Plane Strain and Plane Stress In The Name of God 4/28/2017 12:41 PM 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 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

Outline Introduction Plane strain Plane stress 4/28/2017 12:41 PM Outline Introduction Plane strain Plane stress Plane stress versus plane strain Book examples Summary © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

4/28/2017 12:41 PM 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. Plane Strain and Plane Stress 3 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

4/28/2017 12:41 PM 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 Plane Strain and Plane Stress 4 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

Plane Strain (continued) 4/28/2017 12:41 PM Plane Strain (continued) The displacements and corresponding strain tensor can be approximated by: Plane Strain and Plane Stress 5 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

Plane Strain (continued) 4/28/2017 12:41 PM Plane Strain (continued) The corresponding stress tensor is: Plane Strain and Plane Stress 6 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

Plane Strain (continued) 4/28/2017 12:41 PM 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 Plane Strain and Plane Stress 7 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

Plane Strain (continued) 4/28/2017 12:41 PM 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 𝜙 𝑋 1 , 𝑋 2 , known as the Airy stress function: Plane Strain and Plane Stress 8 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

Plane Strain (continued) 4/28/2017 12:41 PM 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). Plane Strain and Plane Stress 9 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

Plane Strain (continued) 4/28/2017 12:41 PM 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: Plane Strain and Plane Stress 10 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

Plane Strain (continued) 4/28/2017 12:41 PM Plane Strain (continued) The six compatibility equations are Plane Strain and Plane Stress 11 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

Plane Strain (continued) 4/28/2017 12:41 PM 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) Plane Strain and Plane Stress 12 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

4/28/2017 12:41 PM Plane Stress A state of plane stress exists when one of the three principal 𝑇 11 , 𝑇 22 , 𝑇 33 , 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 Plane Strain and Plane Stress 13 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

Plane Stress (continued) 4/28/2017 12:41 PM 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: Plane Strain and Plane Stress 14 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

Plane Stress (continued) 4/28/2017 12:41 PM Plane Stress (continued) Corresponding to this state of plane stress, the strain components are Plane Strain and Plane Stress 15 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

Plane Stress (continued) 4/28/2017 12:41 PM Plane Stress (continued) In order that these strains are compatible, they must satisfy the six compatibility equations. The consequences are: Thus, 𝐸 33 must be a linear function of 𝑋 1 and 𝑋 2 . Since 𝐸 33 =− 𝜈 𝐸 𝑌 𝑇 11 + 𝑇 22 ; 𝑇 11 + 𝑇 22 must be a linear function of 𝑋 1 and 𝑋 2 . Plane Strain and Plane Stress 16 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

Plane Stress versus Plane Strain 4/28/2017 12:41 PM Plane Stress versus Plane Strain Plane stress and plane strain do not ordinarily occur simultaneously. One exception is when 𝑇 33 = 0 and 𝑇 11 = - 𝑇 22 , since Hooke’s Law gives 𝐸 33 = 0. - Plane Stress Plane Strain Stresses 𝑇 33 =0, 𝑇 13 =0, 𝑇 23 =0 𝑇 11 , 𝑇 22 , 𝑇 12 may be non-zero. 𝑇 13 =0, 𝑇 23 =0 𝑇 11 , 𝑇 22 , 𝑇 33 , 𝑇 12 may be non-zero. Strains 𝐸 13 =0, 𝐸 23 =0 𝐸 11 , 𝐸 22 , 𝐸 33 , 𝐸 12 may be non-zero. 𝐸 33 =0, 𝐸 13 =0, 𝐸 23 =0 𝐸 11 , 𝐸 22 , 𝐸 12 may be non-zero. Plane Strain and Plane Stress 17 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

Example 1 Consider the following state of stress in a cylindrical body with 𝑋 3 axis normal to its cross-sections: Show that the most general form of 𝐺 𝑋 1 , 𝑋 2 , which gives rise to a possible state of stress in the body in the absence of body force, is Plane Strain and Plane Stress 18

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

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, G 𝑋 1 , 𝑋 2 must be a linear function of 𝑋 1 and 𝑋 2 . That is, Plane Strain and Plane Stress 20

4/28/2017 12:41 PM Example 2 The airy stress function that satisfies the biharmonic equation is: The stress components from the plane strain are: Plane Strain and Plane Stress 21 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

4/28/2017 12:41 PM Example 2 (continued) The surface tractions (i.e., stress vectors on the surface of the body) on the boundary of the body are Plane Strain and Plane Stress 22 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

Example 2 (continued) If the beam is unconstrained at 𝑋 3 =± 𝑏 2 , 4/28/2017 12:41 PM Example 2 (continued) If the beam is unconstrained at 𝑋 3 =± 𝑏 2 , Plane Strain and Plane Stress 23 © 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries. The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

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

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

Example 3 (continued) On the boundary planes 𝑋 2 =± ℎ 2 , we demand that they are traction-free. Thus, Plane Strain and Plane Stress 26

Example 3 (continued) On the boundary plane 𝑋 1 =0, the surface traction is given by Let the resultant of this distribution be denoted by -P 𝑒 2 ; then Plane Strain and Plane Stress 27

Example 3 (continued) In terms of P , the in-plane stress components are =𝐿 For 𝑋 1 =𝐿 Plane Strain and Plane Stress 28

Example 3 (continued) If the beam is in a plane strain condition , there will be normal compressive stresses on the boundary 𝑋 3 =± 𝑏 2 whose magnitude is given by Plane Strain and Plane Stress 29

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

Example 3 (continued) Since 𝑇 33 is not a linear function of 𝑋 1 and 𝑋 2 , 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 Plane Strain and Plane Stress 31

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. Plane Strain and Plane Stress 32

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). Plane Strain and Plane Stress 33

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

Summary (continued) Slice of unit thickness 1 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 Plane Strain and Plane Stress 35