CIVIL ENGINEERING FORMULAS

52 CHAPTER TWO

Maxwell’s theorem states that if unit loads rest upon a beam at two points, A

andB, the deflection at Adue to the unit load at Bequals the deflection at Bdue

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 Pis the force, fis the deflection, and

Uis the work of deformation, which equals the resilience:

(2.12)

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 Sremains con-

stant, and I/cvaries as M. For rectangular beams of breadth band depth d,I/c

bd^2 /6 and MSbd^2 /6. For a cantilever beam of rectangular cross section, under

a load P,PxSbd^2 /6. If bis constant, d^2 varies withx, and the profile of the

shape of the beam is a parabola,as in Fig. 2.15. If dis constant, bvaries as x,and

the beam is triangular in plan (Fig. 2.16).

dU

dP

f

FIGURE 2.15 Parabolic beam of

uniform strength.

Plan

Elevation

P

d

x

FIGURE 2.16 Triangular beam of

uniform strength.

Plan

b

P

Elevation

x