Equilibrium and Elasticity

Equilibrium and Elasticity

Statics is the branch of mechanics concerned with bodies at rest or in equilibrium under the action of forces. This is fundamental to civil engineering, where structures must be designed to remain stable.

Conditions for Static Equilibrium

A rigid body is in static equilibrium if two conditions are met:

1. Net Force is Zero (Translational Equilibrium): $\sum \mathbf{F} = 0$ $\sum F_x = 0, \quad \sum F_y = 0, \quad \sum F_z = 0$

2. Net Torque is Zero (Rotational Equilibrium): $\sum \boldsymbol{\tau} = 0$ The sum of torques about any axis must be zero.

Engineering Application: These principles are used to calculate reactions at supports (pins, rollers, fixed supports) and internal forces in beams and trusses. The National Structural Code of the Philippines (NSCP) 2015 provides the loads (Dead, Live, Wind, Earthquake) that must be considered in these equilibrium equations.

Elastic Properties of Materials

Solid materials deform under load. If the deformation is reversible, the material is elastic.

Stress and Strain

• Stress ($\sigma$): The force per unit area. $\sigma = \frac{F}{A}$ Unit: Pascal (Pa) or N/m$^2$.
• Strain ($\epsilon$): The fractional deformation. $\epsilon = \frac{\Delta L}{L_0}$ Unitless.

Hooke's Law and Young's Modulus

For small deformations, stress is proportional to strain. $\sigma = Y \epsilon$ where $Y$ (or $E$) is the Young's Modulus (Modulus of Elasticity). It measures the stiffness of a material.

Other Moduli:

• Shear Modulus ($S$ or $G$): Relates shear stress ($\tau$) to shear strain ($\gamma$). $\tau = G \gamma$.
• Bulk Modulus ($B$): Relates volume stress (pressure $P$) to volume strain ($\Delta V/V$). $P = -B (\Delta V/V)$.

Engineering Context: Concrete and steel are the primary materials in civil engineering. Their elastic properties are critical for design.

• Steel: $E \approx 200 \text{ GPa}$
• Concrete: $E \approx 20-30 \text{ GPa}$ (depending on strength)