6.7Stiffness Matrix for a Uniform Beam 191Example 6.2
DeterminetheunknownnodaldisplacementsandforcesinthebeamshowninFig.6.11.Thebeamis
ofuniformsectionthroughout.
The beam may be idealized into two beam elements, 1–2 and 2–3. From Fig. 6.11, we see that
v 1 =v 3 =0,Fy,2=−W,M 2 =+M.Therefore,eliminatingrowsandcolumnscorrespondingtozero
displacementsfromEq.(6.53),weobtain
⎧
⎪⎪
⎪⎨
⎪⎪
⎪⎩Fy,2=−W
M 2 =M
M 1 = 0
M 3 = 0⎫
⎪⎪
⎪⎬
⎪⎪
⎪⎭
=EI
⎡
⎢
⎢
⎢
⎣
27 / 2 L^39 / 2 L^26 /L^2 − 3 / 2 L^2
9 / 2 L^26 /L 2 /L 1 /L
6 /L^22 /L 4 /L 0
− 3 / 2 L^21 /L 02 /L
⎤
⎥
⎥
⎥
⎦
⎧
⎪⎪
⎪⎨
⎪⎪
⎪⎩
v 2
θ 2
θ 1
θ 3⎫
⎪⎪
⎪⎬
⎪⎪
⎪⎭
(i)Equation(i)maybewrittensuchthattheelementsof[K]arepurenumbers
⎧
⎪⎪
⎨
⎪⎪
⎩
Fy,2=−W
M 2 /L=M/L
M 1 /L= 0
M 3 /L= 0⎫
⎪⎪
⎬
⎪⎪
⎭
=
EI
2 L^3
⎡
⎢
⎢
⎣
27 9 12 − 3
912 4 2
12 4 8 0
− 3204
⎤
⎥
⎥
⎦
⎧
⎪⎪
⎨
⎪⎪
⎩
v 2
θ 2 L
θ 1 L
θ 3 L⎫
⎪⎪
⎬
⎪⎪
⎭
(ii)ExpandingEq.(ii)bymatrixmultiplication,wehave
{
−W
M/L
}
=
EI
2 L^3
([
27 9
912
]{
v 2
θ 2 L}
+
[
12 − 3
42
]{
θ 1 L
θ 3 L})
(iii)and
{
0
0
}
=
EI
2 L^3
([
12 4
− 32
]{
v 2
θ 2 L}
+
[
80
04
]{
θ 1 L
θ 3 L})
(iv)Equation(iv)gives
{
θ 1 L
θ 3 L}
=
⎡
⎣
−^32 −^12
−^34 −^12
⎤
⎦
{
v 2
θ 2 L}
(v)Fig.6.11
Beam of Example 6.2.