178 6 Deflections of Elastic StructuresdW 1 W 2Mpy 1 y (^2) M
W 1 W 2 W (^3)
y 1 y 2 y 3 M
Mp
e
W 2
y 1
W 1
M
y 2 Mp
a b
l/2
a c
d
M
Mp
l/2
e
f
b
c
l/2
a c
d
M
Mp
l/2
e
f
b
Fig. 6.20 Multiplication of two bending moment diagrams
In this case, the displacement as a result of the multiplication of two graphs may
be calculated using two following rules:
1.Trapezoid rule allows calculating the required displacement in terms ofextreme
ordinates
D
l
6 EI
.2abC2cdCadCbc/; (6.22)
where the crosswise end ordinates has unity coefficients. This formula is precise.
2.Simpson’s rule allows calculating the required displacement in terms ofextreme
andmiddleordinates
D
l
6 EI
.abC4efCcd/: (6.23)
Equation (6.23) may also be used for calculation of displacements, if the bend-
ing moment diagram in the actual condition is bounded by acurveline. If the
bending moment diagramMpis bounded by quadratic parabola (Fig.6.20c), then
the result of multiplication of two bending moment diagrams by formula (6.23)is
exact; this case occurs if a structure is carrying uniformly distributed load. If the
bending moment diagramMpis bounded by cubic parabola, then the procedure
(6.23) leads to the approximate result.
If a graphMpis bounded by a broken line, then both graphs have to be divided
by several portions as shown in Fig.6.20d. In this case, the result of multiplication
of both graphs is Z
MpMNdxD ̋ 1 y 1 C ̋ 2 y 2 : (6.24)