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Exploring Beam Deflection and Flexure Formulas in Strength of Materials

Understand beam deflection factors, moment-area method, and flexure formula application. Explore manual and formulaic methods for calculating beam deflection. Learn about factors affecting beam deflection and the relationship between load, shear, and moment diagrams.

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Exploring Beam Deflection and Flexure Formulas in Strength of Materials

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  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

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