Mechanical Engineering > LECTURE SLIDES/NOTES > Study Guide > MECH2413 Engineering Mechanics Chapter 06: Deflection of Beams_ The geometric result o (All)

Study Guide > MECH2413 Engineering Mechanics Chapter 06: Deflection of Beams_ The geometric result of a beam being stressed

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In this chapter, we will discuss the geometric result of a beam being stressed: Deflection. • First of all, a summary of important equations are given for the ordinary bending theory. • Then t ... he deflection in different cases are discussed. 1) Single region calculation (single equation) 2) Multiple region calculation (multiple equation) 3) Superposition 2Ordinary Bending Theory • Consider an ordinary (uniaxial) bending, • Under equilibrium conditions, we have ܸ݀ ,ݍ− = ݔ݀ ܯ݀ ܸ = ݔ݀ 3Ordinary Bending Theory • The bending moment M and the shear force V are the resultants of the normal stresses σ (acting in the x-direction) and the shear stresses t (acting in the zdirection), respectively ܣ݀ߪ ; ܰ = නܣ݀߬ ; ܸ = නܣ݀ߪݖ = නܯ 4Ordinary Bending Theory • The strain ε and the shear strain γ describe the deformation of an arbitrary element of the beam with length dx and height dz. = ߝ ݑ߲ ݔ߲ = ߛ ݓ߲ + ݔ߲ ݑ߲ ݖ߲ ܧ = ߝܧ = ߪ ݑ߲ ݔ߲ (ܩ = ߛܩ = ߬ ݓ߲ + ݔ߲ ݑ߲ )ݖ߲ 5Ordinary Bending Theory • The neutral axis is an axis in the cross section of a beam along which there are no longitudinal stresses or strains. • If the section is symmetric, isotropic and is not curved before a bend occurs, then the neutral axis is at the geometric centroid. 6Deflection curve for normal stress 7Deflection curve • To determine the beam deflection, we have the following assumption. w’ 8Assumptions • Every point of a cross section undergoes the same deflection in the z-direction. This implies that the height of the beam does not change due to bending: ݔ ݓ = ݓ ߝ ௭ = ݓ߲ = 0ݖ߲ • Plane cross sections of the beam remain plane during the bending. In addition to the displacement w, a cross section undergoes a rotation. The angle of rotation ψ = ψ(x) is a small angle [Show More]

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