Page 95 - Civil Engineering Formulas

BEAM FORMULAS 49

or if
L kM P
M kM P when x (2.6)
2 wL
A plastic hinge forms at this point when the moment equals kM P . For equilibrium,
w x
kM P x (L x) kM P
2 L
w L kM P L KM P 1 kM P 2
kM P
2 2 wL 2 wL 2 wL
leading to
2 2 2
k M P wL
3kM P 0 (2.7)
wL 2 4
When the value of M P previously computed is substituted,
2 4 4
7k 4k 4 or k (k 7 ) 7
from which k 0.523. The ultimate load is
4M P (1 k) M P
wL 6.1 (2.8)
L L
In any continuous beam, the bending moment at any section is equal to the
bending moment at any other section, plus the shear at that section times its
arm, plus the product of all the intervening external forces times their respective
arms. Thus, in Fig. 2.12
(2.9)
V x R 1 R 2 R 3 P 1 P 2 P 3
M x R 1 (l 1 l 2 x) R 2 (l 2 x) R 3 x
P 1 (l 2 c x) P 2 (b x) P 3 a (2.10)
M x M 3 V 3 x P 3 a (2.11)

Table 2.1 gives the value of the moment at the various supports of a uniformly
loaded continuous beam over equal spans, and it also gives the values of the shears

P 1 P 2 P 3

c b a
R 1 R 2 R 3 x R 4 R 5
l 1 l 2 l 3 l 4

FIGURE 2.12 Continuous beam.