Chapter 7Transverse Shear

Shear stress in a prismatic beam due to shear forceShear flow in a built-up beam

Shear flow in thin-walled beamShear center of a cross-section

SHEAR IN A STRAIGHT BEAM

• Transverse shear stress always has its associated longitudinal shear stress acting along longitudinal planes of the beam.

• Recall 3d stress element… The transverse shear stress and the longitudinal shear stress must be equal!

• Further proof that the transverse shear stress and the longitudinal shear stress must coexist?

SHEAR FORMULA

ydAdx

dM

It

tdxdAI

MydA

I

ydMM

tdxdAdA

Fx

1

0

0''

0

'' where

'

AyydAQ

It

VQ

A

Q is called the first area moment of the section above y’.t is the average shear stress over the area tdx, and it is also the average shear stress on the cross-section y’ above the neutral axis.

What if you take a vertical cut?

What is dF?

s’

SHEAR IN BEAMS

Consider two rectangular cross sections

• The actual shear stress is not uniform.

Rectangular cross section

A

V

yhb

yh

yh

b

yyh

bQ

Qbbh

V

It

VQ

5.1

22

22

2

')2

(

121

max

22

3

SHEAR IN BEAMS

EXAMPLE 7.3

A steel wide-flange beam has the dimensions shown in Fig. 7–11a. If it is subjected to a shear of V = 80kN, plot the shear-stress distribution acting over the beam’s cross-sectional area.

EXAMPLE 1 (cont)

Copyright © 2011 Pearson Education South Asia Pte Ltd

• The moment of inertia of the cross-sectional area about the neutral axis is

• For point B, tB’ = 0.3m, and A’ is the dark shaded area shown in Fig. 7–11c

Solutions

4623

3

m 106.15511.002.03.002.03.012

12

2.0015.012

1

I

MPa 13.13.0106.155

1066.01080

m 1066.002.03.011.0''

6

33

'

''

33'

B

BB

B

It

VQ

AyQ

EXAMPLE 1 (cont)

• For point B, tB = 0.015m, and QB = QB’,

• For point C, tC = 0.015m, and A’ is the dark shaded area in Fig. 7–11d.

• Considering this area to be composed of two rectangles,

• Thus,

Solutions

MPa 6.22

015.0106.155

1066.010806

33

B

BB It

VQ

33 m 10735.01.0015.005.002.03.011.0'' AyQC

MPa 2.25

015.0106.155

10735.010806

33

max

C

cC It

VQ

7.3 SHEAR FLOW IN BUILT-UP BEAM

• Shear flow ≡ shear force per unit length along longitudinal axis of a beam.

I

VQ

dx

dFq

QI

dMydA

I

dMdF

dAdAdF

'

''' q = shear flow (shear force per unit thickness)V = internal resultant shearI = moment of inertia of the entire cross-sectional area

SHEAR FLOW IN BUILT-UP BEAM (cont)

The horizontal (direction parallel to the neutral axis) cuts can be long.The vertical cuts must be short.

EXAMPLE 7.6

Nails having a total shear strength of 40 N are used in a beam that can be constructed either as in Case I or as in Case II, Fig. 7–18. If the nails are spaced at 90 mm, determine the largest vertical shear that can be supported in each case so that the fasteners will not fail.

EXAMPLE 2 (cont)

Copyright © 2011 Pearson Education South Asia Pte Ltd

• Since the cross section is the same in both cases, the moment of inertia about the neutral axis is

Case I

• For this design a single row of nails holds the top or bottom flange onto the web.

• For one of these flanges,

Solutions

433 mm 205833401012

125030

12

1

I

N 1.27mm 205833

mm 3375

mm 90

N 40

mm 33755305.22''

4

3

3

V

V

I

VQq

AyQ

EXAMPLE 2 (cont)

Case II

• Here a single row of nails holds one of the side boards onto the web.

• Thus,

Solutions

(Ans)N 3.81mm 205833

mm 1125

mm 90

N 40

mm 11255105.22''

4

3

3

V

V

I

VQq

AyQ

SHEAR FLOW IN THIN-WALLED BEAM

• Approximation: only the shear-flow component that acts parallel to the walls of the member will be counted.

EXAMPLE 3

The thin-walled box beam in Fig. 7–22a is subjected to a shear of 10 kN. Determine the variation of the shear flow throughout the cross section.

• The moment of inertia is

• For point B, the area thus q’B = 0.

• Also,

• For point C,

• The shear flow at D is

Solutions

0'A

433 mm 1846412

186

12

1I

3

3

cm 304122'

cm 5.17155.3'

AyQ

AyQ

D

C

N/mm 5.91 kN/cm 951.0

184

2/5.1710

I

VQq C

C

N/mm 163 kN/cm 63.1

184

2/3010

I

VQq D

D