MCQ questions on the content of Chapters 7, 8, and 9 of the book, "Aircraft
Structures for engineering students."
Chapter 7: Bending of Thin Plates
1. The flexural rigidity (D) of a thin plate is defined by which formula?
a) D = Et / (1 - ν)
b) D = Et³ / 12
c) D = Et³ / [12(1 - ν²)]
d) D = E / (1 - ν²)
Answer: c
2. For a thin plate subjected to pure bending, what is the primary assumption regarding its
cross-sections?
a) They become curved after bending.
b) They remain plane and normal to the longitudinal fibers after bending.
c) They experience significant shear deformation.
d) They are only valid for thick plates.
Answer: b
3. When a thin plate bends into a surface with curvatures of opposite signs (e.g., like a
saddle), it is known as:
a) Synclastic bending
b) Isotropic bending
c) Planar bending
d) Anticlastic bending
Answer: d
4. The governing differential equation for the small deflection (w) of a thin plate subjected
to a distributed transverse load (q) is:
a) ∇²w = q/D
b) ∇⁴w = q/D
c) ∇²w = qD
d) ∇w = q/D
Answer: b
5. What are the boundary conditions for a simply supported edge of a plate, for example,
along the line x = 0?
a) w = 0 and ∂w/∂x = 0
b) w = 0 and ∂²w/∂x² = 0
c) ∂w/∂x = 0 and ∂²w/∂x² = 0
d) Mx = 0 and Qx = 0
Answer: b
6. For a clamped or built-in edge of a plate (e.g., at x = 0), the boundary conditions are:
a) w = 0 and ∂²w/∂x² = 0
b) Mx = 0 and Mxy = 0
c) w = 0 and ∂w/∂x = 0
d) Qx = 0 and w = 0
Answer: c
7. The relationship between the twisting moment (Mxy) and the plate deflection (w) is given
by:
a) Mxy = -D(1 - ν)(∂²w/∂x∂y)
b) Mxy = -D(∂²w/∂x²)
c) Mxy = D(1 + ν)(∂w/∂x)
d) Mxy = -D(∂w/∂y)
Answer: a
8. When using the energy method for plate bending, the total potential energy of the system
is the sum of:
a) Kinetic energy and potential energy.
b) Strain energy and potential energy of the load.
c) Only the strain energy due to bending.
d) Only the potential energy of the transverse load.
Answer: b
9. In the equation Mx = -D(∂²w/∂x² + ν∂²w/∂y²), what does ∂²w/∂x² represent?
a) The slope of the plate in the x-direction.
b) The curvature of the plate in the xz-plane.
c) The rate of twist of the plate.
d) The shear force in the x-direction.
Answer: b
10. A plate with a small initial curvature (w₀) subjected to in-plane compressive loads will
experience:
a) No additional deflection until the buckling load is reached.
b) Immediate additional bending deflection (w₁).
c) A reduction in the initial curvature.
d) Only shear stresses.
Answer: b
11. The strain energy in a thin plate is primarily due to:
a) Shear deformation only.
b) Axial stretching of the mid-plane only.
c) Bending and twisting moments.
d) The potential energy of the applied loads.
Answer: c
12. What are principal moments in a plate?
a) The moments that cause the maximum shear stress.
b) The moments acting on planes where the twisting moment is zero.
c) The moments acting parallel to the plate edges.
d) The sum of Mx and My.
Answer: b
13. For a thin plate, the potential energy (V) of a transverse load (q) is calculated by the
integral:
a) V = ∫∫ wq dx dy
b) V = -∫∫ wq dx dy
c) V = ∫∫ (1/2)wq² dx dy
d) V = -∫∫ (1/2)w²q dx dy
Answer: b
14. The potential energy due to an in-plane compressive load Nx is proportional to:
a) ∫∫ Nx(∂w/∂x) dx dy
b) ∫∫ Nx(∂²w/∂x²) dx dy
c) ∫∫ (1/2)Nx(∂w/∂x)² dx dy
d) ∫∫ (1/2)Nx w² dx dy
Answer: c
15. The term ∇⁴w in the plate bending equation is known as the:
a) Laplacian operator.
b) Gradient operator.
c) Biharmonic operator.
d) Divergence operator.
Answer: c
Chapter 8: Columns
16. The Euler buckling load (PCR) for a perfectly straight, pin-ended column is given by:
a) PCR = πEI / L²
b) PCR = π²EI / L
c) PCR = 2π²EI / L²
d) PCR = π²EI / L²
Answer: d
17. The slenderness ratio of a column is defined as:
a) L / A (Length / Area)
b) L / I (Length / Second moment of area)
c) Le / r (Effective length / Radius of gyration)
d) A / r (Area / Radius of gyration)
Answer: c
18. For a column fixed at both ends, the effective length (Le) is:
a) Le = L
b) Le = 2L
c) Le = 0.5L
d) Le = 0.7L
Answer: c
19. The point on a load-deflection graph for a perfect column where it can either remain
straight or buckle is known as a:
a) Yield point.
b) Bifurcation point.
c) Proportional limit.
d) Fracture point.
Answer: b
20. When a column's critical stress is above the material's proportional limit, its buckling
behavior is described as:
a) Elastic buckling.
b) Torsional buckling.
c) Inelastic buckling.
d) Plastic collapse.
Answer: c
21. The Tangent Modulus Theory for inelastic buckling uses which modulus in the Euler
formula?
a) Young's Modulus (E)
b) Shear Modulus (G)
c) Tangent Modulus (Et)
d) Reduced Modulus (Er)
Answer: c
22. For predicting the buckling load of real columns with imperfections, which theory is
generally found to give the most accurate results?
a) The Reduced Modulus Theory.
b) The basic Euler Theory.
c) The Tangent Modulus Theory.
d) The Secant Modulus Theory.
Answer: c
23. A Southwell plot is used for:
a) Calculating the yield stress of a material.
b) The experimental determination of the critical buckling load of an imperfect column.
c) Plotting stress-strain curves for brittle materials.
d) Determining the fatigue life of a component.
Answer: b
24. A "beam-column" is a structural member subjected to:
a) Only pure bending.
b) Only pure torsion.
c) Only axial compression.
d) Both transverse loads and axial loads.
Answer: d
25. As the compressive axial load (P) on a beam-column approaches the critical buckling load
(PCR), the bending moment and deflection:
a) Decrease to zero.
b) Remain constant.
c) Increase, theoretically, to infinity.
d) Become unpredictable.
Answer: c
26. The energy method for calculating buckling loads is based on the principle that at the
point of neutral equilibrium, the total potential energy is:
a) At a maximum.
b) Zero.
c) At a stationary value.
d) Equal to the kinetic energy.
Answer: c
27. For a column that is fixed at one end and free at the other, the effective length (Le) is:
a) Le = L
b) Le = 2L
c) Le = 0.5L
d) Le = L / 2
Answer: b
28. The Euler buckling formula is valid only as long as the critical stress (σCR) is:
a) Greater than the ultimate tensile stress.
b) Equal to the yield stress.
c) Less than the proportional limit of the material.
d) Greater than the shear stress.
Answer: c
29. In the reduced modulus theory of inelastic buckling, the cross-section is assumed to
behave as if:
a) It were made of a completely different material.
b) It were nonhomogeneous, with modulus E on the convex side and Et on the concave side.
c) The entire section has a modulus of Et.
d) The column is perfectly plastic.
Answer: b
30. What is the effect of an initial imperfection (e.g., initial crookedness) on the behavior of a
column under compressive load?
a) It has no effect until the Euler load is reached.
b) It increases the critical buckling load.
c) It causes the column to start bending as soon as the load is applied.
d) It prevents the column from buckling.
Answer: c
31. Flexural-torsional buckling is a failure mode most associated with:
a) Solid, circular columns.
b) Short, thick columns.
c) Thin-walled columns of open cross-section.
d) Doubly-symmetrical I-beams.
Answer: c
32. For a column with an eccentric load P, the maximum bending moment in the column is a
function of:
a) P and e only.
b) P, e, and sec(mL/2).
c) P, e, and tan(mL/2).
d) P and L only.
Answer: b
33. The different buckled shapes (modes) a column can take correspond to:
a) Different materials.
b) Different eigenvalues of the governing differential equation.
c) Different loading directions.
d) Different temperatures.
Answer: b
Chapter 9: Thin Plates (Buckling and Tension Fields)
34. The critical buckling stress (σCR) for a thin, flat plate is given by:
The buckling coefficient k depends on:
a) Only the material's Young's Modulus.
b) The plate's aspect ratio (a/b), loading, and boundary conditions.
c) Only the thickness (t) and width (b) of the plate.
d) The temperature of the plate.
Answer: b
35. The minimum value of the buckling coefficient k for a simply supported flat plate under
compression is:
a) 1.0
b) 2.5
c) 4.0
d) 7.2
Answer: c
36. Local instability in thin-walled columns refers to:
a) The entire column buckling as a single unit.
b) The buckling of individual plate elements (flanges and webs) of the column's cross-section.
c) The yielding of the material at the corners.
d) The torsional failure of the column.
Answer: b
37. A "tension field beam" is a beam where:
a) The flanges are in tension and the web is in compression.
b) The web has buckled under shear but continues to carry load through diagonal tension.
c) The entire beam is under a uniform tensile load.
d) The beam is designed to fail in tension.
Answer: b
38. In a complete diagonal tension field, what is the assumed state of the web?
a) It carries only pure shear stress.
b) It is incapable of carrying any load after buckling.
c) It is wrinkled and carries only diagonal tensile stress.
d) It carries only compressive stress.
Answer: c
39. In a tension field beam, what is the primary function of the vertical stiffeners?
a) To carry the applied shear load directly.
b) To resist the compressive forces induced by the diagonal tension in the web.
c) To increase the bending stiffness of the beam.
d) To prevent the flanges from buckling.
Answer: b
40. The angle of the diagonal tension field (α) in a tension field beam adjusts itself to:
a) Always be 45 degrees.
b) Maximize the shear stress in the web.
c) Minimize the total strain energy of the beam.
d) Align with the direction of the applied load.
Answer: c
41. How do longitudinal stiffeners increase the buckling stress of a wide skin panel?
a) By increasing the panel's thickness.
b) By reducing the effective width 'b' of the individual plate elements.
c) By carrying all the compressive load themselves.
d) By increasing the material's Young's Modulus.
Answer: b
42. In a tension field beam, the flanges are subjected to the primary bending moment and
what additional load?
a) A uniform torsional moment.
b) An axial compressive load from the diagonal tension field.
c) A transverse shear load.
d) A uniform tensile load along their length.
Answer: b
43. For a thin-walled column section, what is the term for buckling where the wavelength is
on the order of the element widths, leading to a change in cross-sectional shape?
a) Euler buckling
b) Primary instability
c) Torsional instability
d) Local instability
Answer: d
44. Primary instability in a column is characterized by:
a) Localized wrinkling of the skin.
b) Buckling of the entire element without a change in cross-sectional shape.
c) Changes in cross-sectional area.
d) Inter-rivet buckling.
Answer: b