188 CHAPTER 6 Matrix Methods
wehave,fromEqs.(6.45)and(6.46),
[Kij]=EI⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
12 μ^2 /L^3 SYM
− 12 λμ/L^312 λ^2 /L^3
6 μ/L^2 − 6 λ/L^24 /L
− 12 μ^2 /L^312 λμ/L^3 − 6 μ/L^212 μ^2 /L^3
12 λμ/L^3 − 12 λ^2 /L^36 λ/L^2 − 12 λμ/L^312 λ^2 /L^3
6 μ/L^2 − 6 λ/L^22 /L 6 μ/L^26 λ/L^24 λ/L⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
(6.47)
Again,thestiffnessmatrixforthecompletestructureisassembledfromthememberstiffnessmatrices,
theboundaryconditionsareapplied,andtheresultingsetofequationssolvedfortheunknownnodal
displacementsandforces.
Theinternalshearforcesandbendingmomentsinabeammaybeobtainedintermsofthecalculated
nodaldisplacements.Thus,forabeamjoiningnodesiandj,weshallhaveobtainedtheunknownvalues
ofvi,θiandvj,θj.ThenodalforcesFy,iandMiarethenobtainedfromEq.(6.44)ifthebeamisaligned
withthexaxis.Hence,
Fy,i=EI(
12
L^3
vi−6
L^2
θi−12
L^3
vj−6
L^2
θj)
Mi=EI(
−
6
L^2
vi+4
L
θi+6
L^2
vj+2
L
θj)
⎫
⎪⎪
⎪⎬
⎪⎪
⎪⎭
(6.48)
Similarexpressionsareobtainedfortheforcesatnodej.FromFig.6.6,weseethattheshearforceSy
andbendingmomentMinthebeamaregivenby
Sy=Fy,i
M=Fy,ix+Mi}
(6.49)
SubstitutingEq.(6.48)intoEq.(6.49)andexpressinginmatrixformyield
{
Sy
M}
=EI
⎡
⎢
⎢
⎣
12
L^3
−
6
L^2
−
12
L^3
−
6
L^2
12
L^3
x−6
L^2
−
6
L^2
x+4
L
−
12
L^3
x+6
L^2
−
6
L^2
x+2
L
⎤
⎥
⎥
⎦
⎧
⎪⎪
⎨
⎪⎪
⎩
vi
θi
vj
θj⎫
⎪⎪
⎬
⎪⎪
⎭
(6.50)
The matrix analysis of the beam in Fig. 6.6 is based on the condition that no external forces are
applied between the nodes. Obviously, in a practical case, a beam supports a variety of loads along
itslength,andtherefore,suchbeamsmustbeidealizedintoanumberofbeamelementsforwhichthe
precedingconditionholds.Theidealizationisaccomplishedbymerelyspecifyingnodesatpointsalong
the beam such that any element lying between adjacent nodes carries, at the most, a uniform shear
and a linearly varying bending moment. For example, the beam of Fig. 6.7 would be idealized into
beamelements1–2,2–3,and3–4forwhichtheunknownnodaldisplacementsarev 2 ,θ 2 ,θ 3 ,v 4 ,andθ 4
(v 1 =θ 1 =v 3 =0).