Shear of Beams - Structural and Stress Analysis - page 260

g δ y

τ

B A

τ B A

δ y

τ

C D g

τ

C D

δ x

F IGURE 10.9 Determination of
strain energy due to shear

(a) (b)

does work. If the shear loads producing the shear stress are gradually applied, then
the work done by the shear force on the element and hence the strain energy stored,
δ

U , is given by

1
2 τ t

δ U = δ x γ δ y

or

1
2 τγ t

δ U = δ x δ y

=

τ/ G , where G is the shear modulus and t

δ x δ y is the volume of the element.

Now γ

Hence

τ 2
G ×

1
2

δ U = volume of element

The total strain energy, U , due to shear in a structural member in which the shear
stress, τ , is uniform is then given by

τ 2
2 G ×

U = volume of member (10.20)

10.4 S HEAR S TRESS D ISTRIBUTION IN T HIN-WALLED O PEN S ECTION B EAMS

In considering the shear stress distribution in thin-walled open section beams we shall
make identical assumptions regarding the calculation of section properties as were
made in Section 9.6. In addition we shall assume that shear stresses in the plane of the
cross section and parallel to the tangent at any point on the beam wall are constant

t

τ

Assumed
constant
across t

Assumed
negligible

F IGURE 10.10
Assumptions in thin-
walled open section
beams

Thickness, t

(a) (b)