Stress analysis

Stress analysis is an engineering (e.g., civil engineering and mechanical engineering) discipline that determines the stress in materials and structures subjected to static or dynamic forces or loads. A stress analysis is required for the study and design of structures, e.g., tunnels, dams, mechanical parts, and structural frames among others, under prescribed or expected loads. Stress analysis may be applied as a design step 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 when the determined stress from the applied force(s) is less than the ultimate tensile strength, ultimate compressive strength or fatigue strength the material is known to be able to withstand, though ordinarily a factor 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 in megapascals (MPa) or gigapascals (GPa). In imperial units, stress is expressed in pounds-force per square inch (psi) or kilopounds-force per square inch (ksi).

Analysis methods

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 are applied, 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 loading conditions is termed fatigue analysis and is most often applied to aerodynamic structural systems.

Modelling

To determine the distribution of stress in a structure it is necessary to solve a boundary-value problem by specifying the boundary conditions, i.e. displacements and/or forces on the boundary. Constitutive equations, such as e.g. Hooke's Law for linear elastic materials, are used to describe the stress:strain relationship in these calculations. A boundary-value problem based on the theory of elasticity is applied to structures expected to deform elastically, i.e. infinitesimal strains, under design loads. When the loads applied to the structure induce plastic deformations, the theory of plasticity is implemented.

Approximate solutions for boundary-value problems can be obtained through the use of numerical methods such as the finite element method, the finite difference method, and the boundary element method, which are implemented in computer programs. Analytical or close-form solutions can be obtained 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 a plane stress or a plane strain condition can be assumed.

Experimental testing

Stress analysis can be performed experimentally by applying forces to a test element or structure and then determining the resulting stress using sensors. In this case the process would more properly be known as testing (destructive or non-destructive). Experimental methods may be used in cases where mathematical approaches are cumbersome or inaccurate. Special equipment appropriate 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 fundamental materials 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 an application, for quality 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 as Young's modulus, Poisson's ratio, yield strength, and strain-hardening characteristics can be determined.

Stress in plastic protractor exhibited due to birefringence.

The photoelastic method relies on the physical phenomenon of birefringence. Unlike the analytical methods 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 refractive indices 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 is similar to the state of the stress in the structure.

Dynamic mechanical analysis is a technique used to study and characterize viscoelastic materials, particularly polymers. Polymers composed of long molecular chains have unique viscoelastic properties, which combine the characteristics of elastic solids and Newtonian fluids. The viscoelastic property of polymer is studied by dynamic mechanical analysis where a sinusoidal force (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 purely viscous 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 other experimental conditions (temperature, frequency, or time) can be varied.

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 a 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 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 strength and 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 yield strength 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:

$\text{Factor of safety} = \frac{\text{Ultimate tensile strength}}{\text{Maximum allowable stress}}$

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

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 within structures. The load transfer may be identified visually, or by simple logic for simple structures. For more complex structures, more complex methods such as theoretical solid 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 engineering or failure analysis is performed to identify weakness, where broken parts are analysed for the cause or causes of failure. The method seeks 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 lower tensile strength than it should for example.

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 component and is indistinguishable from a scalar. One-dimensional objects include a piece of wire loaded at the 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 and the reduction in cross-sectional area is very small and can be neglected, i.e., the cross-sectional area is assumed constant during deformation. For this case, the stress is called engineering stress or nominal stress. In some other cases, e.g., elastomers and plastic materials, the change in cross-sectional area is significant, and the stress must be calculated assuming the current cross-sectional area instead of the initial cross-sectional area. This is termed true stress and is expressed as

$\sigma_\mathrm{true} = (1 + \varepsilon_\mathrm e)(\sigma_\mathrm e)\,\!$,

where

$\varepsilon_\mathrm e\,\!$ is the nominal (engineering) strain, and

$\sigma_\mathrm e\,\!$ is nominal (engineering) stress.

The relationship between true strain and engineering strain is given by

$\varepsilon_\mathrm{true} = \ln(1 + \varepsilon_\mathrm e)\,\!$.

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

Plane stress

Figure 7.1 Plane stress state in a continuum.

A state of plane stress exists when one of the three principal $\left(\sigma_1, \sigma_2, \sigma_3 \right)\,\!$, 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-plane loading 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:

$\sigma_{ij} = \begin{bmatrix} \sigma_{11} & \sigma_{12} & 0 \\ \sigma_{21} & \sigma_{22} & 0 \\ 0 & 0 & 0 \end{bmatrix} \equiv \begin{bmatrix} \sigma_{x} & \tau_{xy} & 0 \\ \tau_{yx} & \sigma_{y} & 0 \\ 0 & 0 & 0 \end{bmatrix}\,\!$.

The corresponding strain tensor is:

$\varepsilon_{ij} = \begin{bmatrix} \varepsilon_{11} & \varepsilon_{12} & 0 \\ \varepsilon_{21} & \varepsilon_{22} & 0 \\ 0 & 0 & \varepsilon_{33}\end{bmatrix}\,\!$

in which the non-zero $\varepsilon_{33}\,\!$ term arises from the Poisson'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.

Plane strain

Figure 7.2 Plane strain state in a continuum.

Main article: Infinitesimal strain theory

If one dimension is very large compared to the others, the principal strain in 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 the direction of the longest dimension can be disregarded for calculations. Thus, allowing a two dimensional analysis of stresses, e.g. a dam analyzed at a cross section loaded by the reservoir.

Stress transformation in plane stress and plane strain

Consider a point $P\,\!$ in a continuum under a state of plane stress, or plane strain, with stress components $(\sigma_x, \sigma_y, \tau_{xy})\,\!$ and all other stress components equal to zero (Figure 7.1, Figure 8.1). From static equilibrium of an infinitesimal material element at $P\,\!$ (Figure 8.2), the normal stress $\sigma_\mathrm{n}\,\!$ and the shear stress $\tau_\mathrm{n}\,\!$ on any plane perpendicular to the $x\,\!$-$y\,\!$ plane passing through $P\,\!$ with a unit vector $\mathbf n\,\!$ making an angle of $\theta\,\!$ with the horizontal, i.e. $\cos \theta\,\!$ is the direction cosine in the $x\,\!$ direction, is given by:

$\sigma_\mathrm{n} = \frac{1}{2} ( \sigma_x + \sigma_y ) + \frac{1}{2} ( \sigma_x - \sigma_y )\cos 2\theta + \tau_{xy} \sin 2\theta\,\!$

$\tau_\mathrm{n} = -\frac{1}{2}(\sigma_x - \sigma_y )\sin 2\theta + \tau_{xy}\cos 2\theta\,\!$

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 $\theta\,\!$, if one knows the stress components $(\sigma_x, \sigma_y, \tau_{xy})\,\!$ on any two perpendicular directions at that point. It is important to remember that we are considering a unit area of the infinitesimal element in the direction parallel to the $y\,\!$-$z\,\!$ plane.

Figure 8.1 - Stress transformation at a point in a continuum under plane stress conditions.

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 $\tau_\mathrm{n}\,\!$ equal to zero. Thus we have:

$\tau_\mathrm{n} = -\frac{1}{2}(\sigma_x - \sigma_y )\sin 2\theta + \tau_{xy}\cos 2\theta=0\,\!$

and we obtain

$\tan 2 \theta_\mathrm{p} = \frac{2 \tau_{xy}}{\sigma_x - \sigma_y}\,\!$

This equation defines two values $\theta_\mathrm{p}\,\!$ which are $90^\circ\,\!$ apart (Figure 8.3). The same result can be obtained by finding the angle $\theta\,\!$ which makes the normal stress $\sigma_\mathrm{n}\,\!$ a maximum, i.e. $\frac{d\sigma_\mathrm{n}}{d\theta}=0\,\!$

The principal stresses $\sigma_1\,\!$ and $\sigma_2\,\!$, or minimum and maximum normal stresses $\sigma_\mathrm{max}\,\!$ and $\sigma_\mathrm{min}\,\!$, respectively, can then be obtained by replacing both values of $\theta_\mathrm{p}\,\!$ into the previous equation for $\sigma_\mathrm{n}\,\!$. This can be achieved by rearranging the equations for $\sigma_\mathrm{n}\,\!$ and $\tau_\mathrm{n}\,\!$, first transposing the first term in the first equation and squaring both sides of each of the equations then adding them. Thus we have

$\begin{align} \left[ \sigma_\mathrm{n} - \tfrac{1}{2} ( \sigma_x + \sigma_y )\right]^2 + \tau_\mathrm{n}^2 &= \left[\tfrac{1}{2}(\sigma_x - \sigma_y)\right]^2 + \tau_{xy}^2 \\ (\sigma_\mathrm{n} - \sigma_\mathrm{avg})^2 + \tau_\mathrm{n}^2 &= R^2 \end{align}\,\!$

where

$R = \sqrt{\left[\tfrac{1}{2}(\sigma_x - \sigma_y)\right]^2 + \tau_{xy}^2} \quad \text{and} \quad \sigma_\mathrm{avg} = \tfrac{1}{2} ( \sigma_x + \sigma_y )\,\!$

which is the equation of a circle of radius $R\,\!$ centered at a point with coordinates $[\sigma_\mathrm{avg}, 0]\,\!$, called Mohr's circle. But knowing that for the principal stresses the shear stress $\tau_\mathrm{n} = 0\,\!$, then we obtain from this equation:

$\sigma_1 =\sigma_\mathrm{max} = \tfrac{1}{2}(\sigma_x + \sigma_y) + \sqrt{\left[\tfrac{1}{2}(\sigma_x - \sigma_y)\right]^2 + \tau_{xy}^2}\,\!$

$\sigma_2 =\sigma_\mathrm{min} = \tfrac{1}{2}(\sigma_x + \sigma_y) - \sqrt{\left[\tfrac{1}{2}(\sigma_x - \sigma_y)\right]^2 + \tau_{xy}^2}\,\!$

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

When $\tau_{xy}=0\,\!$ the infinitesimal element is oriented in the direction of the principal planes, thus the stresses acting on the rectangular element are principal stresses: $\sigma_x = \sigma_1\,\!$ and $\sigma_y = \sigma_2\,\!$. Then the normal stress $\sigma_\mathrm{n}\,\!$ and shear stress $\tau_\mathrm{n}\,\!$ as a function of the principal stresses can be determined by making $\tau_{xy}=0\,\!$. Thus we have

$\sigma_\mathrm{n} = \frac{1}{2} ( \sigma_1 + \sigma_2 ) + \frac{1}{2} ( \sigma_1 - \sigma_2 )\cos 2\theta\,\!$

$\tau_\mathrm{n} = -\frac{1}{2}(\sigma_1 - \sigma_2 )\sin 2\theta\,\!$

Then the maximum shear stress $\tau_\mathrm{max}\,\!$ occurs when $\sin 2\theta = 1\,\!$, i.e. $\theta = 45^\circ\,\!$ (Figure 8.3):

$\tau_\mathrm{max} = \frac{1}{2}(\sigma_1 - \sigma_2 )\,\!$

Then the minimum shear stress $\tau_\mathrm{min}\,\!$ occurs when $\sin 2\theta = -1\,\!$, i.e. $\theta = 135^\circ\,\!$ (Figure 8.3):

$\tau_\mathrm{min} = -\frac{1}{2}(\sigma_1 - \sigma_2 )\,\!$

Graphical representation of stress at a point

Mohr's circle, Lame's stress ellipsoid (together with the stress director surface), and Cauchy's stress quadric are two-dimensional graphical representations of the state of stress at a point. They allow 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.

Mohr's circle

Main article: Mohr's circle

Mohr's circle, named after Christian Otto Mohr, is the locus of points that represent the state of stress on individual planes at all their orientations. The abscissa, $\sigma_\mathrm{n}\,\!$, and ordinate, $\tau_\mathrm{n}\,\!$, of each point on the circle are the normal stress and shear stress components, respectively, acting on a particular cut plane with a unit vector $\mathbf n\,\!$ with components $\left(n_1, n_2, n_3 \right)\,\!$.

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 a point 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).

Cauchy's stress quadric

The Cauchy's stress quadric, also called the stress surface, is a surface of the second order that traces the variation of the normal stress vector $\sigma_\mathrm n \,\!$ as the orientation of the planes passing through a given point is changed.

Graphical representation of the stress field

See also: Stress field

The complete state of stress in a body at a particular deformed configuration, i.e., at a particular time during the motion of the body, implies knowing the six independent components of the stress tensor $(\sigma_{11}, \sigma_{22}, \sigma_{33}, \sigma_{12}, \sigma_{23}, \sigma_{13})\,\!$, or the three principal stresses $(\sigma_1, \sigma_2, \sigma_3)\,\!$, at each material point in the body at that time. However, numerical analysis and analytical methods allow 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:

• Isobars are curves along which the principal stress, e.g., $\sigma_1\,\!$ is constant.
• Isochromatics are curves along which the maximum shear stress is constant. This curves are directly determined using photoelasticity methods.
• Isopachs are curves along which the mean normal stress is constant
• Isostatics or stress 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 fixed reference direction. These curves can also be obtained directly by photoelasticity methods.
• Slip lines are curves on which the shear stress is a maximum.

See also

• Forensic engineering
• Structural analysis
• Stress
• Piping
• Worst Case Circuit Analysis
• List of finite element software packages