406 11 Matrix Stiffness Method
RD
ac
bd;thenR^1 Ddet^1 R
d c
ba.In our case the inverse matrix becomesK^1 Dh
EI1
48=h^2
12=h^2 6=h
6=h 7Dh^3
48 EI
12=h^2 6=h
6=h 7Resolving equations:Vector of required displacements (angular of rigid joint and
linear of cross bar) are
ZEDZ 1 .rad/
Z 2 .m/
DK^1 PEDh^3
48 EI
12=h^2 6=h
6=h 7
0
P
DPh^3
48 EI6=h
7We can see that in order to calculate the vector of end displacementsZ, in fact, we
need to have three matrices: they are static matrixA, internal stiffness matrixkQ,and
vector of external loadsEP.
Vector of internal unknowns bending moments isESfinDSE 1 CSE 2 , where moments
of the first stateSE 1 D
000
̆T
because external load is applied at joint only.
Therefore, the final vector of bending moments at sections 1–3 is
ESfinDES 2 DkAQ TZEDEI
h2
42 6=h
4 6=h
303
5„ ƒ‚ ...
QkATPh^3
48 EI6=h
7„ ƒ‚ ...
EZDPh
82
6
6
6 5
3
33
7
7
7Corresponding bending moment diagram is shown in Fig.11.26h. For calculation of
reactions, the following algorithm can be applied: bending moments – shear forces –
axial forces – reactions.
Now we show the application of the matrix displacement method for analysis
of the frame in Fig.11.27a; the relative bending stiffnesses are presented in circle.
Analysis of this frame has been performed early by the both classical method, so
this frame may be treated as the etalonone. Comparing withdisplacement method
in canonical form will allow us to understand a physical meaning of an each matrix
procedure.
The frame has two unknowns of the displacement method. They are the angu-
lar displacement at joint 1 and linear displacement of cross bar 1-C. The primary
system andMP^0 diagram are shown in Fig.11.27b.
AncillaryZ-Pdiagram (Fig.11.27c) shows that the structure has one possible
angular displacement of joint 1 and corresponding possible external joint moment,
as well as one horizontal displacement 2 ofcross bar and corresponding force. Un-
known internal forces (momentsS 1 – S 3 ) and their positive directions are shown on
S-ediagram (Fig.11.27d).
The finite elements areA-1, 1-B, and 1-C. The fixed end moments at joint 1 are