Structure 1 with member BC of length $L / 2$ /2 has force $P_{1}$ applied at joint C. Structure 2 with member BC of length L has force $P_{2}$ applied at C.
Both structures have constant flexural rigidity EI. If the ratio of the
Applied forces $P_{1} / P_{2}=5 / 2$ , the ratio of the joint B reactions $R_{B 1} / R_{B 2}$ is
Approximately:
A) 1
B) 5/4
C) 3/2
D) 2

Question

Quizplus · Accepted Answer

The deflection of a beam (δ) due to a point load P at any point is given by the formula $δ = \frac{P \cdot L^3}{3EI}$ for a simply supported beam, where L is the length of the beam, E is the modulus of elasticity, I is the moment of inertia, and P is the applied force. The reaction at the support B for both structures can be considered proportional to the deflection caused by the applied forces $P_1$ and $P_2$, since both structures have constant flexural rigidity EI.Given that $P_1 / P_2 = 5 / 2$ and the length of member BC in structure 1 is $L/2$, the deflection caused by $P_1$ in structure 1 can be compared to the deflection caused by $P_2$ in structure 2 by adjusting for the length difference. The deflection formula adjusted for length gives us $\delta \proportional P \cdot L^3$, so when the length is halved, the effect on deflection (and thus on the reaction at B) is reduced by a factor of $1/2^3 = 1/8$.Therefore, for structure 1 with force $P_1$ and length $L/2$, the reaction at B compared to structure 2 with force $P_2$ and length L, taking into account the force ratio of $5/2$ and the length effect, gives us a combined factor of $(5/2) \cdot (1/8) = 5/16$. However, since the length effect significantly reduces the deflection and thus the reaction force at B for structure 1, we need to adjust our understanding based on the direct comparison of forces and the structural response, which is more straightforwardly related to the force applied rather than the deflection formula directly.Given the complexity of directly applying the deflection formula to determine the reaction at B without considering the specific structural setup and load distribution, the correct approach involves understanding how the force and length differences affect the reactions at the supports. The force ratio of $5/2$ suggests that for equal lengths, the reaction would directly scale with the force. However, the difference in lengths (and thus the leverage effect) between the two structures complicates this direct comparison.Considering the principles of static equilibrium and beam deflection without overcomplicating the analysis with the detailed deflection formula leads to recognizing that the reaction at B for both structures will indeed be influenced by both the applied force and the length of the beam, but the exact ratio of $R_{B1} / R_{B2}$ requires a more nuanced understanding of structural behavior than provided by a straightforward application of the deflection formula. The correct answer, based on the information given and typical structural analysis principles, would focus on the proportional relationship between the forces and the structural responses, leading to the selection of choice B) 5/4 as an approximation of how the reactions at B compare, acknowledging the simplification in the explanation.

Structure 1 with Member BC of Length L/2L / 2L/2 /2 Has Force

Structure 1 with Member BC of Length $L / 2$ /2 Has Force