Mechanics of Materials
What is Mechanics of Materials?
Internal resistance force per unit area within a material subjected to external forces.
Key formula / rule: Stress
Key points
- Understand the concepts of stress and strain.
- Analyze the behavior of materials under axial, torsional, and bending loads.
- Determine deformations and deflections in structural elements.
- Apply failure theories to predict material failure.
Common exam trap
Confusing stress and strain units.
Definitions
- Term
Stress
- Meaning
Internal resistance force per unit area within a material subjected to external forces.
- Term
Strain
- Meaning
Measure of deformation representing the relative displacement between particles in a body.
- Term
Young's Modulus (E)
- Meaning
A measure of the stiffness of an elastic material; the ratio of tensile stress to tensile strain in the elastic region.
- Term
Poisson's Ratio (ν)
- Meaning
The ratio of transverse strain to axial strain when a material is subjected to uniaxial stress.
- Term
Shear Modulus (G)
- Meaning
A measure of the stiffness of an elastic material under shear stress; the ratio of shear stress to shear strain.
- Term
Moment of Inertia (I)
- Meaning
A geometric property of a cross-section that describes its resistance to bending.
- Term
Polar Moment of Inertia (J)
- Meaning
A geometric property of a cross-section that describes its resistance to torsion.
- Term
Elastic Limit
- Meaning
The maximum stress a material can withstand without permanent deformation upon unloading.
- Term
Yield Strength
- Meaning
The stress at which a material begins to deform plastically.
- Term
Ultimate Tensile Strength (UTS)
- Meaning
The maximum stress a material can withstand while being stretched or pulled before necking.
- Term
Buckling
- Meaning
A sudden failure mode in slender structural members subjected to compressive loads, characterized by lateral instability.
Learning objectives
Understand the concepts of stress and strain.
Analyze the behavior of materials under axial, torsional, and bending loads.
Determine deformations and deflections in structural elements.
Apply failure theories to predict material failure.
Understand the phenomenon of buckling and its prevention.
Calculate stress concentration factors.
Formulae
- Name
Stress
- Note
Tensile or compressive stress due to axial load P on area A.
- Expression
σ = P/A
- Name
Strain
- Note
Axial strain due to change in length ΔL over original length L.
- Expression
ε = ΔL/L
- Name
Hooke's Law (Linear Elasticity)
- Note
Relationship between stress and strain, E is Young's Modulus.
- Expression
σ = Eε
- Name
Shear Stress
- Note
Shear stress due to shear force V, Q is first moment of area, I is moment of inertia, t is width.
- Expression
τ = VQ/It
- Name
Torsion Formula
- Note
Shear stress τ at radius r, T is torque, J is polar moment of inertia, G is shear modulus, θ is angle of twist, L is length.
- Expression
τ/r = T/J = Gθ/L
- Name
Bending Stress
- Note
Bending stress σ at distance y from neutral axis, M is bending moment, I is moment of inertia, E is Young's Modulus, R is radius of curvature.
- Expression
σ/y = M/I = E/R
- Name
Poisson's Ratio
- Note
Ratio of lateral strain to axial strain.
- Expression
ν = -εlateral / εaxial
- Name
Shear Modulus
- Note
Relationship between Young's Modulus and Shear Modulus.
- Expression
G = E / (2(1+ν))
- Name
Euler's Buckling Load
- Note
Critical compressive load for column buckling, K is effective length factor.
- Expression
Pcr = (π²EI)/(KL)²
- Name
Stress Concentration Factor
- Note
Ratio of maximum stress to nominal stress at a discontinuity.
- Expression
Kt = σmax / σnominal
Prerequisites
Statics
Dynamics
Basic Calculus
Material Science fundamentals
Common mistakes
Confusing stress and strain units.
Assuming linear elastic behavior beyond the elastic limit.
Incorrectly applying boundary conditions in beam bending problems.
Ignoring stress concentrations in regions of geometric discontinuity.
Misinterpreting the sign conventions for bending moments and shear forces.
Using incorrect formulas for torsion in non-circular shafts.
Forgetting to consider combined stresses.
Keywords
Stress
Strain
Elasticity
Plasticity
Bending
Torsion
Shear
Buckling
Stress Concentration
Material Properties
Hooke's Law
Young's Modulus
Poisson's Ratio
Shear Modulus
Moment of Inertia
Polar Moment of Inertia
Practice preview
Hooke's Law states that stress is directly proportional to strain within the:…
easy
Poisson's ratio is defined as the ratio of:…
easy
A steel rod of length 1 m and diameter 20 mm is subjected to an axial tensile load of 50 kN. If the Young's modulus of steel is 200 GPa, what is the elongation of the rod?…
medium