INTRODUCTION:
We see that whenever a cantilever or a beam is loaded, it deflects from its
original position. The amount, by which a beam deflects, depends upon its cross-
section and the bending moment. In modern design offices, following are the two
design criteria for the deflection of a cantilever or a beam:
1. Strength
2. Stiffness.
As per the strength criterion of the beam design, it should be strong enough
to resist bending moment and shear force. Or in other words, the beam should
be strong enough to resist the bending stresses and shear stresses. And as per
the stiffness criterion of the beam design, which is equally important, it should
be stiff enough to resist the deflection of the beam. Or in other words, the beam
should be stiff enough not to deflect more than the permissible limit* under the
action of the loading. In actual practice, some specifications are always laid to
limit the maximum deflection of a cantilever or a beam to a small fraction of its
span.
In this chapter, we shall discuss the slope and deflection of the centre line of
beams under the different types of loadings.
CURVATURE OF THE BENDING BEAM:
Consider a beam AB subjected to a bending moment. As a result of loading,
let the beam deflect from ABC to ADB into a circular arc as shown in Fig. 1.1.
Let l = Length of the beam AB,
M = Bending moment,
R = Radius of curvature of the bent up beam,
I = Moment of inertia of the beam section,
E = Modulus of elasticity of beam material,
y = Deflection of the beam (i.e., CD) and
i = Slope of the beam (i.e angle which the tangent at A makes with AB).
From the geometry of a circle, we know that AC × CB = EC × CD
Figure.1.1 Curvature of the beam
As per Indian Standard Specifications, this
limit is Span/325.
NOTES:
1. The above equations for
deflection (y) and slope (i) have been derived
from the bending moment only i.e., the
effect of shear force has been neglected. This
Page 1 of 3
We see that whenever a cantilever or a beam is loaded, it deflects from its
original position. The amount, by which a beam deflects, depends upon its cross-
section and the bending moment. In modern design offices, following are the two
design criteria for the deflection of a cantilever or a beam:
1. Strength
2. Stiffness.
As per the strength criterion of the beam design, it should be strong enough
to resist bending moment and shear force. Or in other words, the beam should
be strong enough to resist the bending stresses and shear stresses. And as per
the stiffness criterion of the beam design, which is equally important, it should
be stiff enough to resist the deflection of the beam. Or in other words, the beam
should be stiff enough not to deflect more than the permissible limit* under the
action of the loading. In actual practice, some specifications are always laid to
limit the maximum deflection of a cantilever or a beam to a small fraction of its
span.
In this chapter, we shall discuss the slope and deflection of the centre line of
beams under the different types of loadings.
CURVATURE OF THE BENDING BEAM:
Consider a beam AB subjected to a bending moment. As a result of loading,
let the beam deflect from ABC to ADB into a circular arc as shown in Fig. 1.1.
Let l = Length of the beam AB,
M = Bending moment,
R = Radius of curvature of the bent up beam,
I = Moment of inertia of the beam section,
E = Modulus of elasticity of beam material,
y = Deflection of the beam (i.e., CD) and
i = Slope of the beam (i.e angle which the tangent at A makes with AB).
From the geometry of a circle, we know that AC × CB = EC × CD
Figure.1.1 Curvature of the beam
As per Indian Standard Specifications, this
limit is Span/325.
NOTES:
1. The above equations for
deflection (y) and slope (i) have been derived
from the bending moment only i.e., the
effect of shear force has been neglected. This
Page 1 of 3
