Simple Stress and Strain

Simple Stress and Strain

Learning Objectives

Understand the fundamental concepts of stress and strain.
Differentiate between normal, shear, and bearing stresses.
Apply Hooke's Law to relate stress and strain in the elastic region.
Calculate deformation of prismatic bars under axial loads.
Understand stress concentration and its effects on structural members.
Analyze the stress-strain diagram and identify key material properties.
Evaluate structural safety using factors of safety and margin of safety.
Understand thermal stresses and statically indeterminate members.

Mechanics of Deformable Bodies, also known as Strength of Materials, deals with the internal effects of forces acting on a body. While Engineering Mechanics (Statics) treats bodies as rigid, this subject acknowledges that all materials deform under load. The fundamental concepts are stress (intensity of force) and strain (intensity of deformation).

Concept of Stress

Stress ( $\sigma$ )

Stress is the internal resistance of a material to an external force, defined as force per unit area. It represents the intensity of the internal force on a specific plane area.

Normal Stress (Axial Stress)

Normal stress acts perpendicular to the cross-sectional area.

Sign Convention:

Tensile Stress (+): Pulls the material apart (elongation).
Compressive Stress (-): Pushes the material together (shortening).

Normal Stress

Formula for normal stress where $P$ is applied axial load and $A$ is the cross-sectional area.

\sigma = \frac{P}{A}

Variables

Symbol	Description	Unit
$\sigma$	Normal stress	Pa
$P$	Applied axial load	N
$A$	Cross-sectional area	$m^2$

Shear Stress ( $\tau$ )

Shear stress acts parallel (tangential) to the cross-sectional area. It tends to slide one layer of material over another.

Shearing Force

The shear stress is caused by a shear force acting on the area.

Shear Stress

Formula for shear stress where $V$ is the shear force and $A$ is the area resisting the shear.

\tau = \frac{V}{A}

Variables

Symbol	Description	Unit
$\tau$	Shear stress	Pa
$V$	Internal shear force	N
$A$	Area parallel to the applied shear force	$m^2$

Bearing Stress ( $\sigma_b$ )

Bearing stress is the contact pressure between two separate bodies. It is a compressive stress.

Bearing Area

For a bolt in a plate, the projected contact area is $A_b = d \times t$ (diameter $\times$ thickness).

Bearing Stress

Formula for bearing stress where $P$ is the force and $A_b$ is the projected contact area.

\sigma_b = \frac{P}{A_b}

Variables

Symbol	Description	Unit
$\sigma_b$	Bearing stress	Pa
$P$	Applied compressive force	N
$A_b$	Projected contact area	$m^2$

Interactive Simulation

Interact with the simulation below to visualize how different stresses occur in structural connections.

Normal Stress Visualizer

50 kN

←

50 kN

→

Area (

A

): 314.2 mm²

Calculated Stress (

\\sigma

)

159.15 MPa

Force (

P

): 50 kN

Radius (

r

): 10 mm

Adjust the force and the radius of the circular cross-section to see how they affect the normal stress. Notice that increasing the area (radius) decreases the stress, while increasing the force increases the stress.

Stress Concentration and Saint-Venant's Principle

Stress Concentration

When a structural member contains discontinuities such as holes, notches, fillets, or sharp corners, the stress lines crowd together around the discontinuity. This results in highly localized stresses that are much greater than the average nominal stress calculated by $\sigma = P/A$ .

Saint-Venant's Principle

Saint-Venant's Principle states that the localized effects of an applied load dissipate rapidly as you move away from the point of application.

At a sufficient distance (usually equal to the largest dimension of the cross-section) away from the loaded ends or points of support, the stress distribution becomes essentially uniform, and equations like $\sigma = P/A$ become highly accurate. Near the load application points, the stress is complex and localized.

Stress Concentration Factor ( $K_t$ )

The maximum stress is defined using a theoretical stress concentration factor determined experimentally or via finite element analysis, depending on the geometry of the discontinuity.

In ductile materials under static loading, yielding at the concentration redistributes the stress safely. However, for brittle materials or parts under fatigue (cyclic) loading, stress concentrations are the primary cause of sudden, catastrophic failure and must be strictly accounted for in design.

Maximum Stress at Discontinuity

Formula to calculate the maximum stress at a stress concentration.

\sigma_{max} = K_t \sigma_{avg}

Variables

Symbol	Description	Unit
$\sigma_{max}$	Maximum localized stress	Pa
$K_t$	Theoretical stress concentration factor	unitless
$\sigma_{avg}$	Nominal average stress at the net cross-section	Pa

Concept of Strain and Deformation

Strain ( $\epsilon$ )

Strain is the measure of deformation representing the displacement between particles in the body relative to a reference length. It is a dimensionless quantity representing the geometric expression of deformation caused by the action of stress on a physical body.

Normal Strain

Normal strain represents the change in length per unit length.

Normal Strain

Formula for normal strain where $\delta$ is total deformation and $L$ is the original length.

\epsilon = \frac{\delta}{L}

Variables

Symbol	Description	Unit
$\epsilon$	Normal strain	m/m or unitless
$\delta$	Total axial deformation	m
$L$	Original length	m

Shear Strain ( $\gamma$ )

Measures the change in angle between two lines that were originally perpendicular. It is caused by shear stress and represents the angular distortion of the body.

Stress-Strain Relationship

For many engineering materials (like steel), the initial relationship between stress and strain is linear.

Hooke's Law

Within the proportional limit (elastic region), stress is directly proportional to strain.

Modulus of Elasticity ( $E$ )

The constant of proportionality is the Modulus of Elasticity (Young's Modulus). It is a measure of the material's stiffness.

Steel: $E \approx 200 \text{ GPa}$ ( $29,000 \text{ ksi}$ )
Aluminum: $E \approx 70 \text{ GPa}$ ( $10,000 \text{ ksi}$ )

Hooke's Law

Linear relation between stress and strain.

\sigma = E \epsilon

Variables

Symbol	Description	Unit
$\sigma$	Normal stress	Pa
$E$	Modulus of Elasticity	Pa
$\epsilon$	Normal strain	unitless

Substituting the definitions of stress and strain ( $\sigma = P/A, \epsilon = \delta/L$ ), we can calculate the axial deformation of a prismatic bar under a constant load:

Axial Deformation

Equation for deformation of a prismatic bar under constant axial load.

\delta = \frac{PL}{AE}

Variables

Symbol	Description	Unit
$\delta$	Total axial deformation	m
$P$	Applied axial load	N
$L$	Original length	m
$A$	Cross-sectional area	$m^2$
$E$	Modulus of Elasticity	Pa

Thermal Stress and Strain

When a material is subjected to a change in temperature, it will expand when heated and contract when cooled. The corresponding change in length is thermal deformation. If this deformation is prevented by supports, internal forces are generated, creating thermal stress.

Thermal Deformation

The unconstrained thermal deformation depends on the coefficient of thermal expansion ( $\alpha$ ), the change in temperature ( $\Delta T$ ), and the original length ( $L$ ).

Thermal Deformation

Equation for temperature-induced change in length.

\delta_T = \alpha \Delta T L

Variables

Symbol	Description	Unit
$\delta_T$	Thermal deformation	m
$\alpha$	Coefficient of thermal expansion	$1/^\circ C$
$\Delta T$	Change in temperature	$^\circ C$
$L$	Original length	m

Statically Indeterminate Members

Members for which the equations of statics ( $\Sigma F = 0$ , $\Sigma M = 0$ ) are not sufficient to determine all the unknown support reactions or internal forces.

Compatibility Equations

To solve statically indeterminate problems, we must supplement the equilibrium equations with relationships involving the deformation of the members. These geometric relationships are called compatibility equations.

Poisson's Ratio ( $\nu$ )

When a material is stretched in one direction (longitudinal), it contracts in the perpendicular directions (lateral). Poisson's ratio is the absolute ratio of lateral strain to longitudinal strain. For most metals, $\nu$ is between $0.25$ and $0.35$ .

Poisson's Ratio

Ratio of lateral to longitudinal strain.

\nu = -\frac{\epsilon_{lateral}}{\epsilon_{longitudinal}}

Variables

Symbol	Description	Unit
$\nu$	Poisson's Ratio	unitless
$\epsilon_{lateral}$	Lateral strain	unitless
$\epsilon_{longitudinal}$	Longitudinal strain	unitless

Relationships Between Elastic Constants

The properties of isotropic materials are interrelated through the Modulus of Elasticity ( $E$ ), Shear Modulus ( $G$ ), Bulk Modulus ( $K$ ), and Poisson's Ratio ( $\nu$ ). These relationships are fundamental for solving complex deformation problems.

Shear Modulus ( $G$ ): Relates shear stress to shear strain ( $\tau = G\gamma$ ). Bulk Modulus ( $K$ ): Measures the material's resistance to uniform compression (volumetric strain).

Shear Modulus

Relationship between $E$ and $\nu$ .

G = \frac{E}{2(1 + \nu)}

Variables

Symbol	Description	Unit
$G$	Shear Modulus	Pa
$E$	Modulus of Elasticity	Pa
$\nu$	Poisson's Ratio	unitless

Bulk Modulus

Relationship between $E$ and $\nu$ .

K = \frac{E}{3(1 - 2\nu)}

Variables

Symbol	Description	Unit
$K$	Bulk Modulus	Pa
$E$	Modulus of Elasticity	Pa
$\nu$	Poisson's Ratio	unitless

Combined Elastic Relationship

Relating $E$ , $K$ , and $G$ .

E = \frac{9KG}{3K + G}

Variables

Symbol	Description	Unit
$E$	Modulus of Elasticity	Pa
$K$	Bulk Modulus	Pa
$G$	Shear Modulus	Pa

Design Philosophies

Working Stress Design (WSD) / Allowable Stress Design (ASD)

In this traditional method, members are designed such that the maximum actual stresses caused by service loads do not exceed a specified allowable stress. This allowable stress is found by applying a Factor of Safety ( $FS$ ) to the material's yield or ultimate strength.

Allowable Stress

Calculating allowable stress using a safety factor.

\sigma_{allow} = \frac{\sigma_{Y}}{FS} \quad \text{or} \quad \frac{\sigma_{U}}{FS}

Variables

Symbol	Description	Unit
$\sigma_{allow}$	Allowable working stress	Pa
$\sigma_{Y}$	Yield stress	Pa
$\sigma_{U}$	Ultimate stress	Pa
$FS$	Factor of Safety	unitless

Ultimate Limit State (ULS) / Load and Resistance Factor Design (LRFD)

Modern codes (like NSCP) increasingly use ULS. Instead of reducing the material's strength by a single large safety factor, ULS applies separate load factors (to increase the expected service loads based on uncertainty) and resistance factors (to slightly reduce the material's theoretical capacity).

ULS Requirement

Design strength must exceed factored load effect.

\text{Factored Load Effect} \le \phi R_n

Variables

Symbol	Description	Unit
$\phi$	Resistance factor	unitless
$R_n$	Nominal resistance or capacity	various

Factor of Safety ( $FS$ )

It is the ratio of the ultimate strength (or yield strength) of the material to the allowable (or working) stress. It is an index of structural capacity beyond the expected loads, accounting for uncertainties in material properties, loading, and analysis.

Factor of Safety

Ratio of failure stress to allowable stress.

FS = \frac{\sigma_{ultimate}}{\sigma_{allowable}} \quad \text{or} \quad FS = \frac{\sigma_{yield}}{\sigma_{allowable}}

Variables

Symbol	Description	Unit
$FS$	Factor of Safety	unitless
$\sigma_{ultimate}$	Ultimate failure stress	Pa
$\sigma_{yield}$	Yield stress	Pa
$\sigma_{allowable}$	Allowable stress	Pa

Margin of Safety ( $MS$ )

It is a measure used particularly in aerospace engineering. A positive $MS$ indicates the design is safe, while a negative $MS$ indicates failure.

Margin of Safety

Alternative index for structural safety.

MS = FS - 1 = \frac{\sigma_{failure}}{\sigma_{applied}} - 1

Variables

Symbol	Description	Unit
$MS$	Margin of Safety	unitless
$FS$	Factor of Safety	unitless
$\sigma_{failure}$	Failure stress	Pa
$\sigma_{applied}$	Applied working stress	Pa

Material Behavior

The Stress-Strain Diagram

The stress-strain diagram is a graphical representation of the behavior of a material when subjected to an increasing load.

Materials are broadly classified into two categories based on their behavior under load:

Ductile Materials: (e.g., mild steel, aluminum) Undergo significant plastic deformation (yielding) before failure. They provide ample warning before fracture.
Brittle Materials: (e.g., concrete, cast iron, glass) Fail suddenly with very little or no plastic deformation. Their stress-strain curve is generally linear up to fracture.

Key Points on the Diagram (for ductile materials like mild steel):

Proportional Limit: The highest stress at which stress is directly proportional to strain (Hooke's Law applies).
Elastic Limit: The highest stress the material can withstand without undergoing permanent deformation. Once the load is removed, the material returns to its original shape.
Yield Point: The point at which there is an appreciable elongation or yielding of the material without any corresponding increase in load.
Ultimate Stress (Tensile Strength): The maximum stress the material can withstand. It is the highest point on the stress-strain curve.
Rupture Strength (Fracture Point): The stress at which the material actually breaks or fractures.

True Stress vs. Engineering Stress:

Engineering Stress: Calculated using the original cross-sectional area ( $A_0$ ). $\sigma = P / A_0$
True Stress: Calculated using the actual, instantaneous cross-sectional area ( $A$ ) at any given load. Since the area decreases (necking) under tension, true stress is always higher than engineering stress. $\sigma_{true} = P / A$

Interactive Simulation

Interact with the simulation below to explore different stress and strain states for various materials.

Material Behavior States

Explore how different materials deform under stress

Loading chart...

Apply Strain (Pull Material)ε = 0.0000

Current Stress0.0 MPa

Current StateElastic Region

Ductile Steel (e.g., Low Carbon Steel)

Exhibits a distinct elastic region, yield point, a plastic plateau, strain hardening, and necking before fracture. Highly ductile and tough.

What is happening now?

Elastic Region

Material deforms reversibly. Stress is directly proportional to strain (Hooke's Law applies). If the load is removed, the material returns exactly to its original shape. The bonds between atoms are stretched but not broken.

Microscopic View

Atoms are stretched uniformly like springs.

Key Takeaways

Stress ( $\sigma = P/A$ ) is the internal intensity of force. Normal stress is perpendicular to the area; Shear stress is parallel.
Bearing Stress is localized compressive stress at the contact surface between two bodies.
Saint-Venant's Principle guarantees that localized load stresses dissipate quickly, becoming uniform at a distance equal to the cross-section's largest dimension.
Stress Concentrations ( $K_t$ ) occur at geometric discontinuities (holes, corners), heavily amplifying local stresses and posing severe risk for brittle/fatigue failure.
Strain ( $\epsilon = \delta/L$ ) is the intensity of deformation.
Thermal Strain ( $\delta_T = \alpha \Delta T L$ ) causes internal thermal stresses if deformation is restricted.
Hooke's Law ( $\sigma = E\epsilon$ ) relates stress and strain linearly via the Modulus of Elasticity ( $E$ ), valid within the elastic range.
Deformation ( $\delta = PL/AE$ ) allows us to calculate how much a member stretches or shortens under load.
Statically Indeterminate Members require compatibility equations in addition to statics for solving unknown forces.
Factor of Safety ( $FS$ ) is often applied to the Yield or Ultimate Stress to determine the Allowable Stress: $\sigma_{allow} = \sigma_{yield} / FS$ .
Margin of Safety ( $MS$ ) is an alternative index: $MS = FS - 1$ .
Stress-Strain Diagram reveals key material properties such as Proportional Limit, Yield Point, Ultimate Stress, and Rupture Strength.
Engineering Stress assumes constant area, whereas True Stress accounts for the reduction in cross-sectional area (necking).

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Learning Objectives

Concept of Stress

Stress (σ\sigmaσ)

Normal Stress (Axial Stress)

Normal Stress

Shear Stress (τ\tauτ)

Shearing Force

Shear Stress

Bearing Stress (σb\sigma_bσb​)

Bearing Area

Bearing Stress

Interactive Simulation

Normal Stress Visualizer

Stress Concentration and Saint-Venant's Principle

Stress Concentration

Saint-Venant's Principle

Stress Concentration Factor (KtK_tKt​)

Maximum Stress at Discontinuity

Concept of Strain and Deformation

Strain (ϵ\epsilonϵ)

Normal Strain

Normal Strain

Shear Strain (γ\gammaγ)

Stress-Strain Relationship

Hooke's Law

Modulus of Elasticity (EEE)

Hooke's Law

Axial Deformation

Thermal Stress and Strain

Thermal Deformation

Thermal Deformation

Statically Indeterminate Members

Compatibility Equations

Poisson's Ratio (ν\nuν)

Poisson's Ratio

Relationships Between Elastic Constants

Shear Modulus

Bulk Modulus

Combined Elastic Relationship

Design Philosophies

Working Stress Design (WSD) / Allowable Stress Design (ASD)

Allowable Stress

Ultimate Limit State (ULS) / Load and Resistance Factor Design (LRFD)

ULS Requirement

Factor of Safety (FSFSFS)

Factor of Safety

Margin of Safety (MSMSMS)

Margin of Safety

Material Behavior

The Stress-Strain Diagram

Interactive Simulation

Material Behavior States

Ductile Steel (e.g., Low Carbon Steel)

What is happening now?

Elastic Region

Microscopic View

Stress ( $\sigma$ )

Shear Stress ( $\tau$ )

Bearing Stress ( $\sigma_b$ )

Stress Concentration Factor ( $K_t$ )

Strain ( $\epsilon$ )

Shear Strain ( $\gamma$ )

Modulus of Elasticity ( $E$ )

Poisson's Ratio ( $\nu$ )

Factor of Safety ( $FS$ )

Margin of Safety ( $MS$ )