Rods between rigid walls: 22-Mec-A7 Advanced Strength of Materials, Undated paper, Q3

Question 3: Rods between rigid walls (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Three rods welded in series between two rigid walls; joints B and C carry the applied loads.

Given data
Rods 1 and 3	E = 50 GPa, A = 0.01 m², L = 1.2 m
Rod 2	E = 30 GPa, A = 0.025 m², L = 1.5 m
Load at joint B	130 kN (toward the left wall)
Load at joint C	300 kN (toward the right wall)

Find. the axial displacements of joints B and C.

Series rods A–B–C–D between rigid walls; 130 kN left at B, 300 kN right at C.

Approach. Each rod is an axial spring of stiffness k = EA/L; assemble the two free-joint equilibrium equations and solve the 2×2 system.

Rod stiffnesses. $$k_1=k_3=\frac{EA}{L}=\frac{(50\times10^9)(0.01)}{1.2}=417\ \text{MN/m},\quad k_2=\frac{(30\times10^9)(0.025)}{1.5}=500\ \text{MN/m}.$$
Joint equilibrium. Taking rightward positive, with walls fixed, $$(k_1+k_2)u_B-k_2u_C=-130\ \text{kN},\qquad -k_2u_B+(k_2+k_3)u_C=+300\ \text{kN}.$$
Solve. Inverting the system, $$u_B=\boxed{+0.052\ \text{mm}},\qquad u_C=\boxed{+0.356\ \text{mm}},$$ both directed toward the right wall.

Although the load at B points left, both joints translate to the right: the 300 kN pull at C dominates and drags joint B with it through the stiff central rod. The signs confirm rod 1 is stretched and rod 3 compressed, consistent with the net rightward drift.

Final results
Displacement of joint B	+0.052 mm (right)
Displacement of joint C	+0.356 mm (right)

Question 3 of 8: Rods between rigid walls

Question 3: Rods between rigid walls (20 marks)