Plane Stress Problems

It is said that a state of plane stress exists when the stress components fulfill the conditions 

According to these equations and using, as above, cylindrical coordinates in the stress-strain relationships, the zz component of the stress tensor can be written as 

After substituting Eq. (16.135) into Eq. (4.76), the remaining stresses are given by 

An inspection of these results permits us to conclude that all the plane stress equations may be converted to the corresponding equations for the state of plane strain if E and v are replaced by E and v, respectively, where 

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In fact, the solution for a plane stress problem can be determined from the solution of the corresponding plane strain problem and vice versa. Note that in contrast to the plane strain case, the remaining stresses in the plane stress are not required to be independent of z. In fact, the three-dimensionality of plane stress is closely linked to the fact that the conditions fulfilled by the stresses no longer lead to a single nontrivial compatibility equation. In other words, if the remaining stresses ctyy, and <5xy are functions of only X and y, the strain-displacement equations cannot in general be satisfied. 

We shall see that for a prismatic or cylindrical body with the same symmetry as in the case of plane strain and loaded normal to the z axis but now with its ends load-free, a plane stress problem is obtained in which the nonzero stresses vary with z. Strictly speaking, a true plane stress state is present only in thin plates with the main surfaces load-free and with external forces z-independent but symmetrically distributed through its thickness. 

These expressions are valid for plane strain. However, we are dealing with a plane stress problem, and consequently the ends of the cylinder are not restricted. For this reason, close to such parts. 

For plane strain or plane stress problems which have proportional stress boundary values it is found that a linear elastic solution for the stresses is a solution for the stress-time distribution whenever the kernel functionals of the constitutive equation can be decomposed into a product form. The strain-time distribution for this case will be given by substituting the linear stress solution into the non-linear constitutive equation which is homogeneous to degree one. That such a solution is applicable is demonstrated in the following discussion. 

For space oostraint it is not possible to show the results associated with the plane stress problem. However it is interesting to note that when a test example is run with fiber length comparable to the element size the deviation in the stiffness con jonents are found to vary over a wide range. As a consequence nodal deflections tend to loose exact symmetry. This may be due to the fact that for most of the fibers part of them are lying outside the element. A comparison with experiment carried out with steel fiber reinforced concrete may be found in reference (2). 

Let us assume that stress gradient in axial direction is present but smooth. Then we can use a perturbation method and expand the solution of equation (30) in a series. The first term of this expansion will be a solution of the plane strain problem and potential N will be equal to zero. The next terms of the stress components will contain potential N also. 

Rather than a plane-stress state, a three-dimensional stress state is considered in the elasticity approach of Pipes and Pagano [4-12] to the problem of Section 4.6.1. The stress-strain relations for each orthotropic layer in principal material directions are... 

Figure 8.2. Stress components in a plane-strain problem.

Consider a plane-strain problem in the x-z plane, as shown in Fig. 8.2. The stress tensor, expressed in Cartesian coordinates, takes the form... 

Let us consider now the deformation and stresses of a cylindrical pipe under two different boundary conditions (Fig. 16.2). In both eases the length of the pipe is considered constant according to the requirements for a plane strain problem. The external and internal radii are R2 and R, respectively. If the applied forces and the displacements are also uniform, the deformation is purely radial, and in cylindrical coordinates = u r). According to the Navier equations, rot u = 0. Hence, Vdiv u = 0, which implies... 

To illustrate the plane stress situation let us consider the problem of a viscoelastic cyUnder rotating uniformly around its axis, with special application to flat geometries (discs) (Fig. 16.3). 

Let us consider now another important problem consisting of an annular disc with internal and external radii represented, respectively, by i i and R2. The equation to be solved, according to the plane stress conditions, is... 

For two-dimensional problems, two special cases are considered namely, p/anc stress and plane strain. For the case of plane stress, only the in-plane (e.g., the xy-plane) components of the stresses are nonzero and for plane strain, only the inplane components of strains are nonzero. In reality, however, only the average values of the z-component stresses are zero in the plane stress cases. As such, this class of problems is designated by the term generalized plane stress. The conditions for each case will be discussed later. It is to be recognized that, in actual crack problems, these limiting conditions are never achieved. References to plane stress and plane strain, therefore, always connote approximations to these well-defined conditions. 

There are nine components of (unknown) stresses at any point in a stressed body, and they generally vary from point to point within the body. These stresses must be in equilibrium with each other and with other body forces (such as gravitational and inertial forces). For elastostatic problems, the body forces are typically assumed to be zero, and are not considered further. For simplicity, therefore, the equilibrium of an element dx, dy, 1) under plane stress (a22 = = xz = = 0) is... 

The main experimental problem is to prepare specimens in which the plane strain/plane stress... 

The three-dimensional properties of a laminate given by Eqns (6.11), (6.12), and (6.32) are needed in situations where out-of-plane stresses develop. Besides the obvious case of out-of-plane loading such as the local indentation and the associated solution of contact stresses in an impact problem, out-of-plane stresses typically arise near free edges of laminates, in the immediate vicinity of plydrops and near matrix cracks or delaminations. Typical examples are shown in Figure 6.4. The red lines indicate regions in the vicinity of which out-of-plane stresses [Pg.132]

To this point, elastic solutions have been found by making some simplifying assumption and the more general elasticity equations have been by-passed. Such general solutions can be complex, but problems in which the geometric simplification of plane stress or plane strain can be made allow a relatively straightforward scientific approach and solution. 

In plane strain problems, the displacements that exist in a particular direction are assumed to be zero. If this direction is Xy it follows from the definition of strain (Eq. (2.14)) that e,3=e23=e33=0, i.e., the strains are two-dimensional. As an example, consider the problem shown in Fig. 4.13 a knife edge indenting a thick block of material. Most of the displacements are occurring in the x and directions, i.e., the material is being pushed downwards or sideways. The only exceptions are in the vicinity of the front and back surfaces, where displacements in the Xj direction are possible. Overall, the components of the displacement vector at any point can be assumed to be independent of Xj. From Hooke s Law, the assumption that 3=e23 33 plies that <7 3=cr23=0. As with plane stress, only the stress components and are needed to define the... 

In Chapter 2, stress and strain were defined, the compatibility and equilibrium equations were introduced and the relationship between stress and strain was defined. Thus, any solution that satisfies all these equations and the appropriate boundary conditions will be the solution that gives the stress and strain distribution for a particular loading geometry. For the most general problems, the scientific process can be difficult but for plane stress and plane strain problems in elastically isotropic bodies the solution involves a single differential equation. 

Clough RW (1960) The finite element method in plane stress analysis. In Proceedings of ASCE 2nd conference on electronic computation. Conference papers American Society of Civil Engineers 2nd conference on electronic computation, 8-9 Sept 1960, Pittsburgh Courant R (1942) Variational methods for the solution of problems of equilibrium and vibrations. Bull Amer Math Soc 49 1-23... 

Results of 2D plane stress simulations are presented using the new procedure in this publication. The 2D plane stress modelling is performed because the core research topic for which the procedure is developed was aplane stress problem (Saharan Mitri, 2009). Also, plane stress modelling provides a convenient mean to undertake and understand fundamental studies for the dynamic rock fracturing processes. In past, the laboratory scale studies were undertaken using this plane stress concept (e. g., see Kutter Fairhurst, 1971 Foumey et al., 1993). The extension ofthe developed procedure for 3D problems is not difficult but will involve enormous computational resources. 

The prepared model for the developed numerical procedure validation tests is shown in Figure 6. The model is the plane stress idealization of a 3D problem. It uses 4,560triangular elements and 2,312 nodes. One more model with 10,276 elements and 5,275 nodes is prepared to illustrate the effect of a free face on the rock fracturing process. In this case, a 38mm diameter blasthole is kept 0.5m away from a free boundary (face) and emulsion type explosive pulse is used to provide the dynamic load. Rest of the model conditions remains the same as earlier. 

For two dimensional problems we can consider two idealized states the plane strain state where s = Sxz = yi = and the plane stress state in which = Oxz = [Pg.52]

This approach is relevant to structures commonly used in pressure vessel construction. What this means is that the problem can be treated as a two-dimensional situation, wherein the stress or the strain components at every point in the body are fimctions only of the reference coordinates parallel to that plane. For example, a long, thick-walled cylinder subjected to internal pressure can be treated as a plane elastic problem. 

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