BCE Structural Analysis 4 — Questions and Answers
Question 1: In matrix stiffness analysis, the element stiffness matrix for a truss member of length L, area A, and modulus E is of size:
- 2×2
- 4×4 (Correct answer)
- 6×6
- 3×3
Correct answer: 4×4
A truss member has 2 DOFs per node (x and y translation) and 2 nodes, giving a 4×4 element stiffness matrix in 2D.
Question 2: The three-moment equation (Clapeyron's theorem) is used to analyze:
- Trusses with multiple panels
- Continuous beams over multiple spans (Correct answer)
- Portal frames under lateral loads
- Arch bridges under moving loads
Correct answer: Continuous beams over multiple spans
Clapeyron's three-moment theorem relates the moments at three successive supports of a continuous beam, enabling indeterminate analysis span by span.
Question 3: A structure is geometrically unstable if:
- It has more reactions than equilibrium equations
- All reactions are parallel or concurrent even if the reaction count is sufficient (Correct answer)
- It is statically indeterminate to the first degree
- Its stiffness matrix has a large determinant
Correct answer: All reactions are parallel or concurrent even if the reaction count is sufficient
Even with the correct number of reactions, a structure is geometrically unstable if all reaction lines of action are parallel or pass through a common point.
Question 4: In the stiffness method, the joint equilibrium equation is written as:
- [K]{F} = {d}
- [K]{d} = {F} (Correct answer)
- [F]{K} = {d}
- [d]{K} = {F}
Correct answer: [K]{d} = {F}
The stiffness method forms [K]{d} = {F}, where [K] is the global stiffness matrix, {d} the displacement vector, and {F} the load vector.
Question 5: Which of the following correctly defines the flexibility coefficient f_ij?
- Force at i due to unit displacement at j
- Displacement at i due to a unit force applied at j (Correct answer)
- Stiffness at i due to unit rotation at j
- Reaction at support i due to unit load at j
Correct answer: Displacement at i due to a unit force applied at j
The flexibility coefficient f_ij is the displacement at coordinate i caused by a unit force applied at coordinate j, with all other loads absent.
Question 6: The plastic hinge in a steel beam forms at the section where:
- Shear force is maximum
- The bending moment reaches the plastic moment capacity M_p (Correct answer)
- Deflection is maximum
- Slope is zero
Correct answer: The bending moment reaches the plastic moment capacity M_p
A plastic hinge forms when the cross-section's bending moment reaches M_p, causing unlimited rotation at constant moment.
Question 7: For a simply supported beam of span L with a central point load P, the maximum bending moment is:
- PL/8
- PL/4 (Correct answer)
- PL/2
- PL
Correct answer: PL/4
Each support reaction is P/2, and taking moments to midspan gives a maximum moment of (P/2)(L/2) = PL/4.
BCE Structural Analysis 4
In matrix stiffness analysis, the element stiffness matrix for a truss member of length L, area A, and modulus E is of size: