Plane Strain and Plane Stress

1. In The Name of God Plane Strain and Plane Stress By Reza Barati Under Guidance of Prof. G. Heidarinejad Continuum Mechanics University of Tarbiat Modares Dec, 12 2011 E-mail: r88barati@gmail.com

2. Outline • Introduction • Plane strain • Plane stress • Plane stress versus plane strain • Book examples • Summary

3. Introduction • In real engineering components, stress and strain are 3-D tensors, but many problems in elasticity may be treated satisfactorily by a two dimensional, or plane theory of elasticity. 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. 3 Plane Strain and Plane Stress

4. Plane Strain • 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. Plane strain state in a continuum 4 Plane Strain and Plane Stress

5. Plane Strain (continued) • The displacements and corresponding strain tensor can be approximated by: 5 Plane Strain and Plane Stress

6. Plane Strain (continued) • The corresponding stress tensor is: 6 Plane Strain and Plane Stress

7. Plane Strain (continued) • For a static stress field associated with a plane strain problem in the absence of body forces, the equilibrium equations reduce to 7 Plane Strain and Plane Stress

8. Plane Strain (continued) • It can be verified that the other two equations of equilibrium are satisfied for the stress components calculated from the following equations for any scalar function , known as the Airy stress function: 8 Plane Strain and Plane Stress

9. Plane Strain (continued) • The Airy stress function (φ): solutions to plane strain and plane stress problems can be obtained by using various stress function techniques which employ the Airy stress function to reduce the generalized formulation to the governing equations with solvable unknowns. • Scalar potential function that can be used to find the stress tensor. • Satisfies equilibrium in the absence of body forces. • Only for two-dimensional problems (plane stress-plane strain). 9 Plane Strain and Plane Stress

10. Plane Strain (continued) • However, not all stress components obtained this way are acceptable as possible elastic solutions, because the strain components derived from them may not be compatible; that is, there may not exist displacement components that correspond to the strain components. To ensure the compatibility of the strain components, we first obtain the strain components in terms of as follows: 10 Plane Strain and Plane Stress

11. Plane Strain (continued) • The six compatibility equations are 11 Plane Strain and Plane Stress

12. Plane Strain (continued) • For plane strain problems, the only compatibility equation that is not automatically satisfied is • Substitution of the strain components into above Equation results in (this relation is called the biharmonic equation) 12 Plane Strain and Plane Stress

13. Plane Stress • A state of plane stress exists when one of the three principal , , , 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. Plane stress state in a continuum 13 Plane Strain and Plane Stress

14. Plane Stress (continued) • The stress and strain tensors are: • The equations of equilibrium can be assured if we again introduce the Airy stress function, which is repeated here: 14 Plane Strain and Plane Stress

15. Plane Stress (continued) • Corresponding to this state of plane stress, the strain components are 15 Plane Strain and Plane Stress

16. Plane Stress (continued) • In order that these strains are compatible, they must satisfy the six compatibility equations. The consequences are: • Thus, must be a linear function of .Since ; must be a linear function of 16 Plane Strain and Plane Stress

17. Plane Stress versus Plane Strain • Plane stress and plane strain do not ordinarily occur simultaneously. One exception is when = 0 and = - , since Hooke’s Law gives = 0. 17 Plane Strain and Plane Stress

18. Example 1 • Consider the following state of stress in a cylindrical body with axis normal to its cross-sections: • Show that the most general form of , which gives rise to a possible state of stress in the body in the absence of body force, is 18 Plane Strain and Plane Stress

19. Example 1 (continued) • The strain components are 19 Plane Strain and Plane Stress

20. Example 1 (continued) • Substituting the preceding into the compatibility equations, we obtain • Thus, for the given stress tensor to be a possible elastic state of stress, Gmust be a linear function of and . That is, 20 Plane Strain and Plane Stress

21. Example 2 • The airy stress function that satisfies the biharmonic equation is: • The stress components from the plane strain are: 21 Plane Strain and Plane Stress

22. Example 2 (continued) • The surface tractions (i.e., stress vectors on the surface of the body) on the boundary of the body are 22 Plane Strain and Plane Stress

23. Example 2 (continued) • If the beam is unconstrained at, 23 Plane Strain and Plane Stress

24. Example 3 • The airy stress function is: • This satisfies the biharmonic equation. 24 Plane Strain and Plane Stress

25. Example 3 (continued) • The in-plane stresses are: 25 Plane Strain and Plane Stress

26. Example 3 (continued) • On the boundary planes, we demand that they are traction-free. Thus, 26 Plane Strain and Plane Stress

27. Example 3 (continued) • On the boundary plane , the surface traction is given by • Let the resultant of this distribution be denoted by -P; then 27 Plane Strain and Plane Stress

28. Example 3 (continued) • In terms of P , the in-plane stress components are • For 28 Plane Strain and Plane Stress

29. Example 3 (continued) • If the beam is in a plane strain condition , there will be normal compressive stresses on the boundary whose magnitude is given by 29 Plane Strain and Plane Stress

30. Example 3 (continued) • This plane strain solution is: 30 Plane Strain and Plane Stress

31. Example 3 (continued) • Since is not a linear function of and , it cannot be simply removed to give a plane stress solution without affecting the other stress components. However, if the beam is very thin (i.e., very small b compared with the other dimensions), then a good approximate solution for the beam is 31 Plane Strain and Plane Stress

32. Summary • Plane strain and Plane stress are two simplification structural models for the modeling of 3D problems, in which: • Plane strain modelling: strain in Z-direction is negligible. • Plane stress modelling: stress in Z-direction is negligible. 32 Plane Strain and Plane Stress

33. Summary (continued) • Under what conditions a problem can be approximated as a plane problem? • In general, if the problem has one dimension is much larger (or smaller) than the other two directions, one should consider plane strain (stress). 33 Plane Strain and Plane Stress

34. Summary (continued) PLANE STRESSExamples: 1. Thin plate with a hole 2. Thin cantilever plate 34 Plane Strain and Plane Stress

35. Summary (continued) Slice of unit thickness PLANE STRAIN Examples: 1 1. Dam subjected to water loading 2. Long cylindrical pressure vessel subjected to internal/external pressure and constrained at the ends 35 Plane Strain and Plane Stress