Part VII — Theory of Structures

Chapter 7: Structural Analysis Basics

7.1 Stress and Strain

Stress is internal resistance force per unit area (σ = P/A), while strain is the ratio of deformation to original dimension (ε = δL/L). Hooke's Law states that, within the elastic limit, stress is directly proportional to strain (σ = Eε), where E (Modulus of Elasticity/Young's Modulus) is a material property indicating stiffness. Steel has a much higher Modulus of Elasticity (approximately 200 GPa / 2×10⁵ N/mm²) than concrete (typically around 25–35 GPa depending on grade), reflecting steel's greater stiffness.

7.2 Types of Loads and Beams

Common load types on structural members include point (concentrated) loads, uniformly distributed loads (UDL), and uniformly varying loads (UVL). Beams are classified by support conditions: a simply supported beam rests on two supports (one typically a pin, one a roller) with no moment resistance at supports; a cantilever beam is fixed at one end and free at the other; a fixed (built-in) beam is rigidly fixed at both ends (resisting moment at both supports); and a continuous beam spans more than two supports.

7.3 Bending Moment and Shear Force

Bending Moment (BM) at a section is the algebraic sum of moments of all forces on one side of the section about that section; Shear Force (SF) is the algebraic sum of all transverse forces on one side of the section. For a simply supported beam with a central point load W and span L, the maximum bending moment occurs at the centre, with magnitude WL/4; for a simply supported beam under a uniformly distributed load w per unit length over span L, the maximum bending moment (also at the centre) is wL²/8. For a cantilever of length L with a point load W at the free end, the maximum bending moment occurs at the fixed support, with magnitude WL.

7.4 Trusses

A truss is a structure composed of straight members connected at joints (idealised as pin joints), designed to carry loads primarily as axial (tension or compression) forces in each member, with no bending — used widely in bridges and roof structures. Common analysis methods include the Method of Joints (analysing equilibrium at each joint sequentially) and the Method of Sections (cutting the truss to analyse a group of members directly via overall equilibrium).

7.5 Practice Set — Theory of Structures (16 MCQs)

1. Stress is defined as:
   (a) Force per unit volume (b) Force per unit area (c) Deformation per unit length (d) Energy per unit area
2. Strain is defined as:
   (a) Force per unit area (b) The ratio of deformation to original dimension (c) Force per unit volume (d) Energy per unit mass
3. Hooke's Law states that, within the elastic limit:
   (a) Stress is inversely proportional to strain (b) Stress is directly proportional to strain (c) Stress is independent of strain (d) Strain is always zero
4. The Modulus of Elasticity of steel is approximately:
   (a) 2×10⁵ N/mm² (200 GPa) (b) 2×10³ N/mm² (c) 2×10⁷ N/mm² (d) 25–35 GPa
5. Compared to concrete, steel's Modulus of Elasticity is:
   (a) Much lower (b) Roughly equal (c) Much higher, reflecting greater stiffness (d) Not comparable
6. A cantilever beam is characterised by being:
   (a) Simply supported at both ends (b) Fixed at one end and free at the other (c) Fixed at both ends (d) Supported at more than two points
7. A simply supported beam typically rests on:
   (a) Two fixed supports resisting moment (b) One pin support and one roller support, with no moment resistance (c) A single central support only (d) Three or more supports
8. For a simply supported beam of span L with a central point load W, the maximum bending moment is:
   (a) WL/2 (b) WL/4 (c) WL²/8 (d) WL
9. For a simply supported beam of span L under a uniformly distributed load w per unit length, the maximum bending moment is:
   (a) wL/4 (b) wL²/8 (c) wL²/4 (d) wL/2
10. For a cantilever of length L with a point load W at the free end, the maximum bending moment occurs at:
    (a) The free end, with magnitude zero (b) The fixed support, with magnitude WL (c) The midpoint, with magnitude WL/2 (d) The free end, with magnitude WL
11. A truss is designed so that its members carry loads primarily as:
    (a) Bending moments (b) Axial (tension or compression) forces (c) Shear forces only (d) Torsional forces
12. The Method of Joints, used in truss analysis, analyses equilibrium at:
    (a) Each joint sequentially (b) Only the support reactions (c) The centroid of the truss (d) Only the top chord members
13. The Method of Sections in truss analysis is particularly useful for:
    (a) Finding forces in specific members directly, without solving the whole truss (b) Finding only the total weight of the truss (c) Determining the modulus of elasticity (d) Measuring deflection only
14. A fixed (built-in) beam differs from a simply supported beam in that it:
    (a) Cannot resist any moment at its supports (b) Rigidly resists moment at both supports (c) Has no supports at all (d) Can only carry point loads
15. Shear Force at a section of a beam is defined as:
    (a) The algebraic sum of all transverse forces on one side of the section (b) The bending moment at that section (c) The axial force in the beam (d) The deflection at that section
16. A continuous beam is defined as a beam that:
    (a) Has only one support (b) Spans more than two supports (c) Is always a cantilever (d) Has no bending moment anywhere

Answer Key

1.(b)

2.(b)

3.(b)

4.(a)

5.(c)

6.(b)

7.(b)

8.(b)

9.(b)

10.(b)

11.(b)

12.(a)

13.(a)

14.(b)

15.(a) 16.(b)