Equilibrium and Elasticity

Equilibrium and Elasticity

Learning Objectives

Define and apply the conditions for static equilibrium.
Calculate the center of mass and understand its relation to the center of gravity.
Define stress, strain, and understand their relationship through Hooke's Law.
Analyze material deformation including tension, compression, shear, and bulk stress.
Interpret the stress-strain curve and identify key material properties such as yield strength and ultimate strength.

For a structure like a bridge or a building to serve its purpose, it must remain stationary and maintain its shape under various loads. This requires the principles of static equilibrium and an understanding of how materials deform (elasticity).

Static Equilibrium

Static Equilibrium Concepts

An object is in static equilibrium if it is completely at rest in our chosen frame of reference. This means it has no linear acceleration and no angular acceleration.

For a rigid body (an object whose size and shape do not change under load), two conditions must be met simultaneously for it to be in equilibrium.

Translational Equilibrium Condition

The vector sum of all external forces must be zero.

\Sigma \vec{F} = 0

Variables

Symbol	Description	Unit
$\Sigma \vec{F}$	Net external force	N

Translational Equilibrium in 2D

In 2D (xy-plane), the translational equilibrium condition breaks down into two scalar equations: $\Sigma F_x = 0$ and $\Sigma F_y = 0$ .

Rotational Equilibrium Condition

The vector sum of all external torques about any axis must be zero.

\Sigma \vec{\tau} = 0

Variables

Symbol	Description	Unit
$\Sigma \vec{\tau}$	Net external torque	$N\cdot m$

Choosing an Axis of Rotation

When applying the torque equation ( $\Sigma \tau = 0$ ), you are free to choose the axis of rotation anywhere you like. A strategic choice of axis (e.g., placing it exactly where an unknown force acts) will eliminate that unknown force from your torque equation, simplifying the math significantly.

Interactive Simulation

Use this torque balance model to see how lever arms and loads must offset each other for rotational equilibrium.

Interactive Physics Simulation

Torque Balance & Rotational Equilibrium Simulator

Place different masses along a balanced beam. Observe CCW and CW torques, pivot reaction forces, and translational/rotational equilibrium states.

✓ Equilibrium (Στ = 0)

Left Block Parameters (CCW)

Mass 1 (m1)20 kg

Position 1 (r1)-3.0 m

Right Block Parameters (CW)

Mass 2 (m2)30 kg

Position 2 (r2)2.0 m

Live Torque Summation Equation

\Sigma\tau = \tau_{CCW} - \tau_{CW} = 588.6 \text{ N}\cdot\text{m} - 588.6 \text{ N}\cdot\text{m} = 0.0 \text{ N}\cdot\text{m}

For **Rotational Equilibrium**, counter-clockwise (CCW) torque must exactly equal clockwise (CW) torque ($\Sigma \tau = 0$).

CCW Torque (τ1 = m1·g·r1)

588.60 N·m

CW Torque (τ2 = m2·g·r2)

588.60 N·m

Fulcrum Reaction Force (FR)

588.60 N

Net Residual Torque (Στ)

0.00 N·m

Center of Gravity (CG)

Center of Gravity (CG) Concepts

The center of gravity is the point at which the entire weight of an object can be considered to act for the purpose of calculating torques due to gravity.

For a uniform object in a uniform gravitational field (like near the Earth's surface), the center of gravity coincides perfectly with the geometric center of mass (CM).

Center of Mass Location ( $x_{cm}$ )

The geometric center of a mass distribution.

Center of Mass for Discrete Masses

Calculates center of mass position for a system of discrete particles.

x_{cm} = \frac{\Sigma m_i x_i}{\Sigma m_i}

Variables

Symbol	Description	Unit
$x_{cm}$	Center of mass position	m
$m_i$	Mass of particle i	kg
$x_i$	Position of particle i	m

Center of Mass for a Continuous Body

Calculates center of mass position for a continuous uniform body.

x_{cm} = \frac{1}{M} \int_0^L x \, dm

Variables

Symbol	Description	Unit
$x_{cm}$	Center of mass position	m
$M$	Total mass	kg
$L$	Length of the body	m
$x$	Position along the length	m
$dm$	Infinitesimal mass element	kg

Elasticity and Deformation

Elasticity and Deformation Concepts

In reality, no object is perfectly "rigid." When forces are applied, all materials deform to some extent. Understanding this deformation is the bridge between basic physics and "Mechanics of Materials," a core engineering subject.

Stress ( $\sigma$ )

Stress characterizes the intensity of the internal forces acting within a deformable body. It is the applied force per unit cross-sectional area. The SI unit is the Pascal (Pa), where $1 \text{ Pa} = 1 \text{ N/m}^2$ .

Stress Equation

Calculates stress from applied force and cross-sectional area.

\sigma = \frac{F}{A}

Variables

Symbol	Description	Unit
$\sigma$	Stress	Pa
$F$	Perpendicular force applied	N
$A$	Cross-sectional area	$m^2$

Strain ( $\epsilon$ )

Strain is the measure of the relative deformation (change in shape or size) of an object in response to stress. It is a dimensionless ratio.

Strain Equation

Calculates strain from change in length relative to original length.

\epsilon = \frac{\Delta L}{L_0}

Variables

Symbol	Description	Unit
$\epsilon$	Strain	dimensionless
$\Delta L$	Change in length	m
$L_0$	Original length	m

Hooke's Law for Continua

Hooke's Law for Continua Concepts

For small deformations, most solid materials exhibit elastic behavior: they return to their original shape when the stress is removed, and the strain is directly proportional to the stress. This is Hooke's Law applied to continuous media.

Hooke's Law

General relationship between stress and strain within the elastic limit.

\text{Stress} = \text{Elastic Modulus} \times \text{Strain}

Variables

Symbol	Description	Unit
$\text{Stress}$	Applied stress	Pa
$\text{Elastic Modulus}$	Material property indicating stiffness	Pa
$\text{Strain}$	Resulting strain	dimensionless

Interactive Simulation

Adjust force, area, and elastic modulus to connect stress, strain, and stiffness before moving into mechanics of materials.

Interactive Physics Simulation

Tensile Specimen & Stress-Strain Curve

Apply axial tensile force on a specimen. Observe Hooke's Law in the linear elastic region, and the horizontal curving path representing plastic flow beyond the yield threshold.

Elastic Region

Material Preset

Axial Tensile Force (P)35 kN

Cross-Section Area (A)250 mm²

Young's Modulus (E)200 GPa

Yield Strength (σ_y)250 MPa

Governing Formulas

Normal Stress

\sigma = \frac{P}{A}

Hooke's Law (Elastic Strain)

\epsilon = \frac{\sigma}{E} \quad (\sigma \le \sigma_y)

Normal Stress (σ)

140.0 MPa

Strain (ε)

7.000e-4

Deformation Mode

ELASTIC

Types of Elastic Moduli

The specific "Elastic Modulus" depends on the type of stress being applied. Common types include Young's Modulus, Shear Modulus, and Bulk Modulus.

Young's Modulus ( $E$ )

Measures resistance to tension (stretching) or compression (squeezing) along one axis. This is crucial for designing columns and cables.

Young's Modulus Stress-Strain Equation

The fundamental relationship between stress and strain for tension or compression.

\sigma = E \epsilon

Variables

Symbol	Description	Unit
$\sigma$	Stress	Pa
$E$	Young's Modulus	Pa
$\epsilon$	Strain	dimensionless

Young's Modulus Force-Area Equation

Relates applied force, area, and length changes for tension or compression.

\frac{F}{A} = E \left(\frac{\Delta L}{L_0}\right)

Variables

Symbol	Description	Unit
$F$	Force applied	N
$A$	Cross-sectional area	$m^2$
$E$	Young's Modulus	Pa
$\Delta L$	Change in length	m
$L_0$	Original length	m

Shear Modulus ( $G$ )

Measures resistance to shear forces (forces acting parallel to a surface, trying to slide layers past one another), where $\gamma$ is the shear strain angle.

Shear Modulus Equation

Hooke's Law applied to shear deformation.

\tau_{shear} = G \gamma

Variables

Symbol	Description	Unit
$\tau_{shear}$	Shear stress	Pa
$G$	Shear Modulus	Pa
$\gamma$	Shear strain	rad

Bulk Modulus ( $B$ )

Measures resistance to uniform compression from all sides (like an object submerged deep in the ocean). It relates pressure ( $P$ ) to volume strain ( $\Delta V/V_0$ ). The negative sign in the equation indicates that increased pressure causes a decrease in volume.

Bulk Modulus Equation

Relates pressure change to volume strain.

\Delta P = -B \left(\frac{\Delta V}{V_0}\right)

Variables

Symbol	Description	Unit
$\Delta P$	Change in pressure	Pa
$B$	Bulk Modulus	Pa
$\Delta V$	Change in volume	$m^3$
$V_0$	Original volume	$m^3$

Thermal Stress

Thermal Stress Concepts

If a structural member is constrained so that it cannot expand or contract when subjected to a temperature change ( $\Delta T$ ), large internal stresses develop. The thermal strain is $\epsilon_{thermal} = \alpha \Delta T$ . Because the member is constrained, the opposing stress developed is defined by the thermal stress equation.

Thermal Stress Equation

Calculates stress caused by constrained thermal expansion or contraction.

\sigma_{thermal} = E \alpha \Delta T

Variables

Symbol	Description	Unit
$\sigma_{thermal}$	Thermal stress	Pa
$E$	Young's Modulus	Pa
$\alpha$	Coefficient of linear expansion	$1/^\circ C$
$\Delta T$	Change in temperature	$^\circ C$

The Stress-Strain Curve

The Stress-Strain Curve Concepts

If you steadily increase the tensile stress on a material (like a steel rod) and plot the resulting strain, you get a characteristic curve.

Regions of the Stress-Strain Curve

STEP-BY-STEP

1. Proportional Limit: The highest stress where Hooke's law is valid (the curve is a straight line). The slope of this line is Young's Modulus ( $E$ ).
1. Elastic Limit (Yield Strength): The maximum stress a material can withstand without permanent (plastic) deformation. Up to this point, if you remove the load, the material snaps back to $L_0$ .
1. Plastic Region: Beyond the yield strength, the material deforms permanently. It behaves more like putty.
1. Ultimate Tensile Strength (UTS): The maximum stress the material can sustain before necking and eventual fracture.
1. Fracture Point: The stress at which the material breaks.

Yield Strength Importance

Engineering designs almost always require materials to stay well below their Yield Strength, ensuring they remain in the elastic region under typical operating loads.

Shear and Bulk Moduli

Deformation Beyond Tension

While Young's Modulus ( $E$ ) handles simple stretching and compression, complex structures experience other types of stress.

Shear Stress and Strain: Forces acting parallel to a surface cause layers of the material to slide past one another. The Shear Modulus ( $G$ ) relates shear stress ( $\tau = F_{parallel}/A$ ) to shear strain ( $\gamma = \Delta x / h$ ). This is critical in analyzing bolts, rivets, and torsion in drive shafts.
Bulk Stress and Strain: Forces acting uniformly from all directions (like hydrostatic pressure underwater) cause volume changes. The Bulk Modulus ( $B$ ) relates the change in pressure ( $\Delta P$ ) to the fractional change in volume ( $\Delta V / V_0$ ).

Shear Modulus Equation

Hooke's Law applied to shear deformation.

\tau = G \gamma

Variables

Symbol	Description	Unit
$\tau$	Shear stress	Pa
$G$	Shear Modulus	Pa
$\gamma$	Shear strain	rad

Key Takeaways

Static Equilibrium requires both zero net force ( $\Sigma \vec{F} = 0$ ) and zero net torque ( $\Sigma \vec{\tau} = 0$ ).
The Center of Gravity is the point where the total weight acts. It coincides with the center of mass for uniform fields.
Stress ( $\sigma = F/A$ ) measures the intensity of internal forces. Strain ( $\epsilon = \Delta L/L_0$ ) measures the resulting deformation.
Hooke's Law states that stress is proportional to strain in the elastic region, governed by an elastic modulus like Young's Modulus ( $E$ ).
Materials have limits (Yield Strength, Ultimate Strength) beyond which they deform permanently or break. Engineering designs must account for these limits.

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Learning Objectives

Static Equilibrium

Static Equilibrium Concepts

Translational Equilibrium Condition

Translational Equilibrium in 2D

Rotational Equilibrium Condition

Choosing an Axis of Rotation

Interactive Simulation

Torque Balance & Rotational Equilibrium Simulator

Center of Gravity (CG)

Center of Gravity (CG) Concepts

Center of Mass Location (xcmx_{cm}xcm​)

Center of Mass for Discrete Masses

Center of Mass for a Continuous Body

Elasticity and Deformation

Elasticity and Deformation Concepts

Stress (σ\sigmaσ)

Stress Equation

Strain (ϵ\epsilonϵ)

Strain Equation

Hooke's Law for Continua

Hooke's Law for Continua Concepts

Hooke's Law

Interactive Simulation

Tensile Specimen & Stress-Strain Curve

Types of Elastic Moduli

Young's Modulus (EEE)

Young's Modulus Stress-Strain Equation

Young's Modulus Force-Area Equation

Shear Modulus (GGG)

Shear Modulus Equation

Bulk Modulus (BBB)

Bulk Modulus Equation

Thermal Stress

Thermal Stress Concepts

Thermal Stress Equation

The Stress-Strain Curve

The Stress-Strain Curve Concepts

Regions of the Stress-Strain Curve

Yield Strength Importance

Shear and Bulk Moduli

Deformation Beyond Tension

Shear Modulus Equation

Center of Mass Location ( $x_{cm}$ )

Stress ( $\sigma$ )

Strain ( $\epsilon$ )

Young's Modulus ( $E$ )

Shear Modulus ( $G$ )

Bulk Modulus ( $B$ )