Strength of Materials · Chapter 2

Simple Stress-strain & Elastic Constants

Stress Tensor · Strain Types · Hooke’s Law · E, G, K, μ · Deflection · Temperature Stresses · Strain Energy

σ = Eε G = τ/φ E = 3K(1−2μ) E = 2G(1+μ) σ = EαT

📋 Table of Contents

2.1 Stress — Normal and Shear
2.2 Matrix Representation — 3D Stress Tensor and Strains
2.3 Differential Form of Strains
2.4 Allowable Stresses and Factor of Safety
2.5 Saint Venant Principal
2.6 Hooke’s Law + 2.7 Four Elastic Constants (E, G, K, μ)
2.7.1 Relationships Between Elastic Constants
2.8 Hooke’s Law Applications + 2.9 Volumetric Strain
2.10 Deflection of Axially Loaded Members
2.11 Statically Indeterminate Structures
2.13 Temperature Stresses
2.15 Strain Energy

2.1 Stress — Normal and Shear

2.1.1 Normal Stress

Normal stresses are always normal to the cross-section. Two types:

Direct / Axial Stress

Produced when axial force acts at CG. Uniform across cross-section for prismatic body. Tensile = +ve, Compressive = −ve.

σ = P / A

Bending Stress

Produced by bending moment. Varies linearly from zero at NA to maximum at extreme fibre. Tensile = +ve, Compressive = −ve.

σ = M·y / I

2.1.2 Shear Stress or Tangential Stress

Resistance offered by material against shearing force. Two types: direct shear (τ = S/A) and torsional shear (due to torque).

Shear Stress Formula

τ = S / A

Unit: N/mm²

Complementary Shear Stress:
τ_xy = τ_yx (from moment equilibrium)

Do you know? A shear stress in a given direction cannot exist without a balancing shear stress of equal intensity in a direction at right angle to it → called Complementary Shear Stress.

2.2 Matrix Representation of Stress and Strain

2.2.1 Stress Tensor — 3D Stress Element

Stress and strain are second order tensor quantities. In a 3D loaded body, at a point three mutually perpendicular planes exist. At each plane there are 3 stress components — 1 normal + 2 shear. Total: 9 components, but due to complementarity only 6 are independent.

3D Stress Matrix (Triaxial)

      [Stress] =

      | σ_xx   τ_xy   τ_xz |

      | τ_yx   σ_yy   τ_yz |

      | τ_zx   τ_zy   σ_zz |

σ = Normal stress (diagonal)
τ = Shear stress (off-diagonal)
τ_xy=τ_yx, τ_xz=τ_zx, τ_yz=τ_zy

2D Stress Matrix (Biaxial/Plane)

      [2D stress] =

      | σ_xx   τ_xy |

      | τ_yx   σ_yy |

Also known as plane stresses
Used in most 2D structural problems

Notation Rules (from source):

Normal stresses: σ_xx, σ_yy, σ_zz — first letter = plane, second letter = direction
Shear stresses: τ_xy — x = plane, y = direction of stress
3D stresses are called Triaxial stresses

2.2.2 Matrix Representation of Strains

Axial Strain (ε)

ε = ΔL / L

Strain in direction of applied force

Lateral Strain (ε_L)

ε_L = ΔB / B

Strain perpendicular to applied force

Shear Strain (φ)

φ = Δ / h

Angular deformation due to shear

3D Strain Matrix

    | ε_xx    φ_xy/2   φ_xz/2 |

    | φ_yx/2   ε_yy    φ_yz/2 |

    | φ_zx/2   φ_zy/2   ε_zz   |

Total shear strain in xy plane = φ_xy/2 + φ_xy/2 = φ_xy
Linear strain of diagonal = φ/2 (half of shear strain in the body)

📌 Key Note: Under pure normal stress → volume changes but shape does NOT change. Under pure shear stress → shape changes but volume does NOT change.

2.3 Differential Form of Strains

If point P moves to position (u, v, w) when force F is applied, linear strains in x, y, z directions are given by partial derivatives of displacements:

Differential Strain Equations

Linear Strains:

        ε_xx = ∂u/∂x

        ε_yy = ∂v/∂y

        ε_zz = ∂w/∂z

Shear Strains:

        φ_xy = ∂u/∂y + ∂v/∂x

        φ_yz = ∂v/∂z + ∂w/∂y

        φ_xz = ∂u/∂z + ∂w/∂x

2.4 Allowable Stresses and Factor of Safety

Factor of Safety (FOS or n)

FOS = Failure Stress / Working Stress

Margin of Safety = n − 1 (used in aircraft design)

Ductile Materials

σ_allow = σ_y / FOS

σ_y = yield stress
Failure by yielding

Brittle Materials

σ_allow = σ_u / FOS

σ_u = ultimate stress
Failure by fracture

2.5 Saint Venant Principal

Except at extreme ends of a bar carrying direct loading, stress distribution over cross-section is uniform.

      σ_max at 1-1 = 1.387 σ_avg

      σ_max at 2-2 = 1.027 σ_avg

      σ_max at 3-3 = σ_avg

2.6 Hooke’s Law + 2.7 Elastic Constants (E, G, K, μ)

2.6 Hooke’s Law

Under direct loading, within proportionality limit, stress is directly proportional to strain.

Hooke’s Law

σ = E · ε

E = constant of proportionality = Young’s Modulus of Elasticity (slope of σ-ε curve)

Valid only up to the limit of proportionality | E has same unit as stress (N/mm²)

2.6.1 Assumptions in Hooke’s Law:

1. Homogeneous (equal properties at all points) 2. Isotropic (equal properties in all directions) 3. Elastic

Note: Orthotropic material (3 mutually perp. directions) → 9 independent elastic constants. Anisotropic / Non-isotropic (e.g. crystal) → 21 independent elastic constants. Homogeneous & Isotropic → 2 independent constants (E and μ).

2.7 The Four Elastic Constants

Figure 2.1 — Four Elastic Constants: Definitions, Diagrams, and Values

Fig. 2.1 — Four elastic constants. E = slope of σ-ε (steel = 2×10⁵ N/mm²). G = shear stress/shear strain. K = inversely proportional to compressibility. μ = 0.25–0.42 for metals. Hooke’s law valid only up to limit of proportionality.

2.7.1 Relationships Between the Four Elastic Constants

(a)

E = 3K(1 − 2μ)

Relates E, K and μ

(b)

E = 2G(1 + μ)

Relates E, G and μ

(c)

E = 9KG / (3K + G)

Relates all three E, K, G

(d)

μ = (3K − 2G) / (6K + 2G)

Relates μ with K and G

Key Notes from Source:

The ratio E/K for most metals is about more than one third
For homogeneous & isotropic material: only 2 independent constants (E and μ)
For orthotropic material: 9 independent constants; Anisotropic: 21
For metals: μ ≤ 0.42 → if ΔV = 0, then σ_x + σ_y + σ_z = 0 (as μ ≠ 0.5)
Pure rubber: μ = 0.5 (perfectly incompressible)

2.8 Applications of Hooke’s Law + 2.9 Volumetric Strain

Case I — Uniaxial Loading (σ_x applied)

    ε_x = σ_x / E    
    (longitudinal strain, in direction of load)

    ε_y = −μ·σ_x/E   
    (lateral strain in y)

    ε_z = −μ·σ_x/E   
    (lateral strain in z)
  

Case II — Triaxial Loading (σ_x, σ_y, σ_z applied)

    ε_x = σ_x/E − μ(σ_y/E) − μ(σ_z/E)

    ε_y = σ_y/E − μ(σ_x/E) − μ(σ_z/E)

    ε_z = σ_z/E − μ(σ_x/E) − μ(σ_y/E)

2.9 Volumetric Strain (ε_V)

Volumetric strain = ratio of change in volume to original volume. Also called Dilatation.

Volumetric Strain Formulae

General 3D

ε_V = ΔV/V
= ε_x + ε_y + ε_z
= (σ_x+σ_y+σ_z)(1−2μ)/E

Cylindrical Rod (dia d, length l)

ε_V = ε_l + 2ε_d
(ε_l = longitudinal
ε_d = diametral)

Spherical Body (dia d)

ε_V = 3ε_d
(ε_d = diametral
strain)

    If equal triaxial: σ_x = σ_y = σ_z = σ →
    ε_V = 3σ(1−2μ)/E = σ/K
  

Note (from source): If change in volume ΔV = 0, either σ_x + σ_y + σ_z = 0 OR 1−2μ = 0 (i.e., μ = 0.5). But for metals μ ≤ 0.42, so if ΔV = 0, then σ_x + σ_y + σ_z = 0.

2.10 Deflection of Axially Loaded Members

General Axial Deflection Formula

Δ = ∫₀ᴸ P_x dx / (A_x · E_x)

Case I: Prismatic Bar (Uniform)

Δ = PL / AE

AE = axial rigidity | AE/L = axial stiffness

Case II: Circular Tapered Bar

Δ = 4PL / (π·D₁·D₂·E)

If D₁ = D₂ = D → Δ = PL/AE (uniform)

Case III: Prismatic Bar — Self Weight

Δ = γL²/2E = WL/2AE

γ = unit weight | W = γAL = total weight

Composite Bar (2 materials, firmly jointed)

      Equilibrium:  P = P₁ + P₂

      Compatibility: δ₁ = δ₂

      P₁ = PA₁E₁/(A₁E₁+A₂E₂)

      P₂ = PA₂E₂/(A₁E₁+A₂E₂)

      Δ = PL/(A₁E₁+A₂E₂)

2.10.1 Principle of Superposition

In a linear elastic structure, total displacements (or stresses) from multiple simultaneous loads = algebraic sum of displacements (or stresses) caused by each load acting independently.

2.11 Statically Indeterminate Axial Loaded Structures

Structures where static equilibrium equations alone are insufficient to find internal forces. Solved using Flexibility Approach (D_k < D_o — deflection compatibility) or Stiffness Approach. Extra equations are written in the form of deflection, slope and rotation — called compatibility equations.

Flexibility Method — Compatibility Equation (for bar fixed at both ends)

    Step 1: Equilibrium → P₁ + P₂ = P (or ΣF = 0)

    Step 2: Compatibility → Total deformation = 0 (fixed-fixed)

       i.e., P₁L₁/A₁E₁ + P₂L₂/A₂E₂ = 0

2.13 Temperature Stresses

If bar is rested over frictionless surface, free thermal expansion/contraction occurs without stress. If bar is constrained between fixed supports, thermal stresses develop.

Figure 2.2 — Temperature Stresses in Constrained Bar

Fig. 2.2 — Temperature stresses in a constrained bar. σ = EαT regardless of case. From source: thermal stresses are independent of member dimensions.

Universal Temperature Stress Formula

σ = E · α · T

E = Young’s modulus

α = Coefficient of thermal expansion

T = Change in temperature (°C)

Note from source: Thermal stresses are independent of member dimensions. Free expansion: ΔL = LαT (x), Δb = bαT (y), Δd = dαT (z). Free expansion → no stress. Constrained → stress = EαT.

2.15 Strain Energy

Energy stored in the body due to deformation against applied load. Denoted by U. Strain energy density = strain energy per unit volume.

Direct Stress

u = σ²/2E per unit vol

= (1/2)·σ·ε per unit vol

Shear Force (2.15.1)

U = ∫k·S²/2GA dx

k = 1.2 (rect.) | k = 10/9 (circular)

Bending Moment (2.15.3)

U = ∫M²/2EI dx

M = bending moment at section

Torque (2.15.4)

U = ∫T²/2GJ dx

T = torque | J = polar MOI

Chapter 2: Simple Stress-strain and Elastic Constants — Strength of Materials · MADE EASY Postal Study Package 2019

Simple Stress-strain & Elastic Constants – Chapter 2

Simple Stress-strain & Elastic Constants

2.1 Stress — Normal and Shear

2.1.1 Normal Stress

2.1.2 Shear Stress or Tangential Stress

2.2 Matrix Representation of Stress and Strain

2.2.1 Stress Tensor — 3D Stress Element

2.2.2 Matrix Representation of Strains

2.3 Differential Form of Strains

2.4 Allowable Stresses and Factor of Safety

2.6 Hooke’s Law + 2.7 Elastic Constants (E, G, K, μ)

2.6 Hooke’s Law

2.7 The Four Elastic Constants

2.7.1 Relationships Between the Four Elastic Constants

2.8 Applications of Hooke’s Law + 2.9 Volumetric Strain

Case I — Uniaxial Loading (σ_x applied)

Case II — Triaxial Loading (σ_x, σ_y, σ_z applied)

2.9 Volumetric Strain (ε_V)

2.10 Deflection of Axially Loaded Members

2.10.1 Principle of Superposition

2.11 Statically Indeterminate Axial Loaded Structures

2.13 Temperature Stresses

2.15 Strain Energy

Leave a Reply Cancel reply

Simple Stress-strain & Elastic Constants

2.1 Stress — Normal and Shear

2.1.1 Normal Stress

2.1.2 Shear Stress or Tangential Stress

2.2 Matrix Representation of Stress and Strain

2.2.1 Stress Tensor — 3D Stress Element

2.2.2 Matrix Representation of Strains

2.3 Differential Form of Strains

2.4 Allowable Stresses and Factor of Safety

2.6 Hooke’s Law + 2.7 Elastic Constants (E, G, K, μ)

2.6 Hooke’s Law

2.7 The Four Elastic Constants

2.7.1 Relationships Between the Four Elastic Constants

2.8 Applications of Hooke’s Law + 2.9 Volumetric Strain

Case I — Uniaxial Loading (σ_x applied)

Case II — Triaxial Loading (σ_x, σ_y, σ_z applied)

2.9 Volumetric Strain (ε_V)

2.10 Deflection of Axially Loaded Members

2.10.1 Principle of Superposition

2.11 Statically Indeterminate Axial Loaded Structures

2.13 Temperature Stresses

2.15 Strain Energy

Leave a Reply Cancel reply

Related News

Shear Stress in Beams – Chapter 6

Bending Stress in Beams – Chapter 5