Mechanics of Solids I

Transverse Loading

Introduction

0 0

0

0

x x x xz xy

y xy y x

z xz z x

F dA M y z dA

F dA V M z dA

F dA M y M

o Distribution of normal and shearing stresses

satisfies

o Transverse loading applied to a beam

results in normal and shearing stresses in

transverse sections.

Shear V is the result of a transverse shear-

stress distribution that acts over the beam’s

x-section.

Introduction

o When shearing stresses are exerted on

the vertical faces of an element, equal

stresses must be exerted on the horizontal

faces.

o Longitudinal shear stress (yx) must exist

o Shear does not occur in a beam subjected to pure bending

Shear on the Horizontal Face of a Beam Element

o Consider prismatic beam

o For equilibrium of beam element

0x D CA

D C

A

F H dA

M MH y dA

I

A

D C

Q y dA

dMM M x V x

dx

o Note,

VQH x

I

H VQq shear flow

x I

o Substituting,

VQq shear flow

I • Shear flow,

where

1

2

'

first moment of area above

second moment of full cross section

A

A A

Q y dA

y

I y dA

• Same result found for lower area

H H

Shear on the Horizontal Face of a Beam Element

Example 6.1

A beam is made of three planks, nailed

together. Knowing that the spacing between

the nails is 25 mm and that the vertical shear

in the beam is V = 500 N, determine the shear

force in each nail.

Shearing Stress in a Beam

o 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: Shear formula.

ave

H q x VQ x

A A I t x

ave

VQ

It

Midterm 2/49

A beam with rectangular cross section

subject to a vertical shear Vy and a horizontal

shear Vz as shown. Determine shear stress at

point A, B, C and D on the beam cross

section. Vz = 30 kN

x

y

z

50 mm

80 mm

Vy = 20 kN

10 mm

10 mm

A

C

D

B

xy and yx exerted on a transverse and a horizontal plane through D’ are equal.

o 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 Stress in a Beam

o On the upper and lower surfaces of the beam,

yx= 0. It follows that xy= 0 on the upper and

lower edges of the transverse sections.

Shearing Stresses xy in Common Types of Beams

• For a narrow rectangular beam,

2

2

max

31

2

3

2

xy

VQ V y

Ib A c

V

A

parabola

• By comparison, max is 50% greater than the average

shear stress determined from avg = V/A.

Shearing Stresses xy in Common Types of Beams

max

ave

web

VQ

It

V

A

Wide-flange beam (W-beam) and Standard beam (S-beam)

A wide-flange beam consists of two (wide) “flanges” and a “web”.

Using analysis similar to a rectangular x-section, the shear stress

distribution acting over x-section is shown

o There is a jump in shear stress at the flange-web junction since x-sectional thickness changes at this point o The web carries significantly more shear force than the flanges

Example 6.2

A timber beam is to support the three

concentrated loads shown. Knowing

that for the grade of timber used,

determine the minimum required

depth d of the beam.

all12 MPa, 0.8 MPaall

Longitudinal Shear on an Arbitrary Shape Beam

Earlier, we learn how to calculate shear flow

along horizontal surfaces

How to calculate q along vertical surfaces?

0x D CA

F H dA

H VQq

x I

Shear flow is calculated by using the same

equation.

But by cutting through the vertical surface!

Example 6.3

A square box beam is constructed from four planks as shown. Knowing that the

spacing between nails is 44 mm. and the beam is subjected to a vertical shear of magnitude V = 2.5 kN, determine the shearing force in each nail.

Shearing Stresses in Thin-Walled Members

o Consider a segment of a wide-flange beam

subjected to the vertical shear V.

o The longitudinal shear force on the element

is VQ

H xI

zx xz

H VQ

t x It

o The corresponding shear stress is

o NOTE: 0xy

0xz

in the flanges

in the web

o Previously found a similar expression for the

shearing stress in the web

xy

VQ

It

• The variation of shear flow across the

section depends only on the variation of

the first moment.

VQq t

I

• For a box beam, q grows smoothly from

zero at A to a maximum at C and C’

and then decreases back to zero at E.

• The sense of q in the horizontal portions

of the section may be deduced from

the sense in the vertical portions or the

sense of the shear V.

Shearing Stresses in Thin-Walled Members

o For a wide-flange beam, the shear flow

increases symmetrically from zero at A

and A’, reaches a maximum at C and

then decreases to zero at E and E’.

o The continuity of the variation in q and

the merging of q from section branches

suggests an analogy to fluid flow.

Shearing Stresses in Thin-Walled Members

Directional sense of q is such that

shear appears to “flow” through the x-section

inward at beam’s top flange

“combining” and then “flowing” downward

through the web

then separating and “flowing” outward at the

bottom flange

Shearing Stresses in Thin-Walled Members

Important notes

If a member is made from segments having thin walls, only the

shear flow parallel to the walls of member is important

Shear flow varies linearly along segments that are perpendicular

to direction of shear V

Shear flow varies parabolically along segments that are inclined

or parallel to direction of shear V

On x-section, shear “flows” along segments so that it contributes

to shear V yet satisfies horizontal and vertical force equilibrium

Shearing Stresses in Thin-Walled Members

Knowing that a given vertical shear V causes a maximum shearing stress of 75 Mpa in the hat-shaped extrusion shown, determine the corresponding shearing stress at (a) point a, (b) point b.

Example 6.4