of the latter, the shear V and bending mo-
ment M' of the conjugate beam are equal,
respectively, to the slope 0 and deflection >>
at the corresponding section of the given
beam.
Assign supports to the conjugate beam
that are compatible with the end conditions
of the given beam. At A 9 the given beam
has a specific slope but zero deflection.
Correspondingly, the conjugate beam has a
specific shear but zero moment; i.e., it is
simply supported at A.
At C, the given beam has a specific
slope and a specific deflection. Corre-
spondingly, the conjugate beam has both a
, ub). ,_ Force diagram ,.. of. conjugate beam.. shear & and a bending moment; i.e., ' ' it has a
fixed support at C.
FIGURE 38. Deflection of overhanging 2. Construct the M/No/) diagram
beam
of the given beam
Load the conjugate beam with this area.
The moment at B is - wcf/2; the moment
varies linearly from A to B and parabolically from C to B.
- Compute the resultant of the load in selected Intervals
Compute the resultant W 1 of the load in interval AB and the resultant W 2 of the load in the
interval BC. Locate these resultants. (Refer to the AISC Manual for properties of the
complement of a half parabola.) Then W 1 = (L/2)[w<P/(2EI)] = ^d
1
LI (4EI)\ X 1 =
2
AL; W 2 =
((M)[Wd
1
I(ZEI)] = w<P/(6El)-, X 2 =
3
Ad. - Evaluate the conjugate-beam reaction
Since the given beam has zero deflection at B, the conjugate beam has zero moment at
this section. Evaluate the reaction R'L accordingly. Thus M'B = - R'LL + W 1 LIl = O; R'L W(Il
= Wd^2 Lf(UEI).
5. Determine the deflection
Determine the deflection at C by computing M'c. Thus yc = Afc = - R'L(L + d)+ W{(d + L/3)
- W 2 (3d/4) = w#(4L + 3d)/(24EI).
UNIT-LOAD METHOD OF COMPUTING BEAM
DEFLECTION
The cantilever beam in Fig. 390 carries a load that varies uniformly from w Ib/lin ft at the
free end to zero at the fixed end. Determine the slope and deflection of the elastic curve at
the free end.
Calculation Procedure:- Apply a unit moment to the beam
Apply a counterclockwise unit moment at A (Fig. 396). (This direction is selected because
it is known that the end section rotates in this manner.) Let x = distance from A to given
(a) Force diagram of given beam