The document discusses common errors that can occur in finite element analysis. It identifies two main categories of errors: idealization errors and discretization errors. Idealization errors relate to how the physical problem is simplified, including establishing boundary conditions and material behavior assumptions. Discretization errors occur during the process of replacing the continuous idealized model with a finite element model, and can include issues with imposing boundary conditions, element distortions, and matrix ill-conditioning during the solution process. Overall, the document emphasizes the importance of properly understanding the physical problem, applying appropriate simplifications and boundary conditions, and ensuring a high-quality mesh to minimize errors in finite element analysis.
General steps of the finite element methodmahesh gaikwad
General Steps used to solve FEA/ FEM Problems. Steps Involves involves dividing the body into a finite elements with associated nodes and choosing the most appropriate element type for the model.
The document discusses two-dimensional finite element analysis. It describes triangular and quadrilateral elements used for 2D problems. The derivation of the stiffness matrix is shown for a three-noded triangular element. Shape functions are presented for triangular and quadrilateral elements. Examples are provided to calculate strains for a triangular element and to determine temperatures at interior points using shape functions.
This document introduces the stiffness method for structural analysis. It begins by discussing degrees of freedom and statical determinacy, explaining how to calculate the number of degrees of freedom and degree of statical indeterminacy for frames. It then introduces the direct stiffness method, using a simple spring system example to demonstrate the basic approach. Key steps include establishing equilibrium equations in matrix form relating applied loads to displacements, and solving these equations to determine member forces and displacements. The chapter concludes by discussing local and global coordinate systems for members and how to establish the member stiffness matrix relating forces and displacements.
- Saint-Venant's principle states that the stress and strain distribution on a cross-section of a loaded material will be independent of the applied load if the cross-section is located away from the point of load application.
- The principle of superposition allows breaking down structures into individual load cases and adding their effects to determine the total stress, strain, or deflection.
- Statically indeterminate structures require additional compatibility equations relating deformations to solve for member forces.
The document discusses the variational formulation and Rayleigh-Ritz method for solving systems. The variational approach involves calculating the total potential of a system and finding the stationary value where the potential is zero. The Rayleigh-Ritz method assumes the form of the unknown solution in terms of known trial functions with adjustable parameters. The functional is expressed in terms of these parameters and differentiated to obtain equations that are set to zero to solve for the parameters.
This document discusses vibration isolation and base isolation techniques. It begins with an introduction to vibration isolation and its objectives. It then describes various types of vibration isolators like elastomeric isolators and lead-rubber bearings that are used to isolate machines from vibrations. The document also discusses the concept and components of base isolation, where flexible isolators are placed under a structure to isolate it from ground motions during an earthquake. Examples of base isolated buildings in India are provided. The conclusion emphasizes that base isolation is an effective seismic-resistant design technique.
constant strain triangular which is used in analysis of triangular in finite element method with the help of shape function and natural coordinate system.
1. The stiffness method is used to analyze the beam by determining its degree of kinematic indeterminacy, selecting unknown displacements, restraining the structure, and generating a stiffness matrix.
2. A 4m beam with supports at 1.5m and 3m is analyzed using a stiffness matrix approach. The displacements selected are the rotations at joints B and C.
3. The stiffness matrix is generated by applying unit rotations at each joint and calculating the actions. This matrix is then used along with the applied loads in a superposition equation to solve for the unknown displacements.
The document describes the formulation of a Linear-Strain Triangular (LST) finite element. The LST element has 6 nodes, 12 degrees of freedom, and a quadratic displacement function, offering advantages over the Constant Strain Triangular (CST) element. The procedure to derive the LST element stiffness equations is identical to that used for the CST element. Key steps include discretizing the element, selecting displacement functions, defining strain-displacement relationships, and deriving the element stiffness matrix using the total potential energy approach.
Introduction to finite element analysisTarun Gehlot
The document provides an introduction to finite element analysis (FEA) or the finite element method (FEM). It describes FEA as a numerical method used to solve engineering and mathematical physics problems that cannot be solved through analytical methods due to complex geometries, loadings, or material properties. FEA involves discretizing a complex model into smaller, simpler elements connected at nodes, then applying the governing equations to obtain a numerical solution for the unknown primary variable (usually displacement) at nodes. Secondary variables like stress are then determined from nodal displacements. The process involves preprocessing, solving, and postprocessing steps.
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2) Knowing the location of the IC allows simplifying velocity analysis of any point on the object, as the object can be treated as rotating about the IC. The IC can be located using various methods depending on the available velocity information.
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The document provides an introduction to the finite element method (FEM). It discusses how FEM can be used to obtain approximate solutions to boundary value problems in engineering. It outlines the general steps involved, including preprocessing (defining the model), solution/processing (computing unknown values), and postprocessing (analyzing results). Examples of FEM applications include structural analysis, fluid flow, heat transfer, and more. The key aspects of FEM include discretizing the domain into simple elements, choosing shape functions to approximate variations within each element, and assembling the element equations into a global system of equations to solve.
1. The document discusses unsymmetrical bending of beams. When a beam bends about an axis that is not perpendicular to a plane of symmetry, it is undergoing unsymmetrical bending.
2. Key aspects discussed include determining the principal axes, direct stress distribution, and deflection of beams under unsymmetrical bending. Equations are provided to calculate stresses and deflections.
3. An example problem is given involving finding the stresses at two points on a cantilever beam subjected to an unsymmetrical loading. The principal moments of inertia and neutral axis orientation are calculated.
ME6503 design of machine elements - question bank.Mohan2405
This document contains questions and problems related to the design of machine elements, specifically regarding shafts and couplings. It includes 20 questions in Part A testing basic recall and understanding, 13 multi-part problems in Part B applying concepts to design scenarios, and 4 complex design problems in Part C. The topics covered include stresses in shafts, hollow vs solid shafts, keys and keyways, rigid and flexible couplings, and the design of shafts and keys based on strength and rigidity considerations.
This document discusses shape functions in finite element analysis. Shape functions are used to approximate quantities like displacements, strains and stresses between nodes in an element. They are used to interpolate between discrete nodal values and discretize continuous quantities into nodal degrees of freedom. Shape functions are derived for specific element types by choosing interpolation polynomials and natural coordinates that relate the physical coordinates to a standard coordinate system. The document outlines the derivation process for shape functions of bar and beam elements.
The document discusses various interpolation methods used in numerical control (NC) machining including linear, circular, helical, and higher order curve interpolation methods. It provides details on circular interpolation approaches and defines machine axes configurations. It also includes tables of important G and M codes used for CNC programming and examples of CNC milling part programs.
Relevance Vector Machines for Earthquake Response Spectra drboon
This study uses Relevance Vector Machine (RVM) regression to develop a probabilistic model for the average horizontal component of 5%-damped earthquake response spectra. Unlike conventional models, the proposed approach does not require a functional form, and constructs the model based on a set predictive variables and a set of representative ground motion records. The RVM uses Bayesian inference to determine the confidence intervals, instead of estimating them from the mean squared errors on the training set. An example application using three predictive variables (magnitude, distance and fault mechanism) is presented for sites with shear wave velocities ranging from 450 m/s to 900 m/s. The predictions from the proposed model are compared to an existing parametric model. The results demonstrate the validity of the proposed model, and suggest that it can be used as an alternative to the conventional ground motion models. Future studies will investigate the effect of additional predictive variables on the predictive performance of the model.
Relevance Vector Machines for Earthquake Response Spectra drboon
This study uses Relevance Vector Machine (RVM) regression to develop a probabilistic model for the average horizontal component of 5%-damped earthquake response spectra. Unlike conventional models, the proposed approach does not require a functional form, and constructs the model based on a set predictive variables and a set of representative ground motion records. The RVM uses Bayesian inference to determine the confidence intervals, instead of estimating them from the mean squared errors on the training set. An example application using three predictive variables (magnitude, distance and fault mechanism) is presented for sites with shear wave velocities ranging from 450 m/s to 900 m/s. The predictions from the proposed model are compared to an existing parametric model. The results demonstrate the validity of the proposed model, and suggest that it can be used as an alternative to the conventional ground motion models. Future studies will investigate the effect of additional predictive variables on the predictive performance of the model.
The document provides an introduction to the Finite Element Method (FEM). It discusses the history and development of FEM from the 1950s to the present. It outlines the basic concepts of FEM including discretization of the domain into finite elements connected at nodes, and the approximation of displacements within each element. The document also discusses minimum potential energy theory, which is the variational principle that FEM is based on. Example problems and a tutorial are mentioned. Advantages of FEM include its ability to model complex geometries and loading, while disadvantages include increased computational time and memory requirements compared to other methods.
The document discusses finite element methods and their applications in microelectromechanical systems (MEMS). It covers the basic formulation of finite element methods, including discretization, selection of displacement functions, derivation of element stiffness matrices, and assembly of global equations. It also discusses specific applications of finite element analysis to problems in MEMS like heat transfer analysis, thermal stress analysis, and static/modal analysis. The finite element method is well-suited for complex geometries and materials and can model irregular shapes, general loads/boundary conditions, and nonlinear behavior.
Boosting ced using robust orientation estimationijma
In this paper, Coherence Enhancement Diffusion (CED) is boosted feeding external orientation using new
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that scale reflects the orientation of local ridge. For this purpose a new scheme is proposed in which pre
calculated orientation, by using local and integration scales. From the experiments it is found the proposed
scheme is working much better in noisy environment as compared to the traditional Coherence
Enhancement Diffusion
Data-Driven Motion Estimation With Spatial AdaptationCSCJournals
The pel-recursive computation of 2-D optical flow raises a wealth of issues, such as the treatment of outliers, motion discontinuities and occlusion. Our proposed approach deals with these issues within a common framework. It relies on the use of a data-driven technique called Generalised Cross Validation to estimate the best regularisation scheme for a given pixel. In our model, the regularisation parameter is a general matrix whose entries can account for different sources of error. The motion vector estimation takes into consideration local image properties following a spatially adaptive approach where each moving pixel is supposed to have its own regularisation matrix. Preliminary experiments indicate that this approach provides robust estimates of the optical flow.
The document discusses three dimensional problems in finite element analysis. It describes the deformation of a point in a body under forces using displacement components u, v and w. It presents the strain-displacement relations and equations for stresses, strains, body forces and tractions. It also discusses the four noded tetrahedral element, shape functions, element stiffness matrix, and dynamic analysis using the Lagrangian and equations of motion. Finally, it briefly covers guidelines for finite element modeling and debugging models.
Structural engineering & softwatre application ce406Saqib Imran
The document discusses the matrix method of structural analysis. It introduces matrix algebra and how it can be used to analyze structures. There are two main approaches discussed - the stiffness matrix method and flexibility matrix method. The stiffness matrix method determines displacements at joints by considering member stiffness. The flexibility matrix method determines redundant forces by considering flexibility properties. Several examples are provided to illustrate analyzing beams and trusses using the force method and displacement method within the matrix framework. Formulas are given for defining member and structure stiffness/flexibility matrices and solving for displacements and forces.
APPLICATION OF PARTICLE SWARM OPTIMIZATION TO MICROWAVE TAPERED MICROSTRIP LINEScseij
This document discusses using Particle Swarm Optimization (PSO) to design a tapered microstrip transmission line to match an arbitrary load to a 50Ω line. PSO was used to optimize the impedances of a three section tapered line to minimize reflections. Simulations found impedances that gave good matching at 5GHz. PSO converged to solutions in under 1000 iterations. This demonstrates PSO's effectiveness in solving multi-objective microwave engineering optimization problems.
Application of particle swarm optimization to microwave tapered microstrip linescseij
Application of metaheuristic algorithms has been of continued interest in the field of electrical engineering
because of their powerful features. In this work special design is done for a tapered transmission line used
for matching an arbitrary real load to a 50Ω line. The problem at hand is to match this arbitray load to 50
Ω line using three section tapered transmission line with impedances in decreasing order from the load. So
the problem becomes optimizing an equation with three unknowns with various conditions. The optimized
values are obtained using Particle Swarm Optimization. It can easily be shown that PSO is very strong in
solving this kind of multiobjective optimization problems.
Finite Element Analysis Lecture Notes Anna University 2013 Regulation NAVEEN UTHANDI
One of the most Simple and Interesting topics in Engineering is FEA. My work will guide average students to score good marks. I have given you full package which includes 2 Marks and Question Banks of previous year. All the Best
For Guidance : Comment Below Happy to Teach and Learn along with you guys
This document discusses a method for predicting the dynamic response and flutter characteristics of structures using experimental modal parameters when the exact system properties like mass and stiffness are unknown. The method uses modal parameters obtained from ground vibration tests in finite element and computational fluid dynamics software to analyze transient response and flutter speeds. It was validated on a tapered aluminum plate structure by comparing results obtained using experimental modal data to those from a finite element model using the actual material properties. Close agreement was observed between the two methods, showing this approach can accurately analyze structures without prior knowledge of system configurations.
An iterative morphological decomposition algorithm for reduction of skeleton ...ijcsit
Shape representation is an important aspect in image processing and computer vision. There are several skeleton transforms that lead to morphological shape representation algorithm. One of the main problems with these algorithms is in selecting the skeleton points that represent the shape component. If the numbers of skeleton subsets are reduced then the reconstruction process will be easy and time consuming. The present paper proposes a skeleton scheme that selects skeleton points based on the largest shape element. By this, overall skeleton subsets will be reduced. The present method is applied on various images and is compared with generalized skeleton transform and octagon-generating decomposition algorithm.
The document describes the implementation of geometrical nonlinearity in finite element analysis software FEASTSMT. It discusses total Lagrangian formulation and the Newton-Raphson method to solve nonlinear finite element equations arising from large deformations. Element formulations accounting for incremental strains, strain-displacement relationships, and stresses are developed. The implementation is validated on a solid element and beam bending problem by comparing results with MARC software and analytical solutions, showing good agreement.
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Implementation Of Geometrical Nonlinearity in FEASTSMTiosrjce
Analysis of the structures used in aerospace applications is done using finite element
method. These structures may face unexpected loads because of variable environmental situations.
These loads could lead to large deformation and inelastic manner. The aim of this research is to
formulate the finite elements considering the effect of large deformation and strain. Here total
Lagrangian method is used to consider the effect of large deformation. After deriving required
relations, implementation of formulated equation is done in FEASTSMT(Finite Element Analysis of
Structures - Substructured and Multi-Threading). .Newton-Raphson method was utilized to solve
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Marc Software.
Boosting CED Using Robust Orientation Estimationijma
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scheme is working much better in noisy environment as compared to the traditional Coherence
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ANATOMY OF SOA - Thomas Erl - Service Oriented Architecture
FEM 10 Common Errors.ppt
1. Common Errors in Finite Element Analysis
Idealization error Discretization error
Idealization Error
•Posing the problem
•Establishing boundary conditions
•Stress-strain assumption
•Geometric simplification
•Specifying simplification
•Specifying material behaviour
•Loading assumptions
Prof .N. Siva Prasad, Indian Institute of Technology Madras Error - 1
2. Discretization Error
•Imposing boundary conditions
•Displacement assumption
•Poor strain approximation due to element distortion
•Feature representation
•Numerical integration
•Matrix ill-conditioning
•Degradation of accuracy during Gaussian elimination
•Lack of inter-element displacement compatibility
•Slope discontinuity between elements
Prof .N. Siva Prasad, Indian Institute of Technology Madras Error - 2
3. Idealization Error
One must be able to understand the physical nature of an
analysis problem well enough to conceive a proper
idealization. Engineering assumptions are always
required in the process of idealization.
Example:
•Establishing boundary conditions
•Specifying Material behaviour
Prof .N. Siva Prasad, Indian Institute of Technology Madras Error - 3
4. Establishing Boundary Conditions
There are innumerable examples of how the specification of
improper boundary conditions can lead to either no results or poor
results.
It is impractical to review every possible manner in which one can
error when establishing boundary conditions. The finite element
analyst must gain a sufficient theoretical understanding of
mechanical idealization principles so that he can understand what
boundary conditions are applicable in any particular case.
Consider a bracket loaded by the weight of a servo motor, as
illustrated in Figure 1, 2
Prof .N. Siva Prasad, Indian Institute of Technology Madras Error - 4
5. Fig 1. Actual Motor Bracket
Fig. 2 Simplified
Prof .N. Siva Prasad, Indian Institute of Technology Madras Error - 5
6. Idealization Error – Specifying Material Behaviour
Elasto-Plastic state (Incompressible in plastic region)
Elastomeric material (= 0.5, Constitutive relations undefined)
Discretization Error
Discretization is the process where the idealization, having
an infinite number of DOF’s, is replaced with a model
having finite number of DOF’s. The following are errors
associated with discretization.
Prof .N. Siva Prasad, Indian Institute of Technology Madras Error - 6
7. No of Points = 8
dof per point = 2
Total Equations = 16
Prof .N. Siva Prasad, Indian Institute of Technology Madras Error - 7
8. Element Errors
•Imposing essential boundary conditions
•Displacement assumption
•Poor strain approximation due to element distortion
•Feature representation
Global Errors
•Numerical integration
•Matrix ill-conditioning
•Degradation of accuracy during Gaussian elimination
•Lack of inter-element displacement compatibility
•Slope discontinuity between elements
Prof .N. Siva Prasad, Indian Institute of Technology Madras Error - 8
9. Element Error – Imposing Essential Boundary
Conditions
Rigid body motion is displacement of a body in such a manner
that no strain energy is induced. When one considers finite
elements in three-dimensional space, with nodes having three
translations DOF’s restraining rigid body motion can be more
complicated. Consider the 8-node brick element in three-
dimensional space depicted in Figure 4. What are the
minimum nodal displacement restraints required to prevent
rigid body motion of this element?
Prof .N. Siva Prasad, Indian Institute of Technology Madras Error - 9
10. Fig 4. 8-node Brick Element in 3-D Space
In 3D space, there exists potential for six rigid body modes
( 3 displacements, (x, y, z) and 3 rotations θx, θy and θz )
Prof .N. Siva Prasad, Indian Institute of Technology Madras Error - 10
11. Fig 5. Restraining One Node
Restraining one node will allow the body to rigidly rotate.
Prof .N. Siva Prasad, Indian Institute of Technology Madras Error - 11
12. Fig 6. Nodal Restraint as a Ball and Socket Joint
Translational motion arrested but the body tend to rotate
Prof .N. Siva Prasad, Indian Institute of Technology Madras Error - 12
13. Fig 7. Inhibiting Rigid Body Motion
Displacement is restrained at 5 and 8. This prevents rotation
about z axis.
Prof .N. Siva Prasad, Indian Institute of Technology Madras Error - 13
14. Most finite element software programs will issue a warning,
“singular system encountered” or “Non-positive definite
system encountered”. The latter warning refers to the fact
that the strain energy is not greater than zero using the
given boundary conditions, suggesting that the structure is
either unstable, as in the case of structural collapse, or not
properly restrained, such that rigid body modes of
displacement are possible.
Prof .N. Siva Prasad, Indian Institute of Technology Madras Error - 14
15. Element Error – Displacement Assumption
The h-method of finite element analysis endeavours to
minimize this error by using lower order displacement
assumptions (typically linear or quadratic) then refining the
model using more, smaller elements. Using more smaller
elements to gain increased accuracy is known as h-
convergence.
Prof .N. Siva Prasad, Indian Institute of Technology Madras Error - 15
16. Element Error – Poor Strain Approximation Due to
Element Distortion
Distorted elements influence the accuracy of the finite
element approximation for strain. For instance, some
bending elements (beams, plated, shells) compute
transverse shear stain. This type of elements may have
difficulty computing shear strain when the element becomes
very thin.
Prof .N. Siva Prasad, Indian Institute of Technology Madras Error - 16
18. Global Errors
Global errors are those associated with the assembled finite
element model. Even if each element exactly represents the
displacement within the boundary of a particular element,
the assembled model may not represent the displacement
within the entire structure, due to global errors.
Global Error – Numerical Integration
The use of numerical integration instead of closed-form
integration introduces error.
Prof .N. Siva Prasad, Indian Institute of Technology Madras Error - 18
19. Global Error – Matrix Ill-Conditioning
The finite elements solution can render poor solutions for
displacement due to round off error. Ill-conditioning errors
typically manifest themselves during the solution phase of
an analysis.
Figure: Structure with an Ill-conditioned Stiffness Matrix
Prof .N. Siva Prasad, Indian Institute of Technology Madras Error - 19
20. A stepped shaft, characterized by two different cross sections,
is depicted in Figure. Two rod elements is used to model the
structure, with the expanded equilibrium equations for
Element I given as:
3
2
1
3
2
1
1
1
1
0
0
0
0
1
1
0
1
1
F
F
F
U
U
U
L
A
E
(1)
The equilibrium equations for Elements 2 are:
3
2
1
3
2
1
2
2
2
1
1
0
1
1
0
0
0
0
F
F
F
U
U
U
L
A
E (2)
Prof .N. Siva Prasad, Indian Institute of Technology Madras Error - 20
22. The element stiffness matrices are added together to
represent the global stiffness:
P
F
U
U
U
00
.
0
00
.
1
00
.
1
00
.
0
00
.
1
01
.
1
0100
.
00
.
0
0100
.
0100
. 1
3
2
1
(1)
Boundary conditions are now imposed upon the global system
of Equations, (1). Since U1 = 0 and F1 is unknown, the 3 X 3
system in (1) is replaced by a 2 X 2 systems:
P
U
U 00
.
0
00
.
1
00
.
1
00
.
1
01
.
1
3
2
(2)
Prof .N. Siva Prasad, Indian Institute of Technology Madras Error - 22
23. Using Gaussian elimination, Equations (2) is manipulated to
yield:
P
U
P
U
U
101
00
.
0
00990
.
00
.
0
00
.
1
01
.
1
3
3
2
(3)
Consider a small error in computation of k11 given by
Equation(2)
P
U
U 00
.
0
00
.
1
00
.
1
00
.
1
02
.
1
3
2
(4)
Prof .N. Siva Prasad, Indian Institute of Technology Madras Error - 23
24. Using Gaussian elimination to solve the system containing
the small error:
2
3
3
1.02 1.00 0.00
51.0
0.00 .0196
U
U P
U P
(5)
It may be surprising to note that a 1% error in one of the
entries of the stiffness matrix is responsible for a 50% error
in displacement.
Prof .N. Siva Prasad, Indian Institute of Technology Madras Error - 24
25. 25
EQUILIBRIUM AND COMPATIBILITY IN THE SOLUTION
1.Equilibrium of nodal forces and moments is satisfied. The
structural equations {F} – [K] {Q}= {0} are nodal equilibrium
equations. Therefore, the solution vector {Q} is such that
nodal forces and moments have a zero resultant at every
node.
2.Compatibility prevails at nodes. Elements connected to one
another have the same displacements at the connection
point. (i.e.) elements are compatible at nodes to the extent of
nodal d.o.f. they share.
26. 26
3. Equilibrium is usually not satisfied across interelement
boundaries. Figure 4.4-1 provides a simple example.
Imagine that the elements are constant-strain triangles
and that node 4 is the only node displaced, as shown.
Then σx2 is the only nonzero stress, and the shaded
differential element is not in equilibrium
27. 27
4. Compatibility may or may not be satisfied across
interlement boundaries. The constant-strain triangle and
the plane bilinear element, compatibility is guaranteed
because element sides remain straight even after the
element is deformed.
Other elements, such as plate elements are
incompatible in rotation about an interlement boundary.
Figure 4.4-2 is another case in point. Stretching of the
right edge causes vertical edges of the right element to
bend, and the interlement gap (shaded) appears. (This
element, called either incompatible or nonconforming)
28. 28
5.Equilibrium is usually not satisfied within elements.
Satisfaction of the differential equations of equilibrium at
every point in an element demands a relation among
element d.o.f. that usually does not result from solution of
the global finite element equations [K]{Q} = {F}.
An important property of any reliable element is that its
displacement field be capable of representing all possible
states of constant strain.
29. 29
6. Compatibility is satisfied within elements. We require only
that the element displacement field be continuous and
single-valued. These properties are automatically
provided by polynomial fields.
CONVERGENCE REQUIREMENTS
1.Within each element, the assumed field for φ must contain
a complete polynomial of degree m.
2.Across boundaries between elements, there must be
continuity of φ and its derivatives through order m-1.
30. 30
3. Let the elements be used in a mesh (rather than tested
individually), and let boundary conditions on the mesh be
appropriate to a constant value of any of the mth derivatives
of φ. Then, as the mesh is refined, each element must
come to display that constant value.
For example, if φ = φ(x,y) and П contains first derivatives of
φ, then the lowest order acceptable field has the form
φ = a1 + a2x + a3y in each element, only φ itself need be
continuous across interelement boundaries, and each
element of an appropriately loaded mesh must display a
constant value of φ,x ( or of φ,y for other appropriate loading),
at least as the mesh is refined.
31. 31
This suggests the following criterion for modeling. In a mesh
of low-order elements (such as the bilinear element), the
ratio of stress variation across the element to mean stress
within the element should be small.
It is a simple test that can be performed numerically, so as to
check the validity of an element formulation and its program
implementation. We assume that the element is stable in the
sense described below. Then, if the element passes the
patch test, we have assurance that all convergence criteria
are met.
THE PATCH TEST
32. 32
Procedure. One assembles a small number of elements
into a “patch,” taking care to place at least one node within
the patch, so that the node is shared by two or more
elements, and so that one or more interelement boundaries
exist. Figure 4.6-1 shows an acceptable two-dimensional
patch, built of four-node elements. Boundary nodes of the
patch are loaded by consistently derived nodal loads
appropriate to a state of constant stress.
34. 34
Stability. At the outset we assumed that the element to be
patch-tested is stable. A stable element is one that admits
no zero-energy deformation states when adequately
supported against rigid-body motion. Unstable elements
should be used with caution. They can produce an unstable
mesh, whose displacements are excessive and quite
unrepresentative of the actual structure.
35. 35
“Weak” Patch Test. An element that fails to display constant
stress in a patch of large elements has not necessarily failed
the patch test. If, as the mesh is repeatedly subdivided,
elements come to display the expected state of constant
stress, then the element is said to have passed a “ weak”
patch test, and convergence to correct results is assured.
36. Convergence
To ensure monotonic convergence of the finite element
solution, both the individual elements and the assemblage
of elements (“the mesh”) must meet certain requirements.
Monotonic Convergence Using the Displacement Based,
h-Method
With mesh refinement, the finite element solution is
expected to convergence, monotonically, to the exact
solution.
Prof .N. Siva Prasad, Indian Institute of Technology Madras Conver - 1
37. Prof .N. Siva Prasad, Indian Institute of Technology Madras Conver - 2
38. Requirements for Monotonic Convergence
(i) Requirements for Each Element’s Displacement Assumption
1. Rigid Body Representation assumption must be able to
account for all rigid body displacement modes of the
element.
2. Uniform Strain Representation: Constant strain states for
all strain components specified in the constitutive
equations of a particular idealization must be represented
within the element as the largest dimension of the element
approaches zero.
Prof .N. Siva Prasad, Indian Institute of Technology Madras Conver - 3
39. (ii) Requirements for the Mesh
1. Compatibility Between Elements: The dependent
variable(s), and p-1 derivatives of the dependent variable,
must be continuous at the nodes and across the inter-
element boundaries of adjacent elements.
2. Mesh Refinement: Each successive mesh refinement must
contain all of the previous nodes and elements in their
original location.
3. Uniform Strain Representation: The mesh must be able to
represent uniform strain when boundary conditions that are
consistent with a uniform strain condition are imposed.
Prof .N. Siva Prasad, Indian Institute of Technology Madras Conver - 4
41. In a one-dimensional element with rectangular coordinates, a
polynomial displacement assumption must have a constant
term to ensure that the type of rigid body motion described
above is allowed.
U(X) = a1 +a2X
2. Convergence and Uniform Strain Representation within an
Element
dx
du
E
E xx
xx
2
2
1
)
( a
dx
du
x
a
a
x
u xx
Prof .N. Siva Prasad, Indian Institute of Technology Madras Conver - 6
42. 3. Convergence and Inter-Element Compatibility
Continuity must be maintained across element boundaries,
as well as at the nodes.
A. Incompatibility due to Elements Not Properly Connected:
Prof .N. Siva Prasad, Indian Institute of Technology Madras Conver - 7
43. Figure shows that Nodes 2 and 3 are coincident nodes,
meaning that they have the same spatial coordinated but
belong to separate elements.
Element Node a Node b
1 1 2
2 3 4
Connectivity table before node merge
Prof .N. Siva Prasad, Indian Institute of Technology Madras Conver - 8
44. Connectivity table after node merge
Element Node a Node b
1 1 2
2 2 3
When creating finite element models, a node merging
procedure is typically invoked, such that each pair of coincident
nodes is replaced by a single node, and the connectivity table
is updated to reflect the new connectivity.
Prof .N. Siva Prasad, Indian Institute of Technology Madras Conver - 9
45. B. Incompatibility due to Differing Order Displacement
Assumptions:
Prof .N. Siva Prasad, Indian Institute of Technology Madras Conver - 10
46. Consider two elements in Figure 14, one having a linear
displacement assumption and other having a quadratic
assumption. A gap between the elements occurs since the
displacement for the linear element can only be represented
by a straight line while displacement in the other element is
a quadratic function.
Prof .N. Siva Prasad, Indian Institute of Technology Madras Conver - 11
47. In figure that the mid-side node of Element 1 is connected to
corner nodes of Element 2 and 3. Higher order elements must
be matched such that the mid-side node of one element is
connected to the mid-side node of the other.
Prof .N. Siva Prasad, Indian Institute of Technology Madras Conver - 12
48. C. Incompatibility due to Differing Nodal Variables:
Prof .N. Siva Prasad, Indian Institute of Technology Madras Conver - 13
49. Incompatibility also occurs when element having differing types of nodal
DOF are joined. Figure shows a 2-node beam element attached to one
node of a 8-node element, which have translational DOF only, while
structural elements have both translational and rotational DOF’s.
When joining continuum and structural elements, a special constraint must
be imposed upon the structural element’s rotational DOF, the least 8-node
brick element shown in Figure is well restrained. Its nodes do not have
rotational DOF’s, therefore, at the node where the beam is attached, there
exits no DOF from the brick to couple with the rotational DOF of the beam.
As shown in figure, the beam element will experience rigid body rotation.
Prof .N. Siva Prasad, Indian Institute of Technology Madras Conver - 14
50. D. Incompatibility due to incompatible elements:
Type of incompatibility to be mentioned occurs when elements in
mesh are of the “incompatible type”
Prof .N. Siva Prasad, Indian Institute of Technology Madras Conver - 15
51. Consider the two elements in Figure both are identically
formulated, 4 node quadrilateral surface elements designed
with incompatible modes of displacement. Assume that two
nodes of Element 2 are given a displacement of Δy as
depicted in figure (a). With no other displacement, the
elements would appear as shown. However, displacement
for Element 2 would be computed as shown in figure (18).
Prof .N. Siva Prasad, Indian Institute of Technology Madras Conver - 16
52. 4. Convergence and the Discretization Process:
Each successive mesh refinement must contain all of the
previous nodes and elements in their original location.
Prof .N. Siva Prasad, Indian Institute of Technology Madras Conver - 17
53. 5. Convergence and Uniform Strain Representation within Mesh
The mesh must be able to represent a uniform strain state,
when suitable boundary conditions are imposed. A Patch
Test has been devised to test the uniform strain condition.
Either displacements or loads can be applied, depending
upon what characteristics of the mesh are to be investigated.
Prof .N. Siva Prasad, Indian Institute of Technology Madras Conver - 18
54. Figure illustrates a mesh for a patch test of two-
dimensional elements although, theoretically, substantial
variation in the geometry (somewhat distorted) of the
elements used for a patch test is permitted.
Prof .N. Siva Prasad, Indian Institute of Technology Madras Conver - 19
55. Elementary Beam, Plate, and Shell Elements
Euler-Beronoulli Beams
Beams are slender structural members, with one dimension
significantly greater that the other two.
Prof .N. Siva Prasad, Indian Institute of Technology Madras Beam - 1
56. Euler-Beronoulli Beam Assumptions
If Euler- Bernoulli (“elementary”) beam theory is to be used,
some restrictions must be imposed to ensure suitable
accuracy. Six items related to the Euler- Beronulli beam
bending assumptions are considered in brief:
1. Slenderness (Transverse shear neglected, L/h ≥ 10,
Orthogonal planes remain plane and orthogonal)
2. Straight, narrow, and uniform beams (Arch structure –
membrane forces.)
3. Small deformations (Second order terms and dropped
from the Green-Lagrange Strain mertic)
Prof .N. Siva Prasad, Indian Institute of Technology Madras Beam - 2
57. 4. Bending loads only (Loads that cause twist or cause x- or
y- displacement of neutral axis are ignored.)
5. Linear, isotropic, homogeneous materials response
6. Only normal stress in the x-coordinate direction (σxx)
significant (σxx vary linearly in the z direction and σzz = 0
Prof .N. Siva Prasad, Indian Institute of Technology Madras Beam - 3
58. Kirchhoff Plate Bending Assumptions
Several restrictions must be imposed when using the
Kirchhoff plate bending formulation if suitable accuracy is
to be obtained; six related items are considered briefly:
1. Thinness
2. Flatness and uniformity
3. Small deformation
4. No membrane deformation
5. Linear, isotropic, homogeneous materials
6. σxx, σyy , σxy the only stress components of significance
Prof .N. Siva Prasad, Indian Institute of Technology Madras Beam - 4
59. 1.Thin Plates:
If the ratio of the in-plane dimensions to thickness is greater than ten, a
plate may be considered thin; in other words, thin plates are such that
.
10
/
/
t
b
t
l
(i) Transverse Shear Stress Does Not Affect Transverse
Displacement:
Although transverse displacement of a plat loaded normal to its neutral
surface is affected by both normal stress and transverse shear stress, it is
assumed that the shear stress does not have a significant impact on the
magnitude of transverse displacement. It can be shown that the magnitude
of transverse displacement due to shear stress is small in thin plates.
Prof .N. Siva Prasad, Indian Institute of Technology Madras Beam - 5
60. (ii) Orthogonal Planes Remain Plane and Orthogonal:
In a thin plate, any cross sectional plane, originally orthogonal
to the neutral surface, is assumed to remain plane and
orthogonal when the plate is loaded. As with deep beams,
cross sections in thick plates, originally orthogonal and
planar, will be neither under transverse load, due to shearing
effects.
Prof .N. Siva Prasad, Indian Institute of Technology Madras Beam - 6
61. 2. Flat and Uniform
Kirchhoff plates are presumed flat, and as such, loads are
not carried by membrane deformation. If initial curvature
were to exist, some of the load could be carried by
membrane action, hence, curved plates (shells) require a
different approach to account for the combination of
simultaneous membrane and bending deformation.
Prof .N. Siva Prasad, Indian Institute of Technology Madras Beam - 7
62. 3. Small Deformation
1. The expression for stain actually contains second order
terms, and these terms are truncated if the strains are
presumed small.
2. Like the Euler-Bernoulli beam, plate curvature can be
expressed in terms of second order derivatives if the
square of the slop is negligible.
3. As a plate deflects its transverse stiffness changes.
Prof .N. Siva Prasad, Indian Institute of Technology Madras Beam - 8
63. Why does the transverse stiffness change with deflection? As
a plate deforms into a curved (or a doubly curved) surface,
transverse loads are resisted by both bending and membrane
deformation. The addition of membrane deformation affects the
transverse stiffness of the plate. Consider figure where a ball is
placed on a very thin “plate”.
Notice that as the amount of plate deflection increases, more
load is carried by membrane tension and less by bending.
Shell theory is required to account for structure that carry loads
through simultaneous bending and membrane deformation
Prof .N. Siva Prasad, Indian Institute of Technology Madras Beam - 9
64. 4. No Membrane Deformation
Kirchhoff plates do not account for membrane deformation.
This means that at the neutral surface, there can be no
displacement in the x- or y-coordinate directions, which
would suggest the existence of membrane deformation
5. Signification Stress Components are σ xx, σ yy , σ xy
As in shallow beams, it is assumed that transverse shear
stress in thin plates is relatively insignificant when compared
to the normal stress in the longitudinal direction of the beam,
hence:
σyz = σzx = 0 and σzz = 0
Prof .N. Siva Prasad, Indian Institute of Technology Madras Beam - 10
65. Shear Stress Affects the Edge Reaction Forces
In plate bending stress affects transverse displacement. In
addition, shear stress affects reaction forces on restrained
edges of plates and shells.
The restrained edges of shell and plate structures develop
reaction forces. The magnitude of the reaction is dependent
upon the behaviour in the vicinity very near the restrained
edge. This boundary layer effect cannot be captured without
accounting for transverse shear stress. In addition, even in
models that do account for shear effects, the boundary layer
effect is difficult to model without special care.
Prof .N. Siva Prasad, Indian Institute of Technology Madras Beam - 11