--- title: "Compute D matrix for plane problems" subject: "Finite Element Method" chapter: 6 key: "semester_8/finite_element_method:elasticity-matrix-plane-stress-strain" source: https://www.bctengineer.com/q/semester_8/finite_element_method/elasticity-matrix-plane-stress-strain site: BCT Engineering --- # Compute D matrix for plane problems Finite Element Method · Chapter 6: Applications in Solid Mechanics - Asked in: year not given, 6 marks - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_8/finite_element_method/elasticity-matrix-plane-stress-strain - Whole chapter: https://www.bctengineer.com/md/semester_8/finite_element_method/chapter-6 ## Question For a steel with $E = 200$ GPa and $\nu = 0.3$, compute the elasticity matrix $[D]$ for (a) plane stress and (b) plane strain. For a strain state $\varepsilon_x = 0.001$, $\varepsilon_y = 0$, $\gamma_{xy} = 0$, find the stresses $\sigma_x$, $\sigma_y$ and $\sigma_z$ in each case. ## Short answer For plane stress, D = [[219780,65934,0],[65934,219780,0],[0,0,76923]] MPa. For plane strain, D = [[269231,115385,0],[115385,269231,0],[0,0,76923]] MPa. With epsilon_x=0.001, plane stress gives sigma_x=219.8, sigma_y=65.9, sigma_z=0 MPa; plane strain gives sigma_x=269.2, sigma_y=115.4, sigma_z=115.4 MPa. ## Answer #### (a) Plane stress $$ \frac{E}{1 - \nu^2} = \frac{200\times10^3}{1 - 0.09} = 219\,780\ \text{MPa} $$ $$ [D] = 219\,780\begin{bmatrix} 1 & 0.3 & 0 \\ 0.3 & 1 & 0 \\ 0 & 0 & 0.35 \end{bmatrix} = \begin{bmatrix} 219\,780 & 65\,934 & 0 \\ 65\,934 & 219\,780 & 0 \\ 0 & 0 & 76\,923 \end{bmatrix}\ \text{MPa} $$ Note that $\dfrac{1 - \nu}{2} = 0.35$ and $219\,780(0.35) = 76\,923 = G = \dfrac{E}{2(1 + \nu)}$. #### (b) Plane strain $$ \frac{E}{(1 + \nu)(1 - 2\nu)} = \frac{200\times10^3}{1.3(0.4)} = 384\,615\ \text{MPa} $$ $$ [D] = 384\,615\begin{bmatrix} 0.7 & 0.3 & 0 \\ 0.3 & 0.7 & 0 \\ 0 & 0 & 0.2 \end{bmatrix} = \begin{bmatrix} 269\,231 & 115\,385 & 0 \\ 115\,385 & 269\,231 & 0 \\ 0 & 0 & 76\,923 \end{bmatrix}\ \text{MPa} $$ #### Stresses for $\varepsilon_x = 0.001$ | Case | $\sigma_x$ (MPa) | $\sigma_y$ (MPa) | $\sigma_z$ (MPa) | |---|---|---|---| | Plane stress | $219\,780(0.001) = 219.8$ | $65\,934(0.001) = 65.9$ | 0 | | Plane strain | $269\,231(0.001) = 269.2$ | $115\,385(0.001) = 115.4$ | $\nu(\sigma_x + \sigma_y) = 0.3(384.6) = 115.4$ | The shear modulus term is the same in both cases. Plane strain gives higher stresses because $\varepsilon_z = 0$ is enforced. **Answer:** plane stress $\sigma_x = 219.8$ MPa, $\sigma_y = 65.9$ MPa, $\sigma_z = 0$; plane strain $\sigma_x = 269.2$ MPa, $\sigma_y = 115.4$ MPa, $\sigma_z = 115.4$ MPa. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. For E = 200 GPa and nu = 0.3, what is the plane stress factor E/(1 - nu^2)? A. 181,818 MPa B. 219,780 MPa C. 200,000 MPa D. 384,615 MPa Answer: B. 200x10^3/(1 - 0.09) = 219,780 MPa; 384,615 is the plane strain factor. (Key point: Plane stress D) 2. What are D11 and D12 for plane strain with E = 200 GPa and nu = 0.3? A. 384,615 and 115,385 MPa B. 269,231 and 65,934 MPa C. 219,780 and 65,934 MPa D. 269,231 and 115,385 MPa Answer: D. Factor 384,615 times (1 - nu) = 0.7 gives 269,231, and times nu = 0.3 gives 115,385. (Key point: Plane strain D) 3. What is true of the shear term D33 in the two cases? A. It is zero in plane stress B. It equals E/2 in both C. It is larger in plane strain D. It equals G = E/[2(1+nu)] = 76,923 MPa in both Answer: D. Both matrices reduce to G in the shear position, 76,923 MPa. (Key point: Shear term) 4. For ex = 0.001 only, what is sigma_z in each case? A. Plane stress 0; plane strain 115.4 MPa B. Plane stress 115.4 MPa; plane strain 0 C. Both 115.4 MPa D. Both zero Answer: A. Plane stress has sigma_z = 0 by definition; plane strain has sigma_z = nu(sigma_x + sigma_y) = 0.3(384.6) = 115.4 MPa. 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