6.4Matrix Analysis of Pin-jointed Frameworks 181wheretheyoverlap;forexample,the[k 11 ]submatrixinEq.(ii)receivescontributionsfrom[K 12 ]and
[K 13 ]. The complete stiffness matrix is then of the form shown in Eq. (iv). It is sometimes helpful,
whenconsideringthestiffnessmatrixseparately,towritethenodaldisplacementabovetheappropriate
column(seeEq.(iv)).Wenotethat[K]issymmetrical,thatallthediagonaltermsarepositive,andthat
thesumofeachrowandcolumniszero
[K 12 ]=
AE
L
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
10
k 11
00− 10
k 12
00− 10
k 21
0010
k 22
00⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
[K 13 ]=
AE
L
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
00
k 11
0100
k 13
0 − 100
k 31
0 − 100
k 33
01⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
(iii)[K 23 ]=
AE
√
2 L
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
1
2
−
1
2
k 22
−1
2
1
2
−
1
2
1
2
k 23
1
2−
1
2
−
1
2
1
2
k 32
1
2−
1
2
1
2
−
1
2
k 33
−1
2
1
2
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎧
⎪⎪
⎪⎪⎪
⎪⎨
⎪⎪
⎪⎪
⎪⎪
⎩
Fx,1
Fy,1
Fx,2
Fy,2
Fx,3
Fy,3⎫
⎪⎪
⎪⎪
⎪⎪⎬
⎪⎪
⎪⎪
⎪⎪
⎭
=
AE
L
⎡u^1 v^1 u^2 v^2 u^3 v^3
⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎢⎢
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
10 −100 0
01 0 0 0 − 1
− 101 +
1
2
√
2
−
1
2
√
2
−
1
2
√
2
1
2
√
2
00 −
1
2
√
2
1
2
√
2
1
2
√
2
−
1
2
√
2
00 −
1
2
√
2
1
2
√
2
1
2
√
2
−
1
2
√
2
0 − 1
1
2
√
2
−
1
2
√
2
−
1
2
√
2
1 +
1
2
√
2
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
⎥⎥
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
⎧
⎪⎪
⎪⎪⎪
⎪⎨
⎪⎪
⎪⎪
⎪⎪
⎩
u 1 = 0
v 1 = 0
u 2
v 2
u 3 = 0
v 3 = 0⎫
⎪⎪
⎪⎪
⎪⎪⎬
⎪⎪
⎪⎪
⎪⎪
⎭
(iv)