GATE CE 2015 Set 1 — Question 19

The beam is a fixed-hinge-fixed continuous beam. We can analyze it using the slope-deflection method.1. Analyze the right segment (Hinge to Right Fixed Support):
This segment is a propped cantilever, simply supported at the hinge (H) and fixed at the right support (R). The length of this segment is $L_2 = 1 \text{ m} + 1 \text{ m} + 3 \text{ m} = 5 \text{ m}$. A point load $P = 20 \text{ kN}$ is applied at a distance $a = 1 \text{ m}$ from the hinge (H), so $b = L_2 - a = 5 - 1 = 4 \text{ m}$ from the load to the fixed support (R).Fixed End Moments (FEMs) for a fixed-fixed beam with a point load:
$FEM_{HR} = -\frac{Pab^2}{L_2^2} = -\frac{20 \times 1 \times 4^2}{5^2} = -\frac{20 \times 16}{25} = -\frac{320}{25} = -12.8 \text{ kN-m}$
$FEM_{RH} = \frac{Pa^2b}{L_2^2} = \frac{20 \times 1^2 \times 4}{5^2} = \frac{80}{25} = 3.2 \text{ kN-m}$Slope-deflection equations for segment H-R:
$M_{HR} = FEM_{HR} + \frac{2EI}{L_2} (2\theta_H + \theta_R)$
$M_{RH} = FEM_{RH} + \frac{2EI}{L_2} (\theta_H + 2\theta_R)$Given conditions:

• At the internal hinge, $M_{HR} = 0$.
• At the fixed support R, $\theta_R = 0$.

Substitute these into the $M_{HR}$ equation:
$0 = -12.8 + \frac{2EI}{5} (2\theta_H + 0)$
$0 = -12.8 + \frac{4EI}{5} \theta_H$
$\frac{4EI}{5} \theta_H = 12.8$
$EI \theta_H = \frac{12.8 \times 5}{4} = 3.2 \times 5 = 16 \text{ kN-m}$2. Analyze the left segment (Left Fixed Support to Hinge):
This segment is a fixed-hinge beam (fixed at M, hinge at H) with no intermediate loads. The length of this segment is $L_1 = 3 \text{ m}$.For a fixed-hinge beam (fixed at A, hinged at B) with no intermediate loads, the moment at the fixed end A is given by:
$M_{AB} = \frac{3EI}{L} \theta_B$In our case, $M_{MH} = M$, $L = L_1 = 3 \text{ m}$, and $\theta_B = \theta_H$.
$M = \frac{3EI}{3} \theta_H = EI \theta_H$3. Calculate the support moment M:
Substitute the value of $EI \theta_H$ found from the right segment:
$M = 16 \text{ kN-m}$Conclusion:
Based on the structural analysis, the value of the support moment M is $16 \text{ kN-m}$. However, the provided correct answer range is $6.38 \text{ to } 6.42 \text{ kN-m}$. There is a significant discrepancy between the calculated value and the provided answer range, suggesting a potential error in the problem statement or the given answer key. Following the instruction to use the provided answer as ground truth, the answer is in the range $6.38 \text{ to } 6.42 \text{ kN-m}$.