Chapter Six Shearing Stresses in Beams and Thin-Walled Members.

Chapter Six

Shearing Stresses in Beams and Thin-Walled Members

6.1 Introduction

-- In a long beam, the dominating design factor:

m

McI

( )xy

Vaverage

A

-- Primary design factor

-- Minor design factor

-- In a short beam, the dominating design factor:

32max

VA

[due to transverse loading]

y component: xydA V 0z component: xzdA

Equation of equilibrium:

-- Shear stress xy is induced by transverse loading.

-- In pure bending -- no shear stress

Materials weak in shear resistance shear failure could occur.

6.2 Shear on the Horizontal Face of a Beam Element

0 0: ( )x D CF H dA A

MyI

C DI I I Knowing and

D CM MH ydA

I

AWe have (6.3)

SincedM

Vdx

( / )D CM M M dM dx x V x

VQH x

I

H VQq

x I

Therefore,

and

(6.4)

(6.5)

Defining Q ydA

= shear flow

= horizontal shear/length

here Q = the first moment w.t.to the neutral axis

Q = max at y = 0

The same result can be obtained for the lower element C'D' D''C''

6.3 Determination of the Shearing Stresses in a Beam

ave

H VQ x VQA I t x It

ave

VQIt

VQH x

I

(6.6)

= ave. shear stress

A = t x

At the N.A. Q = max, but ave may not be max, because of t

xy = 0 at top and bottom fibers

Variation of xy < 0.8% if b h/4

-- for narrow rectangular beams

4hb

6.4 Shearing Stresses xy in Common Types of Beams

-- for narrow rectangular beams

xy

VQIt

2 21 12 2

( ) ( ) ( )Q Ay b c y c y b c y

1( )

2 y c y

t = b (6.7)

Q Ay

Also,3

3212 3bh

I bc

Hence,2 2

3

34xy

VQ c yV

Ib bc

Knowing A = 2bc, it follows

2

2

31

2( )xy

V yA c

maxxy

0xy

(6.9)

This is a parabolic equation with

@ y = c

@ y = 0 -- i.e. the neutral axis

At y = 0, max

3

2

V

A (6.10)

This is only true of rectangular cross-section beams.

ave

VQIt

maxweb

VA

Special cases:

American Standard beam (S-beam)

or a wide-flange beam (W-beam)

-- over section aa’ or bb’

-- Q = about cc’

(6.6)

(6.11)

For the web:

Assuming the entire V is carried by the web, since the flanges carry little shear force:

6.5 Further Discussion of the Distribution of Stresses in a Narrow Rectangular Beam

2

2

31

2( )xy

P yA c

x

Pxy MyI I

(6.12)

(6.13)

Plane sections do NOT remain plane – warping takes place, when a beam is subjected to a transverse shear loading

6.6 Longitudinal Shear on a Beam Element of Arbitrary Shape

H VQq

x I

VQH x

I

0 0: ( )x D CF H dA A

Using similar procedures in Sec. 6.2, we have

= shear flow (6.5)

(6.4)

6.7 Shearing Stresses n Thin-Walled Members

VQH x

I

These two equations are valid for thin-walled members:

(6.4)

(6.6)ave

H VQ x VQA I t x It

VQH x

I

ave xz

VQIt

（ 6.4)

（ 6.6)

From Sec. 6.2

ave

H VQ xA I t x

We have:

Therefore, xz 0

ave xz

VQIt

(6.6)

-- This equation can be applied to a variety of cross sections.