Civil Engineering PE Civil Engineering Theory of Structure 5 — Questions and Answers
Question 1: For a cable carrying a uniformly distributed horizontal load w over span L with a central sag d, the horizontal tension H at the supports is:
- wL²/4d
- wL²/8d (Correct answer)
- wL/2d
- wL²/2d
Correct answer: wL²/8d
For a parabolic cable with UDL, the horizontal component of cable tension is H = wL²/(8d), derived from moment equilibrium at midspan.
Question 2: The upper bound (kinematic) theorem of plasticity states that a load computed by:
- Satisfying equilibrium only is a lower bound on collapse load
- Assuming a valid mechanism is an upper bound on collapse load (Correct answer)
- Satisfying both equilibrium and yield is the true collapse load
- Maximizing internal work is the true collapse load
Correct answer: Assuming a valid mechanism is an upper bound on collapse load
The kinematic (upper bound) theorem states that a collapse load calculated from any assumed valid mechanism is greater than or equal to the true collapse load.
Question 3: For a beam-column subjected to combined axial load P and moment M, the interaction equation used in design checks the condition that:
- P/P_n + M/M_n ≤ 1.0 (Correct answer)
- P/P_n − M/M_n ≤ 1.0
- P·M ≤ P_n·M_n
- P + M ≤ 1.0
Correct answer: P/P_n + M/M_n ≤ 1.0
The linear interaction equation P/P_n + M/M_n ≤ 1.0 ensures combined axial and bending demands remain within the member's capacity.
Question 4: The theorem of three moments assumes that the slope of the elastic curve is continuous at:
- Midspan of each bay
- Each intermediate support (Correct answer)
- The points of inflection
- The free ends of the beam
Correct answer: Each intermediate support
The three-moment equation enforces compatibility by requiring the slope of the deflection curve to be continuous (equal from both sides) at each intermediate support.
Question 5: Which of the following correctly describes the stiffness of a member with a pin at the far end versus a fixed far end?
- Far-end pinned stiffness = (3/4) × far-end fixed stiffness (Correct answer)
- Far-end pinned stiffness = (4/3) × far-end fixed stiffness
- Both have the same stiffness
- Far-end pinned stiffness = (1/2) × far-end fixed stiffness
Correct answer: Far-end pinned stiffness = (3/4) × far-end fixed stiffness
For a prismatic member, releasing the far-end fixity to a pin reduces the near-end rotational stiffness from 4EI/L to 3EI/L, which is 3/4 of the fixed-far-end value.
Question 6: A propped cantilever of span L with UDL w has its prop removed. The released structure is a cantilever. Using compatibility, the prop reaction R is found to satisfy:
- Deflection at prop due to R equals deflection at prop due to w alone
- Deflection at prop due to R plus deflection at prop due to w equals zero (Correct answer)
- Slope at fixed end equals zero
- Moment at free end equals zero
Correct answer: Deflection at prop due to R plus deflection at prop due to w equals zero
The compatibility condition is that the net deflection at the prop location is zero: δ_w (downward from UDL) + δ_R (upward from prop reaction) = 0.
Question 7: In a space truss, the condition for static determinacy is:
- m + r = 2j
- m + r = 3j (Correct answer)
- m + r = 6j
- m = 2j − 3
Correct answer: m + r = 3j
For a space (3D) truss, equilibrium at each joint provides 3 equations, so static determinacy requires m + r = 3j.
Civil Engineering PE Civil Engineering Theory of Structure 5
For a cable carrying a uniformly distributed horizontal load w over span L with a central sag d, the horizontal tension H at the supports is: