Strength of Materials: Stress Analysis

• Study notes on Strength of Materials are provided.

Introduction to Stress

• Stress measures a body's internal resistance to external loads.
• Stress is defined as the internal force per unit area at a given point on any plane, with the SI unit Pascal (Pa or N/m²).

Types of Loads

• Loads can be axial, transverse, bending, or twisting.
• Axial force (Fa) acts along the axis of the body.
• Transverse force (Ft) acts perpendicular to the axis.

Types of Stresses

• Direct stress develops from external force directly acting on the plane, while indirect stress develops from moments when external force does not pass through the centroid.
• Normal stress (σ) develops when internal forces act normal (perpendicular) to the plane.
• Shear stress (τ) develops when internal forces act parallel to the plane.
• When the internal force acts at an angle, both normal and shear stress develop
• Direct axial stress is calculated as σ = P/A.

Stress Analysis Under General Loading

• Stress is a tensor quantity characterized by magnitude, direction, and plane.
• Tensors are quantities characterized by magnitude, direction, and plane; examples include stress, strain, and moment of inertia.
• Stress Tensor at a point has 6 independent stress components and 3 dependent stress components.
• Complimentary shear stress means that if there is a shear stress on one plane, there must be and equal an opposite shear stress on the perpendicular plane.
• Bi-axial stress/plane stress examples are plane stress conditions
• A pure shear state of stress exists with only shear stresses
• Hydrostatic stress has uniform normal stresses in all directions,
• σx = σy = σz = σ

Types of Strain

• Strain is categorized into normal (∈) and shear (γ) types
• Normal strain measures change in size
• Volumetric strain is the sum of strains in three orthogonal directions
• Shear strain measures change in shape and is defined as the change in angle between mutually perpendicular planes
• Strain Tensor relates different strain components.

Types of Material

• Materials can be homogenous, non-homogenous, isotropic, anisotropic, or orthotropic.
• Homogenous material has same material properties at all points in the same direction.
• Non-homogenous material has different material properties at all points in the same direction.
• Isotropic material has same material properties in every direction at a point.
• Anisotropic material has different material properties in every direction at a point.
• Orthotropic material has different material properties in mutually perpendicular directions at a point.

Elastic Constants

• Elastic constants are material properties that relate stress and strain, including Modulus of Elasticity (E), Modulus of Rigidity (G or C), Bulk Modulus (K), and Poisson's Ratio (μ or 1/M).
• Modulus of Elasticity (E) - Ratio of normal stress and normal (longitudinal) strain. 1 GPa = 10^9 Pa = 10^3 MPa. Esteel = 200 GPa, etc
• Modulus of Rigidity (G) - Ratio of shear stress and shear strain.
• Bulk Modulus (K) - Ratio of hydrostatic stress and volumetric strain.
• Poisson's Ratio (μ) - Ratio of magnitude of lateral strain and longitudinal strain. Typically ranges from 0 to 0.5 .

Relation between Elastic Constants and Hooke's Law

• E = 2G(1 + μ).
• E = 3K(1 – 2μ).
• Hooke's Law states stress is directly proportional to strain within the proportional limit, and constants of proportionality are the elastic constants.

Stress-Strain Curve for Ductile and Brittle Materials

• Key points on the curve for Ductile Materials include: Elastic Limit, Yield Strength, Ultimate Strength, and Fracture.
• Key points on the curve for Brittle Materials include: A and B

Strength

• Maximum magnitude of stress that the material can sustain without failure.

Ductility

• Ability of a material to deform plastically.
• % change in length = (Δl/l) × 100%
• % change in area = (ΔA/A) × 100%

Resilience and Toughness

• Resilience - Ability of a material to absorb strain energy without permanent deformation.
• Toughness - Ability of a material to absorb strain energy without fracture, measured by Modulus of toughness = Toughness / Volume.

True Stress-Strain

• True stress (στ) can be calculated by στ = P/Af
• True strain (ετ) can be calculated by ετ = ln(lo/Af)

Power Law

• The Power Law is given by στ = kε^n

Axially Loaded Members

• Material of the bar, constant cross-sectional area, axial load passes through centroid, stresses with in proportional limit.
• Elongation of bar Δl = ∫(P dx/AE)

Statically Indeterminate Bars

• Compatibility Equation: Δl = 0

Thermal Stress

• Altotal = αlΔT, and α is the coefficient of thermal expansion.
• Thermal stress with bars fixed in one direction σ = (P/A) = E α ΔT
• Thermal stress in bars fixed in two directions: σχ = σy = E α ΔT/ (1-ν)
• Thermal stress in bars fixed in all directions: σx = σy =σz= (E α ΔT)/(1-2v)
• Thermal Stress in a Bars when there is a gap/yielding of supports Altotal = δ

Strain Energy due to Axial Load

• Strain energy due to axial load U = P^2L / 2AE
• Axial impact load is shown with associated formulas.

Torsion in Circular Shafts

• Material of shaft homogeneous & isotropic and stresses within proportional limit and transverse sections remain plane and undistorted after twisting.
• Torsion equation: T/J = τ/r = Gθ/l
• J is the polar moment of inertia
• r = the distance from axis
• θ = the angle of twist
• Maximum shear stress on outer surface: Tmax = T(d/2) / J
• Angle of twist θ = Tl/GJ (in radians)
• Polar Section Modulus is the measure of the strength and depends on shape and sze
• It measures resistance to deformation under twisting moment.
• The magnitude of torque required for unit angle of twist
• Torsional rigidity (GJ) measures the resistance to deformation under twisting moment.
• Statically Indeterminate Shaft and composite shaft.

Strain Energy due to Torsion

• Strain energy due to torsion U = T^2l / 2GJ

Shear Force & Bending Moment

• Beams are structural members used to support transverse loads

Types of Beams

• Statically Determinate types include Cantilever, Simply Supported, and Overhang Beams
• Statically Indeterminate types include Propped Cantilever, Continuous, and Fixed (Built-in) Beams
• Distributed loads are measured by:Load intensity = w (KN/m)
• Total load = Area under the loading diagram

Shear Force

• Shear force is the transverse internal force at a section
• Shear force is equal to the sum of total transverse force either on the left or right side of the section.

Bending Moment

• Bending moment is the internal moment at a section.
• Bending moment is equal to the sum of moment of all the forces either on the left or right side of the section.

Relation Between Load Intensity(w), Shear Force (F) and Bending Moment (M)

• ds/dx = w. dM/dx = S. MB-MA = Area of SFD between A and B.
• Bending moment is maximum when shear force is zero or there is couple.

Point of Contra Flexure

• A point where the sign of bending moment changes and the curvature of beam changes from sagging to hogging or hogging to sagging.

Bending Stress in Beams

• The material of the beam is homogeneous and isotropic, Young's modulus in tension and compression is same, stresses with in proportional limit.
• Neutral axis goes through centroid if material is homogeneous and no plastic deformation.
• The bending stress is proportional to the distance from the neutral axis.
• Section Modulus (Z) measures the strength of beam (maximum applicable bending moment) with Z=I/ytax

Shear Stress in Beams

• Shear stress distribution depends on cross-section, with formulas provided for rectangular, circular, triangular, diamond, I, and T sections.
• Shear flow in thin walled members is the shear force per unit length.

Shear Center

• The point on the beam section at which the transverse load can be applied without causing twisting.

Deflection of Beams

• Deflection represents linear deviation of a point and the slope represents the angular deviation Methods to find
• Slope and deflection of beams Double Integration and Macaulay's method.
• Moment - Area method and Strain Energy method.

Double Integration Method

• EI d² y/dx² =Mx
• El dy/dx = ∫ Mx+C₁
• Ely = ∫ ∫ Mx+C₁x + C2, to determine at any location of the beam.

Macaulay's Method

• This method is preferred for simply supported beam unsymmetric loading and cantilever beam subjected to multiple loads.
• Apply Moment – Area Method, solve for Moment with key formulas.

Castigliano's Theorem

• The partial derivative of the total strain energy in a structure with respect to any force at a point is equal to the deflection at that point in the direction of the force

Statically Indeterminate Beams

• deflection at point B and C will be same

Complex Stress

• Point is subjected to triaxial state of stress and point is subjected to biaxial state of stress.
• Stresses on oblique planes with associated sign conventions.
• σθ = (σx+σy)/2+((σx-σy)/2)cos2θ +τxy sin 2θ
• τθ = -((σx-σy)/2)sin2θ +τxy cos 2θ

Mohr's Circle

• Mohr's Circle is used for biaxial state of stress. It is used to find principal stresses and maximum shear stress.

Principal Planes and Stresses

• The formulas for principal stress are:
• σ₁,₂ = (σx+σy)/2 ± √(((σx-σy)/2)² +τxy²)
• Maximum Shear Stress under complex load.

Complex Strain

• Strain analysis is similar to stress analysis, where normal stress is replaced by normal strain, and shear stress is replaced by half the shear strain
• Apply the strains on oblique planes given associated formulas with axis to Mohr's Circle for biaxial state of strain

Strain Rosette

• Combination of three strain gauges arranged in three different directions
• Given three star strain and rectangular strain rosette formulas.

Pressure Vessels

• Pressure vessels are categorized by whether they are thin or thick walled, based on the ratio of the thickness (t) to the diameter (d):
• t ≤ d/20 (thin)
• t > d/20 (thick)
• Formulas are given for longitudinal stress, circumferential/hoop stress, and longitudinal strain in both thin cylinders and thin spheres
• Circumferential/hoop strain given in vessel.

Columns

• Columns are structural members used to support axial compressive loads.
• List when P is unstable, stable and critcal
• Formulas are given for the theory of buckling
• Pcr = π² E Imin / l²