SHEARING STRESSES IN BEAMS AND THIN-WALLED

MEMBERS

Lectured by:

Dr Volkan Esat

Based on:

Mechanics of Materials

Beer, Johnston, DeWolf, Mazurek

McGraw Hill

Contents

2

Introduction

Shear on the Horizontal Face of a Beam Element

Example 21

Determination of the Shearing Stress in a Beam

Shearing Stresses txy in Common Types of Beams

Example 22

Longitudinal Shear on a Beam Element of Arbitrary Shape

Example 23

Example 24

Introduction

3

Transverse loading applied to a beam results in normal and shearing stresses in transverse sections.

Longitudinal shearing stresses must exist in any member subjected to transverse loading.

When shearing stresses are exerted on the vertical faces of an element, equal stresses must be exerted on the horizontal faces.

4Shear on the Horizontal Face of a Beam Element

5Shear on the Horizontal Face of a Beam Element

6Example 21

A beam is made of three planks, nailed together. Knowing that the spacing between nails is 25 mm and that the vertical shear in the beam is V = 500 N, determine the shear force in each nail.

SOLUTION:Determine the horizontal force per

unit length or shear flow q on the lower surface of the upper plank.

Calculate the corresponding shear force in each nail.

7Example 21

A beam is made of three planks, nailed together. Knowing that the spacing between nails is 25 mm and that the vertical shear in the beam is V = 500 N, determine the shear force in each nail.

8Determination of the Shearing Stress in a Beam

The average shearing stress on the horizontal face of the element is obtained by dividing the shearing force on the element by the area of the face.

On the upper and lower surfaces of the beam, tyx = 0. It follows that txy = 0 on the upper and lower edges of the transverse sections.

If the width of the beam is comparable or large relative to its depth, the shearing stresses at D1 and D2 are significantly higher than at D.

9Shearing Stresses txy in Common Types of Beams

For a narrow rectangular beam,

A

V

c

y

A

V

Ib

VQxy

2

3

12

3

max

2

2

t

t

For American Standard (S-beam) and wide-flange (W-beam) beams

web

ave

A

V

It

VQ

maxt

t

10

Example 22

For the beam and loading shown, determine the minimum required

width b, knowing that for the grade of timber used, sall = 12 MPa and tall = 825 kPa.

11

Longitudinal Shear on a Beam Element of Arbitrary Shape

We have examined the distributionof the vertical components txy on atransverse section of a beam. Wenow wish to consider thehorizontal components txz of thestresses.

12

Example 23

The built-up wooden beam shown is subjected to a vertical shear of8 kN. Knowing that the nails are spaced longitudinally every 60 mmat A and every 25 mm at B, determine the shearing force in the nails(a) at A, (b) at B. (Given: Ix = 1.504 10

9 mm4.)

13

Example 24

A beam is constructed from five boards bolted together as shown. Determine the maximum shear force developed in each bolt if the bolts are spaced s = 250 mm apart and the applied shear is V = 35 kN.