--- title: "CST plate: stresses and strains at centroid" subject: "Computational Techniques in Civil Engineering" chapter: 4 key: "semester_8/computational_techniques_in_civil_engineering:cst-plane-stress-s79" source: https://www.bctengineer.com/q/semester_8/computational_techniques_in_civil_engineering/cst-plane-stress-s79 site: BCT Engineering --- # CST plate: stresses and strains at centroid Computational Techniques in Civil Engineering · Chapter 4: Finite element method - Asked in: 2079 Shrawan - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_8/computational_techniques_in_civil_engineering/cst-plane-stress-s79 - Whole chapter: https://www.bctengineer.com/md/semester_8/computational_techniques_in_civil_engineering/chapter-4 ## Question A plate of thickness 20 mm is being loaded as shown in the figure. Considering the plane stress condition, determine the stresses and strains at the centroid of the CST element. Take $E = 2.1\times10^5$ N/mm$^2$, $\nu = 0.3$. Ignore the weight of the plate. [Figure: right triangular plate ABC with A at the bottom-left, B at the bottom-right 4 m from A, C directly above A at 3 m; supports at A and C along the left edge; at B a 50 kN upward load and a 70 kN horizontal load (to the right)] ## Short answer Use one plane-stress CST element with t=20 mm and nodal coordinates A(0,0), B(4000,0), C(0,3000) mm. Solving with only node B free gives u_B=0.0404 mm and v_B=0.0825 mm. At the centroid, strains are epsilon_x=1.01e-5, epsilon_y=0, gamma_xy=2.06e-5 and stresses are sigma_x=2.333, sigma_y=0.700, tau_xy=1.667 N/mm^2. ## Answer The plate is modelled by one CST element ABC in plane stress ($t=20$ mm, $E=2.1\times10^5$ N/mm$^2$, $\nu=0.3$). Node numbers: A = 1, B = 2, C = 3. #### Step 1: Geometry - Node A (1): $(0,\ 0)$ mm - Node B (2): $(4000,\ 0)$ mm - Node C (3): $(0,\ 3000)$ mm Area: $A=\tfrac12[x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)]=6\times10^{6}\ \text{mm}^2$ $b_1=y_2-y_3=-3000,\ b_2=y_3-y_1=3000,\ b_3=y_1-y_2=0$ mm $c_1=x_3-x_2=-4000,\ c_2=x_1-x_3=0,\ c_3=x_2-x_1=4000$ mm #### Step 2: [B] and [D] matrices (plane stress) $$ [B]=\frac{1}{2A}\begin{bmatrix}-3000 & 0 & 3000 & 0 & 0 & 0 \\ 0 & -4000 & 0 & 0 & 0 & 4000 \\ -4000 & -3000 & 0 & 3000 & 4000 & 0\end{bmatrix},\quad 2A=1.2\times10^{7}\ \text{mm}^2 $$ $$ [D]=\frac{E}{1-\nu^2}\begin{bmatrix}1&\nu&0\\ \nu&1&0\\0&0&\frac{1-\nu}{2}\end{bmatrix}=10^{5}\begin{bmatrix}2.308 & 0.692 & 0 \\ 0.692 & 2.308 & 0 \\ 0 & 0 & 0.808\end{bmatrix}\ \text{N/mm}^2 $$ #### Step 3: Element stiffness matrix $$ [k]=t\,A\,[B]^T[D][B]= 10^{6}\begin{bmatrix}2.808 & 1.500 & -1.731 & -0.808 & -1.077 & -0.692 \\ 1.500 & 3.683 & -0.692 & -0.606 & -0.808 & -3.077 \\ -1.731 & -0.692 & 1.731 & 0 & 0 & 0.692 \\ -0.808 & -0.606 & 0 & 0.606 & 0.808 & 0 \\ -1.077 & -0.808 & 0 & 0.808 & 1.077 & 0 \\ -0.692 & -3.077 & 0.692 & 0 & 0 & 3.077\end{bmatrix}\ \text{N/mm} $$ (DOF order: $u_1,v_1,u_2,v_2,u_3,v_3$.) #### Step 4: Load vector The 50 kN upward load ($+y$) and 70 kN horizontal load ($+x$) act at B (node 2): $F_{x2}=70\,000$ N, $F_{y2}=50\,000$ N. Self-weight is ignored (as stated). Coordinates are taken in mm with A at the origin. $$ \{F\}=\begin{Bmatrix}0 \\ 0 \\ 70\,000 \\ 50\,000 \\ 0 \\ 0\end{Bmatrix}\ \text{N} $$ #### Boundary conditions and solution A and C are on the supported left edge, so $u_1=v_1=u_3=v_3=0$. Only $u_2,v_2$ are unknown; use rows and columns 3 and 4 of $[k]$. $$ 10^{6}\begin{bmatrix}1.731 & 0 \\ 0 & 0.606\end{bmatrix}\begin{Bmatrix}u_2 \\ v_2\end{Bmatrix}=\begin{Bmatrix}70\,000 \\ 50\,000\end{Bmatrix} $$ $$ \begin{aligned} u_2&=0.04044\ \text{mm} \\ v_2&=0.08254\ \text{mm} \end{aligned} $$ #### Strains and stresses (constant over the element, so equal at the centroid) Centroid: $(1333.33,\ 1000.00)$ mm. $$ \{\varepsilon\}=[B]\{d\}=\begin{Bmatrix}1.011\times10^{-5} \\ 0 \\ 2.063\times10^{-5}\end{Bmatrix} $$ $$ \{\sigma\}=[D]\{\varepsilon\}=\begin{Bmatrix}2.333 \\ 0.7 \\ 1.667\end{Bmatrix}\ \text{MPa} $$ **Answer:** $u_B=0.0404$ mm, $v_B=0.0825$ mm. Strains at the centroid: $\varepsilon_x=1.01\times10^{-5}$, $\varepsilon_y=0$, $\gamma_{xy}=2.06\times10^{-5}$. Stresses at the centroid: $\sigma_x=2.333$, $\sigma_y=0.700$, $\tau_{xy}=1.667$ N/mm$^2$. Check: reactions from $[k]\{d\}$ at A and C are $(-136.7,-78.0)$ kN and $(66.7,28.0)$ kN; with the loads at B, $\Sigma F_x=0$ and $\Sigma F_y=0$. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is the area of triangle with A(0,0), B(4000,0), C(0,3000)? A. 12x10^6 mm^2 B. 6x10^6 mm^2 C. 3x10^6 mm^2 D. 7000 mm^2 Answer: B. A = 1/2 x 4000 x 3000 = 6x10^6 mm^2. (Key point: Area of triangle) 2. Why are only u2 and v2 unknown? A. A and C are on the supported edge so u1 = v1 = u3 = v3 = 0 B. Only node B has stiffness C. Strains are zero at A and C D. The load is applied at A and C Answer: A. Supports at A and C restrain both components there, leaving node B free. (Key point: Unknown displacements) 3. How are the centroid stresses found? A. {sigma} = [D][B]{d} B. {sigma} = [B]{d}/E C. {sigma} = [k]{d} D. {sigma} = [D]^-1 {F} Answer: A. Strain {eps} = [B]{d}, then stress {sigma} = [D]{eps}; both constant in a CST. (Key point: Stress from strain) 4. Why is eps_y = 0 here? A. Poisson's ratio is zero B. The y-row of [B] uses c1 v1 + c2 v2 + c3 v3 with c2 = 0 and v1 = v3 = 0 C. The load has no y-component D. Plane stress forces eps_y to zero Answer: B. c2 = 0, so v2 does not enter eps_y, and v1 = v3 = 0. 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