Plane strain examples

There are two general types of problems involved in this plane analysis, plane stress and plane strain. These two types will be defined by setting down certain restrictions and assumptions on the stress and displacement fields. They will also be introduced descriptively in terms of . It turns out that, just as the state of plane stress often arises in thin components, a state of plane strain often arises in very thick components. Consider the three dimensional block of material in Fig. The material is constrained from undergoing normal strain in the z direction, for example by preventing movement with . This is an idealized model and thus an approximation.

There are, in actuality, triaxial (σzz. , etc.) stresses that we ignore here as being small relative to the in- plane stresses! Final note: for an orthotropic material, write the tensorial. The resulting FEA models can give valuable insight into local stresses more rapidly and efficiently than a full 3D model. The two FEA methods are called plane stress and plane strain. Both use 2D planar elements that look like thin . A related notion, plane strain , is often applicable to very thick members.

In certain situations, a gently curved thin plate may also be assumed to have plane stress for the purpose of stress analysis. If one of the principal strains (say ? 3) is zero, and if the remaining strains are independent of the dimension along its principal axis, n it is called plane strain. This condition occurs in particular geometries.

For example , if a long, soli prismatic bar is loaded . The stress state under these conditions tends to triaxial and there is zero strain perpendicular to both the stress axis and the direction of crack propagation when a material is loaded in tension. Under plane - strain conditions, materials behave essentially elastic until the fracture stress is reached and then rapid fracture occurs . EXAMPLE FOR 2D PLAIN STRAIN DEFORMATION. The tube is pressurize the outer pressure is neglected. An element of material in plane strain has ε x. Find the principal strains, the (in-plane) maximum shear strains, and the strains on an element oriented at an angle θ=30°.

The stress equilibrium equations derived previously give us only half the picture. To fully solve for the response of a bounded. Plane strain means that ε z. C with the appropriate constitutive relation. Any other material will have . Two-dimensional elastic problems were the first successful examples of the applica- tion of the finite element method.

Indee we have already used this situation to illustrate the basis of the finite element formulation in Chapter where the general relationships were derived . In general, if the problem has one dimension is much larger (or smaller) than the other two directions, one should consider plane strain (stress). More complex behavior (for example : orthotropy, plasticity, viscoelasticity, and fracture). Example : An Inflating Balloon. For a linearly elastic isotropic material, the strain and stress matrices take on the form.

We will derive the CST stiffness matrix by using the principle of minimum potential energy because the energy formulation is the most feasible for the development of the equations for both two- and three-dimensional finite elements. We will then present a simple, thin-plate plane stress example. This screen demo shows the basic procedure of finite element analysis by VisualFEA, using a simple example of plane strain case. The preprocessing, solving and postprocessing stages are explained through the step-by-step progress of defining outlines, generating meshes, assigning attributes, solving .