201
THE SLOPE DEFLECTION METHOD
9 The Slope Deflection Method 1.
INTRODUCTION SIGN CONVENTIONS :
The
slope deflection
method,
presented by Professor G.A.
in
Maney
its
present foim,
was
first
D
of the University of Minnesota. In this method, the joints are considered to be rigid, i.e. the joints rot ate as a whole a nd the angles between the tangents to the~"eia stic curve eeting at the joint do not change du e to deforma(1915)
Fig. 9.1.
a nti -clock wise. If the
m
.of the
tion?THe rotations of the joints are treated as unknowns. A series of simultaneous equations, each expressing the relation between the moments acting at the ends of the members are written in terms of slope and deflection.
The
the decade just prior to the introduction of the
moment
is
based on the direction
get
=;^ " .---C &r A B
I
solution of the slope-deflection equation
are calculated using the slope deflection equations
sign convention
Mb 4-\-Mbd— Mbc=0.
along with the equilibrium equations, gives the values of the unknown rotations of the joints. Knowing these rotations, the end
moments
moment we
new
A
B
1'
M AS
During
distribution
method, nearly all continuous frames were analyzed by the slope deflection method.
The
sign convention used
in the case of bending of simple becomes clumsy if used for the ca.-e of more complex beams and frames where more Than two members meet at a joint. In
beams,
etc.,
our earlier sign convention for simple beams, a moment is considered and negative if it bends the beam convex upwards
to be positive if it
bends the beam concave upwards.
Thus, for the case of structure
shown
in Fig. 9*1, the three moments acting at the rigid joint B, where the three members BA, BC and BD meet are all positive
Fig. 9.2. Sign convention.
new sign convention that will be used in this inethod, a supporj[ moment .acting in the clockwise^directton yvtiUss. Hence
according to the previous sign convention since all the three moments^tend to bend the three corresponding beams convex upwards.
Hence the equilibrium equation conveniently applied ihe joint
B
is in
if
SMu=0
the previous sign convention
is
equilibrium.
However, the examination of joint B (Fig, 9"!) reveals that tbe moments Mba and Mad are clockwise while the moment Mbc is 200
taken as positive and that injhe.witj^cfo^ A corresponding change will have to be
B
cannot be used, though
at the joint
in the
"
made while plotting the moment diagram. For any spanof _a_ beam pr_ member .with rigid joints a positive support" moment (or end moment) at the right hand end will be plotted above the base line and negative support moment below the base. Similarly, for the left hand end, the negative end moment is plotted above the base a'ld positive end moment is plotted below the base line, as shown in Fig. 9*2.
support