Introduction to Structural Analysis

Purpose of Structural Analysis

Structural analysis determines the internal forces (axial, shear, moment) and deformations caused by applied loads. This information is required before any structural member can be designed.

Types of Structures

• Beams – horizontal members carrying transverse loads
• Trusses – triangulated frameworks of axial members
• Frames – beams and columns with rigid joints
• Arches – curved structures in compression

Degrees of Freedom

Static Determinacy:

For beams/frames: Stable if reactions ≥ equations

• 3 equilibrium equations (ΣFx, ΣFy, ΣM = 0)

Degree of indeterminacy = Reactions - 3 (for beams)

Support Types

| Support | Reactions | DOF Restrained | |---------|-----------|----------------| | Pin | Fx, Fy | 2 | | Roller | Fy | 1 | | Fixed | Fx, Fy, M | 3 |

Beam Reactions (Simply Supported)

For a simply supported beam with point load P at distance 'a' from left support (span L):

• R_A = P(L-a)/L
• R_B = Pa/L

For uniform load w over full span:

• R_A = R_B = wL/2

Shear Force and Bending Moment Diagrams

Sign Convention

• Positive shear: left face up, right face down
• Positive moment: sagging (concave up = tension on bottom)

Relationships

• dV/dx = -w(x) (shear changes with distributed load)
• dM/dx = V(x) (moment changes with shear)
• Max moment occurs where V = 0

Example: Simply Supported Beam

Beam with uniform load w, span L:

• V(x) = wL/2 - wx
• V = 0 at x = L/2 → max moment at midspan
• M_max = wL²/8

Summary

Structural analysis provides the forces and moments needed for design. Master support reactions, shear force, and bending moment diagrams — they appear in nearly every structural problem.