Equilibrium and Elasticity

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.

Conditions for Equilibrium

1. First Condition (Translational Equilibrium): The vector sum of all external forces acting on the object must be zero. This prevents the center of mass from accelerating.
ΣF⃗=0 \Sigma \vec{F} = 0
In 2D (xy-plane), this breaks down into two scalar equations: ΣFx=0\Sigma F_x = 0 ΣFy=0\Sigma F_y = 0
2. Second Condition (Rotational Equilibrium): The vector sum of all external torques acting on the object, measured about any arbitrary axis, must be zero. This prevents the object from starting to spin.
Στ⃗=0 \Sigma \vec{\tau} = 0
In 2D, this is a single scalar equation: Στaxis=0\Sigma \tau_{axis} = 0

Important

When applying the torque equation (Στ=0\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.

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 (xcmx_{cm})

For a system of discrete masses mim_i at positions xix_i:
xcm=ΣmixiΣmi x_{cm} = \frac{\Sigma m_i x_i}{\Sigma m_i}
For a continuous uniform body of length LL:
xcm=1M∫0Lx dm x_{cm} = \frac{1}{M} \int_0^L x \, dm

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 (σ\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 Pa=1 N/m21 \text{ Pa} = 1 \text{ N/m}^2.
σ=FA \sigma = \frac{F}{A}
(Where FF is the perpendicular force and AA is the area).

Strain (ϵ\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.
ϵ=ΔLL0 \epsilon = \frac{\Delta L}{L_0}
(Where ΔL\Delta L is the change in length and L0L_0 is the original length).

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

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

Hooke's Law for Continua Concepts

The specific "Elastic Modulus" depends on the type of stress being applied.
  • Young's Modulus (EE): Measures resistance to tension (stretching) or compression (squeezing) along one axis. σ=Eϵ\sigma = E \epsilon, or F/A=E(ΔL/L0)F/A = E (\Delta L/L_0). This is crucial for designing columns and cables.
  • Shear Modulus (GG): Measures resistance to shear forces (forces acting parallel to a surface, trying to slide layers past one another). τshear=Gγ\tau_{shear} = G \gamma, where γ\gamma is the shear strain angle.
  • Bulk Modulus (BB): Measures resistance to uniform compression from all sides (like an object submerged deep in the ocean). It relates pressure (PP) to volume strain (ΔV/V0\Delta V/V_0). ΔP=−B(ΔV/V0)\Delta P = -B (\Delta V/V_0). The negative sign indicates that increased pressure causes a decrease in volume.

Thermal Stress

Thermal Stress Concepts

If a structural member is constrained so that it cannot expand or contract when subjected to a temperature change (ΔT\Delta T), large internal stresses develop. The thermal strain is ϵthermal=αΔT\epsilon_{thermal} = \alpha \Delta T. Because the member is constrained, the opposing stress developed is:
σthermal=EαΔT \sigma_{thermal} = E \alpha \Delta T

Thermal Stress Concepts

(Where α\alpha is the coefficient of linear expansion).

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

    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 (EE).
    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 L0L_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.

Important

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 (EE) 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 (GG) relates shear stress (τ=Fparallel/A\tau = F_{parallel}/A) to shear strain (γ=Δx/h\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 (BB) relates the change in pressure (ΔP\Delta P) to the fractional change in volume (ΔV/V0\Delta V / V_0).

Shear Modulus Equation

Hooke's Law applied to shear deformation.

$$ \tau = G \gamma $$
Key Takeaways
  • Static Equilibrium requires both zero net force (ΣF⃗=0\Sigma \vec{F} = 0) and zero net torque (Στ⃗=0\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 (σ=F/A\sigma = F/A) measures the intensity of internal forces. Strain (ϵ=ΔL/L0\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 (EE).
  • Materials have limits (Yield Strength, Ultimate Strength) beyond which they deform permanently or break. Engineering designs must account for these limits.
PreviousRotational Motion - Examples & Applications
Quiz Me
NextEquilibrium and Elasticity - Examples & Applications