trength of Materials (SOM) – GSSC

Technical Assistant (Civil)

1. Basics of Strength of Materials
 Definition: Study of behavior of materials under loads (tension, compression, shear,
bending, torsion).
 Stress (σ): Force per unit area

σ=PA\sigma = \frac{P}{A}σ=AP

o Units: N/mm² or MPa

 Strain (ε): Deformation per unit length

ε=ΔLL\varepsilon = \frac{\Delta L}{L}ε=LΔL

 Hooke’s Law:

σ=E⋅ε\sigma = E \cdot \varepsilonσ=E⋅ε

o EEE = Modulus of Elasticity

Types of Stress:

 Tensile, Compressive, Shear, Bearing, Bending

 Factor of Safety (FS):

FS=Failure LoadWorking LoadFS = \frac{\text{Failure Load}}{\text{Working

Load}}FS=Working LoadFailure Load

2. Axial Load
 Axial Stress: σ=PA\sigma = \frac{P}{A}σ=AP
 Elongation/Shortening: ΔL=PLAE\Delta L = \frac{PL}{AE}ΔL=AEPL
 Poisson’s Ratio (ν): Lateral strain / Longitudinal strain
 Volumetric Strain:
εv=εx+εy+εz\varepsilon_v = \varepsilon_x + \varepsilon_y + \varepsilon_zεv=εx+εy
+εz

3. Shear Stress & Strain

  Direct Shear:
τ=VA\tau = \frac{V}{A}τ=AV
 Torsion of Circular Shaft:

τmax=T⋅rJ,θ=TLGJ\tau_{\text{max}} = \frac{T \cdot r}{J}, \quad \theta =

\frac{TL}{GJ}τmax=JT⋅r,θ=GJTL

Where:

o J=πd432J = \frac{\pi d^4}{32}J=32πd4 for solid shaft

o G=G = G= Modulus of Rigidity

4. Bending of Beams
 Bending Stress:

σ=MyI\sigma = \frac{M y}{I}σ=IMy

Where:

o MMM = Bending moment

o yyy = Distance from neutral axis
o III = Moment of inertia
 Section Modulus:

Z=IyZ = \frac{I}{y}Z=yI

 Deflection:

δ=PL348EI(for simply supported beam with central load)\delta = \frac{PL^3}{48EI}

\quad \text{(for simply supported beam with central load)}δ=48EIPL3
(for simply supported beam with central load)

 Types of Bending:
o Sagging (concave up)
o Hogging (concave down)

5. Combined Stresses
 Principal Stress (σ₁, σ₂):

σ1,2=σx+σy2±(σx−σy2)2+τxy2\sigma_{1,2} = \frac{\sigma_x + \sigma_y}{2} \pm

\sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^2 + \tau_{xy}^2}σ1,2=2σx+σy±(2σx
−σy)2+τxy2
  Maximum Shear Stress:
τmax=(σx−σy2)2+τxy2\tau_{\text{max}} = \sqrt{\left(\frac{\sigma_x -
\sigma_y}{2}\right)^2 + \tau_{xy}^2}τmax=(2σx−σy)2+τxy2

6. Columns (Slender Members)

 Euler’s Buckling Load:

Pcr=π2EI(KL)2P_{cr} = \frac{\pi^2 EI}{(KL)^2}Pcr=(KL)2π2EI

Where KKK = Effective length factor

 Slenderness Ratio:

λ=Leffr,r=IA\lambda = \frac{L_{\text{eff}}}{r}, \quad r = \sqrt{\frac{I}{A}}λ=rLeff

,r=AI

7. Springs
 Helical Spring (Axial Load):
o Deflection: δ=8WD3nGd4\delta = \frac{8WD^3 n}{G d^4}δ=Gd48WD3n
o Stress: τ=8WDπd3K\tau = \frac{8 W D}{\pi d^3} Kτ=πd38WDK (K =
Wahl’s factor)

8. Energy Methods
 Strain Energy in Axial Members:

U=σ22E⋅V=P2L2AEU = \frac{\sigma^2}{2E} \cdot V = \frac{P^2 L}{2AE}U=2Eσ2

⋅V=2AEP2L

 Castigliano’s Theorem: For deflection:

δ=∂U∂P\delta = \frac{\partial U}{\partial P}δ=∂P∂U

9. Important Formulas – Quick Revision

Topic Formula
Axial Load σ = P/A, ΔL = PL/AE
 Topic Formula
Shear τ = V/A
Torsion τ = Tr/J, θ = TL/GJ
Bending σ = My/I, δ = PL³/48EI
Column Pcr = π² EI/(KL)²
Spring δ = 8WD³n/Gd⁴, τ = 8WD/πd³ K

10. Exam Tips

 Always check units consistency.
 For GSSC TA, focus on formulas + direct application MCQs.
 Draw beam sketches for bending, shear, torsion to avoid mistakes.
 Memorize simple cases for deflection and maximum stress.

Strength of Materials – GSSC TA (Civil)

MCQs – Complete Set

1. Axial Load & Stress

1. Tensile stress formula: σ = P/A ✅
2. Elongation of bar: ΔL = PL/AE ✅
3. A steel rod 2 m long, 20 mm diameter, axial load 50 kN → σ ≈ 159 MPa ✅
4. Poisson’s ratio: ν = lateral strain / longitudinal strain ✅
5. Volumetric strain: εv = εx + εy + εz ✅

2. Shear Stress
6. Direct shear: τ = V/A ✅
7. Maximum shear in solid circular shaft: τmax = 16T/πd³ ✅
8. Maximum shear in hollow circular shaft: τmax = 16T/do³(1-(di/do)⁴) ✅
9. Angle of twist: θ = TL/GJ ✅
10. Hollow shafts preferred → stronger in torsion per unit weight ✅

3. Bending of Beams
11. Bending stress: σ = My/I ✅
 12. Section modulus: Z = I/y ✅
13. Max bending stress: σmax = M/Z ✅
14. Simply supported beam, central load → δ = PL³/48EI ✅
15. Cantilever, end load → δ = PL³/3EI ✅
16. Sagging → top fibers compressed, bottom stretched ✅
17. Hogging → top fibers stretched, bottom compressed ✅
18. Simply supported, uniform load → max bending moment at mid-span ✅
19. Simply supported, point load at mid-span → Mmax = PL/4 ✅
20. Moment of inertia – rectangle: I = bd³/12 ✅
21. Moment of inertia – circle: I = πd⁴/64 ✅

4. Torsion
22. Torsion formula: τ = Tr/J ✅
23. Polar moment of inertia – solid: J = πd⁴/32 ✅
24. Polar moment of inertia – hollow: J = π(do⁴-di⁴)/32 ✅
25. Angle of twist: θ = TL/GJ ✅
26. Torsion constant G → modulus of rigidity ✅
27. Hollow shafts → lighter and stronger in torsion ✅

5. Columns
28. Euler’s formula (pin–pin): Pcr = π² EI / L² ✅
29. General Euler’s formula: Pcr = π² EI / (KL)² ✅
30. K factor:

 Pin–pin → K=1
 Fixed–fixed → K=0.5
 Fixed–free → K=2
 Fixed–pin → K=0.7 ✅

31. Slenderness ratio: λ = L/r, r = √(I/A) ✅

32. Short columns fail by compression, long columns by buckling ✅

6. Springs
33. Helical spring deflection: δ = 8WD³n / Gd⁴ ✅
34. Shear stress in spring: τ = 8WD/πd³ K ✅
35. Wahl factor K → accounts for stress concentration ✅
 36. Spring stiffness (k) → k = F/δ ✅
37. Close-coiled spring → shear stress dominates ✅

7. Combined Stresses
38. Principal stresses: σ1,2 = (σx+σy)/2 ± √[(σx-σy)/2)² + τxy²] ✅
39. Max shear stress: τmax = √[(σx-σy)/2)² + τxy²] ✅
40. Principal plane → zero shear stress ✅
41. Biaxial stress system → σx = 50 MPa, σy = 30 MPa, τxy = 20 MPa → σ1 ≈ 63.25
MPa, σ2 ≈ 16.75 MPa ✅

8. Energy Methods
42. Strain energy in axial member: U = P²L / 2AE ✅
43. Strain energy in bending: U = M²L / 2EI ✅
44. Castigliano theorem → δ = ∂U/∂P ✅
45. Work done by axial load = strain energy stored ✅

9. Miscellaneous / Practical
46. Maximum stress occurs at outermost fiber ✅
47. Neutral axis → plane with zero stress ✅
48. Section modulus – rectangle: Z = bd²/6 ✅
49. Section modulus – circle: Z = πd³/32 ✅
50. Moment of inertia depends on geometry only ✅
51. Bending + axial load → use combined stress formula ✅
52. Shear center → point where load doesn’t twist section ✅
53. Solid vs hollow shaft → same torque, hollow lighter ✅
54. Maximum shear in rectangular section: 1.5V/A approx ✅
55. Columns fail by Euler formula when slenderness ratio > 100–120 ✅
56. Fixed–fixed column → buckling load 4× pin–pin load ✅
57. Energy methods simplify deflection and slope calculations ✅
58. Helical spring energy: U = ½ k δ² ✅
59. Shear in beam → max at supports for simply supported ✅
60. Bending moment diagram → triangular for point load, parabolic for uniform load
✅
Tips for GSSC TA Exam:

 Memorize formulas first, then practice numerical application.

 Draw beam & shaft sketches for visual questions.
 Focus on simply supported & cantilever beams, torsion, springs, and Euler columns
– high probability in exam.
 Use slenderness ratio rules for quick column questions.