Lecture Mechanics of materials (Third edition) - Chapter 6: Shearing stresses in beams and thinwalled members

• Determine the horizontal force per unit length or shear flow q on the lower surface of the upper plank. • Calculate the corresponding shear[r]

MECHANICS OF MATERIALS

Ferdinand P Beer

E Russell Johnston, Jr.

John T DeWolf

Lecture Notes:

J Walt Oler Texas Tech University

CHAPTER

Shearing Stresses in

Beams and Thin-Walled Members

Thin-Walled Members

Introduction

Shear on the Horizontal Face of a Beam Element

Example 6.01

Determination of the Shearing Stress in a Beam

Shearing Stresses τxy in Common Types of Beams

Further Discussion of the Distribution of Stresses in a

Sample Problem 6.2

Longitudinal Shear on a Beam Element of Arbitrary Shape

Example 6.04

Shearing Stresses in Thin-Walled Members

Plastic Deformations

Sample Problem 6.3

Unsymmetric Loading of Thin-Walled Members

Example 6.05

Example 6.06

0

0

0 0

=

∫ −

=

=

∫

=

=

∫

=

−

=

∫

=

=

=

=

∫

=

x z

xz z

x y

xy y

xy xz

x x

x

y M

dA F

dA z

M V

dA F

dA z

y M

dA F

σ τ

σ τ

τ τ

σ

• Distribution of normal and shearing stresses satisfies

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

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

• Longitudinal shearing stresses must exist

in any member subjected to transverse loading

• Consider prismatic beam

• For equilibrium of beam element

∫

−

=

∆

∑ = = ∆ + ∫ −

A

C D

A

D D

x

dA y I

M M

H

dA H

x V x dx

dM M

M

dA y Q

C D

A

∆

=

∆

=

−

∫

=

• Note,

flow

shear I

VQ x

H q

x I

VQ H

=

=

∆

∆

=

∆

=

∆

• Substituting,

Shear on the Horizontal Face of a Beam Element

flow shear

I

VQ x

H

∆

∆

=

• Shear flow,

• where

section cross

full of moment second

above area

of moment first

' 2

1

=

∫

=

=

∫

=

+A A

A

dA y I

y

dA y Q

• Same result found for lower area

Q Q

q I

Q V x

H q

∆

−

=

′

∆

=

=

′ +

′

−

=

′

=

∆

′

∆

=

′

axis neutral to

respect

h moment wit first

0

• 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

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

Example 6.01

4 6

2

3 12

1

3 12

1

3 6

m 10 20 16

] m 060 0 m 100 0 m 020 0

m 020 0 m 100 0 [ 2

m 100 0 m 020 0

m 10 120

m 060 0 m 100 0 m 020 0

−

−

×

=

× +

+

=

×

=

×

=

=

I

y A Q

SOLUTION:

• Determine the horizontal force per

unit length or shear flow q on the

lower surface of the upper plank

m

N 3704

m 10 16.20

) m 10 120 )(

N 500 (

4 6

-3 6

=

×

×

=

I

VQ q

• Calculate the corresponding shear force in each nail for a nail spacing of

25 mm

m N q

F = ( 0 025 m ) = ( 0 025 m )( 3704

N 6 92

=

F

• 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

It VQ

x t

x I

VQ A

x q A

H

ave

=

∆

∆

=

∆

∆

=

∆

∆

=

τ

• On the upper and lower surfaces of the beam,

τyx= 0 It follows that τxy= 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.

Shearing Stresses τxy in Common Types of Beams

• For a narrow rectangular beam,

A V

c

y A

V Ib

VQ

xy

2 3

1 2

3

max

2 2

=

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

−

=

=

τ τ

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

web

ave

A V It VQ

=

=

max

τ τ

Stresses in a Narrow Rectangular Beam

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

−

=

2

2 1

2

3

c

y A

P

xy

τ

I

Pxy

x = +

σ

• Consider a narrow rectangular cantilever beam

subjected to load P at its free end:

• Shearing stresses are independent of the distance from the point of application of the load

• Normal strains and normal stresses are unaffected

by the shearing stresses

• From Saint-Venant’s principle, effects of the load application mode are negligible except in immediate vicinity of load application points

• Stress/strain deviations for distributed loads are negligible for typical beam sections of interest