Building Design and Construction Handbook - Merritt - Ventech!

STRUCTURAL THEORY 5.83
3EI
K � (5.115)
L
This equation indicates that a prismatic beam hinged at only one end has threefourths
the stiffness, or resistance to end rotation, of a beam fixed at both ends.
5.11.4 Fixed-End Moments
A beam so restrained at its ends that no rotation is produced there by the loads is
called a fixed-end beam, and the end moments are called fixed-end moments. Fixedend
moments may be expressed as the product of a coefficient and WL, where W
is the total load on the span L. The coefficient is independent of the properties of
other members of the structure. Thus, any member can be isolated from the rest of
the structure and its fixed-end moments computed.
Assume, for example, that the fixed-end moments for the loaded beam in Fig.
F F
5.63a are to be determined. Let M be the moment at the left end L and M the
L R
moment at the right end R of the beam. Based on the condition that no rotation is
permitted at either end and that the reactions at the supports are in equilibrium with
the applied loads, two equations can be written for the end moments in terms of
the simple-beam end rotations, � L at L and � R,atR for the specific loading.
Let K L be the fixed-end stiffness at L and K R the fixed-end stiffness at R, as
given by Eqs. (5.112) and (5.113). Then, by resolution of the moment diagram into
simple-beam components, as indicated in Fig. 5.63ƒ to h, and application of the
superposition principle (Art. 5.10.6), the fixed-end moments are found to be
F M L ��K L(�L � CR� R)
(5.116)
F M R ��K R(�R � CL� L)
(5.117)
where C L and C R are the carry-over factors to L and R, respectively [Eqs. (5.109)
and (5.110)]. The end rotations � L and � R can be computed by a method described
in Art. 5.5.15 or 5.10.4.
Prismatic Beams. The fixed-end moments for beams with constant moment of
inertia can be derived from the equations given above with the use of Eqs. (5.111)
and (5.114):
FIGURE 5.63 Determination of fixed-end moments in beam LR: (a) Loads on the fixed-end
beam are resolved (b) to(d) into the sum of loads on a simple beam. (e) to(h) Bending-moment
diagrams for conditions (a) to(d), respectively.