Chapter5 Bending Design

Analysis of Beams in Bending*Analysis of Beams in Bending (5.1-5.3)MAE 314 Solid MechanicsYun Jing

Analysis of Beams in Bending

Analysis of Beams in Bending*Bending Moment Along a BeamIn this chapter, we will learn how to find the bending moment M along the beam.M is not necessarily constant; sometimes M is a function of x.We will also solve for the shear force V(x), which will be used in Chapter 6.As before, we need a new FBD every time the loading changes.

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Analysis of Beams in Bending*Review of Beam Supports3 equilibrium equations: FY = 0, FX = 0, M = 0Ignore the horizontal (x-direction) components, because these are axial loading.R1R3R2R1R1R2R2R3M1R1R1R1R2R2R2R3R3R3R4R4M1M1M2


Analysis of Beams in Bending*Sign ConventionRecall the applied loading results in both a bending moment M and a shear force V.Positive shear and bending moment


Analysis of Beams in Bending*


Analysis of Beams in Bending*Example ProblemDraw the shear and bending moment diagrams for the beam andloading shown, and determine the maximum absolute value (a) of theshear and (b) of the bending moment.


Analysis of Beams in Bending*Relations Between F, V, and MFor beams with more complicated loading, it is helpful to develop a relationship between load, shear and bending moment.

Sum forces in the vertical direction.

or


Analysis of Beams in Bending*Relations Between F, V, and MSum moment about C.

Neglect (x)2 term since it is much smaller than x term.or


Analysis of Beams in Bending*Example ProblemDetermine (a) the equations of the shear and bending moment curvesfor the beam and loading shown, and (b) the maximum absolute valueof the bending moment in the beam.


Analysis of Beams in Bending*Example ProblemDraw the shear and bending-moment diagrams for the beam and loading shown.


Design of Beams in Bending*Design of Beams in Bending (5.4)MAE 314 Solid MechanicsYun Jing


Design of Beams in Bending*Design of Beams for BendingRecall the largest normal stress in the beam subject to bending occurs at the surface and can be defined as

A safe design requires the maximum stress is no more than the allowable stress (max all), so

where

Design of Beams in Bending

Design of Beams in Bending*Procedure for DesignDetermine the value for all.Draw shear and moment diagrams.From the diagrams, determine the maximum absolute bending moment.Determine the minimum allowable value Smin.Use Smin to determine best cross section dimensions.Timber beam: Smin = bh2/6Rolled-steel beam: Use Appendix C in textbook


Design of Beams in Bending*Example ProblemFor the beam and loading shown, design the cross section of the beam,knowing that the grade of timber used has an allowable normal stress of12 MPa.


Design of Beams in Bending*Example ProblemKnowing that the allowable stress for the steel used is 160 MPa, selectthe most economical S-shape beam to support the loading shown.


Design of Beams in Bending*Example ProblemA beam is to be made of steel that has an allowable bending stress of 170MPa. Select an appropriate W shape that will carry the loading shown in the figure below


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Chapter5 Bending Design

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