Question 1

Elastic energy terms: identify correct statements about strain energy, proof resilience, and modulus of resilience  Consider the standard definitions used in strength of materials. Select the correct group of statements.

Accepted Answer

Definitions Strain energy: total elastic energy stored in a body under load (recoverable).Proof resilience: maximum strain energy up to elastic limit.Modulus of resilience: proof resilience per unit volume. Final Answer All of the above statements are correct.

Question 2

Beam theory: meaning of point of contraflexure (inflection point)  On a bending moment diagram for a beam, identify what occurs at the point of contraflexure.

Accepted Answer

Concept The point of contraflexure (inflection) is where the curvature changes sign; equivalently, the bending moment diagram crosses zero. Final Answer Bending moment changes sign (passes through zero).

Question 3

Solid circular shaft: torque–diameter relation at allowable shear stress  For a solid circular shaft of diameter D, limited by maximum allowable shear stress τ, state the formula for the maximum torque transmitted.

Accepted Answer

Concept/Formula For a solid circular shaft: T = (τ/J)·(J·R) = τ·(J/R).Polar moment J = (π/32)·D⁴, outer radius R = D/2 ⇒ J/R = (π/16)·D³.Thus, T = (π/16)·τ·D³. Final Answer (π/16)·τ·D³.

Question 4

Stability of columns: applicability of Euler's buckling formula  State for which type of column Euler's critical load formula is valid, based on slenderness.

Accepted Answer

Concept Euler's buckling formula assumes elastic instability and is valid for columns with high slenderness ratio (long/slender), where buckling governs before material yield or crushing. Final Answer Long (slender) columns.

Question 5

Plane stress analysis: compute the maximum normal stress (principal stress) for a biaxial state with shear A body is under σ_x = 1200 MPa (tension) and σ_y = 600 MPa (tension) on mutually perpendicular planes. A shear stress τ

Accepted Answer

Given σx = 1200 MPa, σy = 600 MPa, τxy = 400 MPa. Approach Principal stresses for plane stress: σ1,2 = (σx + σy)/2 ± √{ [ (σx − σy)/2 ]² + τxy² }. Step-by-step calculation σavg = (1200 + 600)/2 = 900 MPa.Δ = (σx − σy)/2 = (1200 − 600)/2 = 300 MPa.Radical = √(300² + 400²) = √(90,000 + 160,000) = √250,000 = 500 MPa.σ1 = 900 + 500 = 1400 MPa; σ2 = 900 − 500 = 400 MPa. Verification Mohr’s circle would have center at 900 MPa and radius 500 MPa, consistent with σ1 and σ2 above. Final Answer 1400 MPa.

Question 6

Thick cylinder (Lame’s theory): Maximum tangential (hoop) stress at the inner surface due to internal pressure.  Given: a thick cylindrical shell with inner radius r_i, outer radius r_o, internal pressure p, and negligible external pressure. Choose …

Accepted Answer

Given data Thick cylinder with inner radius r_i and outer radius r_o. Internal pressure = p; external pressure ≈ 0. Concept/Approach Use Lame’s equations for thick cylinders: σ_r = A − B/r^2 and σ_θ = A + B/r^2. Step-by-step derivation Boundary 1 (at r = r_i): σ_r(ri) = −p ⇒ −p = A − B/r_i^2.Boundary 2 (at r = r_o): σ_r(ro) = 0 ⇒ 0 = A − B/r_o^2 ⇒ A = B/r_o^2.Substitute A into Boundary 1: −p = (B/r_o^2) − (B/r_i^2) = B,[1/r_o^2 − 1/r_i^2].Solve for B: B = p, r_i^2 r_o^2 / (r_o^2 − r_i^2).Then A = B/r_o^2 = p, r_i^2 / (r_o^2 − r_i^2).Hoop stress at the inner surface: σ_θ(ri) = A + B/r_i^2 = p(r_o^2 + r_i^2)/(r_o^2 − r_i^2). Common pitfalls Mixing up radial and hoop stresses or sign conventions leads to wrong expressions. Final Answer σ_θ(ri) = p × (r_o^2 + r_i^2) / (r_o^2 − r_i^2)

Question 7

Composite torsional shaft: Equivalent torsional spring constant for two serial segments between rigid ends.  Given two stepped portions with torsional spring constants k₁ and k₂ in series (same torque, angular twists add). Choose the correct express…

Accepted Answer

Given data Two shaft segments in series with torsional spring constants k₁ and k₂. Concept/Approach For springs in series (same torque T), total twist θ = θ₁ + θ₂ = T/k₁ + T/k₂. Step-by-step Define k_eq by T/θ = k_eq ⇒ 1/k_eq = 1/k₁ + 1/k₂.Therefore, k_eq = (k₁ k₂)/(k₁ + k₂). Verification If k₁ = k₂ = k, then k_eq = k/2 (softer than either alone), which is consistent for series combination. Final Answer k_eq = (k₁ k₂)/(k₁ + k₂)

Question 8

In a simple bending theory, one of the assumption is that the material of the beam is isotropic. This assumption means that the

Accepted Answer

elastic constants are same in all the directions

Question 9

For the beam shown in the below figure, the shear force at A is equal to

Accepted Answer

wl/3

Question 10

A welded joint as compared to a riveted joint has __________ strength.

Accepted Answer

more

Question 11

A body is subjected to two normal stresses 20 kN/m2 (tensile) and 10 kN/m2 (compressive) acting perpendicular to each other. The maximum shear stress is

Accepted Answer

15 kN/m2

Question 12

When a cantilever beam is loaded at its free end, the maximum compressive stress shall develop at

Accepted Answer

bottom fibre

Question 13

A beam of uniform strength has

Accepted Answer

same bending stress at every section

Question 14

In the below figure, the stress corresponding to point D is

Accepted Answer

ultimate stress

Question 15

Two shafts 'A' and 'B' are made of same material. The shaft 'A' is solid and has diameter D. The shaft 'B' is hollow with outer diameter D and inner diameter D/2. The strength of hollow shaft in torsion is __________ as that of solid shaft.

Accepted Answer

15/16

Question 16

The extremeties of any diameter on Mohr's circle represent

Accepted Answer

normal stresses on planes at 45°

Question 17

The ductility of a material __________ with the increase in percentage reduction in area of a specimen under tensile test.

Accepted Answer

increases

Question 18

Principle plane is a plane on which the shear stress is

Accepted Answer

zero

Question 19

The ratio of change in volume to the original volume is called

Accepted Answer

volumetric strain

Question 20

The shear force diagram for a simply supported beam carrying a uniformly distributed load of w per unit length, consists of

Accepted Answer

two right angled triangles

Strength of Materials Questions