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

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.

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

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

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

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