1. CTC / MTC 222 Strength of Materials
Chapter 9
Deflection of Beams

2. Chapter Objectives
Understand the need for considering beam deflection.
List the factors which affect beam deflection.
Understand the relationship between the load, shear, moment, slope and deflection diagrams.
Use standard formulas to calculate the deflection of a beam at selected points.
Use the Principle of Superposition to calculate defection due to combinations of loads.

3. The Flexure Formula
Positive moment – compression on top, bent concave upward
Negative moment – compression on bottom, bent concave downward
Maximum Stress due to bending (Flexure Formula)
σmax = M c / I
Where M = bending moment, I = moment of inertia, and c = distance from centroidal axis of beam to outermost fiber
For a non-symmetric section distance to the top fiber, ct , is different than distance to bottom fiber cb
σtop = M ct / I
σbot = M cb / I
Conditions for application of the flexure formula
Listed in Section 8-3, p. 308

4. Factors Affecting Beam Deflection
Load – Type, magnitude and location
Type of span - Simple span, cantilever, etc
Length of span
Type of supports
Pinned or roller – Free to rotate
Fixed – Restrained against rotation
Material properties of beam
Modulus of Elasticity, E
A measure of the stiffness of a material
The ratio of stress to strain, E = σ / ε
Physical properties of beam
Moment of Inertia, I
A measure of the stiffness of a beam, or of its resistance to deflection due to bending

5. Manual Calculation of Beam Deflection
Successive Integration
Uses the relationships between load, shear, moment, slope and deflection
Moment-Area Method
Uses the M / EI Diagram
Can calculate the change in angle between the tangents to two points A and B on the deflection curve
Can also calculate the vertical deviation of one point from the tangent to another point on the deflection curve

6. Calculation of Beam Deflection by Formula
Equations for the deflection of beams with various loads and support conditions have been developed
See Appendix A-23
Examples
Simply supported beam with a point load at mid-span
Simply supported beam with a uniform load on full span
Deflection at a given point due to a combination of loads can be calculated using the principle of superposition