τ

dx

τ

τ

τdy longitudinal plane

Lecture: Shear Flow

-- Shear Flow along longitudinal planes -- Shear Flow in Thin Walled Members -- Homework: Read Section 7:4-5 Work Problems from Following today's class you should be able to:-- explain what shear flow is and how it relates to shear stress-- be able to calculate shear flow-- explain shear flow distribution in thin walled members

Shear Flow:In the previous lecture, it was explained how the shear stress varies in magnitude along the cross sectional area for bodies subjected to transverse shearing forces. For a rectangular beam it was shown that the shear stress varied along a line that was parabolic in shape.

Looking at an infinitesimal element shows that in order to be in equilibrium similar shear stresses need to act on all four faces in order for the element to be in equilibrium. From this we can imply that there is also a shear stress and corresponding shear force acting along the longitudinal plane of a body subjected to a transverse shearing force.

The shearing force acting per unit length along the longitudinal plane is defined as the shear flow, q.As a quantity, the shear flow may be given as

20 mm

80 mm

120 mm

20 mm

30 mm

V

20 mm

80 mm

20 mm

30 mm

V

where q is the shear flow (shear force per unit length) V is the transverse shear force Q is the first moment of area of section on one side of the longitudinal plane. I is the moment of inertia of the entire cross section

So how do you use this? It's used in two very common ways. 1) Used to determine shearing load for individual connectors along a shear plane.2) Used to determine average shear values in thin walled members.

We'll look at examples of each:

------------------------------------------------------------------------------------------Example 1: Shear Load in Connectors:Suppose we have a beam that is built out of four pieces of lumber nailed together in the box like shape shown below. Nails are placed every 30 mm. The beam is subjected to a constant transverse shear force of 1200 N.

Determine a) the shear flow along the longitudinal plane between boards b) the shear force each nail must support. c) the maximum shear stress in the wood.

Determine a) the shear flow along the longitudinal plane between boards b) the shear force each nail must support.

=50 mm

120 mm

20 mm

y

NA

=30 mm

120 mm

=20 mmyiNA

yo

80 mm

60 mm40mm

for

where V = 1200 N

= 13.87 x 106 mm4

Therefore the shear flow is

since shear flow also represents longitudinal shear force per unit length and there are 2 nails in each 30 mm length of the beam.

so

over 2 nails

Therefore the shear force acting on each nail is

c) What is the maximum shear stress in the wood.

Expect the maximum shear stress to occur along the neutral axis and is

given by

where V = 1200 N I = 13.87 x 106 mm4 t = 2 x 20mm = 40 mm and

Therefore

Shear Flow in a Thin Walled Section:

The concept of shear flow can also be used to find the distribution of shear stress throughout the section of a thin walled member.

The basic assumptions used are --for a thin walled member the shear stress across the thickness, t, will be considered uniform -- on a free surface there can be no shear stress.

Consider an I-beam subjected to a shearing force, V.

If you were to calculate the shearing stress based on the shear stress equationyou would find that the shear stress increases linearly as you move horizontally and it increases along a parabolic curve as you move vertically.

Shear flow for other shapes:

The variation of shear flow across the section depends only on the variation of the first moment.

For a box beam, q grows smoothly from zero atone extreme, building up to a max, and decreasing back down to zero.

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.

Bodies with an axis of symmetry parallel to the direction of shearing force, V, require the shear flow goes to zero as axis of symmetry is approached.

If the shear flow hits a junction, the flow into the node must equal the flow out of the node at the junction.

Example 2:The beam supports a vertical shear of 7 kip. Determine a) the shear stress at section 1-1 b) the shear stress at B

qA1

qA2

qA3

y

A

y

A

c) the shape of the shearflow and shear stressdistribution.

Solution:

Part a)

therefore at section 1-1:

b) Shear stress at B:

therefore at B:

c) Shape of the shear flow and shear stress distribution

Mechanics of Materials HW Set 23 Spring 2013Problem 7

The double T beam is fabricated by welding the three plates together as shown. If the weld can resist a shear stress of 90 MPa, determine the maximum shear V that can be applied to the beam.

Problem

The box beam is constructed from four boards that are fastened together using nails spaced along the beam every 2 in. If a force P = 2 kip is applied to the beam, determine the shear force resisted by each nail at A and B..____________________________________________________________________Solution:

The beam is made from three thin plates welded together as shown. If it is subjected to a shear of V = 48 kN, determine the shear flow at points A and B Also, calculate the maximum shear stress in the beam.

____________________________________________________________________Solution:

The pipe is subjected to a shear force of 8 kip. Determine the shear flow in the pipe at pints A and B.

____________________________________________________________________Solution: