Advanced Methods of Structural Analysis

278 8 The Displacement Method

h

l

EI 1

abEI (^212)
Z 2 =1
r 12
6 EI 1 /h^2
6 EI 1 /h^2
1 2
Z 2 =1
State 2 *
r 22
2 EI 1 /h
Z 1 =1
4 EI 1 /h
3 EI 2 /l
1 2 r 21

• State 1
  r 11
  Elastic curve
  c Extended fibers d
  State 1
  4 EI 1 /h
  r 11
  1
  3 EI 2 /l
  1
  6 EI 1 /h^2
  r 21
  4 EI 1 /h
  2 EI 1 /h
  6 EI 1 /h^2
  2
  State 2
  6 EI 1 /h^2
  r 12
  1
  12 EI 1 /h^2
  (^1) r 22
  6 EI 1 /h
  6 EI 1 /h
  12 EI 1 /h^2
  2
  e
  f
  g
  h
  Fig. 8.5 (a) Design diagram; (b) Primary system of the displacement method; (c,d) Unit states
  and corresponding bending moment diagrams. Calculation of unit reactions: (e,f) Free-body dia-
  grams for joint 1 and crossbar, state 1; (g,h) Free-body diagrams for joint 1 and crossbar, state 2
  ./denotes the inflection points of the elastic curves. For member 1-2, the extended
  fibers are located below the neutral line. Bending moment diagrams are plotted on
  the extended fibers.
  First unit state.Z 1 D1/Rigid joint 1 rotates clockwise (positive direction) through
  angleZ 1 D 1. The reactions for both members (fixed-pinned and fixed-fixed) in
  the case of angular displacement of the fixed supports are presented in TablesA.3
  andA.4. In this case, the bending moment at joint 1 of the horizontal member
  equals 3 EI 2 =l(TableA.3, row 1); for the vertical member, the specified ordinates
  are 4 EI 1 =hfor joint 1 and 2 EI 1 =hat the bottom clamped support (TableA.4). As
  a result of the angular displacement, unit reactive momentr 11 arises in constraint
  1 and reactive forcer 21 arises in constraint 2; all unit reactions are shown in the
  positive direction.
  Second unit state.Z 2 D1/If constraint 2 has horizontal displacementZ 2 D 1
  from left to right (positive direction), then introduced joint 1 has the same displace-
  ment and, as a result, the vertical member is subjected to bending. Unit reactive