Stress Analysis is an Engineering

7/31/2019 Stress Analysis is an Engineering

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Stress analysis is an engineering(e.g., civil engineeringand mechanical engineering) discipline

that determines thestress in materials and structures subjected to static ordynamicforces or loads.

A stress analysis is required for the study and design of structures, e.g., tunnels, dams, mechanicalparts, and structural frames among others, under prescribed or expected loads. Stress analysis may

be applied as adesignstep to structures that do not yet exist.

The aim of the analysis is usually to determine whether the element or collection of elements,

usually referred to as a structure, can safely withstand the specified forces. This is achieved whenthe determined stress from the applied force(s) is less than the ultimate tensile strength, ultimate

compressive strength orfatigue strength the material is known to be able to withstand, though

ordinarily afactor of safety is applied in design.

Analysis may be performed through mathematical modelling or simulation, through experimental

testing procedures, or a combination of techniques.

Engineering quantities are usually measured inmegapascals (MPa) orgigapascals (GPa). In

imperial units, stress is expressed in pounds-force per square inch (psi) or kilopounds-force persquare inch (ksi).

Contents

[hide]

1 Analysis methods

o 1.1 Modelling

o 1.2 Experimental testing

2 Factor of safety

3 Load transfer 4 Uniaxial stress

5 Plane stress

6 Plane strain

7 Stress transformation in plane stress and plane strain

8 Graphical representation of stress at a point

o 8.1 Mohr's circle

o 8.2 Lame's stress ellipsoid

o 8.3 Cauchy's stress quadric

9 Graphical representation of the stress field

10 See also

11 References

[edit] Analysis methods
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The analysis of stress within a body implies the determination at each point of the body of the

magnitudes of the nine stress components. In other words, it is the determination of the internal

distribution of stresses.

A key part of analysis involves determining the type of loads acting on a structure, including

tension, compression, shear, torsion,bending, or combinations of such loads. When forces areapplied, or expected to be applied, repeatedly, nearly all materials will rupture or fail at a lower

stress than they would otherwise. The analysis to determine stresses under these cyclic loadingconditions is termedfatigue analysis and is most often applied to aerodynamic structural systems.

[edit] Modelling

To determine the distribution of stress in a structure it is necessary to solve aboundary-value

problemby specifying the boundary conditions, i.e. displacements and/or forces on the boundary.

Constitutive equations, such as e.g.Hooke's Law forlinear elastic materials, are used to describethe stress:strain relationship in these calculations. A boundary-value problem based on thetheory

of elasticityis applied to structures expected to deform elastically, i.e. infinitesimal strains, underdesign loads. When the loads applied to the structure induce plastic deformations, the theory ofplasticity is implemented.

Approximate solutions for boundary-value problems can be obtained through the use of numerical

methods such as thefinite element method, the finite difference method, and theboundary element

method, which are implemented in computer programs. Analytical or close-form solutions can beobtained for simple geometries, constitutive relations, and boundary conditions.

All real objects occupy a three-dimensional space. The stress analysis can be simplified in cases

where the physical dimensions and the loading conditions allows the structure to be assumed as

one-dimensional or two-dimensional. For a two-dimensional analysis aplane stress or aplanestrain condition can be assumed.

[edit] Experimental testing

Stress analysis can be performed experimentally by applying forces to a test element or structureand then determining the resulting stress using sensors. In this case the process would more

properly be known as testing(destructiveornon-destructive). Experimental methods may be used

in cases where mathematical approaches are cumbersome or inaccurate. Special equipmentappropriate to the experimental method is used to apply the static or dynamic loading.

There are a number of experimental methods which may be used:

Tensile testing is a fundamentalmaterials science test in which a sample is subjected to uniaxial

tension until failure. The results from the test are commonly used to select a material for anapplication, forquality control, and to predict how a material will react under other types of forces.

Properties that are directly measured via a tensile test are ultimate tensile strength, maximum

elongation and reduction in area. From these measurements properties such asYoung's modulus,

Poisson's ratio, yield strength, and strain-hardening characteristics can be determined.
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Stress in plastic protractor exhibited due to birefringence.

Thephotoelastic methodrelies on the physical phenomenon ofbirefringence. Unlike the analyticalmethods of stress determination, photoelasticity gives a fairly accurate picture of stress distribution

even around abrupt discontinuities in a material. The method serves as an important tool for

determining the critical stress points in a material and is often used for determining stress

concentration factors in irregular geometries. Birefringence is exhibited by certain transparent

materials. A ray of light passing through a birefringent material experiences two refractive indices.This double refraction is exhibited by many optical crystals. But photoelastic materials exhibit the

property of birefringence only on the application of stress, and the magnitude of the refractiveindices at each point in the material is directly related to the state of stress at that point. A model

component is created made of photoelastic material with similar geometry to that of the structure

on which stress analysis is to be performed. This ensures that the state of the stress in the model issimilar to the state of the stress in the structure.

Dynamic mechanical analysis is a technique used to study and characterizeviscoelastic materials,

particularly polymers. Polymers composed of long molecular chains have unique viscoelastic

properties, which combine the characteristics ofelastic solids andNewtonian fluids. The

viscoelastic property of polymer is studied by dynamic mechanical analysis where a sinusoidalforce (stress) is applied to a material and the resulting displacement (strain) is measured. For a

perfectly elastic solid, the resulting strain and the stress will be perfectly in phase. For a purelyviscous fluid, there will be a 90 degree phase lag of strain with respect to stress. Viscoelastic

polymers have the characteristics in between where some phase lag will occur during DMA tests.

Analyzers are made for both stress and strain control. In strain control, a probe is displaced and the

resulting stress of the sample is measured. In stress control, a set force is applied and several otherexperimental conditions (temperature, frequency, or time) can be varied.

[edit] Factor of safety

Main article: Factor of safety

The factor of safety is a design requirement for the structure based on the uncertainty in loads,material strength, and consequences of failure. In design of structures, calculated stresses are

restricted to be less than an specified allowable stress, also known as working or designed stress,

that is chosen as some fraction of the yield strength or of the ultimate strength of the material
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which the structure is made of. The ratio of the ultimate stress to the allowable stress is defined as

the factor of safety.

Laboratory test are usually performed on material samples in order to determine the yield strengthand the ultimate strength that the material can withstand before failure. Often a separate factor of

safety is applied to the yield strength and to the ultimate strength. The factor of safety on yieldstrength is to prevent detrimental deformations and the factor of safety on ultimate strength is to

prevent collapse.

The factor of safety is used to calculate a maximum allowable stress:

"Margin of safety" or "design factor" are other ways to express the factor of safety value. .

[edit] Load transfer

The evaluation of loads and stresses within structures is directed to finding the load transfer path.

Loads will be transferred by physical contact between the various component parts and withinstructures. The load transfer may be identified visually, or by simple logic for simple structures.

For more complex structures, more complex methods such as theoreticalsolid mechanics or by

numerical methods may be required. Numerical methods include direct stiffness method which is

also referred to as the finite element method.

The object is to determine the critical stresses in each part, and compare them to the strength of the

material (see strength of materials).

For parts that have broken in service, a forensic engineeringorfailure analysis is performed to

identify weakness, where broken parts are analysed for the cause or causes of failure. The methodseeks to identify the weakest component in the load path. If this is the part which actually failed,

then it may corroborate independent evidence of the failure. If not, then another explanation has to

be sought, such as a defective part with a lowertensile strengththan it should for example.

[edit] Uniaxial stress

If two of the dimensions of the object are very large or very small compared to the others, the

object may be modelled as one-dimensional. In this case the stress tensor has only one componentand is indistinguishable from a scalar. One-dimensional objects include a piece of wire loaded atthe ends and a metal sheet loaded on the face and viewed up close and through the cross section.

When a structural element is subjected to tension or compression its length will tend to elongate or

shorten, and its cross-sectional area changes by an amount that depends on the Poisson's ratio of

the material. In engineering applications, structural members experience small deformations andthe reduction in cross-sectional area is very small and can be neglected, i.e., the cross-sectional
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area is assumed constant during deformation. For this case, the stress is called engineering stress

ornominal stress. In some other cases, e.g., elastomers andplastic materials, the change in cross-

sectional area is significant, and the stress must be calculated assuming the current cross-sectionalarea instead of the initial cross-sectional area. This is termed true stress and is expressed as

,

where

is the nominal (engineering) strain, andis nominal (engineering) stress.

The relationship between true strain and engineering strain is given by

.

In uniaxial tension, true stress is then greater than nominal stress. The converse holds in

compression.

[edit] Plane stress

Figure 7.1 Plane stress state in a continuum.
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A state ofplane stress exists when one of the three principal , stresses is zero. This

usually occurs in structural elements where one dimension is very small compared to the other two,i.e. the element is flat or thin. In this case, the stresses are negligible with respect to the smaller

dimension as they are not able to develop within the material and are small compared to the in-

plane stresses. Therefore, the face of the element is not acted by loads and the structural element

can be analyzed as two-dimensional, e.g. thin-walled structures such as plates subject to in-planeloading or thin cylinders subject to pressure loading. The other three non-zero components remain

constant over the thickness of the plate. The stress tensor can then be approximated by:

.

The corresponding strain tensor is:

in which the non-zero term arises from thePoisson's effect. This strain term can be temporarily

removed from the stress analysis to leave only the in-plane terms, effectively reducing the analysis

to two dimensions.

[edit] Plane strain

Figure 7.2 Plane strain state in a continuum.

Main article: Infinitesimal strain theory
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If one dimension is very large compared to the others, theprincipal strainin the direction of the

longest dimension is constrained and can be assumed as zero, yielding a plane strain condition

(Figure 7.2). In this case, though all principal stresses are non-zero, the principal stress in thedirection of the longest dimension can be disregarded for calculations. Thus, allowing a two

dimensional analysis of stresses, e.g. a damanalyzed at a cross section loaded by the reservoir.

[edit] Stress transformation in plane stress and plane strain

Consider a point in a continuum under a state of plane stress, or plane strain, with stress

components and all other stress components equal to zero (Figure 7.1, Figure 8.1).

From static equilibrium of an infinitesimal material element at (Figure 8.2), the normal stress

and the shear stress on any plane perpendicular to the - plane passing through with a unitvector making an angle of with the horizontal, i.e. is the direction cosine in the

direction, is given by:

These equations indicate that in a plane stress or plane strain condition, one can determine the

stress components at a point on all directions, i.e. as a function of , if one knows the stress

components on any two perpendicular directions at that point. It is important toremember that we are considering a unit area of the infinitesimal element in the direction parallel

to the - plane.

Figure 8.1 - Stress transformation at a point in a continuum under plane stress conditions.
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Figure 8.2 - Stress components at a plane passing through a point in a continuum under plane stress

conditions.

The principal directions (Figure 8.3), i.e. orientation of the planes where the shear stress

components are zero, can be obtained by making the previous equation for the shear stress equalto zero. Thus we have:

and we obtain

This equation defines two values which are apart (Figure 8.3). The same result can be

obtained by finding the angle which makes the normal stress a maximum, i.e.

The principal stresses and , or minimum and maximum normal stresses and ,

respectively, can then be obtained by replacing both values of into the previous equation for .

This can be achieved by rearranging the equations for and , first transposing the first term in

the first equation and squaring both sides of each of the equations then adding them. Thus we have

where
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which is the equation of a circle of radius centered at a point with coordinates , called

Mohr's circle. But knowing that for the principal stresses the shear stress , then we obtain

from this equation:

Figure 8.3 - Transformation of stresses in two dimensions, showing the planes of action ofprincipal stresses, and maximum and minimum shear stresses.

When the infinitesimal element is oriented in the direction of the principal planes, thus

the stresses acting on the rectangular element are principal stresses: and . Then

the normal stress and shear stress as a function of the principal stresses can be determined by

making . Thus we have

Then the maximum shear stress occurs when , i.e. (Figure 8.3):

Then the minimum shear stress occurs when , i.e. (Figure 8.3):
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[edit] Graphical representation of stress at a point

Mohr's circle,Lame's stress ellipsoid(together with thestress director surface), and Cauchy's

stress quadric are two-dimensional graphical representations of thestate of stress at a point. Theyallow for the graphical determination of the magnitude of the stress tensor at a given point for all

planes passing through that point. Mohr's circle is the most common graphical method.

[edit] Mohr's circle

Main article: Mohr's circle

Mohr's circle, named afterChristian Otto Mohr, is the locus of points that represent the state of

stress on individual planes at all their orientations. The abscissa, , and ordinate, , of each

point on the circleare the normal stress and shear stress components, respectively, acting on a

particular cut plane with a unit vector with components .

[edit] Lame's stress ellipsoid

Main article: Lame's stress ellipsoid

The surface of the ellipsoid represents the locus of the endpoints of all stress vectors acting on all

planes passing through a given point in the continuum body. In other words, the endpoints of all

stress vectors at a given point in the continuum body lie on the stress ellipsoid surface, i.e., the

radius-vector from the center of the ellipsoid, located at the material point in consideration, to apoint on the surface of the ellipsoid is equal to the stress vector on some plane passing through the

point. In two dimensions, the surface is represented by an ellipse (Figure coming).

[edit] Cauchy's stress quadric

The Cauchy's stress quadric, also called thestress surface, is a surface of the second order that

traces the variation of the normal stress vector as the orientation of the planes passing through agiven point is changed.

[edit] Graphical representation of the stress fieldSee also: Stress field

The complete state of stress in a body at a particular deformed configuration, i.e., at a particulartime during the motion of the body, implies knowing the six independent components of the stress

tensor , or the three principal stresses , at each

material point in the body at that time. However, numerical analysis and analytical methods allow
http://en.wikipedia.org/w/index.php?title=Stress_analysis&action=edit&section=10http://en.wikipedia.org/wiki/Stress_(mechanics)#Cauchy.27s_stress_principlehttp://en.wikipedia.org/wiki/Stress_(mechanics)#Cauchy.27s_stress_principlehttp://en.wikipedia.org/w/index.php?title=Stress_analysis&action=edit&section=11http://en.wikipedia.org/wiki/Mohr's_circlehttp://en.wikipedia.org/wiki/Otto_Mohrhttp://en.wikipedia.org/wiki/Otto_Mohrhttp://en.wikipedia.org/wiki/Abscissahttp://en.wikipedia.org/wiki/Abscissahttp://en.wikipedia.org/wiki/Ordinatehttp://en.wikipedia.org/wiki/Circlehttp://en.wikipedia.org/wiki/Circlehttp://en.wikipedia.org/wiki/Unit_vectorhttp://en.wikipedia.org/wiki/Unit_vectorhttp://en.wikipedia.org/w/index.php?title=Stress_analysis&action=edit&section=12http://en.wikipedia.org/wiki/Lame's_stress_ellipsoidhttp://en.wikipedia.org/wiki/Ellipsehttp://en.wikipedia.org/w/index.php?title=Stress_analysis&action=edit&section=13http://en.wikipedia.org/w/index.php?title=Stress_analysis&action=edit&section=14http://en.wikipedia.org/wiki/Stress_fieldhttp://en.wikipedia.org/w/index.php?title=Stress_analysis&action=edit&section=10http://en.wikipedia.org/wiki/Stress_(mechanics)#Cauchy.27s_stress_principlehttp://en.wikipedia.org/w/index.php?title=Stress_analysis&action=edit&section=11http://en.wikipedia.org/wiki/Mohr's_circlehttp://en.wikipedia.org/wiki/Otto_Mohrhttp://en.wikipedia.org/wiki/Abscissahttp://en.wikipedia.org/wiki/Ordinatehttp://en.wikipedia.org/wiki/Circlehttp://en.wikipedia.org/wiki/Unit_vectorhttp://en.wikipedia.org/w/index.php?title=Stress_analysis&action=edit&section=12http://en.wikipedia.org/wiki/Lame's_stress_ellipsoidhttp://en.wikipedia.org/wiki/Ellipsehttp://en.wikipedia.org/w/index.php?title=Stress_analysis&action=edit&section=13http://en.wikipedia.org/w/index.php?title=Stress_analysis&action=edit&section=14http://en.wikipedia.org/wiki/Stress_field


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only for the calculation of the stress tensor at a certain number of discrete material points. To

graphically represent in two dimensions this partial picture of the stress field different sets of

contour lines can be used[1]:

Isobars are curves along which the principal stress, e.g., is constant.

Isochromatics are curves along which themaximum shear stress is constant. This curvesare directly determined using photoelasticity methods.

Isopachs are curves along which the mean normal stress is constant Isostatics orstress trajectories are a system of curves which are at each material point

tangent to the principal axes of stress.

Isoclinics are curves on which the principal axes make a constant angle with a given fixedreference direction. These curves can also be obtained directly by photoelasticity methods.

Slip lines are curves on which the shear stress is a maximum.
http://en.wikipedia.org/wiki/Contour_lineshttp://en.wikipedia.org/wiki/Stress_analysis#cite_note-0http://en.wikipedia.org/wiki/Stress_(mechanics)#Maximum_and_minimum_shear_stresseshttp://en.wikipedia.org/wiki/Stress_(mechanics)#Maximum_and_minimum_shear_stresseshttp://en.wikipedia.org/wiki/Stress_(mechanics)#Stress_deviator_tensorhttp://en.wikipedia.org/wiki/Contour_lineshttp://en.wikipedia.org/wiki/Stress_analysis#cite_note-0http://en.wikipedia.org/wiki/Stress_(mechanics)#Maximum_and_minimum_shear_stresseshttp://en.wikipedia.org/wiki/Stress_(mechanics)#Stress_deviator_tensor

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Stress Analysis is an Engineering