Bar: nodal displacements, stresses, reactions

Data (from the figure): bar fixed at both ends; element 1: $A_1=250$ mm$^2$, $L_1=150$ mm; element 2: $A_2=300$ mm$^2$, $L_2=150$ mm (total 300 mm); $P=300$ kN at the junction node 2; $E=200$ GPa $=200\times10^3$ N/mm$^2$. Nodes: 1 (left wall), 2 (load), 3 (right wall). Units: N, mm.

Element stiffness

$$[k]=\frac{AE}{L}\begin{bmatrix}1&-1\\-1&1\end{bmatrix}$$ $$\frac{A_1E}{L_1}=\frac{250\times200\times10^3}{150}=333333.3\ \text{N/mm},\qquad \frac{A_2E}{L_2}=\frac{300\times200\times10^3}{150}=400000\ \text{N/mm}$$

Assembly

$$\begin{bmatrix}k_1&-k_1&0\\-k_1&k_1+k_2&-k_2\\0&-k_2&k_2\end{bmatrix}\begin{Bmatrix}u_1\\u_2\\u_3\end{Bmatrix}=\begin{Bmatrix}R_1\\P\\R_3\end{Bmatrix}$$

With $u_1=u_3=0$, the second equation gives $(k_1+k_2)u_2=P$:

$$u_2=\frac{300\,000}{333333.3+400000}=0.40909\ \text{mm}$$

Element stresses

$$\sigma_1=E\,\frac{u_2-u_1}{L_1}=200\,000\times\frac{0.40909}{150}=545.45\ \text{MPa (tension)}$$ $$\sigma_2=E\,\frac{u_3-u_2}{L_2}=-545.45\ \text{MPa (compression)}$$

Support reactions

$$R_1=-k_1u_2=-136.36\ \text{kN},\qquad R_3=-k_2u_2=-163.64\ \text{kN}$$

Check: $R_1+R_3+P=-136.36-163.64+300=0$.

Answer: $u_1=0$, $u_2=0.4091$ mm, $u_3=0$; $\sigma_1=+545.5$ MPa, $\sigma_2=-545.5$ MPa; $R_1=136.36$ kN and $R_3=163.64$ kN, both acting opposite to $P$.