214 CHAPTER 6 Matrix Methods
Usethedirectstiffnessmethodtofindallthedisplacementsandhencecalculatetheforcesinallthemembers.
Formember123plottheshearforceandbendingmomentdiagrams.
Brieflyoutlinethesequenceofoperationsinatypicalcomputerprogramsuitableforlinearframeanalysis.
Ans. S 29 =S 28 =
√
2 P/6(tension)
M 3 =−M 1 =PL/9(hogging),M 2 = 2 PL/9(sagging)
SF 12 =−SF 23 =P/ 3Twistingmomentin37,PL/18(anticlockwise).P.6.6 Giventhattheforce–displacement(stiffness)relationshipforthebeamelementshowninFig.P.6.6(a)may
beexpressedinthefollowingform:
⎧
⎪⎪
⎪⎪⎪
⎨
⎪⎪⎪
⎪⎪
⎩Fy,1
M 1 /L
Fy,2
M 2 /L⎫
⎪⎪
⎪⎪⎪
⎬
⎪⎪⎪
⎪⎪
⎭=
EI
L^3⎡
⎢⎢
⎢
⎢⎢
⎣12 − 6 − 12 − 6
−64 62
−12 6 12 6
−62 64⎤
⎥⎥
⎥
⎥⎥
⎦⎧
⎪⎪
⎪⎪⎪
⎨
⎪⎪⎪
⎪⎪
⎩v 1
θ 1 L
v 2
θ 2 L⎫
⎪⎪
⎪⎪⎪
⎬
⎪⎪⎪
⎪⎪
⎭Obtaintheforce–displacement(stiffness)relationshipforthevariablesectionbeam(Fig.P.6.6(b)),composed
ofelements12,23,and34.
SuchabeamisloadedandsupportedsymmetricallyasshowninFig.P.6.6(c).Bothendsarerigidlyfixed,and
thetiesFB,CHhaveacross-sectionalareaa 1 ,andthetiesEB,CGhaveacross-sectionalareaa 2 .Calculatethe
deflections under the loads, the forces in the ties, and all other information necessary for sketching the bending
momentandshearforcediagramsforthebeam.
Neglectaxialeffectsinthebeam.Thetiesaremadefromthesamematerialasthebeam.
Ans. vB=vC=− 5 PL^3 / 144 EI,θB=−θC=PL^2 / 24 EI,
S 1 = 2 P/3,S 2 =
√
2 P/3,
Fy,A=P/3,MA=−PL/4.P.6.7 Thesymmetricalrigid-jointedgrillageshowninFig.P.6.7isencastréat6,7,8,and9andrestsonsimple
supportsat1,2,4,and5.ItisloadedwithaverticalpointloadPat3.
Usethestiffnessmethodtofindthedisplacementsofthestructureandhencecalculatethesupportreactions
andtheforcesinallthemembers.Plotthebendingmomentdiagramfor123.Allmembershavethesamesection
propertiesandGJ=0.8EI.
Ans. Fy,1=Fy,5=−P/ 16
Fy,2=Fy,4= 9 P/ 16
M 21 =M 45 =−Pl/16(hogging)
M 23 =M 43 =−Pl/12(hogging)
Twistingmomentin62,82,74,and94isPl/96.P.6.8 It is required to formulate the stiffness of a triangular element 123 with coordinates (0, 0), (a, 0), and
(0,a),respectively,tobeusedfor“planestress”problems.