5. Stresses in Beams-Flexural Formula

8/11/2019 5. Stresses in Beams-Flexural Formula

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Mechanics of Solids (VDB1063)

Stresses in Beams

Flexural Formula

Lecturer: Dr. Montasir O. Ahmed


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Learning Outcomes

To evaluate the

bending stress by applying the

flexural formula


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Bending Deformation of a Straight Member

Assumptions regarding the way the stress deforms the material:1. Length of longitudinal axis remains unchanged.

2. All cross sections remain plane and perpendicular to the longitudinal axis3. Any deformation of the cross section within its own plane will be neglected (Fig. 6-19b).

Deformation that occur when a

straight beam is subjected to bending

M causes the material within the bottom portion to stretch and within the top portion to compress.Consequently, between these regions there must be a surface ( neutr al sur face ) in which longitudinalfibers of the material will not undergo change in length.


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Bending Deformation of a Straight Member


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The Flexural Formula

Assumptions:

The material behaves in a linear-elastic manner and therefore a linear variation of normal strainmust then be the result of a linear variation in normal stress .

Developing an equation that relates the stress distribution in a beam to the internal resultant bending moment acting on the beams cross section

= -

By applying Hook, we can obtain from


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The Flexural Formula

=

where

max =

max. nor mal str ess in the member, which occur at distance c from the NA.

M = the resul tant internal moment , determined from the method of section and the EOE and

calculated about the NA.

c = the perpendicul ar distance from the NA to a point farthest away from the NA. this is

where max act.

I = the moment of inerti a of the cross section area about the NA. = + 2

= nor mal str ess in the member, which occur at distance y from the NA.

The flexural formula is: OR = -


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EXAMPLE

The simply supported beam in Fig. 6 26a has the cross-sectionalarea shown in Fig. 6 26b. Determine the absolute maximum

bending stress in the beam and draw the stress distribution over thecross section at this location.

10Copyright 2011 Pearson Education South Asia Pte Ltd


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EXAMPLE 4 (cont.)

The maximum internal moment in the beam, 22.5 kN.m, occurs atthe center.

By reasons of symmetry, the neutral axis passes through the

centroid C at the mid-height of the beam, Fig. 6

26b.

11Copyright 2011 Pearson Education South Asia Pte Ltd

Solutions

(Ans) MPa7.12103.301

17.0105.22 ;

m103.301

3.002.016.002.025.002.025.02

6

3

46

312123

121

2

b B

B I My

Ad I I


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EXAMPLE 4 (cont.)

A three-dimensional view of the stress distribution is shown in Fig. 6 26d .

At point B,

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Solutions

MPa2.11103.301

15.0105.22 ; 6

3

B B

B I My

Copyright 2011 Pearson Education South Asia Pte Ltd


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Next Class

Shear in Straight Members

The Shear Formula


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Thank You

5. Stresses in Beams-Flexural Formula

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