Analysis of Beams in Bending 1

Analysis of Beams in Bending

(5.1-5.3)

MAE 314 – Solid Mechanics

Yun Jing

Analysis of Beams in Bending 2

Bending Moment Along a Beam In 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.

1 2 3 12

Analysis of Beams in Bending 3

Review of Beam Supports 3 equilibrium equations: Σ FY = 0, Σ FX = 0, Σ M = 0 Ignore the horizontal (x-direction) components, because these are axial loading.

R1 R3

R2

R1R1

R2R2

R3 M1

R1R1R1 R2R2R2

R3R3

R3R4

R4

M1 M1 M2

Analysis of Beams in Bending 4

Sign Convention Recall 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 5

Analysis of Beams in Bending 6

Analysis of Beams in Bending 7

Analysis of Beams in Bending 8

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 9

Relations Between F, V, and M For beams with more complicated loading, it is helpful to develop a relationship between load, shear and bending moment.

Sum forces in the vertical direction. 0)( xwVVVFy xwV

wdx

dV dxwVV

C

C

x

x

CC '

'or

Analysis of Beams in Bending 10

Relations Between F, V, and M Sum moment about C’.

Neglect (Δx)2 term since it is much smaller thanΔx term.

02

xxwxVMMMMC

22

1xwxVM

Vdx

dM dxVMM

C

C

x

x

CC '

'or

Analysis of Beams in Bending 11

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 12

Example ProblemDraw the shear and bending-moment diagrams for the beam and loading shown.

Design of Beams in Bending 13

Design of Beams in Bending

(5.4)

MAE 314 – Solid Mechanics

Yun Jing

Design of Beams in Bending 14

Design of Beams for Bending Recall 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 (σmax ≤ σall), soS

M

I

cMmaxmax

max c

IS where

all

MS

max

min

Design of Beams in Bending 15

Procedure for Design Determine the value for σall. 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 = bh2/6 Rolled-steel beam: Use Appendix C in textbook

Design of Beams in Bending 16

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 17

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 18

Example Problem A 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