0.02630 -0.03039 -0.03072 -0.03095 -0.03107 -0.03111 -0.00179 -0.00183 -------- local ----------- ------ element ---------displ.x displ.y rotation axial shear moment 1 1.03149 0.03456 -0.00466 37.40 -0.01 0.01 2 1.03149 0.01254 -0.00469 -37.40 322.86 -761.98 2 1.03149 0.01254 -0.00469 95.50 -322.85 761.94 3 1.03148 -0.00170 -0.00482 -95.50 528.05 -2038.30 3 0.00170 1.03148 -0.00482 639.94 353.65 16359.82 4 0.00158 1.01183 -0.00679 -639.94 -353.65 -15163.66 4 0.00158 1.01183 -0.00679 639.98 353.64 15163.00 5 0.00115 0.85856 -0.01829 -639.98 -353.64 -10944.11 5 0.00115 0.85856 -0.01829 639.98 353.66 10944.14 6 0.00072 0.59020 -0.02608 -639.98 -353.66 -6725.06 6 0.00072 0.59020 -0.02608 639.98 353.64 6725.13 7 0.00029 0.25113 -0.03015 -639.98 -353.64 -2506.10 7 0.00029 0.25113 -0.03015 639.98 353.51 2506.99 8 0.00022 0.18897 -0.03047 -639.98 -353.51 -1782.15 8 0.00022 0.18897 -0.03047 639.98 333.29 1782.11 9 0.00015 0.12627 -0.03069 -639.98 -333.29 -1099.10 9 0.00015 0.12627 -0.03069 639.98 289.29 1099.54 10 0.00007 0.06321 -0.03081 -639.98 -289.29 -506.62 10 0.00007 0.06321 -0.03081 639.98 247.19 506.68 11 0.00000 0.00000 -0.03085 -639.98 -247.19 0.00 3 1.03148 -0.00170 -0.00482 -211.27 111.89 -14322.39 12 1.03150 -0.01419 -0.00350 211.27 93.31 14350.28 12 1.03150 -0.01419 -0.00350 -133.82 -93.18 -14350.24 13 1.03152 -0.02840 -0.00064 133.82 571.98 12022.15 13 1.03152 -0.02840 -0.00064 -24.49 -571.98 -12022.14 14 1.03153 -0.02512 0.00137 24.49 1050.78 6342.45 14 1.03153 -0.02512 0.00137 84.84 -1050.79 -6342.45 15 1.03151 -0.01281 0.00182 -84.84 1529.59 -2688.86 15 1.03151 -0.01281 0.00182 194.14 -1529.59 2688.86 16 1.03148 -0.00492 -0.00001 -194.14 2008.39 -15071.78 16 1.03148 -0.00492 -0.00001 273.35 -2008.39 15071.76 17 1.03145 -0.00729 -0.00167 -273.35 2213.59 -21404.73 Chapter 21 – Seismic Design of Concrete Bridges 21-124 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 17 C02-01 rn 17 18 0.00729 0.00677 18 C02-02 rs 18 19 19 20 20 21 21 22 22 23 23 24 24 25 17 26 26 27 0.00677 0.00493 0.00493 0.00310 0.00310 0.00126 0.00126 0.00095 0.00095 0.00063 0.00063 0.00032 0.00032 0.00000 1.03145 1.03146 1.03146 1.03147 19 C02-03 rn 20 C02-04 rn 21 P02-01 rn 22 P02-02 rn 23 P02-03 rn 24 P02-04 rn 25 S03-01 rn 26 S03-02 rn 1.03145 -0.00167 1.02247 -0.00363 1.02247 0.86615 0.86615 0.59507 0.59507 0.25323 0.25323 0.19057 0.19057 0.12734 0.12734 0.06376 0.06376 0.00000 -0.00729 -0.01250 -0.01250 -0.02108 Chapter 21 – Seismic Design of Concrete Bridges -0.00706 -0.01853 -0.01853 -0.02630 -0.02630 -0.03039 -0.03039 -0.03072 -0.03072 -0.03095 -0.03095 -0.03107 -0.03107 -0.03111 -0.00167 -0.00179 -0.00179 -0.00183 2741.74 -2741.74 417.33 19367.77 -417.33 -17956.42 2741.69 -2741.69 2741.69 -2741.69 2741.69 -2741.69 2741.69 -2741.69 2741.69 -2741.69 2741.69 -2741.69 2741.69 -2741.69 -96.61 96.61 -37.19 37.19 417.18 17957.00 -417.18 -12980.05 417.18 12980.12 -417.18 -8003.12 417.17 8003.17 -417.17 -3026.31 416.81 3026.30 -416.81 -2172.40 396.55 2172.30 -396.55 -1359.19 353.00 1359.84 -353.00 -635.98 310.33 636.35 -310.33 -0.06 528.16 2038.61 -322.96 -761.93 322.85 761.91 0.00 0.01 21-125 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 APPENDIX 21.3-15 wFRAME Select Output File – To Determine Superstructure Forces due to Column Hinging, Case 1 10/26/2013, 09:39 Design Academy Example No: 1 (Superstructure Right Push) ************************************************************ * * * wFRAME * * * * PUSH ANALYSIS of BRIDGE BENTS and FRAMES. * * * * Indicates formation of successive plastic hinges. * * * * VER._1.12,_JAN-14-95 * * * * Copyright (C) 1994 By Mark Seyed. * * * * This program should not be distributed under any * * condition. This release is for demo ONLY (beta testing * * is not complete). The author makes no expressed or * * implied warranty of any kind with regard to this program.* * In no event shall the author be held liable for * * incidental or consequential damages arising out of the * * use of this program. * * * ************************************************************ Node Point Information: Fixity condition definitions: s=spring and value r=complete release f=complete fixity with imposed displacement node # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 name S01.00 S01.01 C01.01 P01.01 S02.01 S02.02 S02.03 S02.04 S02.05 S02.06 S02.07 S02.08 S02.09 S02.10 S02.11 S02.12 S02.13 C02.01 C02.02 C02.03 C02.04 P02.01 P02.02 P02.03 P02.04 S03.01 S03.02 S03.03 S03.04 S03.05 S03.06 S03.07 S03.08 S03.09 coordinates -----------fixity -------X Y X-dir. Y-dir. Rotation 0.00 0.00 r r r 2.00 0.00 r r r 2.00 -1.00 r r r 2.00 -2.00 f 0.0000 f 0.0000 r 12.57 0.00 r r r 23.14 0.00 r r r 33.71 0.00 r r r 44.28 0.00 r r r 54.85 0.00 r r r 65.42 0.00 r r r 75.99 0.00 r r r 86.56 0.00 r r r 97.13 0.00 r r r 107.70 0.00 r r r 115.70 0.00 r r r 123.70 0.00 r r r 127.96 0.00 r r r 127.96 -3.38 r r r 127.96 -15.31 r r r 127.96 -27.24 r r r 127.96 -39.17 r r r 127.96 -41.22 s 2.7e+002 r r 127.96 -43.27 s 8.3e+002 r r 127.96 -45.32 s 1.3e+003 r r 127.96 -47.37 f 0.0000 f 0.0000 r 132.22 0.00 r r r 140.22 0.00 r r r 148.22 0.00 r r r 160.97 0.00 r r r 173.72 0.00 r r r 186.47 0.00 r r r 199.22 0.00 r r r 211.97 0.00 r r r 224.72 0.00 r r r Chapter 21 – Seismic Design of Concrete Bridges 21-126 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 35 S03.10 237.47 36 S03.11 250.22 37 S03.12 262.97 38 S03.13 275.72 39 S03.14 283.72 40 S03.15 291.72 41 S03.16 295.98 42 C03.01 295.98 43 C03.02 295.98 44 C03.03 295.98 45 C03.04 295.98 46 P03.01 295.98 47 P03.02 295.98 48 P03.03 295.98 49 P03.04 295.98 50 P03.05 295.98 51 S04.01 300.24 52 S04.02 308.24 53 S04.03 316.24 54 S04.04 326.01 55 S04.05 335.78 56 S04.06 345.55 57 S04.07 355.32 58 S04.08 365.09 59 S04.09 374.86 60 S04.10 384.63 61 S04.11 394.40 62 S04.12 404.17 63 S04.13 413.94 64 C04.01 413.94 65 P04.01 413.94 66 S05.01 415.94 0.00 r 0.00 0.00 0.00 0.00 0.00 0.00 -3.38 -15.33 -27.28 -39.23 -41.46 -43.69 -45.92 -48.15 -50.38 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 -1.00 -2.00 0.00 r r r r r r r r r r r s s s s f r r r r r r r r r r r r r r f s r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r 3.2e+002 r 9.2e+002 r 1.5e+003 r 2e+003 r 0.0000 f 0.0000 r r r r r r r r r r r r r r 0.0000 f 0.0000 7.1e+003 r Spring Information at node points: k's = k/ft or ft-k/rad.; d's = ft or rad. node spring k1 d1 k2 d2 # name 22 P02X01 272.74 0.149 0.00 23 P02X02 828.36 0.105 0.00 24 P02X03 1326.91 0.106 0.00 46 P03X01 317.46 0.149 0.00 47 P03X02 919.77 0.110 0.00 48 P03X03 1476.21 0.109 0.00 49 P03X04 2038.47 0.109 0.00 66 S05X01 7061.00 0.272 0.00 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1000.000 1000.000 1000.000 1000.000 1000.000 1000.000 1000.000 1000.000 Structural Setup: Spans= 5, Columns= 4, Piles= 4, Link Beams= 0 Element Information: element nodes # name fix i j 1 S01-01 rn 1 2 2 C01-01 rs 2 3 3 P01-01 rn 3 4 4 S02-01 rn 2 5 5 S02-02 rn 5 6 6 S02-03 rn 6 7 7 S02-04 rn 7 8 8 S02-05 rn 8 9 9 S02-06 rn 9 10 10 S02-07 rn 10 11 11 S02-08 rn 11 12 12 S02-09 rn 12 13 13 S02-10 rn 13 14 14 S02-11 rn 14 15 15 S02-12 rn 15 16 depth L d area Ei Ef 2.00 6.8 103.5 629528 60480 1.00 6.0 56.5 629528 62107 1.00 6.0 56.5 629528 62107 10.57 6.8 103.5 629528 60480 10.57 6.8 103.5 629528 60480 10.57 6.8 103.5 629528 60480 10.57 6.8 103.5 629528 60480 10.57 6.8 103.5 629528 60480 10.57 6.8 103.5 629528 60480 10.57 6.8 103.5 629528 60480 10.57 6.8 103.5 629528 60480 10.57 6.8 103.5 629528 60480 10.57 6.8 103.5 629528 60480 8.00 6.8 109.6 629528 60480 8.00 6.8 109.6 629528 60480 Chapter 21 – Seismic Design of Concrete Bridges Icr 826.75 94.88 47.44 731.10 731.10 731.10 731.10 731.10 731.10 731.10 731.10 731.10 731.10 778.93 778.93 q Mpp Mpn tol status -0.01 99999 99999 0.02 e 0.00 99999 99999 0.02 e 0.00 99999 99999 0.02 e -0.01 99999 99999 0.02 e -0.01 99999 99999 0.02 e -0.01 99999 99999 0.02 e -0.01 99999 99999 0.02 e -0.01 99999 99999 0.02 e -0.01 99999 99999 0.02 e -0.01 99999 99999 0.02 e -0.01 99999 99999 0.02 e -0.01 99999 99999 0.02 e -0.01 99999 99999 0.02 e -0.01 99999 99999 0.02 e -0.01 99999 99999 0.02 e 21-127 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 16 S02-13 rn 16 17 4.26 6.8 115.6 629528 60480 826.75 -0.01 17 C02-01 rn 17 18 3.38 6.0 56.5 629528 62107 94.88 0.00 18 C02-02 rn 18 19 11.93 6.0 56.5 629528 62107 47.44 0.00 19 C02-03 rn 19 20 11.93 6.0 56.5 629528 62107 47.44 0.00 20 C02-04 rn 20 21 11.93 6.0 56.5 629528 62107 47.44 0.00 21 P02-01 rn 21 22 2.05 6.0 56.5 629528 62107 47.44 0.00 22 P02-02 rn 22 23 2.05 6.0 56.5 629528 62107 47.44 0.00 23 P02-03 rn 23 24 2.05 6.0 56.5 629528 62107 47.44 0.00 24 P02-04 re 24 25 2.05 6.0 56.5 629528 62107 47.44 0.00 25 S03-01 rn 17 26 4.26 6.8 115.6 629528 60480 826.75 -0.01 26 S03-02 rn 26 27 8.00 6.8 109.6 629528 60480 778.93 -0.01 27 S03-03 rn 27 28 8.00 6.8 109.6 629528 60480 778.93 -0.01 28 S03-04 rn 28 29 12.75 6.8 103.5 629528 60480 731.10 -0.01 29 S03-05 rn 29 30 12.75 6.8 103.5 629528 60480 731.10 -0.01 30 S03-06 rn 30 31 12.75 6.8 103.5 629528 60480 731.10 -0.01 31 S03-07 rn 31 32 12.75 6.8 103.5 629528 60480 731.10 -0.01 32 S03-08 rn 32 33 12.75 6.8 103.5 629528 60480 731.10 -0.01 33 S03-09 rn 33 34 12.75 6.8 103.5 629528 60480 731.10 -0.01 34 S03-10 rn 34 35 12.75 6.8 103.5 629528 60480 731.10 -0.01 35 S03-11 rn 35 36 12.75 6.8 103.5 629528 60480 731.10 -0.01 36 S03-12 rn 36 37 12.75 6.8 103.5 629528 60480 731.10 -0.01 37 S03-13 rn 37 38 12.75 6.8 103.5 629528 60480 731.10 -0.01 38 S03-14 rn 38 39 8.00 6.8 109.6 629528 60480 778.93 -0.01 39 S03-15 rn 39 40 8.00 6.8 109.6 629528 60480 778.93 -0.01 40 S03-16 rn 40 41 4.26 6.8 115.6 629528 60480 826.75 -0.01 41 C03-01 rn 41 42 3.38 6.0 56.5 629528 62107 94.44 0.00 42 C03-02 rn 42 43 11.95 6.0 56.5 629528 62107 47.22 0.00 43 C03-03 rn 43 44 11.95 6.0 56.5 629528 62107 47.22 0.00 44 C03-04 rn 44 45 11.95 6.0 56.5 629528 62107 47.22 0.00 45 P03-01 rn 45 46 2.23 6.0 56.5 629528 62107 47.22 0.00 46 P03-02 rn 46 47 2.23 6.0 56.5 629528 62107 47.22 0.00 47 P03-03 rn 47 48 2.23 6.0 56.5 629528 62107 47.22 0.00 48 P03-04 rn 48 49 2.23 6.0 56.5 629528 62107 47.22 0.00 49 P03-05 re 49 50 2.23 6.0 56.5 629528 62107 47.22 0.00 50 S04-01 rn 41 51 4.26 6.8 115.6 629528 60480 826.75 -0.01 51 S04-02 rn 51 52 8.00 6.8 109.6 629528 60480 778.93 -0.01 52 S04-03 rn 52 53 8.00 6.8 109.6 629528 60480 778.93 -0.01 53 S04-04 rn 53 54 9.77 6.8 103.5 629528 60480 731.10 -0.01 54 S04-05 rn 54 55 9.77 6.8 103.5 629528 60480 731.10 -0.01 55 S04-06 rn 55 56 9.77 6.8 103.5 629528 60480 731.10 -0.01 56 S04-07 rn 56 57 9.77 6.8 103.5 629528 60480 731.10 -0.01 57 S04-08 rn 57 58 9.77 6.8 103.5 629528 60480 731.10 -0.01 58 S04-09 rn 58 59 9.77 6.8 103.5 629528 60480 731.10 -0.01 59 S04-10 rn 59 60 9.77 6.8 103.5 629528 60480 731.10 -0.01 60 S04-11 rn 60 61 9.77 6.8 103.5 629528 60480 731.10 -0.01 61 S04-12 rn 61 62 9.77 6.8 103.5 629528 60480 731.10 -0.01 62 S04-13 rn 62 63 9.77 6.8 103.5 629528 60480 731.10 -0.01 63 C04-01 rs 63 64 1.00 6.0 56.5 629528 62107 94.44 0.00 64 P04-01 rn 64 65 1.00 6.0 56.5 629528 62107 47.22 0.00 65 S05-01 rn 63 66 2.00 6.8 103.5 629528 60480 826.75 -0.01 bandwidth of the problem = 11 Number of rows and columns in strage = 198 x 33 ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------Cumulative Results of analysis at end of stage 8 99999 99999 32060 32060 32060 32060 32060 32060 32060 99999 99999 99999 99999 99999 99999 99999 99999 99999 99999 99999 99999 99999 99999 99999 99999 99999 34512 34512 34512 34512 34512 34512 34512 34512 99999 99999 99999 99999 99999 99999 99999 99999 99999 99999 99999 99999 99999 99999 99999 99999 99999 0.02 e 99999 0.02 e 34566 0.02 e 34566 0.02 e 34566 0.02 e 34566 0.02 e 34566 0.02 e 34566 0.02 e 34566 0.02 e 99999 0.02 e 99999 0.02 e 99999 0.02 e 99999 0.02 e 99999 0.02 e 99999 0.02 e 99999 0.02 e 99999 0.02 e 99999 0.02 e 99999 0.02 e 99999 0.02 e 99999 0.02 e 99999 0.02 e 99999 0.02 e 99999 0.02 e 99999 0.02 e 99999 0.02 e 31835 0.02 e 31835 0.02 e 31835 0.02 e 31835 0.02 e 31835 0.02 e 31835 0.02 e 31835 0.02 e 31835 0.02 e 99999 0.02 e 99999 0.02 e 99999 0.02 e 99999 0.02 e 99999 0.02 e 99999 0.02 e 99999 0.02 e 99999 0.02 e 99999 0.02 e 99999 0.02 e 99999 0.02 e 99999 0.02 e 99999 0.02 e 99999 0.02 e 99999 0.02 e 99999 0.02 e Plastic Action at: Element/ Stage/ Code/ S05X01 1 2 P03X02 2 2 P03X01 3 2 P02X01 4 2 P02X02 5 2 P03X03 6 2 C02-02 7 rs C03-02 8 rs Lat. Force *g (DL= 4.2) 574.4611 693.4553 697.6155 773.7037 790.2726 800.7841 823.1016 826.2495 Chapter 21 – Seismic Design of Concrete Bridges / Deflection / (in) 3.3364 6.8393 6.9630 9.2504 9.7509 10.0718 10.7641 10.9793 21-128 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 node# name 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 S01.00 S01.01 C01.01 P01.01 S02.01 S02.02 S02.03 S02.04 S02.05 S02.06 S02.07 S02.08 S02.09 S02.10 S02.11 S02.12 S02.13 C02.01 C02.02 C02.03 C02.04 P02.01 P02.02 P02.03 P02.04 S03.01 S03.02 S03.03 S03.04 S03.05 S03.06 S03.07 S03.08 S03.09 S03.10 S03.11 S03.12 S03.13 S03.14 S03.15 S03.16 C03.01 C03.02 C03.03 C03.04 P03.01 P03.02 P03.03 P03.04 P03.05 S04.01 S04.02 S04.03 S04.04 S04.05 S04.06 S04.07 S04.08 S04.09 S04.10 S04.11 S04.12 S04.13 C04.01 P04.01 S05.01 ---------- GLOBAL Displ.x Displ.y 0.91494 -0.00139 0.91494 0.00001 0.45747 0.00000 0.00000 0.00000 0.91493 0.00733 0.91490 0.01435 0.91487 0.02074 0.91481 0.02619 0.91474 0.03040 0.91466 0.03304 0.91457 0.03381 0.91446 0.03238 0.91433 0.02846 0.91419 0.02172 0.91409 0.01461 0.91397 0.00569 0.91391 0.00017 0.90584 0.00016 0.78568 0.00012 0.54645 0.00007 0.23382 0.00003 0.17604 0.00002 0.11766 0.00001 0.05892 0.00001 0.00000 0.00000 0.91389 -0.00523 0.91386 -0.01338 0.91381 -0.01907 0.91372 -0.02355 0.91360 -0.02325 0.91347 -0.01929 0.91332 -0.01282 0.91314 -0.00498 0.91294 0.00309 0.91273 0.01025 0.91249 0.01537 0.91223 0.01729 0.91195 0.01487 0.91178 0.01070 0.91160 0.00425 0.91150 -0.00020 0.90448 -0.00018 0.79901 -0.00014 0.58202 -0.00009 0.29496 -0.00004 0.23698 -0.00003 0.17826 -0.00003 0.11906 -0.00002 0.05959 -0.00001 0.00000 0.00000 0.91145 -0.00476 0.91133 -0.01206 0.91121 -0.01776 0.91104 -0.02267 0.91086 -0.02546 0.91067 -0.02639 0.91047 -0.02567 0.91025 -0.02355 0.91002 -0.02025 0.90978 -0.01601 0.90953 -0.01107 0.90926 -0.00565 0.90898 -0.00001 0.45449 0.00000 0.00000 0.00000 0.90892 0.00116 --------Rotation 0.00070 0.00070 -0.45747 -0.45747 0.00068 0.00064 0.00057 0.00046 0.00033 0.00017 -0.00003 -0.00025 -0.00050 -0.00078 -0.00100 -0.00123 -0.00136 -0.00339 -0.01570 -0.02377 -0.02801 -0.02835 -0.02858 -0.02871 0.00000 -0.00118 -0.00086 -0.00057 -0.00015 0.00018 0.00042 0.00058 0.00064 0.00061 0.00050 0.00029 0.00000 -0.00039 -0.00066 -0.00096 -0.00113 -0.00301 -0.01407 -0.02167 -0.02580 -0.02619 -0.02646 -0.02662 -0.02671 0.00000 -0.00102 -0.00081 -0.00062 -0.00039 -0.00019 -0.00001 0.00015 0.00028 0.00039 0.00047 0.00053 0.00057 0.00058 -0.45449 -0.45449 0.00058 Chapter 21 – Seismic Design of Concrete Bridges 21-129 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 element node # name fix 1 S01-01 rn 2 C01-01 rs 3 P01-01 rn 4 S02-01 rn 5 S02-02 rn 6 S02-03 rn 7 S02-04 rn 8 S02-05 rn 9 S02-06 rn 10 S02-07 rn 11 S02-08 rn 12 S02-09 rn 13 S02-10 rn 14 S02-11 rn 15 S02-12 rn 16 S02-13 rn 17 C02-01 rn 18 C02-02 rs 19 C02-03 rn 20 C02-04 rn 21 P02-01 rn 22 P02-02 rn 23 P02-03 rn 24 P02-04 re 25 S03-01 rn 26 S03-02 rn 27 S03-03 rn 28 S03-04 rn 29 S03-05 rn 30 S03-06 rn 31 S03-07 rn 32 S03-08 rn 33 S03-09 rn -------- local ----------- ------ element ---------displ.x displ.y rotation axial shear moment 1 0.91494 -0.00139 0.00070 7.84 -0.01 -0.02 2 0.91494 0.00001 0.00070 -7.84 0.03 -0.12 2 -0.00001 0.91494 -0.45747 -121.46 -6.52 -0.08 3 0.00000 0.45747 -0.45747 121.46 6.52 -7.69 3 0.00000 0.45747 -0.45747 -121.46 2.15 2.75 4 0.00000 0.00000 -0.45747 121.46 -2.15 -0.30 2 0.91494 0.00001 0.00070 66.87 -121.49 0.11 5 0.91493 0.00733 0.00068 -66.87 121.59 -1284.78 5 0.91493 0.00733 0.00068 154.24 -121.59 1284.77 6 0.91490 0.01435 0.00064 -154.24 121.70 -2570.58 6 0.91490 0.01435 0.00064 241.60 -121.69 2570.61 7 0.91487 0.02074 0.00057 -241.60 121.80 -3857.45 7 0.91487 0.02074 0.00057 328.86 -121.79 3857.49 8 0.91481 0.02619 0.00046 -328.86 121.90 -5145.39 8 0.91481 0.02619 0.00046 416.22 -121.90 5145.38 9 0.91474 0.03040 0.00033 -416.22 122.01 -6434.50 9 0.91474 0.03040 0.00033 503.69 -122.01 6434.51 10 0.91466 0.03304 0.00017 -503.69 122.12 -7724.70 10 0.91466 0.03304 0.00017 590.80 -122.12 7724.80 11 0.91457 0.03381 -0.00003 -590.80 122.23 -9016.16 11 0.91457 0.03381 -0.00003 678.33 -122.23 9016.11 12 0.91446 0.03238 -0.00025 -678.33 122.33 -10308.64 12 0.91446 0.03238 -0.00025 765.50 -122.32 10308.56 13 0.91433 0.02846 -0.00050 -765.50 122.43 -11602.05 13 0.91433 0.02846 -0.00050 852.83 -122.42 11602.07 14 0.91419 0.02172 -0.00078 -852.83 122.53 -12896.67 14 0.91419 0.02172 -0.00078 929.37 -122.52 12896.78 15 0.91409 0.01461 -0.00100 -929.37 122.60 -13877.32 15 0.91409 0.01461 -0.00100 995.44 -122.61 13877.38 16 0.91397 0.00569 -0.00123 -995.44 122.69 -14858.62 16 0.91397 0.00569 -0.00123 1045.39 -122.66 14858.62 17 0.91391 0.00017 -0.00136 -1045.39 122.71 -15381.25 17 -0.00017 0.91391 -0.00136 -129.65 802.62 37277.89 18 -0.00016 0.90584 -0.00339 129.65 -802.62 -34564.80 18 -0.00016 0.90584 -0.00381 -129.70 803.15 34566.00 19 -0.00012 0.78568 -0.01570 129.70 -803.15 -24984.42 19 -0.00012 0.78568 -0.01570 -129.70 803.20 24984.50 20 -0.00007 0.54645 -0.02377 129.70 -803.20 -15402.33 20 -0.00007 0.54645 -0.02377 -129.70 803.19 15402.42 21 -0.00003 0.23382 -0.02801 129.70 -803.19 -5820.25 21 -0.00003 0.23382 -0.02801 -129.70 802.91 5819.92 22 -0.00002 0.17604 -0.02835 129.70 -802.91 -4173.36 22 -0.00002 0.17604 -0.02835 -129.70 763.19 4173.73 23 -0.00001 0.11766 -0.02858 129.70 -763.19 -2608.48 23 -0.00001 0.11766 -0.02858 -129.70 675.43 2608.88 24 -0.00001 0.05892 -0.02871 129.70 -675.43 -1224.02 24 -0.00001 0.05892 -0.02871 -129.70 596.96 1223.94 25 0.00000 0.00000 -0.02876 129.70 -596.96 0.19 17 0.91391 0.00017 -0.00136 274.71 -252.37 -21895.31 26 0.91389 -0.00523 -0.00118 -274.71 252.41 20820.07 26 0.91389 -0.00523 -0.00118 323.11 -252.43 -20820.31 27 0.91386 -0.01338 -0.00086 -323.11 252.51 18800.50 27 0.91386 -0.01338 -0.00086 389.44 -252.52 -18800.52 28 0.91381 -0.01907 -0.00057 -389.44 252.60 16780.05 28 0.91381 -0.01907 -0.00057 475.44 -252.60 -16780.07 29 0.91372 -0.02355 -0.00015 -475.44 252.73 13558.60 29 0.91372 -0.02355 -0.00015 580.52 -252.73 -13558.60 30 0.91360 -0.02325 0.00018 -580.52 252.86 10335.48 30 0.91360 -0.02325 0.00018 685.74 -252.86 -10335.48 31 0.91347 -0.01929 0.00042 -685.74 252.98 7110.72 31 0.91347 -0.01929 0.00042 791.29 -252.99 -7110.76 32 0.91332 -0.01282 0.00058 -791.29 253.11 3884.34 32 0.91332 -0.01282 0.00058 896.76 -253.11 -3884.34 33 0.91314 -0.00498 0.00064 -896.76 253.24 656.30 33 0.91314 -0.00498 0.00064 1001.66 -253.24 -656.33 34 0.91294 0.00309 0.00061 -1001.66 253.37 -2573.29 Chapter 21 – Seismic Design of Concrete Bridges 21-130 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 34 S03-10 rn 35 S03-11 rn 36 S03-12 rn 37 S03-13 rn 38 S03-14 rn 39 S03-15 rn 40 S03-16 rn 41 C03-01 rn 42 C03-02 rs 43 C03-03 rn 44 C03-04 rn 45 P03-01 rn 46 P03-02 rn 47 P03-03 rn 48 P03-04 rn 49 P03-05 re 50 S04-01 rn 51 S04-02 rn 52 S04-03 rn 53 S04-04 rn 54 S04-05 rn 55 S04-06 rn 56 S04-07 rn 57 S04-08 rn 58 S04-09 rn 59 S04-10 rn 60 S04-11 rn 61 S04-12 rn 62 S04-13 rn 63 C04-01 rs 64 P04-01 rn 65 S05-01 rn 34 0.91294 35 0.91273 35 0.91273 36 0.91249 36 0.91249 37 0.91223 37 0.91223 38 0.91195 38 0.91195 39 0.91178 39 0.91178 40 0.91160 40 0.91160 41 0.91150 41 0.00020 42 0.00018 42 0.00018 43 0.00014 43 0.00014 44 0.00009 44 0.00009 45 0.00004 45 0.00004 46 0.00003 46 0.00003 47 0.00003 47 0.00003 48 0.00002 48 0.00002 49 0.00001 49 0.00001 50 0.00000 41 0.91150 51 0.91145 51 0.91145 52 0.91133 52 0.91133 53 0.91121 53 0.91121 54 0.91104 54 0.91104 55 0.91086 55 0.91086 56 0.91067 56 0.91067 57 0.91047 57 0.91047 58 0.91025 58 0.91025 59 0.91002 59 0.91002 60 0.90978 60 0.90978 61 0.90953 61 0.90953 62 0.90926 62 0.90926 63 0.90898 63 0.00001 64 0.00000 64 0.00000 65 0.00000 63 0.90898 66 0.90892 0.00309 0.01025 0.01025 0.01537 0.01537 0.01729 0.01729 0.01487 0.01487 0.01070 0.01070 0.00425 0.00425 -0.00020 0.91150 0.90448 0.90448 0.79901 0.79901 0.58202 0.58202 0.29496 0.29496 0.23698 0.23698 0.17826 0.17826 0.11906 0.11906 0.05959 0.05959 0.00000 -0.00020 -0.00476 -0.00476 -0.01206 -0.01206 -0.01776 -0.01776 -0.02267 -0.02267 -0.02546 -0.02546 -0.02639 -0.02639 -0.02567 -0.02567 -0.02355 -0.02355 -0.02025 -0.02025 -0.01601 -0.01601 -0.01107 -0.01107 -0.00565 -0.00565 -0.00001 0.90898 0.45449 0.45449 0.00000 -0.00001 0.00116 Chapter 21 – Seismic Design of Concrete Bridges 0.00061 0.00050 0.00050 0.00029 0.00029 0.00000 0.00000 -0.00039 -0.00039 -0.00066 -0.00066 -0.00096 -0.00096 -0.00113 -0.00113 -0.00301 -0.00301 -0.01407 -0.01407 -0.02167 -0.02167 -0.02580 -0.02580 -0.02619 -0.02619 -0.02646 -0.02646 -0.02662 -0.02662 -0.02671 -0.02671 -0.02673 -0.00113 -0.00102 -0.00102 -0.00081 -0.00081 -0.00062 -0.00062 -0.00039 -0.00039 -0.00019 -0.00019 -0.00001 -0.00001 0.00015 0.00015 0.00028 0.00028 0.00039 0.00039 0.00047 0.00047 0.00053 0.00053 0.00057 0.00057 0.00058 -0.45449 -0.45449 -0.45449 -0.45449 0.00058 0.00058 1106.93 -1106.93 1211.99 -1211.99 1317.09 -1317.09 1422.49 -1422.49 1508.34 -1508.34 1574.11 -1574.11 1624.06 -1624.06 139.21 -139.21 139.37 -139.37 139.37 -139.37 139.37 -139.37 139.37 -139.37 139.37 -139.37 139.37 -139.37 139.37 -139.37 139.37 -139.37 935.80 -935.80 983.99 -983.99 1049.77 -1049.77 1122.99 -1122.99 1203.85 -1203.85 1284.17 -1284.17 1364.86 -1364.86 1445.46 -1445.46 1526.44 -1526.44 1607.02 -1607.02 1687.98 -1687.98 1768.74 -1768.74 1849.31 -1849.31 116.10 -116.10 116.10 -116.10 1913.55 -1913.55 -253.37 253.49 -253.49 253.62 -253.62 253.75 -253.74 253.87 -253.87 253.95 -253.95 254.03 -254.03 254.07 721.18 -721.18 721.63 -721.63 721.60 -721.60 721.60 -721.60 721.93 -721.93 675.41 -675.41 573.88 -573.88 412.94 -412.94 291.44 -291.44 -114.87 114.91 -114.90 114.98 -114.98 115.06 -115.04 115.14 -115.16 115.26 -115.24 115.34 -115.36 115.46 -115.47 115.56 -115.57 115.67 -115.68 115.78 -115.78 115.88 -115.89 115.99 -115.99 116.09 3.57 -3.57 -8.59 8.59 0.03 -0.01 2573.29 -5804.54 5804.56 -9037.39 9037.42 -12271.85 12271.87 -15507.92 15507.96 -17539.28 17539.29 -19571.26 19571.27 -20653.43 34272.36 -31834.75 31835.00 -23211.55 23211.56 -14588.34 14588.36 -5965.18 5964.92 -4354.71 4356.28 -2849.53 2849.93 -1570.10 1570.30 -649.82 649.96 0.02 -13619.84 13130.47 -13130.79 12211.27 -12211.27 11291.09 -11291.09 10166.65 -10166.64 9041.13 -9041.08 7914.66 -7914.64 6787.06 -6787.07 5658.43 -5658.45 4528.83 -4528.86 3398.19 -3398.23 2266.51 -2266.52 1133.80 -1133.79 0.10 -4.40 -2.28 -4.71 -3.97 -0.06 0.02 21-131 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 APPENDIX 21.3-16 PSSECx Input File PSSEC300,_OCT_26_2005 Bridge Design Academy - Prototype Supestructure Capacity S1 1.0NEG Number of different types of concrete 1 For each concrete type input: Type number; Model code= 0 simple(unconfined/confined), 1 Mander's (unconfined) strength f'c0 (ksi), strain ec0, strength fcu (ksi), ult. strain ecu, conc. density 1 1 5.200 .002 0.5 0.0025 150 Number of different types of P/S steel 1 For each type, 1st line for tensile parameters,2nd line for cmpressive parameters type#;E;fy;strain hard. factor;fu;ult. strain;PS-code: 0 tendons, 1 otherwise E;fy;strain hard. factor;fu;ult. strain 1 28500 245 2 270 0.030 0 0 0 0 0 0 Number of different types of mild steel 1 For each steel type input: Type number;Model code= 0 simple, 1 complex E(ksi);fy(ksi);strain hard. factor;fu(ksi);ultimate strain 1 1 29000 68 6.41 95 0.09 Number of Conc. Subsections 1 For each Subsec.:Subsection #,Section shape type, Concrete type, No. of fibers Subsec. Dim.(in):(See Manual for input parameters.) Subsec. Dim.(in):(See Manual for input parameters.) Global coord. of the center of Subsec.: Xg, Yg 1 I-shaped, 1 200 706.0 48.0 517.0 81.0 9.125 8.25 0 -5.26 Number of P/S steel groups 1 For each group:group#;P/S type;x-coord.(in);y-coord.(in);area(in^2);P/S force 1 1 0 25.4412 38.28 6157 Number of mild steel rebar cages (rebar distributed around the perimeter) 0 cage#;steel type;cage shape;#of bars;x(in) of 1st bar(y=0);area(in^2)of bar Number of mild steel groups (no logical pattern for distribution) 2 n group#;steel type;x-coord.(in); y-coord.(in); area(in^2) 1 1 0 31.80 47.40 2 1 0 -42.13 34.76 Non P/S Axial load on mid-depth of section (Kips)(+ sign=compression) 0 Numerical Computation Factor (1 to 10) 5 Computer Graphics Card identifier: 0 none; 2 CGA; 3 Hercules; 9 EGA; 12 VGA 12 Output control: 0 short; 1 long output 1 X-Sec. plot control (0=no plot, 1=each stage, 2=every iteration of each step) 0 Analysis Control: p - Positive moment, n - Negative moment n Chapter 21 – Seismic Design of Concrete Bridges 21-132 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 APPENDIX 21.3-17 PSSECx Model for Superstructure Chapter 21 – Seismic Design of Concrete Bridges 21-133 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 APPENDIX 21.3-18 Partial Output from PSSECx Run 05-15-2006 ******** SECx ******** DUCTILITY and STRENGTH of Rectangular, T-, I-, Hammer, Octagonal, Circular, Ring, and Hollowed shaped Prestressed and Reinforced Concrete Sections using fiber models Ver. 3.00, OCT-26-2005 Copyright (C) 2005 By Mark Seyed and Don Lee. This program should not be distributed under any condition. This release is for demo ONLY (beta testing is not complete). Caltrans or the author make no expressed or implied warranty of any kind with regard to this program. In no event shall the author or Caltrans be held liable for incidental or consequential damages arising out of the use of this program. JOB TITLE: Bridge Design Academy - Prototype Supestructure Capacity S1 1.0NEG Concrete Data, Complex Model, Mander's unconfined Concrete Type Compressive Strength (max.) Strain at max. Strength Strength at Ultimate Strain Ultimate strain Unit Weight (pcf) (ksi) (ksi) = 1 = 5.200 = .00200 = 0.000 = .00500 = 150.00 Prestressing Steel Data Material No. Yield Strain Hardening Strain Yield Ultimate Modulus Stress Stress of Elasticity ksi ksi ksi 0.03000 245.10 270.00 28500.00 Tensile prop. 0.00000 0.00 0.00 0.00 Compressive prop. 1 is 7-wire and Low-Relaxation Tendon 1 Ultimate Strain 0.00860 0.00860 0.00000 0.00000 Prestress element type # with 270 ksi strands. ( Refer to PCI Design Handbook 4th Edition.) Mild Steel Reinforcing Data Material No. 1 Yield Strain 0.00234 Hardening Strain 0.01503 Ultimate Strain 0.09000 Yield Stress ksi 68.00 Ultimate Stress ksi 95.00 Rectangular, T-, or I-shaped section information Depth of Section Top Flange width Top Flange thickness Bot Flange width Bot Flange thickness Web thickness (in.) (in.) (in.) (in.) (in.) (in.) = = = = = = 81.00 706.00 9.13 517.00 8.25 48.00 Concrete fiber information Fiber Material x y # # (in) (in) 1 1.0 0.00 -45.56 2 1.0 0.00 -45.17 ……………………………………………… ……………………………………………… ……………………………………………… 197 1.0 0.00 33.79 199 1.0 0.00 34.62 200 1.0 0.00 35.03 area (in^2) 203.11 203.11 292.83 292.83 292.83 Chapter 21 – Seismic Design of Concrete Bridges 21-134 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 Prestressing Steel Fiber Data Fiber No. 1 Material No. 1 x (in) y (in) 0.00 25.44 area (in^2) 38.28 P/S force Kips 6157.00 Total P/S force on the section = 6157.0 kips Total moment due to P/S about point (0, 0) = 13053.5 ft-kip Mild Steel Fiber Data Fiber No. 1 2 Material No. 1 1 x (in) 0.00 0.00 y (in) 31.80 -42.13 area (in^2) 47.40 34.76 Axial load at mid-depth of section (kip)(positive means compression) = 0.0 *************************************************** * Analysis Results --Negative Moment Capacity * *************************************************** Initial state due to P/S without non-P/S axial force: N.A. Loc. Curvature Conc. Strain @ max. compressed fiber -41.50 0.0000023 0.00017950 Undeformed P/S element position w.r.t. reference plane P/S Fiber Loc.(y) Undef. pos. Conc. Strain @ same loc. 1 25.44 -0.0058006 -0.0001570 Force Equilibrium Condition of the x-section: Max. Max. Conc. Neutral Steel Strain Axis Strain Conc. step epscmax in. Tens. Comp. 0 -.00001 -41.50 -.00000 5923. 1 -.00001 -42.26 0.00000 5923. 2 -.00001 -43.05 0.00000 5923. 3 -.00000 -43.86 0.00000 5923. 4 -.00000 -44.70 0.00000 5924. 5 0.00000 -45.56 0.00000 5925. 6 0.00010 9055.25 0.00000 5983. 7 0.00011 362.50 0.00000 5990. 8 0.00013 174.76 0.00000 5997. 9 0.00014 110.77 0.00000 6006. 10 0.00016 78.67 0.00000 6017. 11 0.00018 59.45 0.00000 6028. 12 0.00020 46.72 0.00000 6041. 13 0.00022 37.74 0.00000 6055. 14 0.00025 29.97 -.00001 6079. 15 0.00028 14.77 -.00008 6224. 16 0.00032 2.23 -.00020 6470. 17 0.00035 -6.69 -.00035 6806. 18 0.00040 -12.96 -.00055 7231. 19 0.00045 -17.40 -.00078 7745. 20 0.00050 -20.61 -.00105 8346. 21 0.00056 -22.97 -.00136 9034. 22 0.00063 -24.75 -.00171 9811. 23 0.00071 -26.11 -.00210 10680. 24 0.00079 -27.79 -.00266 11498. 25 0.00089 -30.09 -.00356 12094. 26 0.00100 -32.67 -.00499 12356. 27 0.00112 -34.65 -.00682 12476. 28 0.00126 -36.06 -.00897 12528. Chapter 21 – Seismic Design of Concrete Bridges Steel force P/S Comp. Tens. force 236. -1. -6157. 235. 0. -6158. 236. 0. -6158. 236. 0. -6159. 237. 0. -6160. 237. 0. -6161. 237. 0. -6220. 237. 0. -6227. 237. 0. -6235. 237. 0. -6244. 237. 0. -6254. 238. 0. -6265. 238. 0. -6278. 238. 0. -6292. 242. -8. -6312. 268. -109. -6383. 296. -269. -6496. 326. -483. -6648. 359. -751. -6840. 395. -1072. -7069. 435. -1445. -7336. 480. -1872. -7642. 530. -2354. -7987. 587. -2893. -8372. 645. -3223. -8920. 698. -3223. -9568. 739. -3223. -9872. 774. -3223.-10027. 809. -3223.-10115. Net Curvature Moment force in/in (K-ft) -0.8 0.000002 -4. -0.4 0.000002 -147. -0.5 0.000002 -307. 0.3 0.000002 -486. 0.2 0.000002 -683. -0.7 0.000002 -899. -0.4 -.000000 -13142. -0.3 -.000000 -14634. 0.8 -.000001 -16309. 0.9 -.000001 -18186. -0.1 -.000001 -20287. -0.7 -.000002 -22643. -0.3 -.000002 -25286. -1.0 -.000003 -28243. -0.1 -.000003 -31443. 0.4 -.000005 -33995. -0.7 -.000007 -36442. -0.9 -.000009 -39119. 0.5 -.000012 -42153. 0.4 -.000016 -45615. 0.3 -.000020 -49549. -0.4 -.000025 -53987. -0.2 -.000030 -58960. -1.0 -.000036 -64494. 0.4 -.000045 -69683. -0.5 -.000058 -73488. 0.3 -.000077 -75410. 0.9 -.000103 -76502. 0.1 -.000132 -77210. 21-135 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 29 30 31 32 33 34 35 36 37 38 39 40 0.00141 0.00158 0.00178 0.00199 0.00223 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 -37.05 -37.76 -38.26 -38.64 -38.94 0.00 0.00 0.00 0.00 0.00 0.00 0.00 -.01141 -.01411 -.01704 -.02027 -.02388 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 12543. 12533. 12639. 12782. 12893. 0. 0. 0. 0. 0. 0. 0. 848. 893. 948. 1012. 1084. 0. 0. 0. 0. 0. 0. 0. -3223.-10168. -3223.-10204. -3360.-10228. -3546.-10247. -3716.-10261. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.6 0.9 0.9 -0.3 0.6 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -77720. -78121. -79244. -80600. -81787. 0. 0. 0. 0. 0. 0. 0. -.000166 -.000203 -.000243 -.000288 -.000338 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 Prestress Tendon Strain on the x-section: Max. Neutral P/S Steel Conc. Axis Strain No. Strain in. step epscmax 0 -.00001 -41.50 1 -.005644 1 -.00001 -42.26 1 -.005644 2 -.00001 -43.05 1 -.005645 3 -.00000 -43.86 1 -.005646 4 -.00000 -44.70 1 -.005647 5 0.00000 -45.56 1 -.005648 6 0.00010 9055.25 1 -.005701 7 0.00011 362.50 1 -.005708 ……………………………………………… ……………………………………………… ……………………………………………… 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 0.00063 0.00071 0.00079 0.00089 0.00100 0.00112 0.00126 0.00141 0.00158 0.00178 0.00199 0.00223 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 -24.75 -26.11 -27.79 -30.09 -32.67 -34.65 -36.06 -37.05 -37.76 -38.26 -38.64 -38.94 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Strain No. Strain No. Strain No. Strain No. Strain -.007321 -.007674 -.008176 -.008996 -.010301 -.011970 -.013932 -.016152 -.018616 -.021292 -.024243 -.027534 -.005801 -.005801 -.005801 -.005801 -.005801 -.005801 -.005801 Recommended value of 'effective moment of inertia' based on initial slope of moment-curvature diagram (ft^4) = 211.8303 Yield pt. is defined as the First mild steel yields. 23 and The first mild steel yields between the following Steps: The computation of mild steel yield point IS within 2% tolerance. 24 and The first P/S steel yields between the following Steps: The computation of P/S steel yield point IS NOT within 2% tolerance. Yield Nominal Ultimate 24 25 Moments (ft-K) Curvature(rad/in) 67871 0.000040 See force equilibrium table at concrete strain of .003 0 0.000000 End Chapter 21 – Seismic Design of Concrete Bridges 21-136 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 NOTATION AASHTO = AASHTO LRFD Bridge Design Specifications with Interims and California Amendments Ab = area of individual reinforcing steel bar (in.2) top Acap = area of bent cap top flexural steel (in.2) bot Acap = area of bent cap bottom flexural steel (in.2) Acv = area of concrete engaged in interface shear transfer (in.2) Ae = effective shear area; effective abutment wall area (in.2) Ag = gross cross section area (in.2) Ajh = effective horizontal area of a moment resisting joint (in.2) Ajv = effective vertical area for a moment resisting joint (in.2) Aps = prestressing steel area (in.2) As = area of supplemental non-prestressed tension reinforcement (in.2) Asjh = area of horizontal joint shear reinforcement required at moment resisting joints (in.2) Asjhc = total area of horizontal ties placed at the end of the bent cap in Case 1 Knee joints (in.2) Asjv = area of vertical joint shear reinforcement required at moment resisting joints (in.2) Asj bar = area of vertical “J” bar reinforcement required at moment resisting joints with a skew angle > 20 (in.2) Assf = area of bent cap side face steel required at moment resisting joints (in.2) Ask = area of interface shear reinforcement crossing the shear plane (Vertical shear key reinforcement) (in.2) Ast ,max = maximum longitudinal reinforcement area (in.2) Ast ,min = minimum longitudinal reinforcement area (in.2) Ast = total area of column longitudinal reinforcement anchored in the joint; total area of column/pier wall longitudinal reinforcement (in.2) Chapter 21 – Seismic Design of Concrete Bridges 21-137 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 Asu bar = Ash = area of horizontal shear key reinforcement (hanger bars) (in.2) area of bent cap top and bottom reinforcement bent in the form of “U” bars in Knee joints (in.2) AskIso( provided) = area of interface shear reinforcement provided for isolated shear key (in.2) iso AskNon area of interface shear reinforcement provided for non-isolated shear key ( provided) = (in.2) Av = area of shear reinforcement perpendicular to flexural tension reinforcement (in.2) Bcap = bent cap width (in.) BDD = Caltrans Bridge Design Details Beff = effective width of the superstructure for resisting longitudinal seismic moments (in.) Dc = column cross sectional dimension in the direction of interest (in.) Dftg = depth of footing (in.) Ds = depth of superstructure at the bent cap (in.) DSH = D' = cross-sectional dimension of confined concrete core measured between the centerline of the peripheral hoop or spiral (in.) Dc' = confined column cross-section dimension, measured out to out of ties, in the direction parallel to the axis of bending (in.) Ec = modulus of elasticity of concrete (ksi) EDA = Elastic Dynamic Analysis ESA = Elastic Static Analysis Fsk = abutment shear key force capacity; Shear force associated with column Design Seismic Hazards overstrength moment, including overturning effects (ksi) Ieff , I e = effective moment of inertia for computing member stiffness (in.4) ISA = Inelastic Static Analysis K abut = abutment backwall stiffness (kip/in./ft) Keff = effective abutment backwall stiffness (kip/in./ft) Ki = Initial abutment backwall stiffness (kip/in./ft) Chapter 21 – Seismic Design of Concrete Bridges 21-138 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 L = member length from the point of maximum moment to the point of contraflexure (in); length of bridge deck between adjacent expansion joints Lmin,headed = minimum horizontal length from the end of the lowest layer of headed hanger bar to the intersection with the shear key vertical reinforcement (in.) Lmin,hooked = minimum horizontal length from the end of the lowest layer of hanger bar hooks to the intersection with the shear key vertical reinforcement (in.) Lp = equivalent analytical plastic hinge length (in.) Mdl = moment attributed to dead load (kip-ft) M eqcol = column moment when coupled with any existing Mdl & Mp/s will equal the column’s overstrength moment capacity, Mocol (kip-ft) M eqR , L = portion of Meqcol distributed to the left or right adjacent superstructure spans (kip-ft) Mn = nominal moment capacity based on the nominal concrete and steel strengths when the concrete strain reaches 0.003 (kip-ft) Mne = nominal moment capacity based on the expected material properties and a concrete strain, c = 0.003 (kip-ft) sup R , L M ne = expected nominal moment capacity of the right and left superstructure spans utilizing expected material properties (kip-ft) M ocol = column overstrength moment (kip-ft) Mpcol = Idealized plastic moment capacity of a column calculated by M- analysis (kip-ft) Mp/s = moment attributed to secondary prestress effects (kip-ft) My = Moment capacity of a ductile component corresponding to the first reinforcing bar yielding (kip-ft) M- = moment curvature analysis MTD = Caltrans Memo To Designers NH = minimum hinge seat width normal to the centerline of bent (in.) NA = abutment support width normal to centerline of bearing (in.) P = absolute value of the net axial force normal to the shear plane (kip) Pb = beam axial force at the center of the joint including prestressing (kip) Pc = column axial force including the effects of overturning (kip) Chapter 21 – Seismic Design of Concrete Bridges 21-139 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 Pdia = passive pressure force resisting movement at diaphragm abutment (ksf) Pdl = superstructure dead load reaction at the abutment plus weight of the abutment and its footing (kip) Pdlsup = superstructure axial load resultant at the abutment (kip) P/S = prestressed concrete; prestressing strand Pjack = total prestress jacking force (kip) Pn = nominal axial resistance (kip) Pbw = passive pressure force resisting movement at seat abutment (ksf) RA = abutment displacement coefficient S = cap beam short stub length (ft) SDC = Seismic Design Criteria T = natural period of vibration, (seconds), T = 2 m k Tc = total tensile force in column longitudinal reinforcement associated with Mocol (kip) Ti = natural period of the stiffer frame (sec.) Tj = natural period of the more flexible frame (sec.) Vc = nominal shear strength provided by concrete (kip) Vn = nominal shear strength (kip) Vo = overstrength shear associated with the overstrength moment Mo (kip) Vocol = column overstrength shear, typically defined as Mocol /L (kip) V pcol Vs = = column plastic shear, typically defined as Mpcol/L (kip) nominal shear strength provided by shear reinforcement (kip) Vww = shear capacity of one wingwall (kip) a = demand spectral acceleration bv = effective web width taken as the minimum web width within the shear depth d v (in.) dbl = nominal bar diameter of longitudinal column reinforcement (in.2) Chapter 21 – Seismic Design of Concrete Bridges 21-140 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 dv = effective shear depth defined as the distance between resultants of tensile and compressive forces due to flexural, but need not be taken less than 0.9d e or 0.72h (in.) fh = average normal stress in the horizontal direction within a moment resisting joint (ksi) fv = average normal stress in the vertical direction within a moment resisting joint (ksi) fy = nominal yield stress for A706 reinforcement (ksi) fye = expected yield stress for A706 reinforcement (ksi) fyh = nominal yield stress of transverse column reinforcement, hoops/spirals (ksi) f c' = compressive strength of unconfined concrete (psi) f cc' = confined compression strength of concrete (psi) f ce' = expected compressive strength of unconfined concrete (psi) f1 , f 2 = concrete shear factors for ductile members g = acceleration due to gravity, 32.2 ft sec2 h = distance from the center of gravity of the tensile force to the center of gravity of the compressive force of the column section (in.) hdia = backwall height for diaphragm abutment (in.) hbw = backwall height for seat abutment (in.) k ie , k ej = smaller and larger effective bent or column stiffness, respectively (kip/in.) lac = minimum length of column longitudinal reinforcement extension into the bent cap (in.) lac,provided = actual length of column longitudinal reinforcement embedded into the bent cap (in.) ld = development length of the main reinforcement (in.) l dh = development length in tension of standard hooked bars (in.) mi = tributary mass of column or bent i, m = W/g (kip-sec2/ft) mj = tributary mass of column or bent j, m = W/g (kip-sec2/ft) pbw = maximum abutment backwall soil pressure (ksf) pc = nominal principal compression stress in a joint (psi) Chapter 21 – Seismic Design of Concrete Bridges 21-141 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 pt = nominal principal tension stress in a joint (psi) s = spacing of shear/transverse reinforcement (in.) t = top or bottom slab thickness (in.) vjv = nominal vertical shear stress in a moment resisting joint (psi) vc = permissible shear stress carried by concrete (psi) w = width of the backwall or diaphragm, as appropriate (in.)  = factor defining the range over which Fsk is allowed to vary  = factor indicating ability of diagonally cracked concrete to transmit tension and shear  suR = reduced ultimate tensile strain for A706 reinforcement c = local member displacement capacity (in.) col = displacement attributed to the elastic and plastic deformation of the column (in.) C = global displacement capacity (in.) cr+sh = displacement due to creep and shrinkage (in.) d = local member displacement demand (in.) D = global system displacement (in.)  eff = effective longitudinal abutment displacement at idealized yield (in.) eq = relative longitudinal displacement demand at an expansion joint due to earthquake (in.) p = idealized plastic displacement capacity due to rotation of the plastic hinge (in.) p/s = displacement due to prestress shortening (in.) r = relative lateral offset between the point of contra-flexure and the base of the plastic hinge (in.) tem = displacement due to temperature variation (in.) Y = idealized yield displacement of the subsystem at the formation of the plastic hinge (in.) Y(i) = idealized yield displacement of the subsystem at the formation of plastic hinge (i) (in.) Chapter 21 – Seismic Design of Concrete Bridges 21-142 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 col Y = idealized yield displacement of a column at the formation of the plastic hinge (in.)  = angle of inclination of diagonal compressive stresses (radians) p = plastic rotation capacity (radians)  sk = skew angle (degree) s = amount of transverse reinforcement expressed as volumetric ratio  = resistance factor p = idealized plastic curvature (1/in.) u = ultimate curvature capacity (1/in.) y = yield curvature corresponding to the first yield of the reinforcement in a ductile component (1/in.) Y = idealized yield curvature (1/in.) d = local displacement ductility demand D = global displacement ductility demand c = local displacement ductility capacity Chapter 21 – Seismic Design of Concrete Bridges 21-143 BRIDGE DESIGN PRACTICE ● FEBRUARY 2015 REFERENCES 1. AASHTO, (2012). “AASHTO LRFD Bridge Design Specifications,” U.S. Customary Units 2012 (6th Edition), American Association of State Highway and Transportation Officials, Washington, D.C. 2. Caltrans, (2014). California Amendments to the AASHTO LRFD Bridge Design Specifications – 6th Edition, California Department of Transportation, Sacramento, CA. 3. Caltrans, (2013). “Seismic Design Criteria,” Version 1.7, California Department of Transportation, Sacramento, CA. 4. Caltrans (2009). Bridge Memo To Designers 6-1 Column Analysis Consideration, California Department of Transportation, Sacramento, CA, February 2009. 5. Caltrans, (2007). CTBridge Help System, Version 1.3 (Online), Caltrans Bridge Analysis and Design, California Department of Transportation, Sacramento, CA. 6. Caltrans, (2001a). Bridge Memo To Designers 20-6 Seismic Strength of Concrete Bridge Superstructures, California Department of Transportation, Sacramento, CA. 7. Caltrans, (2001b). Memo To Designers 20-9 Splices in Bar Reinforcing Steel, California Department of Transportation, Sacramento, CA, August 2001. 8. CSI, (1976-2007). SAP2000 Advanced 11.0.8, Static and Dynamic Finite Element Analysis of Structures, Computers and Structures, Inc., Berkeley, CA. 9. Mahan, M., (2006). User’s Manual for “xSECTION,” Cross Section Analysis Program, Version 4.00, California Department of Transportation, Sacramento, CA. 10. Mahan, M., (1995). User’s Manual for “wFRAME,” 2-D Push Analysis Program, Version 1.13, California Department of Transportation, Sacramento, CA. Chapter 21 – Seismic Design of Concrete Bridges 21-144