Analysis of Structures

Why it matters

Understanding the analysis of structures is crucial for ensuring the safety and stability of buildings, bridges, and other constructions. It helps engineers design structures that can withstand various loads and forces, preventing failures and ensuring longevity.

Key ideas

• Types of Structures: Structures can be classified into beams, trusses, frames, and arches, each with unique characteristics and analysis methods.
• Loads and Reactions: Structures are subjected to different types of loads such as point loads, distributed loads, and moment loads. Reactions are the forces and moments developed at the supports or connections.
• Equilibrium: For a structure to be in equilibrium, the sum of all forces and moments must be zero. This is a fundamental principle used in structural analysis.
• Method of Joints: Used primarily for truss analysis, this method involves isolating a joint and applying equilibrium equations to solve for unknown forces.
• Method of Sections: Another technique for analyzing trusses, where a section of the truss is isolated to solve for internal forces.
• Bending Moment and Shear Force: These are internal forces within a beam that need to be calculated to ensure the beam's design is safe.

Ideal truss assumptions

An ideal planar truss has straight members connected by frictionless pins, with loads applied at joints. Members act as two-force members carrying axial tension or compression. The method of joints uses two force-balance equations at each joint. The method of sections uses equilibrium of a cut portion to find selected member forces. Frames may also carry shear and bending moments, so they cannot automatically be treated as trusses.

For a simple stable determinate planar truss, m + r = 2j, where m is member count, r is independent reaction count, and j is joint count. This count alone does not prove geometric stability.

Formulas

Use positive sagging moment and shear V = dM/dx throughout the beam example.

• ΣF_x = 0: Sum of horizontal forces must be zero for equilibrium.
• ΣF_y = 0: Sum of vertical forces must be zero for equilibrium.
• ΣM = 0: Sum of moments about any point must be zero for equilibrium.
• M = F·d: Moment (M) is the product of force (F) and perpendicular distance (d) from the point of rotation. Units: Nm.
• V = dM/dx: Shear force (V) is the derivative of the bending moment (M) with respect to distance (x). Units: N.

Worked example

Given: A simply supported beam of length 6 m is subjected to a point load of 10 kN at the center.

1. Calculate reactions at supports:

   • By symmetry, reactions at both supports (A and B) are equal.
   • R_A + R_B = 10 kN
   • R_A = R_B = 5 kN

2. Calculate bending moment at the centre:

   • Take a cut immediately to the left of the central load and use the left-hand free body.
   • With sagging moment positive, M_center = R_A(L/2) = 5 × 3 = 15 kN·m.
   • The central load has zero lever arm about a cut at its own location, so it does not contribute a 10 × 3 term there.

Final Answer: +15 kN·m, a sagging moment. Shear changes from +5 kN to −5 kN across the load; moment is continuous.

Common mistakes

• Forgetting to consider all forces and moments in equilibrium equations.
• Incorrectly assuming symmetry in non-symmetrical structures.
• Miscalculating distances for moment calculations.
• Ignoring the sign convention for moments and shear forces.

For GATE ME

Questions often involve calculating reactions, internal forces, and moments in beams and trusses. Practice solving equilibrium equations, using the method of joints and sections, and drawing shear force and bending moment diagrams.

Quick check

1. What is the primary method used for analyzing trusses?
2. What must be zero for a structure to be in equilibrium?
3. How is the bending moment related to shear force?

Answers: 1. Method of Joints, 2. Sum of forces and moments, 3. Shear force is the derivative of the bending moment.