Structural Analysis
Comprehensive guide to structural analysis principles, methods, and applications in civil engineering.
Beam Analysis
Beams are structural elements that primarily resist loads applied laterally to their axis. The relationship between load, shear force, and bending moment is:
dV/dx = -w(x)
dM/dx = V(x)
Where:
- w(x) = distributed load (N/m)
- V(x) = shear force (N)
- M(x) = bending moment (N·m)
- x = position along the beam (m)
The deflection of a beam is related to the bending moment by:
EI × (d²y/dx²) = M(x)
Where:
- E = Young's modulus (Pa)
- I = moment of inertia (m⁴)
- y = deflection (m)
Truss Analysis
Trusses are structures composed of members connected at joints, designed to carry loads primarily in tension or compression. Two common methods for analyzing trusses are:
1. Method of Joints:
Analyzes the equilibrium of forces at each joint. For a joint in equilibrium:
∑F_x = 0
∑F_y = 0
2. Method of Sections:
Involves cutting the truss into sections and analyzing the equilibrium of each section. For a section in equilibrium:
∑F_x = 0
∑F_y = 0
∑M = 0
Assumptions in truss analysis:
- All members are connected by frictionless pins
- All loads are applied at the joints
- The weight of the members is negligible compared to the applied loads
- Members can only experience axial forces (tension or compression)
Structural Dynamics
Structural dynamics deals with the behavior of structures subjected to dynamic (time-varying) loads. The equation of motion for a single-degree-of-freedom system is:
m(d²x/dt²) + c(dx/dt) + kx = F(t)
Where:
- m = mass (kg)
- c = damping coefficient (N·s/m)
- k = stiffness (N/m)
- x = displacement (m)
- F(t) = external force as a function of time (N)
The natural frequency and damping ratio are important parameters:
ω_n = √(k/m)
ζ = c/(2√(km))
Where:
- ω_n = natural frequency (rad/s)
- ζ = damping ratio (dimensionless)