Page 99 - Civil Engineering Formulas

52 CHAPTER TWO

Maxwell’s theorem states that if unit loads rest upon a beam at two points, A
and B, the deflection at A due to the unit load at B equals the deflection at B due
to the unit load at A.

Castigliano’s Theorem
This theorem states that the deflection of the point of application of an external
force acting on a beam is equal to the partial derivative of the work of defor-
mation with respect to this force. Thus, if P is the force, f is the deflection, and
U is the work of deformation, which equals the resilience:
dU
f (2.12)
dP
According to the principle of least work, the deformation of any structure
takes place in such a manner that the work of deformation is a minimum.

BEAMS OF UNIFORM STRENGTH

Beams of uniform strength so vary in section that the unit stress S remains con-
stant, and I/c varies as M. For rectangular beams of breadth b and depth d, I/c
2
2
bd /6 and M Sbd /6. For a cantilever beam of rectangular cross section, under
2
2
a load P, Px Sbd /6. If b is constant, d varies with x, and the profile of the
shape of the beam is a parabola, as in Fig. 2.15. If d is constant, b varies as x, and
the beam is triangular in plan (Fig. 2.16).

b
Plan x

Plan

P
P
d
x

Elevation Elevation

FIGURE 2.15 Parabolic beam of FIGURE 2.16 Triangular beam of
uniform strength. uniform strength.