Strength of Materials and Structural Analysis Notes for SSC JE Civil

Strength of materials (SOM) and structural analysis are the base of the structural engineering and the highest-weightage part of the SSC JE Civil paper. The questions are about the stress and strain, the elastic constants, the bending moment of the beams, the moment of inertia, the deflection, the torsion, the buckling of the columns and the basics of the trusses and the indeterminate structures. These strength of materials and structural analysis notes for SSC JE Civil give all the formulas in tables with the standard results and 20 practice questions. Read them with the Soil Mechanics Notes for SSC JE, Fluid Mechanics and Hydraulics Notes for SSC JE and Concrete Technology Notes for SSC JE notes.

Stress, Strain and Hooke's Law

• Stress = force / area (N per square mm or MPa). Types: tensile, compressive and shear stress.
• Strain = change in length / original length (no unit). Types: linear (longitudinal), lateral, shear and volumetric strain.
• Hooke's law: within the elastic limit, stress is proportional to the strain. Young's modulus E = stress / strain. E of the steel is about 2 x 10 to the power 5 N per square mm (200 GPa).
• Elongation of a bar under the axial load P: delta = P L / (A E). Under its own weight W: delta = W L / (2 A E). For a tapering circular bar: delta = 4 P L / (pi E d1 d2).
• Poisson's ratio mu = lateral strain / longitudinal strain; its value is about 0.3 for the steel, 0.1 to 0.2 for the concrete, and the theoretical maximum is 0.5.
• Volumetric strain = ex + ey + ez; for a bar under an axial load it is e (1 - 2 mu).
• Thermal stress in a restrained bar = E x alpha x change in the temperature; there is no stress if the bar is free to expand.
• Factor of safety = ultimate stress / working (permissible) stress.

Elastic Constants and Their Relations

Constant	Definition	Relation
Young's modulus E	Stress / linear strain	E = 2 G (1 + mu) = 3 K (1 - 2 mu)
Modulus of rigidity (shear modulus) G	Shear stress / shear strain	G = E / [2 (1 + mu)]
Bulk modulus K	Pressure / volumetric strain	K = E / [3 (1 - 2 mu)]
Combined		E = 9 K G / (3 K + G)

• For mu = 0.5 (rubber, an incompressible material), K is infinite and E = 3G; for mu = 0, E = 2G.
• Mild steel in the stress-strain curve: the limit of proportionality, elastic limit, yield point (upper and lower), ultimate stress, breaking (fracture) stress and the strain hardening. The steel shows the necking before the failure.
• High-strength steel does not have a clear yield point, so the 0.2 percent proof stress is used.

Strain Energy and Types of Loading

Term	Formula or fact
Strain energy (resilience)	U = (stress squared / 2E) x volume = (P squared L) / (2 A E)
Proof resilience	The maximum strain energy stored up to the elastic limit
Modulus of resilience	Proof resilience per unit volume
Gradual load	Stress = P / A
Suddenly applied load	Stress = 2 P / A (twice the gradual load)
Impact load (falling through a height h)	Stress = (P / A) [1 + square root of (1 + 2 h A E / (P L))]

Principal Stresses and Mohr's Circle

• Principal stresses: s1, s2 = (sx + sy) / 2 plus or minus square root of [((sx - sy) / 2) squared + t squared]. The shear stress on the principal planes is zero.
• Maximum shear stress = (s1 - s2) / 2 = the radius of the Mohr circle; it acts on the planes at 45 degrees to the principal planes.
• The angle of the principal plane: tan 2 theta = 2 t / (sx - sy).
• Mohr's circle is the graph that gives the stresses on any plane: its centre is at (sx + sy) / 2 on the normal-stress axis.
• In pure shear, the principal stresses are +t and -t at 45 degrees.

Moment of Inertia and Section Modulus

Section	Moment of inertia about the centroidal axis	Remarks
Rectangle (b wide, d deep)	I = b d cubed / 12	About the base: b d cubed / 3
Solid circle (diameter d)	I = pi d to the power 4 / 64	Polar moment J = pi d to the power 4 / 32
Hollow circle	I = pi (D to the power 4 - d to the power 4) / 64	Annulus
Triangle (base b, height h)	I = b h cubed / 36	About the base: b h cubed / 12
Parallel axis theorem	I = I_centroid + A h squared	h is the distance between the axes
Perpendicular axis theorem	Iz = Ix + Iy	Only for a plane area

• Section modulus Z = I / y_max. For a rectangle Z = b d squared / 6 and for a circle Z = pi d cubed / 32.
• Radius of gyration r = square root of (I / A).
• The strength of a beam in bending is proportional to the section modulus; the stiffness is proportional to the moment of inertia (EI).

Shear Force and Bending Moment

Beam and load	Maximum shear force	Maximum bending moment
Cantilever with a point load W at the free end	W	W L (at the fixed end)
Cantilever with a UDL w per unit length	w L	w L squared / 2 (at the fixed end)
Simply supported beam with a central point load W	W / 2	W L / 4 (at the centre)
Simply supported beam with a UDL w	w L / 2	w L squared / 8 (at the centre)

• The relations: dM / dx = F (the slope of the bending moment diagram is the shear force) and dF / dx = -w (the slope of the shear force diagram is the load).
• The bending moment is maximum where the shear force is zero (or changes its sign).
• The point of contraflexure is the point where the bending moment changes its sign.
• For a point load the SFD is a rectangle and the BMD is a triangle; for a UDL the SFD is a straight inclined line and the BMD is a parabola.
• The sign convention: the sagging moment is positive and the hogging moment is negative.

Bending and Shear Stresses in Beams

• Bending equation: M / I = sigma / y = E / R. The bending stress is zero on the neutral axis and maximum at the extreme fibres: sigma = M y / I.
• The neutral axis passes through the centroid for the symmetrical section.
• Shear stress formula: tau = V A y_bar / (I b). The shear stress is zero at the extreme fibres and maximum at the neutral axis.
• Maximum shear stress = 1.5 times the average shear stress for a rectangular section and 4/3 times for a circular section; for an I-section the web carries most of the shear.

Deflection of Beams

Beam and load	Maximum deflection	Maximum slope
Cantilever with a point load W at the free end	W L cubed / (3 E I)	W L squared / (2 E I)
Cantilever with a UDL w	w L to the power 4 / (8 E I)	w L cubed / (6 E I)
Simply supported beam with a central point load W	W L cubed / (48 E I)	W L squared / (16 E I)
Simply supported beam with a UDL w	5 w L to the power 4 / (384 E I)	w L cubed / (24 E I)

• The differential equation of the elastic curve: E I (d squared y / dx squared) = M.
• Methods: double integration, Macaulay's method, moment-area method (Mohr's theorems) and the conjugate beam method.
• For the same load, a fixed beam has less deflection than a simply supported beam.

Torsion of Shafts

• Torsion equation: T / J = tau / r = G theta / L.
• For a solid shaft J = pi d to the power 4 / 32; the polar modulus Zp = J / r = pi d cubed / 16, so T = (pi / 16) tau d cubed.
• Power transmitted P = 2 pi N T / 60 (N in rpm, T in N m, P in watts).
• The hollow shaft is more economical (more strength per unit weight) than the solid shaft of the same material and weight.
• Torsional rigidity = G J.

Columns and Struts (Buckling)

End condition	Effective length Le	Euler's load
Both ends hinged	L	pi squared E I / L squared
One end fixed and the other free	2 L	pi squared E I / (4 L squared)
Both ends fixed	L / 2	4 pi squared E I / L squared
One end fixed and the other hinged	L / square root of 2 (about 0.7 L)	2 pi squared E I / L squared

• Euler's formula: Pcr = pi squared E I / (Le squared). It is valid for the long columns only.
• Slenderness ratio = Le / r_min, where r_min is the least radius of gyration.
• Rankine's formula is used for the short and the intermediate columns: 1 / Pr = 1 / Pc + 1 / Pe.
• A column buckles about the axis of the least moment of inertia.
• The core (kern) of a section: for a rectangle the middle third (b / 6 and d / 6); for a circle the radius is d / 8.

Thin Cylinders and Spheres

Shell	Hoop (circumferential) stress	Longitudinal stress
Thin cylinder (internal pressure p, diameter d, thickness t)	sigma_c = p d / (2 t)	sigma_l = p d / (4 t)
Thin sphere	p d / (4 t) in all directions	p d / (4 t)

• The hoop stress in a thin cylinder is twice the longitudinal stress, so the cylinder fails by splitting along its length.
• The cylinder is called thin if the thickness is less than d / 20.

Structural Analysis Basics (Theory of Structures)

Topic	Key facts
Static determinacy of a beam	Degree of indeterminacy = reactions - 3 (for a plane structure with no internal hinge); a simply supported beam and a cantilever are determinate; a fixed beam is indeterminate to the 3rd degree (2nd degree for vertical loads only) and a propped cantilever to the 1st degree
Plane truss	Perfect truss: m = 2 j - 3 (m members, j joints); redundant if m is more than 2 j - 3 and deficient if less
Analysis of a truss	Method of joints (for all members) and the method of sections (for a few members); the zero force members are found by the joint rules
Fixed-end moments of a fixed beam	UDL: w L squared / 12; central point load: W L / 8
Moment distribution (Hardy Cross)	An iterative method for the continuous beams and frames; distribution factor = stiffness of the member / the sum of stiffnesses; stiffness 4 E I / L for the far end fixed and 3 E I / L for the far end hinged; the carry-over factor is 1/2
Slope-deflection method	Uses the slope and the deflection as the unknowns
Influence line	The graph of the effect (reaction, shear or moment) at a point as a unit load moves over the structure; for the bending moment at a section of a simply supported beam, the maximum ordinate is a b / L
Three-hinged arch	A determinate structure; the horizontal thrust H = M / y; the bending moment at a point = beam moment minus H x y
Castigliano's theorem	The partial derivative of the strain energy with respect to a load gives the deflection at that load

Important Questions Explained

Question 1: What is the relation between E, G and Poisson's ratio?
Answer: E = 2 G (1 + mu).

Question 2: What is the maximum bending moment of a simply supported beam of span 6 m with a UDL of 10 kN per m?
Answer: M = w L squared / 8 = 10 x 36 / 8 = 45 kN m.

Question 3: What is the maximum deflection of a cantilever of length L with a point load W at the free end?
Answer: W L cubed / (3 E I).

Question 4: What is the effective length of a column with both ends fixed?
Answer: L / 2.

Question 5: What is the hoop stress in a thin cylinder of diameter 1 m, thickness 10 mm and pressure 2 N per square mm?
Answer: sigma_c = p d / (2 t) = 2 x 1000 / (2 x 10) = 100 N per square mm.

Question 6: What is the moment of inertia of a rectangular section of 100 mm width and 200 mm depth about its centroidal axis?
Answer: I = b d cubed / 12 = 100 x 200 cubed / 12 = 6.67 x 10 to the power 7 mm to the power 4.

Question 7: What is the degree of static indeterminacy of a fixed beam?
Answer: 3 for the general loading (2 if there are only vertical loads).

Quick Tips to Remember

• Memorise the five beam results: WL, wL squared / 2, WL / 4, wL squared / 8 and the deflections with 3, 8, 48 and 384.
• Remember the relation among the elastic constants and that Poisson's ratio is 0.3 for the steel and 0.5 at the maximum.
• Remember that the effective length ratios are 1, 2, 0.5 and 0.7.
• Remember that the maximum bending stress is at the extreme fibre and the maximum shear stress is at the neutral axis.
• Remember I = b d cubed / 12 and I = pi d to the power 4 / 64 and the parallel axis theorem.
• Remember the perfect truss condition m = 2j - 3 and the fixed-end moments wL squared / 12 and WL / 8.

Common Mistakes to Avoid

• Using the width and the depth in a wrong order in the moment of inertia.
• Mixing the maximum bending moment of the cantilever (WL) with the simply supported beam (WL / 4).
• Forgetting the 5 in the deflection of the simply supported beam with a UDL.
• Writing the effective length of the cantilever column as L (it is 2L).
• Using the cross-section area instead of the least radius of gyration for the slenderness ratio.

Quick Revision Points

• E = 2G (1 + mu) = 3K (1 - 2mu); sigma = E x epsilon.
• SS beam: central point load M = WL / 4; UDL M = wL squared / 8; deflections W L cubed / 48EI and 5wL to the power 4 / 384EI.
• Cantilever: M = WL or wL squared / 2; deflections WL cubed / 3EI and wL to the power 4 / 8EI.
• Bending: M / I = sigma / y = E / R; shear: tau = VAy / Ib.
• Torsion: T / J = tau / r = G theta / L; P = 2 pi N T / 60.
• Euler: Pcr = pi squared EI / Le squared; Le = L, 2L, L / 2 and 0.7L.
• Perfect truss: m = 2j - 3.

Practice Questions With Answers

1. What is the unit of stress?
   Answer: N per square mm (MPa) (Force per unit area).
2. What is the formula of Young's modulus?
   Answer: Stress / strain (Hooke's law).
3. What is the typical Poisson's ratio of steel?
   Answer: 0.3 (Maximum 0.5).
4. What is the relation of E, G and mu?
   Answer: E = 2G (1 + mu) (Elastic constants).
5. What is the stress in a bar under a suddenly applied load?
   Answer: Twice the gradual stress (2P / A).
6. What is the maximum shear stress in terms of the principal stresses?
   Answer: (s1 - s2) / 2 (Mohr circle radius).
7. What is the moment of inertia of a circle about its diameter?
   Answer: pi d to the power 4 / 64 (Solid circle).
8. What is the section modulus of a rectangle?
   Answer: b d squared / 6 (I / y).
9. What is the maximum bending moment of a cantilever with a point load W at the free end?
   Answer: W L (At the fixed end).
10. What is the maximum bending moment of a simply supported beam with a central load W?
    Answer: W L / 4 (At the centre).
11. What is the maximum bending moment of a simply supported beam with a UDL?
    Answer: w L squared / 8 (At the centre).
12. What is the maximum deflection of a simply supported beam with a UDL?
    Answer: 5 w L to the power 4 / (384 E I) (Centre).
13. Where is the bending stress zero in a beam?
    Answer: On the neutral axis (Maximum at the extreme fibre).
14. Where is the shear stress maximum in a rectangular beam?
    Answer: At the neutral axis (1.5 times the average).
15. What is the formula of the power transmitted by a shaft?
    Answer: P = 2 pi N T / 60 (N in rpm).
16. What is the polar moment of inertia of a solid shaft?
    Answer: pi d to the power 4 / 32 (J).
17. What is the Euler's buckling load of a column with both ends hinged?
    Answer: pi squared E I / L squared (Le = L).
18. What is the effective length of a column fixed at one end and free at the other?
    Answer: 2 L (Cantilever column).
19. What is the hoop stress in a thin cylinder?
    Answer: p d / (2 t) (Twice the longitudinal).
20. What is the condition of a perfect plane truss?
    Answer: m = 2 j - 3 (Members and joints).

How to Practise This Topic

1. Make a one-page formula sheet of the beams, the deflections and the columns.
2. Solve the SFD and BMD problems of the point load and the UDL by yourself.
3. Revise the elastic constants, the torsion formulas and the truss rules the next day.
4. Attempt the practice questions without looking at the answers.
5. Solve the SSC practice papers and the previous year papers to practise the JE pattern.

Related Notes and Practice

• Soil Mechanics Notes for SSC JE
• Fluid Mechanics and Hydraulics Notes for SSC JE
• Concrete Technology Notes for SSC JE
• Surveying Notes for SSC JE
• Civil Engineering (SSC JE) Notes
• SSC practice papers

Frequently Asked Questions

What is strength of materials?

Strength of materials is the study of how the solid materials and the structural members behave under the loads, in terms of the stress, the strain and the deformation.

What is Hooke's law?

Hooke's law states that within the elastic limit, the stress in a material is directly proportional to the strain.

What are the elastic constants?

The elastic constants are Young's modulus, the shear modulus, the bulk modulus and Poisson's ratio, which are related to each other.

What is the bending equation?

The bending equation is M / I = sigma / y = E / R, which gives the bending stress in a beam.

What is the point of contraflexure?

The point of contraflexure is the point in a beam where the bending moment changes its sign.

What is the slenderness ratio of a column?

The slenderness ratio is the effective length of the column divided by its least radius of gyration.

What is Euler's formula?

Euler's formula gives the critical buckling load of a long column: Pcr = pi squared E I / Le squared.

What is the difference between the hoop stress and the longitudinal stress?

In a thin cylinder, the hoop stress p d / (2 t) acts around the circumference and is twice the longitudinal stress p d / (4 t) that acts along the length.

What is a statically determinate structure?

A structure is statically determinate if all its reactions and the internal forces can be found from the equations of the equilibrium alone.

What is the section modulus?

The section modulus is the moment of inertia divided by the distance of the extreme fibre from the neutral axis (Z = I / y); it measures the bending strength of a section.

How many questions come from strength of materials in the SSC JE exam?

Usually 5 to 8 questions come from strength of materials and the structural analysis in the SSC JE Civil paper.

How can I prepare strength of materials for SSC JE?

Learn the formulas and the standard results in tables, practise the beam and column problems, and solve the previous year questions.