182 CHAPTER 6 Matrix Methods
Ifwenowdeleterowsandcolumnsinthestiffnessmatrixcorrespondingtozerodisplacements,we
obtain the unknown nodal displacementsu 2 andv 2 in terms of the applied loadsFx,2(=0) andFy,2
(=−W).Thus,
{
Fx,2
Fy,2}
=
AE
L
⎡
⎢
⎢
⎣
1 +
1
2
√
2
−
1
2
√
2
−
1
2
√
2
1
2
√
2
⎤
⎥
⎥
⎦
{
u 2
v 2}
(v)InvertingEq.(v)gives
{
u 2
v 2}
=
L
AE
[
11
11 + 2
√
2
]{
Fx,2
Fy,2}
(vi)fromwhich
u 2 =L
AE
(Fx,2+Fy,2)=−WL
AE
(vii)v 2 =L
AE
[Fx,2+( 1 + 2√
2 )Fy,2]=−WL
AE
( 1 + 2
√
2 ) (viii)The reactions at nodes 1 and 3 are now obtained by substituting foru 2 andv 2 from Eq. (vi) into
Eq.(iv).Thus,
⎧
⎪⎪
⎪⎨
⎪⎪
⎪⎩
Fx,1
Fy,1
Fx,3
Fy,3⎫
⎪⎪
⎪⎬
⎪⎪
⎪⎭
=
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
− 10
00
−
1
2
√
2
1
2
√
2
1
2
√
2
−
1
2
√
2
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
[
11
11 + 2
√
2
]{
Fx,2
Fy,2}
=
⎡
⎢
⎢
⎢
⎣
− 1 − 1
00
01
0 − 1
⎤
⎥
⎥
⎥
⎦
{
Fx,2
Fy,2}
giving
Fx,1=−Fx,2−Fy,2=WFy,1= 0Fx,3=Fy,2=−WFy,3=W