Solid Mechanics — Strength of Materials

Solid Mechanics (Strength of Materials) is the highest-weightage subject in GATE Civil and ESE (IES), and appears heavily in SSC JE too. This chapter covers every topic — from material properties and elastic constants, through stress transformation and Mohr's circle, theories of failure, bending and shear stresses, deflection of beams, torsion, thick/thin cylinders, columns, retaining walls and shear centre — with all formulae, diagrams, solved examples and exam-pattern analysis.

Learning Objectives

After studying this chapter you will be able to:

• Describe the mechanical properties of engineering materials and read the stress-strain curve for mild steel, identifying every key point
• Apply the elastic-constant inter-relations (\(G\), \(K\), \(E\), \(\nu\)) and generalised Hooke's law to 3D stress states
• Draw SFD and BMD for standard beams and apply the \(\dfrac{dV}{dx} = -w\), \(\dfrac{dM}{dx} = V\) relationships
• Construct Mohr's circle of stress/strain, compute principal stresses, and apply the five classical theories of failure
• Compute beam deflection using double integration, Macaulay's method, moment-area, conjugate beam and unit-load methods
• Apply the flexure and shear-stress formulae (\(\sigma = \dfrac{My}{I}\), \(\tau = \dfrac{VQ}{Ib}\)) to standard and composite sections
• Analyse thin and thick (Lamé) pressure vessels, and design columns using Euler's and Rankine's formulae
• Locate the shear centre of standard thin-walled sections and compute principal moments of inertia

Exam Relevance

Concept Map

Where This Fits

Prerequisite for: Theory of Structures (analysis methods build directly on SFD/BMD and deflection here), and both Steel Structures and RCC/PCC Design (which apply these stress and section-property formulae to real member design).

Mechanical Properties

Key Points on the Stress–Strain Curve

Factor of Safety and Proof Stress

• Factor of Safety (FOS) = Ultimate Stress / Permissible (Working) Stress
• Proof Stress (0.2% offset): stress at which 0.2% permanent strain occurs — used for materials without a well-defined yield point (Al alloys, HYSD bars)
• Resilience (Modulus) = \(\dfrac{\sigma_y^2}{2E}\) — energy stored per unit volume at yield
• Toughness \(\approx\) area under complete \(\sigma\)–\(\varepsilon\) curve \(\approx \sigma_{avg} \times \varepsilon_{fracture}\)

Material Comparison

Stress and Strain Definitions

• Normal Stress: \(\sigma = \dfrac{P}{A}\) (Pa = N/m²; MPa = N/mm²)
• Shear Stress: \(\tau = V/A\) (shear force / area)
• Normal Strain: \(\varepsilon = \dfrac{\delta}{L}\) (dimensionless; δ = deformation, L = original length)
• Shear Strain: \(\gamma = \dfrac{\tau}{G}\) (angular distortion in radians)
• Volumetric Strain: \(e = \Delta V/V = \epsilon_x + \epsilon_y + \epsilon_z\) (for triaxial loading)

Elastic Constants

Inter-Relations Between Elastic Constants

• \(G = \dfrac{E}{2(1+\nu)}\)
• \(K = \dfrac{E}{3(1-2\nu)}\)
• \(E = \dfrac{9KG}{3K+G}\)
• \(\nu = \dfrac{3K-2G}{2(3K+G)}\)

Limits of ν:

• \(\nu = 0 \to K = \dfrac{E}{3}, G = \dfrac{E}{2}\) (cork — no lateral strain)
• \(\nu = 0.5 \to K \to \infty\) (incompressible material, rubber)
• \(\nu < 0\) → auxetic material (rare, expands laterally under tension)
• Theoretical range: \(-1 \le \nu \le 0.5\) (for isotropic elastic materials)

Generalised Hooke's Law (3D)

• \(\gamma_{xy} = \dfrac{\tau_{xy}}{G}\); \(\gamma_{yz} = \dfrac{\tau_{yz}}{G}\); \(\gamma_{zx} = \dfrac{\tau_{zx}}{G}\)
• Volumetric strain: \(e = (1-2\nu)\dfrac{\sigma_x+\sigma_y+\sigma_z}{E} = \dfrac{\sigma_{hydro}}{K}\)

Thermal Stress

• Free expansion: \(\delta_T = \alpha \cdot \Delta T \cdot L\) (no stress if unconstrained)
• Thermal stress (fully constrained bar): \(\sigma_T = E \cdot \alpha \cdot \Delta T\) (compressive if temperature rises)
• Partially constrained (support settles by δ < δ_T): \(\sigma = \dfrac{E(\alpha\Delta T L - \delta)}{L}\)

Composite Bars and Statically Indeterminate Axial Problems

• Compatibility (same deformation): \(\delta_1 = \delta_2 \to P_1 L/(A_1 E_1) = P_2 L/(A_2 E_2) \to \dfrac{P_1}{P_2} = A_1 \dfrac{E_1}{A_2 E_2}\)
• Equilibrium: \(P_{external} = P_1 + P_2\) (if parallel)
• Modular ratio: \(m = \dfrac{E_{steel}}{E_{concrete}}\) (used in RCC transformed section); \(m = \dfrac{280}{3\sigma_{cbc}}\) per IS 456 [WSM]

Plane Stress and Plane Strain

Sign Convention

Relationship between load, SF and BM:

• \(\dfrac{dV}{dx} = -w(x)\) [rate of change of SF = negative of distributed load]
• \(\dfrac{dM}{dx} = V(x)\) [rate of change of BM = SF]
• \(\dfrac{d^2M}{dx^2} = -w(x)\)
• Consequence: Where V = 0 → BM is maximum (or minimum)

Standard SFD/BMD Results

SFD/BMD Shapes for Standard Loads

Load type → SFD shape → BMD shape:

• No load (span between loads) → Constant SF → Linear BM
• Point load → Step (jump) in SF → Kink in BM
• UDL → Linear SF → Parabolic (2nd degree) BM
• UVL → Parabolic SF → Cubic (3rd degree) BM
• Applied moment M₀ at a point → No change in SF → Step (jump) in BM

Point of Contraflexure

The point of contraflexure is where the bending moment changes sign (BM = 0, and the beam changes from sagging to hogging). It exists only in beams where the BMD crosses the zero axis.

• For propped cantilever (fixed at A, roller at B), UDL w: \(M_A = -\dfrac{wL^2}{8}\) (hogging), \(R_A = \dfrac{3wL}{8}\), \(R_B = \dfrac{5wL}{8}\)
• BM = 0 (at support B) and at x = 3L/4 from fixed end A → point of contraflexure

Relationship Between Load, SF, and BM — Key Rules

• The SFD between two point loads is horizontal (constant)
• Under a UDL, SFD is linear, BMD is parabolic
• At a free end → V = 0 and M = 0 (unless a point load/moment is applied)
• At a roller/pin → M = 0 (but V may be non-zero)
• At a fixed support → V and M are both non-zero reactions
• A couple (moment M₀) applied at a point → SFD unchanged, BMD jumps by M₀

Stress Transformation (2D)

On an inclined plane at angle θ to the x-face:

• \(\sigma_\theta = \dfrac{\sigma_x+\sigma_y}{2} + \dfrac{\sigma_x-\sigma_y}{2} \cdot \cos2\theta + \tau_{xy}\cdot\sin2\theta\)
• \(\tau_\theta = -\dfrac{\sigma_x-\sigma_y}{2} \cdot \sin2\theta + \tau_{xy}\cdot\cos2\theta\)

Principal stresses (\(\tau=0\)):

• Max shear stress: \(\tau_{max} = \sqrt{(\sigma_x-\sigma_y)\dfrac{^2}{4} + \tau_{xy}^2} = \dfrac{\sigma_1-\sigma_2}{2}\)
• Angle of principal planes: \(\tan(2\theta_p) = 2\dfrac{\tau_{xy}}{\sigma_x-\sigma_y}\)
• Principal planes are at 90° to each other; max shear planes at 45° to principal planes

Mohr's Circle Construction — Step by Step

1. Draw σ-axis (horizontal) and τ-axis (vertical)
2. Plot point X = (σ_x, +τ_xy) and point Y = (σ_y, −τ_xy)
3. Centre C = midpoint of XY = ((σ_x+σ_y)/2, 0)
4. Radius \(R = |CX| = \sqrt{(\sigma_x-\sigma_y)\dfrac{^2}{4} + \tau_{xy}^2}\)
5. Principal stresses: \(\sigma_1 = \sigma_{avg}+R\), \(\sigma_2 = \sigma_{avg}-R\) (on σ-axis)
6. Max shear stress: \(\tau_{max} = R\) (at top/bottom of circle)
7. Angle 2θ_p on circle = angle from X to horizontal (right); → θ_p on actual element

Mohr's Circle of Strain

Replace σ with ε and τ with γ/2 in all formulae:

• Centre: \(\epsilon_{avg} = \dfrac{\epsilon_x+\epsilon_y}{2}\)
• Radius: \(R_\epsilon = \sqrt{(\epsilon_x-\epsilon_y)\dfrac{^2}{4} + (\dfrac{\gamma_{xy}}{2})^2}\)
• Principal strains: \(\epsilon_1, \epsilon_2 = \epsilon_{avg} \pm R_\epsilon\)
• Max shear strain: \(\gamma_{max} = 2R_\epsilon = \epsilon_1-\epsilon_2\)

Strain rosette (0°–45°–90° rosette):

Theories of Failure

Failure theories predict when a material transitions from safe to unsafe under multiaxial stress, by comparison with the uniaxial yield stress σ_y (or UTS σ_u).

Failure Criteria for Biaxial Stress (σ₃ = 0)

• Rankine (Max Principal Stress): \(\sigma_1 = \sigma_y\)
• Tresca (Max Shear Stress): \(\sigma_1-\sigma_2 = \sigma_y\) (if σ₁ and σ₂ have opposite signs); \(\sigma_1 = \sigma_y\) (if same sign — Tresca = Rankine in same-sign quadrants)
• von Mises (Distortion Energy): \(\sigma_1^2 - \sigma_1\sigma_2 + \sigma_2^2 = \sigma_y^2\) (ellipse in σ₁−σ₂ space)

For pure shear (\(\sigma_1 = -\sigma_2 = \tau\)):

• Tresca: \(\tau_y = \dfrac{\sigma_y}{2}\)
• von Mises: \(\tau_y = \dfrac{\sigma_y}{\sqrt3} \approx 0.577\sigma_y\)
• von Mises is more accurate (agrees with experiments); Tresca is conservative (safe side)

Elastic Curve Equation

Euler-Bernoulli beam equation:

Boundary conditions:

• Fixed end: y = 0, dy/dx = 0
• Simply supported: \(y=0, M=0\ (\dfrac{d^2y}{dx^2}=0)\)
• Free end: M = 0, V = 0 (unless load applied)
• Internal hinge: M = 0 at hinge; y and dy/dx continuous

Standard Deflection Formulae

Methods for Computing Deflections

Double Integration Method

• \(EI\cdot y'' = M(x)\) → integrate twice → apply BCs to find constants C₁, C₂
• Slope: \(\theta = \dfrac{dy}{dx}\) | Deflection: y (positive downward convention or per problem)

Macaulay's Method (Singularity Functions)

• Use <x−a>ⁿ brackets — integrated as (x−a)ⁿ⁺¹/(n+1) only when (x−a) > 0
• Very efficient for multiple point loads / partial UDLs
• Example: SS beam, load P at x = a: \(EIy'' = R_A\cdot x - P\langle x-a\rangle^1\); integrate with the same bracket rule for slope and deflection

Moment Area Method (Mohr's Theorems)

• Theorem I: Change in slope between two points A and B = \((\dfrac{1}{EI}) \times\) Area of \(\dfrac{M}{EI}\) diagram between A and B
• Theorem II: Tangential deviation of B from tangent at A: \(t_{BA} = (\dfrac{1}{EI}) \times\) (Area of \(\dfrac{M}{EI}\) diagram between A and B) \(\times \bar{x}_B\), where \(\bar{x}_B\) = distance of centroid of M/EI diagram from B
• Best used for: cantilevers, overhanging beams, non-prismatic beams

Conjugate Beam Method

• In the conjugate beam: load = M/EI diagram of real beam
• Shear in conjugate beam = slope in real beam; Moment in conjugate beam = deflection in real beam
• Real free end → Conjugate fixed end; Real fixed end → Conjugate free end; Real pin/roller → Conjugate pin/roller

Virtual Work / Unit Load Method

• M = real bending moment (under actual loads); \(\bar m\) = virtual bending moment (due to unit load at C in desired direction)
• Slope: \(\theta_C = \int(M\cdot\bar{m}_\dfrac{\theta}{EI})dx\) (\(\bar m_\theta\) due to unit couple at C)

Flexure Formula (Bending Stress)

Euler-Bernoulli Bending Formula:

• \(\sigma = M\cdot \dfrac{y}{I}\) (bending stress at distance y from NA)
• \(\sigma_{max} = M\cdot \dfrac{y_{max}}{I} = \dfrac{M}{Z}\)
• M = bending moment at section; I = second moment of area about NA; y = distance from neutral axis (NA)
• \(Z = \dfrac{I}{y_{max}}\) = section modulus (m³ or mm³); R = radius of curvature of neutral axis; E = Young's modulus

Section Modulus and Moment of Resistance

Shear Stress in Beams

Shear stress at distance y from NA:

• V = shear force at section; Q = A'·ȳ' = first moment of area of section above (or below) y about NA
• I = moment of inertia of full section about NA; b = width of section at distance y

Maximum shear stress for standard sections:

• Rectangle: \(\tau_{max} = 1.5\times V/A\) (at NA) — parabolic distribution
• Solid Circle: \(\tau_{max} = \dfrac{4V}{3A}\) (at NA)
• I-section: \(\tau_{max} \approx V/A_{web}\) (approximately uniform in web)
• Thin-walled hollow circle: \(\tau_{max} = \dfrac{2V}{A}\)

Shear Flow in Thin-Walled Sections

• Shear flow: \(q = V\cdot \dfrac{Q}{I}\) (force per unit length along the wall)
• In open thin-walled sections (channels, angles): q varies along the wall; zero at free edges. Shear centre is the point through which V must pass for no twisting
• In closed thin-walled sections (box, tube): \(q = V\cdot \dfrac{Q}{I} + q_0\) (q₀ = statically indeterminate shear flow)

Composite Beams (Transformed Section)

• Transform all materials to equivalent area of one material (usually concrete or steel): \(A_{transformed} = A_{original} \times (\dfrac{E_{material}}{E_{reference}})\)
• Modular ratio \(m = \dfrac{E_1}{E_2}\); Neutral axis: \(\Sigma A_t\cdot\bar{y} = 0\) (centroid of transformed section)
• Bending stress in material 1: \(\sigma_1 = M\cdot \dfrac{y}{I_t}\); in material 2: \(\sigma_2 = m\cdot M\cdot \dfrac{y}{I_t}\)

Flitched Beams

A timber beam reinforced with steel plates on two faces. Both materials undergo same curvature (1/R), same strain at same y from NA:

• Equal curvature: \(\dfrac{\sigma_{steel}}{E_s} = \dfrac{\sigma_{timber}}{E_t}\) at same y
• \(\therefore \sigma_{steel} = m\times\sigma_{timber}\) (m = E_s/E_t ≈ 20 for steel-timber)
• Moment of resistance: \(M = M_{steel}+M_{timber}\)

Thin Cylinders (t/D < 1/20 or t/r < 1/10)

Internal pressure p, mean radius r, thickness t:

• Hoop (circumferential) stress: \(\sigma_H = \dfrac{pd}{2t} = \dfrac{pr}{t}\)
• Longitudinal stress: \(\sigma_L = \dfrac{pd}{4t} = \dfrac{pr}{2t} = \dfrac{\sigma_H}{2}\)

Volumetric strain of cylinder:

• \(e_v = (\dfrac{pd}{2tE})[\dfrac{5}{2}-2\nu] = (\dfrac{pd}{2tE})[\dfrac{5-4\nu}{2}]\)
• Hoop strain: \(\epsilon_H = \dfrac{\sigma_H-\nu\sigma_L}{E} = (\dfrac{pd}{2tE})(1-\dfrac{\nu}{2})\)
• Longitudinal strain: \(\epsilon_L = \dfrac{\sigma_L-\nu\sigma_H}{E} = (\dfrac{pd}{2tE})(\dfrac{1}{2}-\nu)\)
• Volumetric strain: \(e_v = 2\epsilon_H+\epsilon_L = (\dfrac{pd}{2tE})(\dfrac{5}{2}-2\nu)\)

Thin Spherical Shell

Both hoop and longitudinal stresses are equal:

• Volumetric strain: \(e_v = 3\epsilon = (\dfrac{3pd}{4tE})(1-\nu)\)
• Comparison: Sphere is more efficient — same stress in all directions; needs ~half the steel of a cylinder for same p and d

Thick Cylinders — Lamé's Equations

When t/d ≥ 1/20, stress variation through thickness is significant. Lamé's equations apply:

• Radial stress: \(\sigma_r = A-\dfrac{B}{r^2}\); Hoop stress: \(\sigma_H = A+\dfrac{B}{r^2}\)
• For internal pressure \(p_i\) only (\(p_o=0\)), \(r_i\) = inner radius, \(r_o\) = outer radius: \(A = p_i \dfrac{r_i^2}{r_o^2-r_i^2}\), \(B = p_i r_i^2 \dfrac{r_o^2}{r_o^2-r_i^2}\)

Hoop stress (max at inner radius):

Boundary conditions:

• At \(r=r_i\): \(\sigma_r = -p_i\) (negative = compressive, pointing inward)
• At \(r=r_o\): \(\sigma_r = 0\) (free outer surface, assuming no external pressure)

Compound Cylinders and Shrink-Fit

A compound cylinder (outer cylinder shrunk onto inner) introduces residual compressive hoop stress at the inner surface, allowing higher working pressure.

• Junction pressure \(p_j\) is found from the interference \(\delta = r_j \times (e_{outer,inner}-e_{inner,outer})\)
• Net hoop stress = prestress (compressive) + operating pressure stress

Torsion Formula

• T = applied torque (N·mm or N·m); J = polar moment of inertia of cross-section
• τ = shear stress at radius r from axis; G = modulus of rigidity (shear modulus)
• φ = angle of twist (radians); L = length of shaft

Maximum shear stress (at outer surface, r = R) and angle of twist:

Polar Moment of Inertia J

Power Transmission

• Power: \(P = T\cdot\omega = T\times2\pi \dfrac{N}{60}\) (P in Watts, T in N·m, N = rpm, ω = rad/s)
• \(T = P\dfrac{\times60}{2\pi N} = 9550\times \dfrac{P(kW)}{N(rpm)}\) [in N·m]

Shafts in Series and Parallel

• Series shafts (same torque T, different φ): \(\phi_{total} = \phi_1+\phi_2 = T \dfrac{L_1}{G_1J_1} + T \dfrac{L_2}{G_2J_2}\)
• Parallel shafts (same φ, shared T): \(T = T_1+T_2\); \(\phi_1=\phi_2 \to \dfrac{T_1L_1}{G_1J_1} = \dfrac{T_2L_2}{G_2J_2}\)
• \(\to T_1/T_2 = \dfrac{G_1J_1L_2}{G_2J_2L_1}\)

Combined Bending and Torsion

For a circular shaft under BM = M and Torque = T simultaneously:

• Equivalent bending moment (max normal stress, Rankine/Tresca): \(M_e = \tfrac12[M+\sqrt{M^2+T^2}]\)
• Equivalent torque (max shear stress): \(T_e = \sqrt{M^2+T^2}\) (both normal and shear stresses combined)
• Design: \(\sigma_{max} = M_e\dfrac{\times32}{\pi D^3} \le \sigma_{permissible}\) or \(\tau_{max} = T_e\dfrac{\times16}{\pi D^3} \le \tau_{permissible}\)

Close-Coiled Helical Springs

• Shear stress in wire: \(\tau = \dfrac{8WD}{\pi d^3}\times K_w\)
• Wahl's stress correction factor: \(K_w = \dfrac{4C-1}{4C-4} + \dfrac{0.615}{C}\); Spring index: \(C = \dfrac{D}{d}\) (D = mean coil dia, d = wire dia)
• Axial deflection: \(\delta = \dfrac{8WD^3n}{Gd^4} = \dfrac{WL}{GJ}\) (n = number of active coils)
• Stiffness: \(k = \dfrac{W}{\delta} = \dfrac{Gd^4}{8D^3n}\)
• Strain energy stored: \(U = \dfrac{W^2D^3n}{Gd^4} = W\dfrac{\delta}{2} = \dfrac{\tau^2}{4G}\times Volume_{wire}\)

Springs in Series and Parallel

• Series springs (same load, deflections add): \(\dfrac{1}{k_{eq}} = \dfrac{1}{k_1}+\dfrac{1}{k_2}+...\), so \(\delta_{total}=\delta_1+\delta_2\)
• Parallel springs (same deflection, loads add): \(k_{eq}=k_1+k_2+...\), so \(W_{total}=W_1+W_2\)

Open-Coiled and Closely Coiled Springs

Classification of Columns

Euler's Buckling Load

Effective length \(L_{eff}\) for various end conditions:

• Both ends pinned: \(L_{eff}=L\)
• One fixed, one free: \(L_{eff}=2L\)
• Both ends fixed: \(L_{eff}=\dfrac{L}{2}\)
• One fixed, one pinned: \(L_{eff}=\dfrac{L}{\sqrt2} \approx 0.707L\)

• Critical stress: \(\sigma_{cr} = \dfrac{P_{cr}}{A} = \dfrac{\pi^2E}{\lambda^2}\) where \(\lambda=\dfrac{L_{eff}}{r}\)
• r = radius of gyration = \(\sqrt{\dfrac{I}{A}}\)
• Euler's formula valid only when \(\sigma_{cr} <\) proportional limit (\(\sigma_p\))

Rankine-Gordon Formula (Intermediate Columns)

• \(\sigma_c\) = crushing stress (yield or compressive strength); \(a\) = Rankine's constant = \(\dfrac{\sigma_c}{\pi^2E}\) [dimensionless]
• For steel: \(a=\dfrac{1}{7500}\) (approx); For cast iron: \(a=\dfrac{1}{1600}\); For timber: \(a=\dfrac{1}{3000}\)

Eccentrically Loaded Columns (Secant Formula)

For a column with eccentric load P at eccentricity e:

• k = radius of gyration; c = distance from centroid to extreme fibre
• Core of section (no tension): load must be within kern. Rectangular section: kern is a rhombus with diagonals b/3 and d/3. Circular section: kern is a circle of diameter D/4

Retaining Walls — Lateral Earth Pressure

Rankine's Theory

• Active earth pressure coefficient: \(K_a = \dfrac{1-\sin\phi}{1+\sin\phi} = \tan^2(45°-\dfrac{\phi}{2})\)
• Passive earth pressure coefficient: \(K_p = \dfrac{1+\sin\phi}{1-\sin\phi} = \tan^2(45°+\dfrac{\phi}{2})\)
• \(K_a\times K_p = 1\)
• Active pressure at depth z: \(p_a = K_a\gamma z - 2c\sqrt{K_a}\) (c-φ soil); for cohesionless soil (c=0): \(p_a = K_a\gamma z\)
• Total active force: \(P_a = \tfrac12 K_a\gamma H^2\) (per unit length, acting at H/3 from base)
• Total passive force: \(P_p = \tfrac12 K_p\gamma H^2\) (acting at H/3 from base)

Coulomb's Theory (Wedge Theory)

• α = inclination of back of wall to horizontal; β = inclination of backfill surface to horizontal
• δ = wall friction angle (0 to 2φ/3); φ = angle of internal friction of soil
• Rankine = Coulomb with α=90°, β=0°, δ=0° (smooth vertical wall, horizontal backfill)

Stability of Retaining Walls

Moment of Inertia (Second Moment of Area)

• \(I_x = \int y^2\,dA\) (about x-axis / NA)
• \(I_y = \int x^2\,dA\) (about y-axis)
• \(I_{xy} = \int xy\,dA\) (product of inertia)
• \(J = I_x+I_y\) (polar moment of inertia = I_p)

Standard MI Values

Parallel Axis Theorem

• \(I_{\bar x}\) = MI about centroidal axis parallel to x; A = area of section; d = distance between the two parallel axes

Perpendicular Axis Theorem (2D)

For a planar lamina: \(I_z = I_x+I_y = J\) (polar MI). This relates out-of-plane MI to in-plane MIs.

Principal Moments of Inertia and Principal Axes

Just as principal stresses are the extreme values of normal stress at a point, principal moments of inertia are the extreme (maximum and minimum) values of MI achieved by rotating the axes. The product of inertia I_xy = 0 about principal axes.

• Angle of principal axes: \(\tan(2\theta_p) = -\dfrac{2I_{xy}}{I_x-I_y}\)
• Check: \(I_{max}+I_{min} = I_x+I_y\) (sum invariant)
• \(I_{max}\times I_{min} - I_{xy}^2 = I_x\cdot I_y - I_{xy}^2\) (product invariant)

Mohr's Circle of MI

Construct exactly like Mohr's stress circle:

1. Plot point X = (I_x, +I_xy) and Y = (I_y, −I_xy)
2. Centre = ((I_x+I_y)/2, 0)
3. Radius \(R = \sqrt{(I_x-I_y)\dfrac{^2}{4} + I_{xy}^2}\)
4. I_max = centre + R, I_min = centre − R

Shear Centre (Flexural Centre)

The shear centre (SC) is the point in the cross-section through which the transverse shear force must pass to produce pure bending (no twisting). If load passes through SC → bending only. If load does not pass through SC → combined bending + torsion.

Shear centre location found by equating external moment of shear force about SC to sum of moment of shear flows in all elements about SC: \(e = \Sigma Q_{flange}\times h/V\) (for open thin-walled sections)

Shear Centre Locations — Summary

Unsymmetric Bending

When the bending moment is not in a plane of symmetry, or when I_xy ≠ 0 (oblique bending), the neutral axis is not perpendicular to the applied moment. The beam deflects in the direction perpendicular to the NA, which is not the direction of loading.

• For oblique loading at angle α to principal x-axis: \(\sigma = M\cos\alpha\cdot \dfrac{y}{I_x} - M\sin\alpha\cdot \dfrac{x}{I_y}\) (if I_xy = 0, i.e., x-y are principal axes)
• Neutral axis angle β: \(\tan\beta = (\dfrac{I_x}{I_y})\tan\alpha\)
• NA is NOT perpendicular to load unless \(I_x=I_y\) (circular) or loading in plane of symmetry

All Key Formulae — Solid Mechanics

Key Mnemonics

Exam-Angle Comparison

Mini Quiz

Practice Questions

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