Basic concepts - The Design of Prestressed Concrete Bridges - page 26

2.4 Bending moments, shear force and torque
Consider the cantilever shown in Figure 2.1 (a), of length L metres and carrying a load
of W kN at its extremity. The cantilever consists of thin top and bottom fl anges, and
of a web joining them together.
The load W creates a bending moment in the cantilever, which at a distance l from
the end is W × l and which reaches a maximum of W × L at the root of the cantilever,
Figure 2.1 (b). The bottom fl ange of the cantilever is stressed in compression, and the
top fl ange is stressed in tension, Figure 2.1 (c). The top and bottom surfaces of the
cantilever are called the top and bottom extreme fi bres, or the extrados and the intrados,
respectively. The bending moment is designated as hogging, as it causes tension on the
top fi bre. Bending moments that cause tension on the bottom fi bre, for instance due to
an upwards load on the end of the cantilever, are designated as sagging. The distance
between the centres of the compression and tension fl anges is h . The compression and
tension forces in the fl anges, ± F , create an internal couple which must balance the
external applied moment (ignoring very small longitudinal forces in the thin web).
Consequently at the root of the cantilever, Fh = WL . As stress = force/area, the stress
in each fl ange

= F / A , where A is the cross-section area of each fl ange.
The cantilever tip defl ects downwards as the top fl ange extends and the bottom
fl ange compresses. The amount of defl ection depends on the height and thickness of
the web, on the width and thickness of the fl anges and on the stiffness of the material
of which the cantilever is made.

σ

The defl ection δ = WL 3 /3 EI

where

W is the load on the cantilever end;

L is the length of the cantilever;

I is the 'moment of inertia' of the cross section of the cantilever, and is a measure of
the strength given by its geometry;

E is the Young's modulus (also called the 'modulus of elasticity' or just the 'modulus')
of the constituent material of the cantilever, and is a measure of the stiffness of the
material. For instance, concrete has a modulus of, typically, 30,000 MPa, while steel,
which is much stiffer, has a modulus of 200,000 MPa.

If the load W was applied suddenly, the cantilever would vibrate as well as
defl ecting.
There are two ways in which the cantilever may collapse. The fi rst is in bending;
either the top or bottom fl anges may not be strong enough, when they would fail by
extension or crushing, respectively, Figure 2.1 (d). The other is if the web joining them
is not strong enough to hold the two fl anges together, when the failure would be in
shear, Figure 2.1 (e).
Shear force is always proportional to the slope of the bending moment diagram,
shown in Figure 2.1 (b). As here this slope is constant, the shear force is also constant,
Figure 2.1 (f), and equal to the load W . The action of shear is best represented by an
analogy, in which the web is considered to be an 'N' truss, whose diagonal members