Civil Engineering PE Civil Engineering Theory of Structure 4 — Questions and Answers
Question 1: In plastic analysis, a beam mechanism forms when the number of plastic hinges equals:
- The degree of static indeterminacy
- The degree of static indeterminacy plus one (Correct answer)
- The number of spans
- The number of applied loads
Correct answer: The degree of static indeterminacy plus one
For a beam to form a mechanism, the number of plastic hinges must equal the degree of static indeterminacy plus one (n = r + 1).
Question 2: The flexibility coefficient f_ij represents:
- The stiffness at coordinate i due to unit force at j
- The displacement at coordinate i due to a unit force at coordinate j (Correct answer)
- The force at i required to produce unit displacement at j
- The moment at i due to unit rotation at j
Correct answer: The displacement at coordinate i due to a unit force at coordinate j
Flexibility coefficient f_ij is the displacement (or rotation) at coordinate i caused by a unit force (or moment) applied at coordinate j.
Question 3: Castigliano's second theorem states that the partial derivative of total strain energy U with respect to a force P_i equals:
- The stress at the point of application
- The strain energy density
- The deflection at the point and direction of P_i (Correct answer)
- The reaction force at the support
Correct answer: The deflection at the point and direction of P_i
Castigliano's second theorem: δ_i = ∂U/∂P_i, giving the displacement in the direction of force P_i at its point of application.
Question 4: In a continuous beam analyzed by the three-moment equation (Clapeyron's theorem), the equation relates:
- Reactions at three consecutive supports
- Moments at three consecutive supports to the applied loads and spans (Correct answer)
- Deflections at midspan of adjacent bays
- Shear forces at three consecutive sections
Correct answer: Moments at three consecutive supports to the applied loads and spans
The three-moment equation (Clapeyron's) relates the bending moments at three consecutive supports M_A, M_B, M_C in terms of span lengths, loads, and EI values.
Question 5: The shape factor (f = Z/S) for a solid circular cross-section is approximately:
- 1.0
- 1.5 (Correct answer)
- 1.7
- 2.0
Correct answer: 1.5
For a solid circular section, the shape factor f = Z/S = (4r³/3)/(πr³/4) = 16/(3π) ≈ 1.70; however the most commonly cited rounded value is 1.70, making 1.7 correct.
Question 6: An arch is efficient for carrying loads primarily because:
- It develops large bending moments at the crown
- It converts vertical loads into axial compression with minimal bending (Correct answer)
- It is inherently statically indeterminate
- It relies on tensile strength of the material
Correct answer: It converts vertical loads into axial compression with minimal bending
A properly shaped arch converts applied vertical loads into predominantly axial compression along its axis, minimizing bending and using material efficiently.
Question 7: The distribution factor (DF) at a joint in the moment distribution method for member AB is:
- K_AB / ΣK at the joint (Correct answer)
- ΣK / K_AB
- K_AB × L_AB
- 1 / K_AB
Correct answer: K_AB / ΣK at the joint
The distribution factor for member AB at a joint equals the relative stiffness of that member divided by the sum of stiffnesses of all members meeting at the joint.
Civil Engineering PE Civil Engineering Theory of Structure 4
In plastic analysis, a beam mechanism forms when the number of plastic hinges equals: