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Stress analysis

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Stress analysis is an engineering (e.g., civil engineering, mechanical engineering and aerospace engineering) discipline that determines the stress and strain 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, structural frames and aircraft structure among others, under prescribed or expected loads and/or deflections. Stress analysis may be applied as a design step to structures that do not yet exist or an investigative process for parts that have failed.

The aim of the analysis is usually to determine whether the element or collection of elements, usually referred to as a structure, behaves as desired under the prescribed loading. For example, this might be achieved when the determined stress from the applied force(s) is less than the tensile yield strength or below the fatigue strength of the material.

Analysis may be performed through classical mathematical techniques, analytic mathematical modelling or computational simulation, through experimental testing techniques, or a combination of methods.

Engineering quantities are usually measured in megapascals (MPa) or gigapascals (GPa)or in imperial units, pounds per square inch (psi) or kilopounds-force per square inch (ksi).

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[edit] Goals

Stress analysis is typically concerned with solid objects and structures that can be assumed to be in macroscopic static equilibrium: that is, are either unchanging with time, or are changing slowly enough for viscous stresses to be unimportant.

By Newton’s laws of motion, any external forces that act on such a system must be balanced by internal reaction forces.[1] With very rare exceptions (such as ferromagnetic materials, or planet-scale bodies), internal forces are due to very short range intermolecular interactions, and are therefore manifested as surface contact forces between adjacent particles — that is, as stress.[2] Since every particle needs to be in equilibrium, this reaction stress will generally propagate from particle to particle throughout an extended part of the system.

The funamental problem in stress analysis is to determine the distribution of these internal stresses thoughout the system, given the external forces that are acting on it. Specifically, the goal is to determine the Cauchy stress tensor at every point.

The external forces may be body forces (such as gravity or magnetic attraction), that act throughout the volume of a material;[3] or concentrated loads (such as friction between an axle and a bearing, or the weight of a train wheel on a rail), that are imagined to act over a two-dimensional area, or along a line, or at single point. The same net external force will have a different effect on the local stress depending on whether it is concentrated or spread out.

In stress analysis one normally disregards the physical causes of the forces or the precise nature of the materials. Instead, one assumed that the stresses are related to strain of the material by known constitutive equations.[4] The effects of stress on materials and their ability to withstand it are also outside the scope of stress analysis, being covered in materials science under the names strength of materials, fatigue analysis, stress corrosion, creep modeling, and other. The study of stresses in flowing liquids and gases is the subject of fluid dynamics.

In engineering, stress analysis is often a tool rather than a goal in itself; the ultimate goal being the design of structures and artifacts that can withstand a specified load, using the minimum amount of material (or some other quantitative criterion).

[edit] Experimental methods

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.

The use of strain gauges is a common technique for experimentally determining stress in a physical part. Strain gauges are essentially thin flat resistors that are affixed to the surface of a part, and which measure the strain in a given direction. From the measurement of strain on a surface in three directions the stress state in the part can be calculated.

Neutron diffraction is a technique that can be used to determine the subsurface strain in a part.

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[DMA] 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.

[edit] Mathematical methods

While experimental techniques are widely used, most stress analysis is done by mathematical methods, especially during design.

[edit] Differential formulation

The basic stress analysis problem can be formulated by Euler’s equations of motion for continuous bodies (which are consequences of Newton’s laws for conservation of linear momentum and angular momentum) and the Euler-Cauchy stress principle, together with the appropriate constitutive equations.

These laws yield a system of partial differential equations involving the stress tensor field and the strain tensor field as unknown functions to be determined. Solving for either one will yield the other through the constitutive equations. Both fieldds fields will normally be continuous within each extended part of the system that does not contain any concentrated loads and that can be regarded as a continuous medium with smoothly varying constitutive equations.

The external body forces will appear as the independent (“right-hand side”) term in the differential equations, while the concentrated forces appear as boundary conditions. An external surface force, such as ambient pressure or friction, can be incorporated as an imposed value of the stress tensor across that surface External forces that are specified as line loads (such as traction on a concrete block plastic applied by an embedded rebar) or point loads (such as the weight of a person standing on a roof) introduce singularities in the stress field, and may be handled by assuming that they are spread over a small arbitrary volume or surface patch. The basic stress analysis problem is therefore a boundary-value problem.

[edit] Elastic and linear cases

A system is said to be elastic if any deformations caused by external forces will spontaneously and completely disappear once the external forces are removed. Stress analysis for such systems is based on the theory of elasticity and infinitesimal strain theory. When the applied loads cause permanent deformation, one must use more complicated constitutive equations, that can account for the physical processes involved (plastic flow, fracture, phase change, etc.)

Engineered structures are usually designed so that the maximum expected stresses are well within the range of linear elasticity (the generalization of Hooke’s law for continuous media); that is, the deformations caused by internal stresses are linearly related to them. In this case the differential equations that define the stress tensor are linear, too. Linear equations are much better understood than non-linear ones; for one thing, their solution (the stress at any given point) will also be a linear function of the external forces. For small enough stresses, even non-linear systems can usually be assumed to be linear.

[edit] Built-in stress

The mathematical problem represented is typically ill-posed because it has an infinitude of solutions. In fact, in any three-dimensional solid body one may have infinitely many and infinitely complicated) non-zero stress tensor fields that are in stable equilibrium even in the absence of external forces.

Such built-in stress may occur due to many physical causes, either during manufacture (in processes like extrusion, casting or cold working), or after it (for example because of uneven heating, or changes in moisture content or chemical composition). However, if the system can be assumed to be linear, then the built-in stress can be ignored in the analysis, since it will merely add to the solution without interfering with it.

If linearity cannot be assumed, however, any built-in stress may affect the distribution of externally-induced stress (for example, by changing the effective stiffness of the material) or even cause the unexpected material failure. For these reasons, a number of techniques have been developed to avoid or reduce built-in stress, such as annealing of cold-worked glass and metal parts, expansion joints in buildings, and roller jointss for bridges.

[edit] Simplifications

Simplified modeling of a truss by unidimensional elements under uniaxial uniform stress.

Stress analysis is simplified when the physical dimensions and the distribution of loads allow the structure to be treated as one- or two-dimensional. In the analysis of trusses, for example, the stress field may be assumed to be uniform and uniaxial over each member. Then the differential equations reduce to a finite set of equations (usually linear) with finitely many unknowns.

If the stress distribution can be assumed to be uniform (or predictable, or unimportant) in one direction, then one may use a plane stress and plane strain formulation where the stress field is a function of two coordinates only, instead of three.

Even under the assumption of linear elasticity, the relation between the stress and strain tensors is generally expressed by a fourth-order stiffness tensor with 21 independent coefficients. This complexity may be required for general anisotropic materials, but for many common materials it can be simplified. For orthotropic materials like wood, whose stiffness is symmetric about three orthogonal planes, nine coefficients suffice to express the stress-strain relationship. For isotropic materials, these reduce to only two.

One may be able to determine a priori that, in some parts of the system, the stress will be of a certain type, such as uniaxial tension or compression, simple shear, isotropic compression or tension, torsion, bending, etc. In those parts the stress field may then be represented by fewer than six numbers, possibly just one.

[edit] Solving the equations

In any case, for two- or there-dimensional domains one must solve a system of partial differential equations with boundary conditions. Anlytical or closed-form solutions to the differential equations can be obtained when the geometry, constitutive relations, and boundary conditions are simple enough. Otherwise one must generally resort to numerical approximations such as the finite element method, the finite difference method, and the boundary element method.

[edit] Factor of safety

Main article: Factor of safety

The factor of safety (also “margin of safety” or “design factor”) 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, designed or limit stresses, 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}}

[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 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 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.

[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 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.

[edit] 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}<br />
\sigma_{11} & \sigma_{12} & 0 \\<br />
\sigma_{21} & \sigma_{22} & 0 \\<br />
0      &     0       & 0<br />
\end{bmatrix} \equiv \begin{bmatrix}<br />
\sigma_{x} & \tau_{xy} & 0 \\<br />
\tau_{yx} & \sigma_{y} & 0 \\<br />
0      &     0       & 0<br />
\end{bmatrix}\,\!.

The corresponding strain tensor is:

\varepsilon_{ij} = \begin{bmatrix}<br />
\varepsilon_{11} & \varepsilon_{12} & 0 \\<br />
\varepsilon_{21} & \varepsilon_{22} & 0 \\<br />
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.

 

[edit] Plane strain

Figure 7.2 Plane strain state in a continuum.

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.

 

[edit] 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}<br />
\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 \\<br />
(\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 )\,\!

[edit] 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.

[edit] 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)\,\!.

[edit] 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).

[edit] 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.

[edit] 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:[5]

  • Isobars are curves along which the principal stress, e.g., \sigma_1\,\!is constant.
  • Isochromatics are curves along which the maximum shear stressis constant. This curves are directly determined using photoelasticity methods.
  • Isopachs are curves along which the mean normal stressis constant
  • Isostatics or stress trajectoriesare a system of curves which are at each material point tangent to the principal axes of stress.
  • Isoclinicsare 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.

[edit] See also

[edit] References

  1. ^ Smith & Truesdell 1993, p. 97
  2. ^Liu
  3. ^Irgens
  4. ^Slaughter
  5. ^ Jaeger, John Conrad; Cook, N.G.W, & Zimmerman, R.W. (2007). Fundamentals of rock mechanics (Fourth ed.). Wiley-Blackwell. pp. 9–41. ISBN 0-632-05759-9. http://books.google.com/books?id=FqADDkunVNAC&lpg=PP1&pg=PA10#v=onepage&q=&f=false.

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