WIKI STRESS|February 25, 2013 1:28 pm

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.

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