The document discusses stress concentration and fatigue failure in machine elements. It defines stress concentration as the localization of high stresses due to irregularities or abrupt changes in cross-section. Stress concentration can be reduced by avoiding sharp changes in cross-section and providing fillets and chamfers. Fatigue failure occurs when fluctuating stresses cause cracks over numerous load cycles. The endurance limit is the maximum stress amplitude that causes failure after an infinite number of cycles. Factors like stress concentration, surface finish, size, and mean stress affect the endurance limit. Designs should minimize stress raisers and protect against corrosion to prevent fatigue failures.
The various forces acts on the reciprocating parts of an engine.
The resultant of all the forces acting on the body of the engine due to inertia forces only is known as unbalanced force or shaking force.
This document discusses various types of governors used to regulate engine speed. It describes centrifugal governors that use rotating balls to control engine speed based on centrifugal force. Specific governors discussed include the Watt, Porter, Proell, Hartnell, Hartung, Wilson-Hartnell, and Pickering governors. Equations are provided for each governor relating factors like ball mass, radius of rotation, spring stiffness, and centrifugal force to the governor's operation and ability to control engine speed under varying loads.
Design involves formulating a plan to satisfy a particular need and create something with physical reality. When designing a chair, factors like purpose, intended user (adult or child), material strength and cost, aesthetics, and ergonomics must be considered. Machine design uses technical information, scientific principles, and imagination to design machines to perform specific functions with maximum economy and efficiency. This document discusses various machine design considerations and principles like types of loads, material selection, and theories of failure.
lecture 4 (design procedure of journal bearing)ashish7185
This document provides information about the design procedure for sliding contact bearings. It defines key terms used in hydrodynamic journal bearings such as diametral clearance, radial clearance, eccentricity, minimum oil film thickness, and short/long bearings. It discusses bearing characteristic number and bearing modulus, and how they relate to the coefficient of friction. Equations are provided for critical pressure, heat generated in bearings, and heat dissipated by bearings. The design procedure involves selecting bearing dimensions, material properties, operating parameters, and verifying thermal equilibrium conditions.
The taylor hobson talysurf surface roughness testervaibhav tailor
This instrument measures surface roughness using a stylus attached to an armature. Variations in the surface profile are sensed by the stylus and cause the gap between the armature and an E-shaped arm to vary, modulating the AC current in a coil. This modulation is demodulated so the output is directly proportional to the vertical displacement of the stylus, allowing a recorder to produce a record of the surface roughness. The instrument provides a more rapid and accurate measurement of surface roughness compared to the Tomlinson surface tester.
1. Interchangeability refers to parts that can be substituted for similar parts from other manufacturers without issues in assembly. It was pioneered by Eli Whitney who demonstrated interchangeable parts for firearms to Congress in 1801.
2. Selective assembly involves measuring and sorting parts into groups based on dimensions before assembly to achieve tighter tolerances not possible through interchangeability alone. It allows for assembling parts from within tolerance ranges that ensure proper fit and function.
3. The advantages of interchangeability and selective assembly include easier assembly, higher production rates, lower assembly costs, simplified repairs and replacements, and the ability to achieve mass production. Selective assembly also reduces waste and increases quality by avoiding needlessly tight tolerances.
This document discusses various types of machine balancing. It begins by defining static and dynamic balancing. Static balancing deals with balancing forces when a machine is at rest, while dynamic balancing deals with balancing forces during motion. It then discusses balancing of single and multiple rotating masses, as well as reciprocating masses. Methods for analytically and graphically balancing multiple masses are provided. The document also covers balancing of engines with different cylinder configurations, including inline, V-shaped, radial, and locomotive engines. Partial balancing techniques are discussed for reducing unbalanced forces in locomotives.
This document discusses the fundamentals and types of mechanisms. It covers topics such as statics, dynamics, kinematics, kinetics, links, kinematic pairs, constrained motions, inversions of mechanisms, and common mechanisms. Examples are provided to illustrate concepts like the four bar chain, slider crank chain, Geneva mechanism, Ackermann steering, and rear wheel sprocket of a bicycle. Mechanisms are analyzed based on their motion, forces, components, and ability to transform input energy into useful work.
This document provides an overview of dynamics of machines including:
1. It defines force, applied force, constraint forces, and types of constrained motions like completely, incompletely, and successfully constrained motions.
2. It discusses static force analysis, dynamic force analysis, and conditions for static and dynamic equilibrium.
3. It covers concepts like inertia, inertia force, inertia torque, D'Alembert's principle, and principle of superposition.
4. It derives expressions for forces acting on the reciprocating parts of an engine while neglecting the weight of the connecting rod.
This document provides an overview of machine design concepts including the basic design process, factors to consider in design, and design of simple machine elements. It discusses the definition of machine design as using scientific principles and imagination to design machines to perform functions efficiently. The basic design process involves understanding requirements, analyzing loads, selecting materials, choosing dimensions, and specifying tolerances. Simple elements discussed include cotter joints, knuckle joints, levers, and components under eccentric loading. Design of these elements involves calculating stresses and selecting dimensions to prevent failure under various loading conditions like tension, shear, bending, and crushing. Standards and preferred sizes are also important considerations in efficient machine design.
The document discusses static force analysis and equilibrium of mechanisms. It covers topics like static equilibrium, equilibrium of two and three force members, members with two forces and torque, free body diagrams, and the principle of virtual work. Examples of static force analysis of four bar and slider-crank mechanisms are presented. Methods to determine the forces and torques required for static equilibrium are demonstrated through graphical techniques like force triangles and the principle of virtual work.
The document discusses stress concentration and fatigue failure in machine elements. It defines stress concentration as irregular stress distribution caused by abrupt changes in cross-section shape. Stress concentration factors are introduced to quantify the maximum stress compared to nominal stress. The document also discusses endurance limit and fatigue strength testing methods. Factors that affect fatigue strength like material properties, surface finish, size and temperature are summarized. Methods to evaluate and reduce stress concentration in designs are provided.
This presentation contains basic idea regarding spur gear and provides the best equations for designing of spur gear. One can Easily understand all the parameters required to design a Spur Gear
ME6601 - DESIGN OF TRANSMISSION SYSTEM NOTES AND QUESTION BANK ASHOK KUMAR RAJENDRAN
This document contains the question bank for the subject ME6601 - Design of Transmission Systems for the sixth semester Mechanical Engineering students of RMK College of Engineering and Technology. It is prepared by R. Ashok Kumar and S. Arunkumar, faculty of the Mechanical Engineering department.
The question bank contains 190 questions divided into two parts: Part A containing conceptual questions and Part B containing design/numerical problems. The questions cover the five units of the subject - Design of Flexible Elements, Spur Gears and Parallel Axis Helical Gears, Bevel, Worm and Cross Helical Gears, Gear Boxes, and Cams, Clutches and Brakes. Most questions are related
1. A shaft transmits power and rotational motion and has machine elements like gears and pulleys mounted on it.
2. Press fits, keys, dowel pins, and splines are used to attach machine elements to the shaft.
3. The shaft rotates on rolling contact or bush bearings and uses features like retaining rings to take up axial loads.
4. Couplings are used to transmit power between drive and driven shafts like between a motor and gearbox.
Kinematics of machines can involve either analyzing an existing mechanism's motion or synthesizing a new mechanism to achieve a desired motion. Kinematic synthesis involves selecting the type of mechanism, determining the number of links needed, and defining the link dimensions. Dimensional synthesis aims to develop link dimensions such that the mechanism's output motion matches the desired motion at select precision points, often spaced using Chebyshev's method to minimize error between points. Slider-crank mechanisms can be synthesized by relating the slider displacement to crank angle at precision points defined using Chebyshev spacing.
Gear measurements:- MECHANICAL MEASUREMENTS AND METROLOGYJaimin Patel
This document provides information about gear measurement and metrology. It discusses various gear profiles like involute and cycloidal profiles. It also defines important gear terminology like pitch circle, pressure angle, addendum, etc. Several methods for measuring gear tooth thickness are described, including using a gear tooth Vernier caliper, constant chord method, base tangent method, and dimension over pins. The document also discusses gear inspection and a working method that uses two carriages and a dial gauge to measure variations when rotating meshed gears.
The document discusses design considerations for machine elements subjected to fluctuating loads. It covers topics such as stress concentration, fatigue failure, endurance limit, factors affecting fatigue strength, and methods to reduce stress concentration and improve fatigue life. Stress concentration occurs due to discontinuities and can be reduced by avoiding abrupt changes in cross-section and providing fillets. Fatigue failure is caused by fluctuating stresses and depends on factors like the number of cycles and mean stress. The endurance limit is the maximum stress amplitude a material can withstand without failure under completely reversed loading. Surface finish, size, and mean stress affect the endurance limit.
This document provides an overview of basic design considerations for machine components. It discusses general design procedures and considerations, types of loads, stress-strain diagrams, types of stresses including tensile, compressive, shear, crushing, bearing, torsional, and bending stresses. It also covers concepts related to stress concentration, creep, fatigue, endurance limit, factor of safety, and theories of failure under static loads. Standard classifications and designations of various steel and alloy types are also presented.
Effect of punch profile radius and localised compressioniaemedu
This document discusses springback in V-bending of high strength steel sheets. It presents results from an experimental investigation and finite element analysis simulation of the effects of punch profile radius and localized compression on springback. The experimental results showed that increasing punch radius or decreasing sheet thickness increases springback, while applying localized compressive stress through bottoming the punch can compensate for springback. The finite element analysis validated the experimental findings. The document provides background on springback in bending, methods to compensate for it, and details of the materials testing and modeling approach used in the study.
The document summarizes a mechanical engineering project involving modeling and programming an expansion joint using Pro/E software. It describes modeling bellows, sleeves, and other expansion joint components in the software. Finite element analysis was performed on a bellows model to analyze stress and displacement under various pressures. C-programming and Pro/PROGRAM tools were used to automate design variations and iterations.
Analysis of failure behavior of shear connection in push-out specimen by thre...IJERDJOURNAL
ABSTRACT:- This study analyzes the failure mechanism of shear connection by three-dimensional finite element analysis (FEA) of push-out specimens that was practically unaffordable experimentally or by twodimensional FEA. For the analysis of the failure behavior of the compression strut formed in the loaded concrete member, the three-dimensional principal stress space is transformed into two-dimensional space by means of the relation between the hydrostatic stress and the deviatoric stress. The analysis of the stress state in the compression strut revealed that the deviatoric stress increases with larger load particularly in the concrete surrounding the lower part of the shear stud. Accordingly, bearing failure of concrete occurred locally within a limited region in the slab. The steep increase of the deviatoric stress accompanying the increase of the load resulted in the failure of concrete around the lower part of the shear stud, which in turn provoked the deformation and the development of bending moment of the shear stud. Finally, plastic hinge formed in the shear stud leading it to reach its limit state. The proposed finite element model can also be used to model the shear connection of the composite beam and, the proposed stress analysis method can be applied to analyze its composite action behavior.
A column is a vertical structural member subjected to compression and bending forces. Short columns fail through crushing or splitting, while slender columns fail through buckling. The document provides examples of calculating required reinforcement area and diameter for a short reinforced concrete column. It also provides examples of calculating the critical buckling load of a rod and determining a suitable universal column section for a given load based on its effective length and slenderness ratio.
Comparative Study on Anchorage in Reinforced Concrete Using Codes of Practice...IJERA Editor
This paper (Part II) reports a comparative study for BS8110 and EC2 of practice and those expressions by Batayneh and Neilsen on tests from literature. These have been treated under straight bar anchorages with transverse pressure. The aim of this study is to evaluate the reliability of the existing equations for bond strength of straight bars by applying to the available tests in the literature .The most important parameters were examined in these tests are concrete strength, anchorage length, concrete covers, bar diameter and transverse pressure. 264 tests from the literature have been chosen, which are all for straight bars with transverse pressure. The specimens are pull-out specimens with small concrete covers, beams ends and slabs. For both comparative studies in Part I and Part II, the conclusions and recommendations are presented here together.
PARAMETRIC STUDIES ON THE EFFECT OF FOUR TYPES OF FASTENER MODELING IN CHANNE...ijmech
In this paper, some parametric studies on four types of Channel type tension fitting’s fasteners’ stiffness
modeling is presented. Tension fittings are commonly classified into five types. They are Bathtub fitting,
Channel fitting, Angle fitting, ‘PI’ fitting and Double angle fitting. Tension fittings are conservatively sized
as their weight is usually small relative to their importance. In the previous studies, the channel fitting was
considered to be fixed at all the fastener locations. Thus, the results obtained were conservative because
the load was getting reacted at the first line of fasteners only. In order to study the effect of fastener’s
flexibility and hence the load flow inside the tension fitting two methods (Tate & Swift) of fastener modeling
were employed in the previous study. It observed that, the flexible boundary condition allow for a better
load flow into the channel fitting as compared to the fixed boundary condition. In this study, fastener
flexibility with two more methods (Grumann and Huth) is performed on the distribution of internal stresses
in the channel fitting as compared to the fixed boundary conditions. Also comparison of previous results
(Tate and Swift) is made with Grumann and Huth methods of modeling of fastener. Aluminum alloy 7050-
T7452 is selected for the study.
Optimization of tube-flange welded joints under Torsional loadingIRJET Journal
This document summarizes research optimizing the geometry of tube-flange welded joints under torsional loading. Finite element analysis was conducted using ANSYS to analyze stresses and deformation on welded joints made of carbon steel and aluminum alloy. Response surface methodology and Taguchi design of experiments were used to generate design points and optimize shape parameters. The analysis found that corner points experience the highest shear stresses and are most prone to failure. Shear stress decreases further from corners. While deformation is greatest at open ends. The h parameter was found to have the highest sensitivity to stresses, so should be a key consideration in joint design.
This document provides a review of limit load solutions for structural components containing defects. It summarizes existing limit load solutions for plates, cylinders, spheres, pipe bends, and shell/nozzle intersections with various defect geometries under different loading conditions. The solutions are useful for failure assessment methods that utilize the limit load or reference stress concept. Experimental verification of some solutions is also discussed, along with factors such as material and geometric hardening effects.
ELASTO-PLASTIC ANALYSIS OF A HEAVY DUTY PRESS USING F.E.M AND NEUBER’S APPR...IAEME Publication
Heavy duty presses are subjected to extreme load conditions especially during operations like bending, shearing, drawing etc. It generates very high stresses in the punch and die of the press tool. As a sequel to this, failure of the press tool occurs, sometimes prematurely. Hence estimation of the stresses under severe load conditions is of paramount importance.
A simulation study of stress induced in pressure vessels during plate rolling...IRJET Journal
1. Finite element simulations were performed to model the plate rolling process used in pressure vessel manufacturing and analyze resulting residual stresses.
2. Dynamic explicit simulations were conducted for four rolling speeds and residual stresses increased with higher rolling speeds due to increased shear stresses.
3. While residual stresses were not large compared to yield strengths, they are important to consider in pressure vessel design as they contribute to overall hoop and longitudinal stresses experienced in the vessel, especially for thinner walls or higher pressures.
4. The study highlights the need to assess residual stresses early in pressure vessel design to understand their impact on critical stresses within the finished product.
This document provides information about flexural testing of materials including steel, pine, and Douglas fir. It includes the experimental setup, procedures, formulas used to calculate flexural properties, graphs of load vs deformation, and tables of test data for each material. The key results are the ultimate flexural strengths of 2.2 kips for steel, 1.05 kips for pine, and still to be determined for Douglas fir. Comparisons are made between the flexural properties of the different materials.
Conventional Design Calculation &3D Modeling of Metal Forming Heavy duty Hydr...IJERA Editor
This document summarizes the design and 3D modeling of a heavy-duty hydraulic press with a 300-ton capacity. It first describes performing conventional design calculations to size the main structural components, including beams, hydraulic cylinder, end caps, and glands. Finite element analysis is then conducted on a 3D model of the press to optimize the design and validate the stress levels calculated through conventional methods. The summary concludes that the conventional calculations estimated a stress of 150 kg/cm2 in the cylinder mounting plate, and further finite element analysis will be used to optimize the design.
Finite Element Analysis of Obround Pressure VesselsIJMER
This document summarizes a finite element analysis of obround pressure vessels conducted using ANSYS software. Circular and obround pressure vessel models were created and analyzed under internal pressure loading. For the circular vessel, hoop stress results from finite element analysis matched analytical solutions. Higher stresses and deformations were found in the obround vessel compared to the circular vessel. A parametric study examined the effect of varying the radius of the curved portion, vessel thickness, and internal pressure on hoop stresses. In all cases, stresses decreased as dimensions increased and pressure. The analysis provides valuable information on stress distributions in obround pressure vessels.
Influence of contact friction conditions on thin profile simulationVan Canh Nguyen
The paper presents the development of the Finite Element model for simulation of thin
aluminium profile extrusion of both solid and hollow shapes. The analysis has shown that the material
flow in simulation is very dependent on the friction model. Experimental and theoretical studies show
that friction traction on the interface between the tool and the deformed material can be represented as
a combination of adhesive friction force and the force that is required to deform surface asperities. In
aluminium extrusion we can clearly distinguish two different areas with respect to friction conditions
such as sticking and sliding and transient zones between them. The lengths of these zones are also
dependent on variation of the choke angle and actual thickness of the profile. To get these values the
material flow problem is to be coupled with the simulation of the tools deformation. A series of
experiments with specially designed tools have been done to investigate how the bearing length and
choke angle may influence the extension of different friction zones and by these means vary the
material flow pattern. The friction models have also been tested with industrial profiles of complex
shapes and have shown good correspondence to reality.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
The document summarizes concepts related to fatigue in welded steel structures. It discusses the mechanism of fatigue failure, factors influencing fatigue behavior, effects of fatigue loading on structural members and weld connections, fatigue analysis methods including the S-N approach and fracture mechanics approach, Indian standard practices, techniques to improve fatigue strength, and conclusions.
Deflections in PT elements pt structure for all pt slabs in civil industry.pdfvijayvijay327286
The document discusses factors that influence deflections in prestressed concrete members and methods for predicting deflections. It covers:
- Short term deflections of unracked members which can be estimated using Mohr's theorem.
- How the tendon profile affects deflections, providing formulas for straight, trapezoidal, parabolic, and other tendon types.
- Downward deflections due to self-weight and imposed loads that can be calculated using formulas provided.
- Estimation of long-term deflections accounting for creep and shrinkage effects, discussing various methods like those of Busemann, McHenry, and Neville.
Artificial intelligence and machine learning are discussed. AI is defined as making computers intelligent like humans through understanding, reasoning, planning, communication and perception. Machine learning is a subset of AI that allows machines to learn from experience without being explicitly programmed. The document provides background on AI and ML, including definitions, history, and discussions of intelligence and applications.
This document discusses various modes of gear failure including abrasive wear, corrosive wear, pitting, and scoring. It provides details on the causes and remedies for each type of failure. The key points covered include:
- Mild and severe abrasive wear based on abrasive particle size and concentration. Remedies include oil filters and increasing surface hardness.
- Corrosive wear caused by lubricant chemistry or additives. Remedies include enclosure, additive selection, and oil changes.
- Initial and progressive pitting defined and causes explained as exceeding surface fatigue strength. Remedies include precision machining and load/stress reduction.
- Initial, moderate, and destructive scoring defined based on lubrication
Basic types of screw fasteners, Bolts of uniform
strength, I.S.O. Metric screw threads, Bolts under
tension, eccentrically loaded bolted joint in shear,
Eccentric load perpendicular and parallel to axis of
bolt, Eccentric load on circular base, design of Turn
Buckle.
This document discusses machine design and engineering materials. It defines machine design as using scientific principles, technical information and imagination to design machines to perform specific functions efficiently. Some basic requirements of machine elements are listed, such as strength, wear resistance, manufacturability and reliability. Common engineering materials like cast iron, carbon steel and alloy steels are described. The document also discusses standardization, preferred numbers, and manufacturing considerations in machine design such as primary shaping processes, machining processes, surface finishing processes and joining processes.
This document discusses the design of various types of levers, including hand levers, foot levers, and bell crank levers. It begins with an introduction to levers and their uses. The main types of levers are then described: one arm lever, two arm lever, and angular/bell crank lever. Design considerations for hand levers and foot levers are provided, including calculating the diameter of the shaft, dimensions of keys and bosses, and cross-sectional dimensions of the lever arm. Design of bell crank levers is also covered. Examples and problems for calculating lever dimensions are given. The document concludes with a brief mention of designing C-clamps and offset links.
This document discusses machine design and the basic procedures and requirements for designing machine elements. It defines machine design as using scientific principles, technical information, and imagination to describe machines that perform functions with maximum economy and efficiency. The basic requirements for machine elements are then listed, including strength, rigidity, wear resistance, manufacturability, safety, and more. The basic procedure for designing machine elements is then outlined in 6 steps: specification of function, determination of forces, selection of material, failure criterion, determination of dimensions, and preparation of working drawings. Materials that could be used like cast iron, plain carbon steel, and alloy steels are then described in more detail.
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This lecture will equip students with basic electrical engineering knowledge on various types of electrical and electronics drawings, different types of drawing papers, different ways of producing a good drawing and the importance of electrical engineering drawing to both engineers and the users.
By the end of this lecture, students will be to differentiate between different electrical diagrams like, block diagrams, schematic diagrams, circuit diagrams among others.
Presentation slide on DESIGN AND FABRICATION OF MOBILE CONTROLLED DRAINAGE.pptxEr. Kushal Ghimire
To address increased waste dumping in drains, a low-cost drainage cleaning robot controlled via a mobile app is designed to reduce human intervention and improve automation. Connected via Bluetooth, the robot’s chain circulates, moving a mesh with a lifter to carry solid waste to a bin. This project aims to clear clogs, ensure free water flow, and transform society into a cleaner, healthier environment, reducing disease spread from direct sewage contact. It’s especially effective during heavy rains with high water and garbage flow.
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I am Dr. T.D. Shashikala, an Associate Professor in the Electronics and Communication Engineering Department at University BDT College of Engineering, Davanagere, Karnataka. I have been teaching here since 1997. I prepared this manual for the VTU MTech course in Digital Communication and Networking, focusing on the Advanced Digital Signal Processing Lab (22LDN12). Based on, 1.Digital Signal Processing: Principles, Algorithms, and Applications by John G. Proakis and Dimitris G. Manolakis, Discrete-Time Signal Processing by Alan V. Oppenheim and Ronald W. Schafer, 3.Digital Signal Processing: A Practical Guide for Engineers and Scientists" by Steven W. Smith. 4.Understanding Digital Signal Processing by Richard G. Lyons. 5.Wavelet Transforms and Time-Frequency Signal Analysis" by Lokenath Debnath . 6. MathWorks (MATLAB) - MATLAB Documentation
AFCAT STATIC Genral knowledge important CAPSULE.pdf
Unit 3 design against fluctuation load
1. DESIGN OF MACHINE ELEMENTS - I
UNIT 3
Design for Fluctuating Load
Dr. Somnath G Kolgiri (ME, PhD, Mechanical Engg.)
SBPCOE, Indapur
2. Content
•Stress concentration - causes & remedies, fluctuating
stresses, fatigue failures, S-N curve, endurance limit, notch
sensitivity, endurance strength modifying factors, design for
finite and infinite life,
•Cumulative damage in fatigue failure, Soderberg, Gerber,
Goodman, Modified Goodman diagrams,
•Fatigue design of components under combined stresses:-
Theoretical treatment only.
3. STRESS CONCENTRATION
•In design of machine elements, the following three
fundamental equations are used,
•The above equations are called elementary equations.
These equations are based on a number of assumptions.
•One of the assumptions is that there are no discontinuities
in the cross-section of the component.
•However, in practice, discontinuities and abrupt changes in
cross-section are unavoidable due to certain features of the
component such as oil holes and grooves, keyways and
splines, screw threads and shoulders.
• Therefore, it cannot be assumed that the cross-section of
the machine component is uniform. Under these
circumstances, the ‘elementary’ equations do not give
correct results.
4. STRESS CONCENTRATION
Whenever a machine component changes the shape of its cross-section, the
simple stress distribution no longer holds good. This irregularity in the stress
distribution caused by abrupt changes of form is called stress concentration.
A stress concentration (stress raisers or stress risers) is a location in an object
where stress is concentrated. An object is strongest when force is evenly
distributed over its area, so a reduction in area, e.g., caused by a crack, results
in a localized increase in stress.
A material can fail, via a propagating crack, when a concentrated stress exceeds
the material's theoretical cohesive strength. The real fracture strength of a
material is always lower than the theoretical value because most materials
contain small cracks or contaminants that concentrate stress.
It occurs for all kinds of stresses in the presence of fillets, notches, holes,
keyways, splines, surface roughness or scratches etc.
5. •A plate with a small circular hole, subjected to tensile stress is
shown in Fig. The distribution of stresses near the hole can be
observed by using the Photo-elasticity technique.
•In this method, an identical model of the plate is made of epoxy
resin. The model is placed in a circular polariscope and loaded at the
edges.
•It is observed that there is a sudden rise in the magnitude of stresses
in thevicinity of the hole.
Definition: Stress concentration is
defined as the localization of high
stresses due to the irregularities present
in the component and abrupt changes of
the cross-section. Stress concentration
factor is used. It is denoted by Kt a
6. THEORETICAL OR FORM STRESS
CONCENTRATION FACTOR
The theoretical or form stress concentration factor is defined as the
ratio of the maximum stress in a member (at a notch or a fillet) to the
nominal stress at the same section based upon net area.
Mathematically, theoretical or form stress concentration factor,
The value of Kt depends upon the material and geometry of the part.
7. Mechanical& Aerospace Engr., SJSU
CONCEPT OF STRESS CONCENTRATION
Theoretical stress
concentration factor, Kt
Maximum stress at the discontinuity
Nominal stress, max stress
with no discontinuity
Kt is used for normal
stresses and Kts for
shear stresses.
8. THE CAUSES OF STRESS CONCENTRATION
ARE AS FOLLOWS:
1. Variation in Properties of Materials In design of machine components, it is assumed
that the material is homogeneous throughout the component. In practice, there is
variation in material properties from one end to another due to the following
factors:
(a) internal cracks and flaws like blow holes;
(b) cavities in welds;
(c) air holes in steel components; and
(d) nonmetallic or foreign inclusions.
These variations act as discontinuities in the component and cause stress concentration.
2. Load Application Machine components are subjected to forces. These forces act
either at a point or over a small area on the component. Since the area is small,
the pressure at these points is excessive. This results in stress concentration. The
examples of these load applications are as follows:
(a) Contact between the meshing teeth of the driving and the driven gear
(b) Contact between the cam and the follower
(c) Contact between the balls and the races of ball bearing
(d) Contact between the rail and the wheel
(e) Contact between the crane hook and the chain
9. 3. Abrupt Changes in Section In order to mount gears, sprockets, pulleys and ball
bearings on a transmission shaft, steps are cut on the shaft and shoulders are
provided from assembly considerations. Although these features are essential,
they create change of the cross-section of the shaft. This results in stress
concentration at these cross-sections.
4. Discontinuities in the Component Certain features of machine components such
as oil holes or oil grooves, keyways and splines, and screw threads result in
discontinuities in the cross-section of the component. There is stress
concentration in the vicinity of these discontinuities.
5. Machining Scratches Machining scratches, stamp marks or inspection marks are
surface irregularities, which cause stress concentration
10. METHODS TO REDUCE STRESS
CONCENTRATION
• The presence of stress concentration can not be totally eliminated but it
may be reduced to some extent.
• A device or concept that is useful in assisting a design engineer to visualize
the presence of stress concentration and how it may be mitigated is that of
stress flow lines.
• The mitigation of stress concentration means that the stress flow lines shall
maintain their spacing as far as possible.
• Some of the changes adopted in the design in order to reduce the stress
concentration are as follows:
1. Avoid abrupt changes in cross section
2. Place additional smaller discontinuities adjacent to discontinuity
3. Improve surface finish
11. In Fig. (a), we see that stress lines tend to bunch up and cut very close to
the sharp re-entrant corner. In order to improve the situation, fillets may
be provided, as shown in Fig. (b) and (c) to give more equally spaced flow
lines.
It may be noted that it is not practicable to use large radius fillets as in case
of ball and roller bearing mountings. In such cases, notches may be cut as
shown in Fig. (d).
12. • Following figures show the several ways of reducing the stress concentration in
shafts and other cylindrical members with shoulders, holes and threads :
• The stress concentration effects of a press fit may be reduced by making more
gradual transition from the rigid to the more flexible shaft.
13. The stress concentration factors are determined by two methods, viz., the
mathematical method based on the theory of elasticity and experimental
methods like photo-elasticity. For simple geometric shapes, the stress
concentration factors are determined by photo-elasticity. The charts for
stress concentration factors for different geometric shapes and conditions
of loading were originally developed by RE Peterson. At present, FEA
packages are used to find out the stress concentration factor for any
geometric shape.
The chart for the stress concentration factor for a rectangular plate with a
transverse hole loaded in tension or compression is shown in Fig. 5.2.
The nominal stress so in this case is given by, where t is
the plate
thickness.
The values of stress concentration factor for a flat plate with a shoulder
fillet subjected to tensile or compressive force are determined from Fig.
5.3. The nominal stress so for this case is given by,
14. Flat plate with a hole
Flat Plate with Shoulder Fillet in Tension or
Compression
15. The charts for stress concentration factor for a round shaft with shoulder
fillet subjected to tensile force, bending moment, and torsional moment
are shown in Fig. 5.4, 5.5 and 5.6 respectively. The nominal stresses in
these three cases are as follows:
(i) Tensile Force
(ii) Bending Moment
(iii) Torsional Moment
18. Q1.A flat plate subjected to a tensile force of 5 KN is shown in Fig. The
plate material is grey cast iron FG 200 and the factor of safety is 2.5.
Determine the thickness of the plate.
Solution
Given P = 5 kN Sut = 200 N/mm2 (fs) = 2.5
19. Q1.A rectangular plate, 15 mm thick, made of a brittle material is shown
in Fig. Calculate the stresses at each of three holes of 3, 5 and 10 mm
diameter. [161.82, 167.33 and 200 N/mm2]
20. Q2. A plate, 10 mm thick, subjected to a tensile load of 20 kN is shown
in Fig. The plate is made of cast iron (Sut = 350 N/mm2) and the factor
of safety is 2.5. Determine the fillet radius. [2.85 or 3 mm]
21. Q2. A non-rotating shaft supporting a load of 2.5 kN is shown in Fig.
The shaft is made of brittle material, with an ultimate tensile strength of
300 N/mm2. The factor of safety is 3. Determine the dimensions of the
shaft.
Solution
Given P = 2.5 kN Sut = 300 N/mm2 (fs) = 3
23. Q1. A round shaft made of a brittle material and subjected to a bending
moment of 15 N-m is shown in Fig. The stress concentration factor at the
fillet is 1.5 and the ultimate tensile strength of the shaft material is 200
N/mm2. Determine the diameter d, the magnitude of stress at the fillet
and the factor of safety. [11.76 mm, 140.91 N/mm2, and 1.42]
Q2. A shaft carrying a load of 5 kN midway between two bearings is
shown in Fig. Determine the maximum bending stress at the fillet
section. Assume the shaft material to be brittle. [20.39 N/mm2]
24. FLUCTUATING STRESSES
.
•In the previous chapters, the external forces acting on a machine
component were assumed to be static.
•In many applications, the components are subjected to forces, which are
not static, but vary in magnitude with respect to time.
•The stresses induced due to such forces are called fluctuating stresses.
•It is observed that about 80% of failures of mechanical components are
due to ‘fatigue failure’ resulting from fluctuating stresses.
•There are three types of mathematical models for cyclic stresses—
fluctuating or alternating stresses, repeated stresses and reversed stresses
•Stress–time relationships for these models are illustrated in Fig
25. •The fluctuating or alternating stress varies in a sinusoidal manner with
respect to time. It has some mean value as well as amplitude value. It
fluctuates between two limits—maximum and minimum stress. The
stress can be tensile or compressive or partly tensile and partly
compressive.
•The repeated stress varies in a sinusoidal manner with respect to time,
but the variation is from zero to some maximum value. The minimum
stress is zero in this case and therefore, amplitude stress and mean
stress are equal
•The reversed stress varies in a sinusoidal manner with respect to time,
but it has zero mean stress. In this case, half portion of the cycle consists
of tensile stress and the remaining half of compressive stress. There is a
complete reversal from tension to compression between these two halves
and therefore, the mean stress is zero.
are maximum and minimum stresses, while are
called mean stress and stress amplitude respectively. It can be proved that
26. FATIGUE FAILURE
It has been observed that materials fail under fluctuating stresses at a
stress magnitude which is lower than the ultimate tensile strength of the
material. Sometimes, the magnitude is even lower than the yield strength.
Further, it has been found that the magnitude of the stress causing fatigue
failure decreases as the number of stress cycles increase. This
phenomenon of decreased resistance of the materials to fluctuating
stresses is the main characteristic of fatigue failure.
•Fatigue failure is defined as time delayed fracture under cyclic loading.
Examples of parts in which fatigue failures are common are transmission
shafts, connecting rods, gears, vehicle suspension springs and ball
bearings.
•The fatigue failure, however, depends upon a number of factors, such as
the number of cycles, mean stress, stress amplitude, stress concentration,
residual stresses, corrosion and creep.
27. ENDURANCE LIMIT
•The fatigue or endurance limit of a material is defined as the
maximum amplitude of completely reversed stress that the standard
specimen can sustain for an unlimited number of cycles without fatigue
failure. Since the fatigue test cannot be conducted for unlimited or
infinite number of cycles, cycles is considered as a sufficient number
of cycles to define the endurance limit.
•There is another term called fatigue life, which is frequently used with
endurance limit. The fatigue life is defined as the number of stress cycles
that the standard specimen can complete during the test before the
appearance of the first fatigue crack.
28. ENDURANCE LIMIT AND FATIGUE FAILURE
It has been found experimentally that when a material is
subjected to repeated stresses, it fails at stresses below the
yield point stresses. Such type of failure of a material is
known as fatigue.
The failure is caused by means of a progressive crack
formation which are usually fine and of microscopic size. The
failure may occur even without any prior indication.
The fatigue of material is effected by the size of the
component, relative magnitude of static and fluctuating loads
and the number of load reversals.
29. FACTORS TO BE CONSIDERED WHILE
DESIGNING MACHINE PARTS TO AVOID
FATIGUE FAILURE
• The following factors should be considered while designing
machine parts to avoid fatigue failure:
• The variation in the size of the component should be as gradual
as possible.
• The holes, notches and other stress raisers should be avoided.
• The proper stress de-concentrators such as fillets and notches
should be provided wherever necessary.
• The parts should be protected from corrosive atmosphere.
• A smooth finish of outer surface of the component increases the
fatigue life.
• The material with high fatigue strength should be selected.
• The residual compressive stresses over the parts surface
increases its fatigue strength.
30. A standard mirror polished specimen, as shown in figure is rotated in a fatigue
testing machine while the specimen is loaded in bending.
As the specimen rotates, the bending stress at the upper fibers varies from
maximum compressive to maximum tensile while the bending stress at the
lower fibers varies from maximum tensile to maximum compressive.
In other words, the specimen is subjected to a completely reversed stress cycle.
This is represented by a time-stress diagram as shown in Fig. (a).
31. Endurance or Fatigue limit (σe) is defined as maximum value of the
completely reversed bending stress which a polished standard specimen can
withstand without failure, for infinite number of cycles.
It may be noted that the term endurance limit is used for reversed bending
only while for other types of loading, the term endurance strength may be
used when referring the fatigue strength of the material.
It may be defined as the safe maximum stress which can be applied to the
machine part working under actual conditions.
We have seen that when a machine member is subjected to a completely
reversed stress, the maximum stress in tension is equal to the maximum
stress in compression as shown in Fig.(a). In actual practice, many machine
members undergo different range of stress than the completely reversed
stress.
The stress verses time diagram for fluctuating stress having values σmin and
σmax is shown in Fig. (c). The variable stress, in general, may be considered
as a combination of steady (or mean or average) stress and a completely
reversed stress component σv.
32. The following relations are derived from Fig. (c):
a =
max min
2
Alternating stress
Mean stress
m =
max min
2
+
33. FACTORS AFFECTING ENDURANCE LIMIT
1) SIZE EFFECT:
• The strength of large members is lower than that of small specimens.
• This may be due to two reasons.
• The larger member will have a larger distribution of weak points than the
smaller one and on an average, fails at a lower stress.
• Larger members have larger surface Ares. This is important because the
imperfections that cause fatigue failure are usually at the surface.
Effect of size:
• Increasing the size (especially section thickness) results in larger surface
area and creation of stresses.
• This factor leads to increase in the probability of crack initiation.
• This factor must be kept in mind while designing large sized components.
34. 2) SURFACE ROUGHNESS:
• Almost all fatigue cracks nucleate at the surface of the members.
• The conditions of the surface roughness and surface oxidation or corrosion
are very important.
• Experiments have shown that different surface finishes of the same material
will show different fatigue strength.
• Methods which Improve the surface finish and those which introduce
compressive stresses on the surface will improve the fatigue strength.
• Smoothly polished specimens have higher fatigue strength.
• Surface treatments. Fatigue cracks initiate at free surface, treatments can be
significant
• Plating, thermal or mechanical means to induce residual stress.
3) EFFECT OF TEMPERATURE:
• When the mechanical component operates above the room temperature, its
ultimate tensile strength, and hence endurance limit decrease with increase
in temperature.
35. 4) Effect of metallurgical variables;
• Fatigue strength generally increases with increase in UTS
• Fatigue strength of quenched & tempered steels (tempered martensitic
structure) have better fatigue strength
• Finer grain size show better fatigue strength than coarser grain size.
• Non-metallic inclusions either at surface or sub-surface reduces' the
fatigue strength.
36. S-N DIAGRAM
Fatigue strength of material is determined by R.R. Moore rotating beam
machine. The surface is polished in the axial direction. A constant bending
load is applied.
37. The S–N curve is the graphical representation of stress amplitude (Sf )
versus the number of stress cycles (N) before the fatigue failure on a log-
log graph paper. The S–N curve for steels is illustrated in Fig. The S–N
diagram is also called Wöhler diagram, after August Wöhler, a German
engineer who published his fatigue research in 1870. The S–N diagram is
a standard method of presenting fatigue data.
38. A record is kept of the number of cycles required to produce failure at a given
stress, and the results are plotted in stress-cycle curve as shown in figure.
A little consideration will show that if the stress is kept below a certain value the
material will not fail whatever may be the number of cycles.
This stress, as represented by dotted line, is known as endurance or fatigue
limit (σe).
It is defined as maximum value of the completely reversed bending stress which
a polished standard specimen can withstand without failure, for infinite number
of cycles (usually 107 cycles).
39. FATIGUE STRESS CONCENTRATION
FACTOR
• When a machine member is subjected to cyclic or fatigue
loading, the value of fatigue stress concentration factor
shall be applied instead of theoretical stress
concentration factor.
• Mathematically, fatigue stress concentration factor,
40. NOTCH SENSITIVITY
Notch sensitivity is defined as the susceptibility of a material to
succumb to the damaging effects of stress raising notches in
fatigue loading.
Notch Sensitivity: It may be defined as the degree to which the
theoretical effect of stress concentration is actually reached.
Notch Sensitivity Factor “q”: Notch sensitivity factor is defined
as the ratio of increase in the actual stress to the increase in the
nominal stress near the discontinuity in the specimen.
Where, Kf and Kt are the fatigue stress concentration factor and
theoretical stress concentration factor.
The stress gradient depends mainly on the radius of the notch,
hole or fillet and on the grain size of the material.
44. There is an approximate relationship between the endurance limit and the
ultimate tensile strength (Sut) of the material. These relationships are
based on 50% reliability.
47. REVERSED STRESSES—DESIGN FOR
FINITE AND INFINITE LIFE
•There are two types of problems in fatigue design—(i) components
subjected to completely reversed stresses, and (ii) components subjected
to fluctuating stresses, the mean stress is zero in case of completely
reversed stresses.
•The design problems for completely reversed stresses are further
divided into two groups—(i) design for infinite life, and (ii) design for
finite life.
Case I: When the component is to be designed for infinite life, the
endurance limit becomes the criterion of failure. The amplitude stress
induced in such components should be lower than the endurance limit in
order to withstand the infinite number of cycles. Such components are
designed with the help of the following equations:
48. Case II: When the component is to be designed for finite life, the S–N
curve as shown in Fig. 5.27 can be used. The curve is valid for steels. It
consists of astraight line AB drawn from cycles to
cycles on a log-log paper. The design procedure for such
problems is as follows:
The fatigue strength corresponding to N cycles. The value of the fatigue
strength (Sf) obtained by the above procedure is used for the design
calculations.
50. INFINITE-LIFE PROBLEMS (REVERSED LOAD)
Example 1. A plate made of steel 20C8 (Sut = 440 N/mm2) in hot rolled
and normalised condition is shown in Fig. It is subjected to a completely
reversed axial load of 30 kN. The notch sensitivity factor q can be taken
as 0.8 and the expected reliability is 90%. The size factor is 0.85. The
factor of safety is 2. Determine the plate thickness for infinite life.
52. Q2. A rod of a linkage mechanism made of steel 40Cr1 (Sut = 550
N/mm2) is subjected to a completely reversed axial load of 100 kN. The
rod is machined on a lathe and the expected reliability is 95%. There is
no stress concentration. Determine the diameter of the rod using a factor
of safety of 2 for an infinite life condition.
53. Q3.A component machined from a plate made of steel 45C8 (Sut = 630
N/mm2) is shown in Fig. It is subjected to a completely reversed axial
force of 50 kN. The expected reliability is 90% and the factor of safety is
2. The size factor is 0.85. Determine the plate thickness t for infinite life,
if the notch sensitivity factor is 0.8.
54. Q4. A 25 mm diameter shaft is made of forged steel 30C8 (Sut = 600
N/mm2). There is a step in the shaft and the theoretical stress
concentration factor at the step is 2.1. The notch sensitivity factor is 0.84.
Determine the endurance limit of the shaft if it is subjected to a reversed
bending moment. [59.67 N/mm2]
Q5. A 40 mm diameter shaft is made of steel 50C4 (Sut = 660 N/mm2)
and has a machined surface. The expected reliability is 99%. The
theoretical stress concentration factor for the shape of the shaft is 1.6 and
the notch sensitivity factor is 0.9. Determine the endurance limit of the
shaft. [112.62 N/mm2]
55. FINITE-LIFE PROBLEMS (REVERSED LOAD)
Q1. A rotating bar made of steel 45C8 (Sut = 630 N/mm2) is subjected to
a completely reversed bending stress. The corrected endurance limit of
the bar is 315 N/mm2. Calculate the fatigue strength of the bar for a life
of 90,000 cycles.
56. Q2. A forged steel bar, 50 mm in diameter, is subjected to a reversed
bending stress of 250 N/mm2. The bar is made of steel 40C8 (Sut = 600
N/mm2). Calculate the life of the bar for a reliability of 90%.
Solution:-
Given:- Sf = Sb = 250 N/mm2 , Sut = 600 N/mm2, R = 90%
57. Q3. A rotating shaft, subjected to a non rotating force of 5 kN and simply
supported between two bearings A and E is shown in Fig. 5.32(a). The
shaft is machined from plain carbon steel 30C8 (Sut = 500 N/mm2) and
the expected reliability is 90%. The equivalent notch radius at the fillet
section can be taken as 3 mm. What is the life of the shaft?
Solution :- Given P = 5 kN Sut = 500
N/mm2, R = 90%, r = 3 mm
Step I Selection of failure-section
Taking the moment of the forces about
bearings A and E, the reactions at A
and E are 2143 and 2857 N
respectively. The bending moment
diagram is shown in Fig. 5.32(b). The
values of the bending moment shown
in the figure are in N-m. The
possibility of a failure will be at the
three sections B, C and D. The failure
will probably occur at the section B rather than at C or D. At the section
58. there is no stress concentration. At the section D, the diameter is more
and the bending moment is less compared with that of section B.
Therefore, it is concluded that failure will occur at the section B.
59. Q4. The section of a steel shaft is shown in Fig. 5.34. The shaft is
machined by a turning process. The section at XX is subjected to a
constant bending moment of 500 kN-m. The shaft material has ultimate
tensile strength of 500 MN/m2, yield point of 350 MN/m2 and
endurance limit in bending for a 7.5 mm diameter specimen of 210
MN/m2. The notch sensitivity factor can be taken as 0.8. The theoretical
stress concentration factor may be interpolated from following tabulated
values: where rf is the fillet radius and d is the
shaft diameter. The reliability is 90%.
Determine the life of the shaft.
61. Q5. A cantilever beam made of cold drawn steel 20C8 (Sut = 540 /mm2)
is subjected to a completely reversed load of 1000 N as shown in Fig.
The notch sensitivity factor q at the fillet can be taken as 0.85 and the
expected reliability is 90%. Determine the diameter d of the beam for a
life of 10000 cycles.
Step I Selection of failure section The failure will occur either at the
section A or at the section B. At section A, although the bending moment
is maximum, there is no stress concentration and the diameter is also
more compared with that of the section B. It is, therefore, assumed that
the failure will occur at the section B.
63. CUMULATIVE DAMAGE IN FATIGUE
In certain applications, the mechanical component is subjected to
different stress levels for different parts of the work cycle. The life of
such a component is determined by Miner’s equation. Suppose that a
component is subjected to completely reversed stresses
cycles, and so on. Let N1 be the number of stress cycles before fatigue
failure, if only the alternating stress is acting. One stress cycle will
consume of the fatigue life and since there are n1 such cycles at this
stress level, the proportionate damage of fatigue life will be
Similarly, the proportionate damage at stress level will be
Adding these quantities, we get
64. Q1. The work cycle of a mechanical component subjected to completely
reversed bending stresses consists of the following three elements:
(i) ± 350 N/mm2 for 85% of time (ii) ± 400 N/mm2 for 12% of time
(iii) ± 500 N/mm2 for 3% of time The material for the component is
50C4 (Sut = 660 N/mm2) and the corrected endurance limit of the
component is 280 N/mm2. Determine the life of the component.
Solution :- Given Sut = 660 N/mm2 Se = 280 N/mm2
Step II Calculation of N1, N2 and N3 From above Fig.
65. Q2.A solid circular shaft made of steel Fe 620 (Sut = 620 N/mm2 and Syt
= 380 N/mm2) is subjected to an alternating torsional moment, which
varies from –200 N-m to + 400 N-m. The shaft is ground and the
expected reliability is 90%. Neglecting stress concentration, calculate the
shaft diameter for infinite life. The factor of safety is 2. Use the
distortion energy theory of failure. [29.31 mm]
66. • A straight line connecting the endurance limit (σe) and the
ultimate strength (σu), as shown by line AB in figure given below
follows the suggestion of Goodman.
• A Goodman line is used when the design is based on ultimate
strength and may be used for ductile or brittle materials.
GOODMAN METHOD FOR COMBINATION
OF STRESSES:
68. • A straight line connecting the endurance limit (σe) and the
yield strength (σy), as shown by the line AB in following
figure, follows the suggestion of Soderberg line.
• This line is used when the design is based on yield
strength. the line AB connecting σe and σy, as shown in
following figure, is called Soderberg's failure stress line.
SODERBERG METHOD FOR COMBINATION
OF STRESSES
69. If a suitable factor of safety (F.S.) is applied to the endurance limit
and yield strength, a safe stress line CD may be drawn parallel to
the line AB.
70. • In the design of components subjected to fluctuating
stresses, the Goodman diagram is slightly modified to
account for the yielding failure of the components,
especially, at higher values of the mean stresses.
• The diagram known as modified Goodman diagram and is
most widely used in the design of the components
subjected to fluctuating stresses.
MODIFIED GOODMAN DIAGRAM:
71. MODIFIED GOODMAN DIAGRAM FOR
FLUCTUATING AXIAL AND BENDING STRESSES
+m
a
Sut
Safe zone
- m
C
Sy
Safe zone
Se
- Syc
Finite life
Sn
1=
Sut
a m
+
Fatigue, m > 0Fatigue, m ≤ 0
a =
Se
nf
a + m =
Sy
ny
Yield
a + m =
Sy
ny
Yield
nfSe
1
=
Sut
a m
+ Infinite life
72. COMBINED LOADING
All four components of stress exist,
xa alternating component of normal stress
xm mean component of normal stress
xya alternating component of shear stress
xym mean component of shear stress
Calculate the alternating and mean principal stresses,
1a, 2a = (xa /2) ± (xa /2)2
+ (xya)2
1m, 2m = (xm /2) ± (xm /2)2
+ (xym)2
73. COMBINED LOADING
Calculate the alternating and mean von Mises stresses,
a′ = (1a + 2a - 1a2a)1/22 2
m′ = (1m + 2m - 1m2m)1/22 2
Fatigue design equation
nfSe
1
=
Sut
′a ′m
+ Infinite life
74. MODIFIED GOODMAN DIAGRAM:
• In the design of components subjected to fluctuating stresses,
the Goodman diagram is slightly modified to account for the
yielding failure of the components, especially, at higher
values of the mean stresses.
• The diagram known as modified Goodman diagram and is
most widely used in the design of the components subjected
to fluctuating stresses.