Chapter 16

5 53355 Member Stiffness Matrices.

Member Stiffness Matrices. The

The orgin of orgin of the global the global coordinate coordinate system will system will be sbe set at et at joint joint .. For

For member member and and ,,

F Foor r mmeemmbbeer r ,, aannd d TThhuuss,, F Foor r mmeemmbbeer r ,, aannd d ..TThhuuss,, 1 1 2 2 3 3 4 4 5 5 6 6  (10  (1066))

F

F

1111.11.2255 2200 2222.33.55 --111441..2255 5500 2222.66.55 0 0 550000 00 00 --550000 00 2 222..55 00 6600 -2-222..55 00 3300 --1111..2255 00 --2222..55 1111..2255 00 --22.522.5 0 0 --550000 00 00 550000 00 2 222..55 00 3300 --2222..55 00 6600

V

V

k k₂₂ == l l y y == --44 -- 00 4 4 = = --11 l l x x == 4 4 -- 44 4 4 == 00

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22

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7 7 8 8 9 9 1 1 2 2 3 3  (10  (1066))

F

F

7 8 7 8 99 11 22 33 5 50000 00 00 --550000 00 00 0 0 1111..2255 2222..55 00 --1111..2255 2222..55 0 0 2222..55 6600 00 --2222..55 3300 --550000 00 00 550000 00 00 0 0 --11.2511.25 --2222..55 00 1111..2255 --22.522.5 0 0 2222..55 3300 00 --2222..55 6600

V

V

k k₁₁ == l l_yy == 00 -- 00 4 4 == 0.0. l l_xx == 44 -- 00 4 4 == 11

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11

ƒƒ

2 2EIEI L L == 2 233200(10200(1099))4343300(10300(10--66))44 4 4 == 30(1030(10 6 6_{) N) N}

#

#

_mm 4 4EIEI L L == 4 433200(10200(1099))4343300(10300(10--66))44 4 4 == 60(1060(10 6 6_{) N) N}

#

#

_mm 6 6EIEI L L22 == 6 633200(10200(1099))4343300(10300(10--66))44 4 422 == 22.5(1022.5(10 6 6_{) N) N} 12 12EIEI L L33 == 12 1233200(10200(1099))4343300(10300(10--66))44 4 433 == 11.25(1011.25(10 6 6_{) N/m) N/m} AE AE L L == 0.01 0.0133200(10200(1099))44 4 4 == 500(10500(10 6 6_{) N/m) N/m} L L == 4m4m

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22

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11

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1 1 16–1.

16–1. Determine the structure stiffness matrixDetermine the structure stiffness matrix KK forfor the

the frame.frame.Assume Assume and and are are fixed.fixed.TTake ake ,, , ,AA == 1011011010332 mm2 mm22ffoor r eeaacch h mmeemmbbeerr.. I I == 3003001110106622 mm mm44 E E == 200 GPa200 GPa 3 3 1 1 8 8 1 1 22 3 3 1 1 2 2 2 2 12 kN 12 kN_/ / mm 1 1 3 3 4 4 6 6 4 m 4 m 9 9 7 7 5 5 2 m 2 m 2 m 2 m 10 kN 10 kN

5 53366  Structure Stiffness Matrix.

 Structure Stiffness Matrix.It iIt is a 9s a 9 9 ma9 matritrix six since nce the the highigheshest cot code nde numbumber ier is 9.Thus 9.Thuss,,

Ans. Ans. 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9  (10  (1066))

I

I

1 1 22 33 44 55 66 77 88 99 5 51111..2255 00 2222..55 --1111..2255 00 2222..55 --550000 00 00 0 0 551111..2255 --2222..55 00 --550000 00 00 --11.2511.25 --22.2522.25 22.5 22.5 --2222..55 112200 --2222..55 00 3300 00 2222..55 3300 --1111..2255 00 --2222..55 1111..2255 00 --2222..55 00 00 00 0 0 --550000 00 00 550000 00 00 00 00 2 222..55 00 3300 --2222..55 00 6600 00 00 00 --550000 00 00 00 00 00 550000 00 00 0 0 --1111..2255 2222..55 00 00 00 00 1111..2255 2222..55 0 0 --2222..55 3300 00 00 00 00 2255..55 6600

Y

Y

K K == * * 16–

16–1. 1. ConContintinuedued

16–2.

16–2. Determine the support reactions at the fixedDetermine the support reactions at the fixed s

suuppppoorrtts s aannd d ..TTaakke e ,, ,, for each member.

for each member. A A == 10101110103322 mm mm22 I I == 30030011101066_22 mm mm44 E E == 200 GPa200 GPa 3 3 1 1

 Known Nodal Loads and Deflections.

 Known Nodal Loads and Deflections.The nodal load acting on theThe nodal load acting on the unconstrained degree of freedom (code number 1,

unconstrained degree of freedom (code number 1,2 and 3) are shown in2 and 3) are shown in Fig.

Fig.aa_andandbb_..

and and

 Loads-Displacement Relation.

 Loads-Displacement Relation.Applying Applying ,,

I

I

DD₁₁ D D₂₂ D D₃₃ 0 0 0 0 0 0 0 0 0 0 0 0

Y

Y

(10 (1066))

I

I

--5(105(1033)) --24(1024(1033)) 11(10 11(1033)) Q Q₄₄ Q Q55 Q Q₆₆ Q Q₇₇ Q Q₈₈ Q Q₉₉

Y

Y

= =

I

I

5 51111..2255 00 2222..55 --1111..2255 00 2222..55 --550000 00 00 0 0 551111..2255 --2222..55 00 --550000 00 00 --11.2511.25 --22.522.5 22.5 22.5 --2222..55 112200 --2222..55 00 3300 00 2222..55 3300 --1111..2255 00 --2222..55 1111..2255 00 --2222..55 00 00 00 0 0 --550000 00 00 505000 00 00 00 00 2 222..55 00 3300 --2222..55 00 6600 00 00 00 --550000 00 00 00 00 00 550000 00 00 0 0 --1111..2255 2222..55 00 00 00 00 1111..2255 2222..55 0 0 --2222..55 3300 00 00 00 00 2222..55 6600

Y

Y

Q Q == KDKD D Dkk ==

F

F

0 0 0 0 0 0 0 0 0 0 0 0

V

V

4 4 5 5 6 6 7 7 8 8 9 9 1 1 2 2 3 3 Q Qkk ==

C

C

--5(105(1033)) --24(1024(1033)) 11(10 11(1033))

S

S

8 8 1 1 22 3 3 1 1 2 2 2 2 12 kN 12 kN_/ / mm 1 1 3 3 4 4 6 6 4 m 4 m 9 9 7 7 5 5 2 m 2 m 2 m 2 m 10 kN 10 kN

5 53377 F Frroom m tthhe e mmaattrriix x ppaarrttiittiioonn,, ,, (1) (1) (2) (2) (3) (3) Solving Eqs Solving Eqs..(1) to (1) to (3),(3), rad rad Using

Using these these results results and and applying applying ,,

Superposition these results to

Superposition these results to those of FEM shown in those of FEM shown in Fig.Fig.aa_,,

Ans. Ans. Ans. Ans. b b    Ans.Ans. Ans. Ans. Ans. Ans. d d    Ans.Ans. R R99 == 3.5553.555 ++ 1616 == 19.55 kN19.55 kN

#

#

mm == 19.6 kN19.6 kN

#

#

mm R R₈₈ == 2.4232.423 ++ 2424 == 26.42 kN26.42 kN == 26.4 kN26.4 kN cc R R77 == 6.7856.785 ++ 00 == 6.785 kN6.785 kN == 6.79 kN6.79 kN :: R R₆₆ == 2.2782.278 -- 55 = = -2.722 kN-2.722 kN

#

#

mm == 2.72 kN2.72 kN

#

#

mm R R₅₅ == 21.5821.58 ++ 00 == 21.58 kN21.58 kN == 21.6 kN21.6 kN cc R R₄₄ = = --1.7851.785 ++ 55 == 3.214 kN3.214 kN == 3.21 kN3.21 kN :: Q Q₉₉ = = --22.5(1022.5(1066)()(--43.15)(1043.15)(10--66)) ++ 30(1030(1066)(86.12)(10)(86.12)(10--66)) == 3.555 kN3.555 kN

#

#

mm Q Q88 = = --11.25(1011.25(1066)()(--43.15)(1043.15)(10--66)) ++ 22.5(1022.5(1066)(86.12)(10)(86.12)(10--66)) == 2.423 kN2.423 kN Q Q77 = = --500(10500(1066)()(--13.57)(1013.57)(10--66)) == 6.785 kN6.785 kN Q Q66 == 22.5(1022.5(1066)()(--13.57)(1013.57)(10--66)) ++ 30(1030(1066)(86.12)(10)(86.12)(10--66)) == 2.278 kN2.278 kN

#

#

mm Q Q₅₅ = = --500(10500(1066)()(--43.15)(1043.15)(10--66)) == 21.58 kN21.58 kN Q Q44 = = --11.25(1011.25(1066)()(--13.57)(1013.57)(10--66)) ++ ((--22.5)(1022.5)(1066)(86.12)(10)(86.12)(10--66)) = = --1.785 kN1.785 kN Q Q_uu == KK₂₁₂₁DD_uu ++ KK₂₂₂₂DD_kk D D₃₃ == 86.12(1086.12(10--66)) D D₂₂ = = --43.15(1043.15(10--66)m)m D D₁₁ = = --13.57(1013.57(10--66) m) m 11(10 11(1033)) == (22.5(22.5DD₁₁ -- 22.522.5DD₂₂ ++ 120120DD₃₃)(10)(1066)) --24(1024(1033_{)) == (511.25(511.25DD} 2 2 -- 22.522.5DD33)(10)(1066)) --5(105(1033)) == (511.25(511.25DD₁₁ ++ 22.522.5DD₃₃)(10)(1066)) Q Q_kk == KK₁₁₁₁DD_uu ++ KK₁₂₁₂DD_kk 16–2

5 53388 For member 1 For member 1 For member 2 For member 2 k k₂₂ ==

F

F

1 111225500 00 --2250022500 --1111225500 00 --2250022500 0 0 11005500000000 00 00 --11005500000000 00 --2222550000 00 6600000000 2222550000 00 3300000000 --1111225500 00 2222550000 1111225500 00 2222550000 0 0 --11005500000000 00 00 11005500000000 00 --2222550000 00 3300000000 2222550000 00 6600000000

V

V

4 4EIEI L L == 4(200)(10 4(200)(1066)(300)(10)(300)(10--66)) 4 4 == 6000060000 2 2EIEI L L == 2(200)(10 2(200)(1066)(300)(10)(300)(10--66)) 4 4 == 3000030000 6 6EIEI L L22 == 6(200)(10 6(200)(1066)(300)(10)(300)(10--66)) 4 422 == 2250022500 12 12EIEI L L33 == (12)(200)(10 (12)(200)(1066)(300)(10)(300)(10--66)) 4 433 == 1125011250 AE AE L L == (0.021)(200)(10 (0.021)(200)(1066)) 5 5 == 10500001050000 l l_{xx == 00} ll_{yy ==} 00 -- ((--4)4) 4 4 == 11 k k₁₁ ==

F

F

8 84400000000 00 00 -8-84400000000 00 00 0 0 55776600 1144440000 00 --55776600 1144440000 0 0 1144440000 4488000000 00 --1144440000 2244000000 --884400000000 00 00 884400000000 00 00 0 0 --57605760 --1144440000 00 55776600 --1440014400 0 0 1144440000 2244000000 00 --1144440000 4488000000

V

V

4 4EIEI L L == 4(200)(10 4(200)(1066)(300)(10)(300)(10--66)) 5 5 == 4800048000 2 2EIEI L L == 2(200)(10 2(200)(1066)(300)(10)(300)(10--66)) 5 5 == 2400024000 6 6EIEI L L22 == 6(200)(10 6(200)(1066)(300)(10)(300)(10--66)) 5 522 == 1440014400 12 12EIEI L L33 == (12)(200)(10 (12)(200)(1066_{)(300)(10)(300)(10}--66₎₎ 5 533 == 57605760 AE AE L L == (0.021)(200)(10 (0.021)(200)(1066₎₎ 5 5 == 840000840000 l l_xx == 55 -- 00 5 5 == 11 llyy == 00 16–3.

16–3. Determine the structure stiffness matrixDetermine the structure stiffness matrix KK forfor th

the e frframame.e. AsAssusume me .. is is pipinned nned anand d .. is is fifixedxed.. TTakakee , , ,, ffoorr each member. each member. A A == 21211110103322 mm mm22 I I == 3003001110106622 mm mm44 E E == 200 MPa200 MPa 1 1 3 3 2 2 1 1 2 2 3 3 2 2 300 kN 300 kN m m 1 1 9 9 5 m 5 m 4 m 4 m 3 3 6 6 4 4 5 5 1 1 8 8 7 7

5 53399  Structure Stiffness Matrix.

 Structure Stiffness Matrix.

Ans. Ans. K K ==

I

I

8 85511225500 00 2222550000 2222550000 --1111225500 00 -8-84400000000 00 00 0 0 11005555776600 --1144440000 00 00 --11005500000000 00 --57605760 --1440014400 22500 22500 --1144440000 110088000000 3300000000 --2222550000 00 00 114444000000 2244000000 2 222550000 00 3030000000 6600000000 --2222550000 00 00 00 00 --1111225500 00 --2250022500 --2222550000 1111225500 00 00 00 00 0 0 -- 11005500000000 00 00 00 11005500000000 00 00 00 --884400000000 00 00 00 00 00 884400000000 00 00 0 0 --55776600 1144440000 00 00 00 00 55776600 1144440000 0 0 --1144440000 2424000000 00 00 00 00 1144440000 4488000000

Y

Y

16–

16–3. 3. ConContintinuedued

*16–4.

*16–4. DeterminDetermine e the the supporsupport t reactireactions ons at at .. and and .. T

Taakkee ,, ,, for each member.

for each member.

A A == 21211110103322 mm mm22 I I == 3003001110106622 mm mm44 E E == 200 MPa200 MPa 3 3 1 1

I

I

DD₁₁ D D₂₂ D D33 D D₄₄ 0 0 0 0 0 0 0 0 0 0

Y

Y

I

I

885511225500 00 2222550000 2222550000 --1111225500 00 -8-84400000000 00 00 0 0 11005555776600 --1144440000 00 00 --11005500000000 00 --57605760 --1440014400 22500 22500 --1144440000 110088000000 3300000000 --2222550000 00 00 1144440000 2244000000 2 222550000 00 3030000000 6600000000 --2222550000 00 00 00 00 --1111225500 00 --2250022500 --2222550000 1111225500 00 00 00 00 0 0 --11005500000000 00 00 00 11005500000000 00 00 00 --884400000000 00 00 00 00 00 884400000000 00 00 0 0 --55776600 1144440000 00 00 00 00 55776600 1144440000 0 0 --1144440000 2244000000 00 00 00 00 11444400 4488000000

Y

Y

I

I

00 0 0 300 300 0 0 Q Q₅₅ Q Q₆₆ Q Q₇₇ Q Q₈₈ Q Q99

Y

Y

== Q Q_kk ==

D

D

0 0 0 0 300 300 0 0

T

T

D D_kk ==

E

E

0 0 0 0 0 0 0 0 0 0

U

U

2 2 1 1 2 2 3 3 2 2 300 kN 300 kN m m 1 1 9 9 5 m 5 m 4 m 4 m 3 3 6 6 4 4 5 5 1 1 8 8 7 7

5 54400 Partition matrix Partition matrix Solving. Solving. Ans. Ans. Ans. Ans. Ans. Ans. Ans. Ans. Ans. Ans. Check equilibrium Check equilibrium a a (Check)(Check) (Check) (Check) a a++

_a

_a

MM₁₁ == 0;0; 300300 ++ 77.0777.07 -- 36.30(4)36.30(4) -- 46.37(5)46.37(5) == 00 (Check)(Check) + +cc

_a

_a

FF_yy == 0;0; 46.3746.37 -- 46.3746.37 == 00

a

a

FFxx == 0;0; 36.3036.30 -- 36.3036.30 == 00 Q Q99 == 77.1 kN77.1 kN

#

#

mm Q Q₈₈ == 46.4 kN46.4 kN Q Q₇₇ == 36.3 kN36.3 kN Q Q₆₆ = = --46.4 kN46.4 kN Q Q₅₅ = = --36.3 kN36.3 kN + +

E

E

0 0 0 0 0 0 0 0 0 0

U

U

D

D

--0.000043220.00004322 0.00004417 0.00004417 0.00323787 0.00323787 --0.001602730.00160273

T

T

E

E

--1111225500 00 --2250022500 --2250022500 0 0 --11005500000000 00 00 --884400000000 00 00 00 0 0 --55776600 1144440000 00 0 0 --1144440000 2424000000 00

U

U

E

E

QQ₅₅ Q Q₆₆ Q Q77 Q Q88 Q Q₉₉

U

U

= = D D₄₄ = = --0.00160273 rad0.00160273 rad D D₃₃ == 0.00323787 rad0.00323787 rad D D₂₂ == 0.00004417 m0.00004417 m D D₁₁ = = --0.00004322 m0.00004322 m 0 0 == 2250022500DD11 ++ 3000030000DD33 ++ 6000060000DD44 300 300 == 2250022500DD11 -- 1440014400DD22 ++ 108000108000DD33 ++ 3000030000DD44 0 0 == 10557601055760DD₂₂ -- 1440014400DD₃₃ 0 0 == 851250851250DD₁₁ ++ 2250022500DD₃₃ ++ 2250022500DD₄₄ + +

D

D

0 0 0 0 0 0 0 0

T

T

D

D

DD11 D D22 D D33 D D₄₄

T

T

D

D

885511225500 00 2222550000 2222550000 0 0 11005555776600 --1144440000 00 22500 22500 --1144440000 110088000000 3300000000 2 222550000 00 3300000000 6600000000

T

T

D

D

00 0 0 300 300 0 0

T

T

= = 16–4

5 54411 Member Stiffness Matrices.

Member Stiffness Matrices.The The origin of origin of the the global coordglobal coordinate system inate system will be will be set set at joint at joint .. F Foor r mmeemmbbeer r aannd d ,, .. F Foor r mmeemmbbeer r ,, aannd d ..TThhuuss,, (10 (1066)) F Foor r mmeemmbbeer r ,, ,,aannd d ..TThhuuss,, (10 (1066₎₎ 1 1 2 2 3 3 6 6 7 7 4 4

F

F

1 2 1 2 33 66 77 44 1 133..112255 00 2266..2255 -1-133..112255 00 2266..2255 0 0 775500 00 00 --775500 00 2 266..2255 00 7700 -2-266..2255 00 3355 --1133..112255 00 --2266..2255 1133..112255 00 --26.2526.25 0 0 --775500 00 00 775500 00 2 266..2255 00 3355 --2266..2255 00 7700

V

V

k k₂₂ == l l_{yy ==} --44 -- 00 4 4 = = --11 l l_{xx ==} 44 -- 44 4 4 == 00

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22

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8 8 9 9 5 5 1 1 2 2 3 3

F

F

8 8 99 55 11 22 33 7 75500 00 00 --775500 00 00 0 0 1133..112255 2266..2255 00 --1133..112255 2266..2255 0 0 2266..2255 7700 00 --2266..2255 3355 --775500 00 00 775500 00 00 0 0 --13.12513.125 --2266..2255 00 1133..112255 --26.2526.25 0 0 2266..2255 3355 00 --2266..2255 7700

V

V

k k₁₁ == l l_yy == 00 -- 00 4 4 == 00 l l_xx == 44 -- 00 4 4 == 11

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11

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2 2EIEI L L == 2[200(10 2[200(1099)][350(10)][350(10--66)])] 4 4 == 35(1035(10 6 6_{) N) N}

#

#

_mm 4 4EIEI L L == 4[200(10 4[200(1099)][350(10)][350(10--66)])] 4 4 == 70(1070(10 6 6_{) N) N}

#

#

_mm 6 6EIEI L L22 == 4[200(10 4[200(1099)][350(10)][350(10--66)])] 4 422 == 26.25(1026.25(10 6 6_{) N) N} 12 12EIEI L L33 == 12[200(10 12[200(1099)][350(10)][350(10--66)])] 4 433 == 13.125(1013.125(10 6 6_{) N) N>>mm} AE AE L L == 0.015[200(10 0.015[200(1099)])] 4 4 == 750(10750(10 6 6_{) N) N>>mm} L L == 4m4m

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22

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ƒƒ

11

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1 1 16–5.

16–5. Determine the structure stiffness matrixDetermine the structure stiffness matrixKKfor the frame.for the frame. T

Taakkee ,, ,, for

for each each member.member.Joints Joints at at 11 and and 33 are are pins.pins. A A == 15151110103322 mm mm22 I I == 3503501110106622 mm mm44 E E == 200 GPa200 GPa 9 9 1 1 22 3 3 1 1 2 2 2 2 1 1 3 3 6 6 4 4 4 m 4 m 2 m 2 m 2 m2 m 60 kN 60 kN 5 5 8 8 7 7

5 54422  Structure Stiffness Matrix.

 Structure Stiffness Matrix.It iIt is a s a 99 9 ma9 matritrix six since nce the the highigheshest ct code ode numnumber ber is 9is 9..ThThusus

(10 (1066_{))    Ans.Ans.} 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9

I

I

1 1 22 33 44 55 66 77 88 99 7 76633..112255 00 2626..2255 2266..2255 00 --1133..112255 00 --775500 00 0 0 776633..112255 --2266..2255 00 --2266..2255 00 --775500 00 --13.12513.125 26.25 26.25 --2266..2255 114400 3355 3355 --2266..2255 00 00 2266..2255 2 266..2255 00 3355 7700 00 --2266..2255 00 00 00 0 0 --2266..2255 3355 00 7070 00 00 00 2266..2255 --1133..112255 00 --26.2526.25 --2266..2255 00 1133..112255 00 00 00 0 0 --775500 00 00 00 00 775500 00 00 --775500 00 00 00 00 00 00 775500 00 0 0 --1133..112255 2266..2255 00 2266..2255 00 00 00 1133..112255

Y

Y

K K == * * 16–6.

16–6. Determine Determine the the support support reactions reactions at at pins pins and and .. T

Taakkee ,, ,,

for each member. for each member.

A A == 15151110103322 mm mm22 I I == 3503501110106622 mm mm44 E E == 200 GPa200 GPa 3 3 1 1 16–5

16–5. . ConContintinuedued

 Known Nodal Loads and Deflections.

 Known Nodal Loads and Deflections.The nodal load acting onThe nodal load acting on the unco

the unconstrainstrained degrened degree of freedoe of freedom (code num (code numbers 1,mbers 1,2,2,3,3,4,4,and 5)and 5) are shown in Fig.

are shown in Fig.aa_{and Fig.and Fig.}bb_..

Q

Q_kk andand DD_kk

 Loads-Displacement Relation.

 Loads-Displacement Relation.Applying Applying ,,

(10 (1066))

I

I

D D₁₁ D D22 D D33 D D44 D D₅₅ 0 0 0 0 0 0 0 0

Y

Y

I

I

776633..112255 00 2266..2255 2266..2255 00 --1133..112255 00 --775500 00 0 0 776633..112255 --2266..2255 00 --2266..2255 00 -7-75500 00 --13.12513.125 26.25 26.25 --2266..2255 114400 3355 3355 --2266..2255 00 00 2266..2255 2 266..2255 00 3355 7700 00 --2266..2255 00 00 00 0 0 --2266..2255 3355 00 7070 00 00 00 2266..2255 --1133..112255 00 --26.2526.25 --2266..2255 00 1133..112255 00 00 00 0 0 --775500 00 00 00 00 775500 00 00 --775500 00 00 00 00 00 00 775500 00 0 0 --1133..112255 2266..2255 00 2266..2255 00 00 00 1133..112255

Y

Y

I

I

00 --41.25(1041.25(1033)) 45(10 45(1033)) 0 0 0 0 Q Q₆₆ Q Q₇₇ Q Q₈₈ Q Q99

Y

Y

== Q Q == KDKD = =

D

D

0 0 0 0 0 0 0 0

T

T

6 6 7 7 8 8 9 9 = =

E

E

0 0 --41.25(1041.25(1033)) 45(10 45(1033)) 0 0 0 0

U

U

1 1 2 2 3 3 4 4 5 5 9 9 1 1 22 3 3 1 1 2 2 2 2 1 1 3 3 6 6 4 4 4 m 4 m 2 m 2 m 2 m2 m 60 kN 60 kN 5 5 8 8 7 7

5 54433 F Frroom m tthhe e mmaattrriix x ppaarrttiittiioonn,, ,, (1) (1) (2) (2) (3) (3) (4) (4) (5) (5) Solving Eqs Solving Eqs..(1) to (5)(1) to (5) Using

Using these these results results and and applying applying ,,

Superposition these results to

Superposition these results to those of FEM shown in those of FEM shown in Fig.Fig.aa_,,

Ans. Ans. Ans. Ans. Ans. Ans. Ans. Ans. R R₉₉ == 5.7155.715 ++ 18.7518.75 == 24.5 kN24.5 kN R R₈₈ == 5.5355.535 ++ 00 == 5.54 kN5.54 kN R R77 == 35.53535.535 ++ 00 == 35.5 kN35.5 kN R R66 = = --5.535 kN5.535 kN ++ 00 == 5.54 kN5.54 kN Q Q99 = = --13.125(1013.125(1066)) -- 47.3802(10 47.3802(10--66)) ++ 26.25(1026.25(1066)) ++ 423.5714(10423.5714(10--66)) ++ 26.25(1026.25(1066)) -- 229.5533(10229.5533(10--66)) ++ 00 == 5.715 kN5.715 kN Q Q₈₈ = = --750(10750(1066)) -- 7.3802(107.3802(10--66)) ++ 00 == 5.535 kN5.535 kN Q Q₇₇ = = --750(10750(1066)) -- 47.3802(1047.3802(10--66)) ++ 00 == 35.535 kN35.535 kN Q Q₆₆ == ((--13.125)(1013.125)(1066)) -- 7.3802(107.3802(10--66)) -- 26.25(1026.25(1066)423.5714(10)423.5714(10--66)) -- 26.25(1026.25(1066)) -- 209.0181(10209.0181(10--66)) ++ 00 = = --5.535 kN5.535 kN Q Q_uu == KK₂₁₂₁DD_uu ++ KK₂₂₂₂DD_kk D D₄₄ = = --209.0181(10209.0181(10--66)) DD₅₅ = = --229.5533(10229.5533(10--66)) D D₃₃ == 423.5714(10423.5714(10--66₎₎ D D₂₂ = = --47.3802(1047.3802(10--66₎₎ D D₁₁ = = --7.3802(107.3802(10--66₎₎ 0 0 == ((--26.2526.25DD22 ++ 3535DD33 ++ 7070DD55)(10)(1066)) 0 0 == (26.25(26.25DD₁₁ ++ 3535DD₃₃ ++ 7070DD₄₄)(10)(1066)) 45(10 45(1033)) == (26.25(26.25DD₁₁ -- 26.2526.25DD₂₂ ++ 140140DD₃₃ ++ 3535DD₄₄ ++ 3535DD₅₅)(10)(1066)) --41.25(1041.25(1033)) == (763.125(763.125DD₂₂ -- 26.2526.25DD₃₃ -- 26.2526.25DD₅₅)(10)(1066)) 0 0 == (763.125(763.125DD₁₁ ++ 26.2526.25DD₃₃ ++ 26.2526.25DD₄₄)(10)(1066)) Q Q_kk == KK₁₁₁₁DD_uu ++ KK₁₂₁₂DD_kk 16–

5 54444 Member 1. Member 1. Member 2. Member 2. k k₂₂ ==

F

F

7 755..7755 00 55445544..2288 -7-755..7755 00 55445544..2288 0 0 44002277..7788 00 00 --44002277..7788 00 5 5445544..2288 00 552233661111..1111 -5-5445544..2288 00 226611880055..5555 --7755..7755 00 --55445544..2288 7755..7755 00 --5454.285454.28 0 0 --44002277..7788 00 00 44002277..7788 00 5 5445544..2288 00 262611880055..5555 --55445544..2288 00 552233661111..1111

V

V

2 2EIEI L L22 == 2(29)(10 2(29)(1033)(650))(650) (12)(12) (12)(12) == 261805.55261805.55 4 4EIEI L L == 4(29)(10 4(29)(1033)(650))(650) (12)(12) (12)(12) == 523611.11523611.11 6 6EIEI L L == 6(29)(10 6(29)(1033)(650))(650) (12) (12)22(12)(12)22 == 5454.285454.28 12 12EIEI L L33 == 12(29)(10 12(29)(1033)(650))(650) (12) (12)33(12)(12)33 == 75.7575.75 AE AE L L == (20)(29)(10 (20)(29)(1033)) (12)(12) (12)(12) == 4027.784027.78 l l y y == --1212 -- 00 12 12 = = --11 l l x x == 00 k k₁₁ ==

F

F

4 4883333..3333 00 00 --44883333..3333 00 00 0 0 113300..9900 77885544..1177 00 --113300..9900 77885544..1177 0 0 77885544..1177 662288333333..3333 00 --78785454.1.177 313141416666..6767 --44883333..3333 00 00 44883333..3333 00 00 0 0 --130.90130.90 --77885544..1177 00 113300..9900 --7854.177854.17 0 0 77885544..1177 331144116666..6677 00 --78785454.1.177 626283833333..3333

V

V

2 2EIEI L L == 2(29)(10 2(29)(1033)(650))(650) (10)(12) (10)(12) == 314166.67314166.67 4 4EIEI L L == 4(29)(10 4(29)(1033)(650))(650) (10)(12) (10)(12) == 628333.33628333.33 6 6EIEI L L22 == 6(29)(10 6(29)(1033)(650))(650) (10) (10)22(12)(12)22 == 7854.177854.17 12 12EIEI L L33 == 12(29)(10 12(29)(1033_)(650))(650) (10) (10)33(12)(12)33 == 130.90130.90 AE AE L L == 20(29)(10 20(29)(1033₎₎ 10(12) 10(12) == 4833.334833.33 l l_yy == ₀₀ l l_xx == 1010 -- 00 10 10 == 11 16–7.

16–7. Determine the structure stiffness matrixDetermine the structure stiffness matrix KK forfor t

thhe e ffrraammee. T. Taakke e ,, ,, for each member.

for each member.

A A == 20 in20 in22 I I == 650 in650 in44 E E == 29291110103322 ksi ksi 2 2 1 1  6  6 5 5 4 4 2 2 1 1 33 2 2 1 1 6 k 6 k 4 k 4 k 3 3 99 8 8 7 7 12 ft 12 ft 10 ft 10 ft

5 54455  Structure Stiffness Matrix.

 Structure Stiffness Matrix.

Ans. Ans. K K ==

I

I

4 4883333..3333 00 00 --44883333..3333 00 00 00 00 00 0 0 113300..9900 77885544..1177 00 --113300..9900 77885544..1177 00 00 00 0 0 77885544..1177 662288333333..3333 00 --77885544..1177 331144116666..6677 00 00 00 --44883333..3333 00 00 44990099..0088 00 54545544..2288 --7755..7755 00 55445544..2288 0 0 --130.90130.90 --77885544..1177 00 44115588..6688 --77885544..3377 00 --44002277..7788 00 0 0 77885544..1177 331144116666..6677 55445544..2288 --78785454.1.177 1111515194944.4.4444 --55445544..2288 00 226611880055..5555 0 0 00 00 --7755..7755 00 --55445544..2288 7755..7755 00 --5454.285454.28 0 0 00 00 00 --44002277..7788 00 00 44002277..7788 00 0 0 00 00 55445544..2288 00 226611880055..5555 --55445544..2288 00 552233661111..1111

Y

Y

16–

16–7. 7. ConContintinuedued

*16–8.

*16–8. Determine Determine the the components components of of displacement displacement at at .. T Taakke e ,, ,, ffoor r eeaacchh member. member. A A == 20 in20 in22 I I == 650 in650 in44 E E == 29291110103322 ksi ksi 1 1 D Dkk QQkk

I

I

DD11 D D22 D D₃₃ D D₄₄ D D₅₅ D D₆₆ 0 0 0 0 0 0

Y

Y

I

I

44883333..3333 00 00 --48483333..3333 00 00 00 00 00 0 0 113300..9900 77885544..1177 00 --113300..9900 77885544..1177 00 00 00 0 0 77885544..1177 662288333333..3333 00 --77885544..1177 331144116666..6677 00 00 00 --44883333..3333 00 00 44990099..0088 00 54545544..2288 --7755..7755 00 55445544..2288 0 0 --130.90130.90 --77885544..1177 00 44115588..6688 --77885544..1177 00 --44002277..7788 00 0 0 77885544..1177 331144116666..6677 55445544..2288 --78785454.1.177 1111515194944.4.4444 --55445544..2288 00 226611880055..5555 0 0 00 00 --7755..7755 00 --55445544..2288 7755..7755 00 --5454.285454.28 0 0 00 00 00 --44002277..7788 00 00 44002277..7788 00 0 0 00 00 55445544..2288 00 226611880055..5555 --55445544..2288 00 552233661111..1111

Y

Y

I

I

--44 --66 0 0 0 0 0 0 0 0 Q Q₇₇ Q Q88 Q Q₉₉

Y

Y

= = = =

F

F

--44 --66 0 0 0 0 0 0 0 0

V

V

= =

C

C

0 0 0 0 0 0

S

S

2 2 1 1  6  6 5 5 4 4 2 2 1 1 33 2 2 1 1 6 k 6 k 4 k 4 k 3 3 99 8 8 7 7 12 ft 12 ft 10 ft 10 ft

5 54466  Partition Matrix.

 Partition Matrix.

Solving the above equations yields Solving the above equations yields

Ans. Ans. Ans. Ans. Ans. Ans. D D66 == 0.007705 rad0.007705 rad D D55 = = --0.001490 in.0.001490 in. D D₄₄ = = --0.6076 in.0.6076 in. D D₃₃ == 0.0100 rad0.0100 rad D D₂₂ = = --1.12 in.1.12 in. D D₁₁ = = --0.608 in.0.608 in. 0 0 == 7854.177854.17DD₂₂ ++ 314166.67314166.67DD₃₃ ++ 5454.285454.28DD₄₄ -- 7854.177854.17DD₅₅ ++ 1151944.441151944.44DD₆₆ 0 0 = = --130.90130.90DD22 -- 7854.177854.17DD33 ++ 4158.684158.68DD55 -- 7854.177854.17DD66 0 0 = = --4833.334833.33DD11 ++ 4909.084909.08DD44 ++ 5454.285454.28DD66 0 0 == 7854.177854.17DD22 ++ 628333.33628333.33DD33 -- 7854.177854.17DD55 ++ 314166.67314166.67DD66 --66 == 130.90130.90DD₂₂ ++ 7854.177854.17DD₃₃ -- 130.90130.90DD₅₅ ++ 7854.177854.17DD₆₆ --44 == 4833.334833.33DD₁₁ -- 4833.334833.33DD₄₄ + +

F

F

0 0 0 0 0 0 0 0 0 0 0 0

V

V

F

F

DD₁₁ D D22 D D33 D D₄₄ D D₅₅ D D₆₆

V

V

F

F

44883333..3333 00 00 --44883333..3333 00 00 0 0 113300..9900 77885544..1177 00 --113300..9900 77885544..1177 0 0 77885544..1177 662288333333..3333 00 --77885544..1177 331144116666..6677 --44883333..3333 00 00 44990099..0088 00 55445544..2288 0 0 --130.90130.90 --77885544..1177 00 44115588..6688 --7854.177854.17 0 0 77885544..1177 313144116666..6677 55445544..2288 --78785454.1.177 1111515194944.4.4444

V

V

F

F

--44 --66 0 0 0 0 0 0 0 0

V

V

+ + 16–9.

16–9. Determine the stiffness matrixDetermine the stiffness matrixKKfor the frame.for the frame.TTakeake ,

,II == 300 in300 in44,,AA == 10 in10 in22ffoor r eeaacch h mememmbbeerr.. E

E == 29291110103322 ksi ksi 16–8

16–8. . ConContintinuedued

Member Stiffness Matrices.

Member Stiffness Matrices.The origin of the global coordinate system will be set atThe origin of the global coordinate system will be set at  joint

 joint ..For member For member ,, ,, andand

4 4EIEI L L == 4[29(10 4[29(1033)](300))](300) 10(12) 10(12) == 290000 k290000 k

#

#

inin 6 6EIEI L L22 == 6[29(10 6[29(1033)](300))](300) [10(12)] [10(12)]22 == 3625 k3625 k 12 12EIEI L L33 == 12[29(10 12[29(1033)](300))](300) [10(12)] [10(12)]33 == 60.4167 k/in60.4167 k/in AE AE L L == 10[29(10 10[29(1033)])] 10(12) 10(12) == 2416.67 k/in2416.67 k/in l l_{yy ==} 1010 -- 00 10 10 == 11 l l_{xx ==} 00 -- 00 10 10 == 00 L L == 10 ft10 ft

ƒƒ

11

ƒƒ

1 1 10 ft 10 ft 20 ft 20 ft 2 2 1 1 3 3 2 2 ₇₇ 1 1 4 4 6 6 9 9 8 8 5 5 2 2 1 1 3 3 2 k 2 k_/ / ftft

5 54477 F

Foor r mmeemmbbeer r ,, ,, aannd d ..

 Structure Stiffness Matrix.

 Structure Stiffness Matrix.It iIt is a 9s a 9 9 ma9 matrix trix sinsince tce the hhe higheighest code st code numnumber ber is 9is 9..ThThusus,,

K K ==

I

I

1 1226688..7755 00 36362255 00 33662255 --11220088..3333 00 --6600..44116677 00 0 0 22442244..2222 909066..2255 990066..2255 00 00 --77..55552211 00 --2416.672416.67 3 3662255 990066..2255 443355000000 7722550000 114455000000 00 -906.25-906.25 --33662255 00 0 0 990066..2255 7272550000 114455000000 00 00 --990066..2255 00 00 3 3662255 00 114455000000 00 229900000000 00 00 --33662255 00 --11220088..3333 00 00 00 00 11220088..3333 00 00 00 0 0 --7.55217.5521 --906.25906.25 --990066..2255 00 00 77..55552211 00 00 --6600..44116677 00 --33662255 00 --33662255 00 00 6600..44116677 00 0 0 --22441166..6677 00 00 00 00 00 00 22441166..6677

Y

Y

* * 1 1 2 2 3 3 6 6 7 7 4 4

F

F

1 2 3 6 7 4 1 2 3 6 7 4 1 1220088..3333 00 00 --11220088..3333 00 00 0 0 77..55552211 990066..2255 00 --77..55552211 990066..2255 0 0 990066..2255 114455000000 00 --990066..2255 7722550000 --11220088..3333 00 00 11220088..3333 00 00 0 0 --7.55217.5521 --990066..2255 00 77..55552211 --906.25906.25 0 0 990066..2255 7722550000 00 --990066..2255 114455000000

V

V

k k₂₂ == 2 2EIEI L L == 2[29(10 2[29(1033)](300))](300) 20(12) 20(12) == 72500 k72500 k

#

#

inin 4 4EIEI L L == 4[29(10 4[29(1033_{)](300))](300)} 20(12) 20(12) == 145000 k145000 k

#

#

inin 6 6EIEI L L22 == 6[29(10 6[29(1033_{)](300))](300)} [20(12)] [20(12)]22 == 906.25 k906.25 k 12 12EIEI L L33 == 12[29(10 12[29(1033)](300))](300) [20(12)] [20(12)]33 == 7.5521 k/in7.5521 k/in AE AE L L == 10[29(10 10[29(1033)])] 20(12) 20(12) == 1208.33 k/in1208.33 k/in l l_{yy ==} 1010 -- 1010 20 20 == 00 l l_{xx ==} 2020 -- 00 20 20 == 11 L L == 20 ft20 ft

ƒƒ

22

ƒƒ

8 8 9 9 5 5 1 1 2 2 3 3

F

F

8 8 99 55 11 22 33 6 600..44116677 00 --36253625 --6600..44116677 00 --36253625 0 0 22441166..6677 00 00 --22441166..6677 00 --33662255 00 229900000000 33662255 00 114455000000 --6600..44116677 00 33662255 6600..44116677 00 33662255 0 0 --22441166..6677 00 00 22441166..6677 00 --33662255 00 114455000000 36362255 00 229900000000

V

V

k k₁₁ == 2 2EIEI L L == 2[29(10 2[29(1033)](300))](300) 10(12) 10(12) == 145000 k145000 k

#

#

inin 16–

5 54488  Known Nodal Loads and Deflections.

 Known Nodal Loads and Deflections.The nodal loads acting on the unconstrainedThe nodal loads acting on the unconstrained degree

degree of freeof freedom (cdom (code nuode number 1,mber 1,2,2,3,3,4,4,5,5,and 6) are shand 6) are shown in own in FigFig..aa_andandbb_..

Q

Q_kk andand DD_kk

 Loads–Displacement Relation.

 Loads–Displacement Relation.ApplyingApplyingQQ==KDKD..

From the matrix partition,

From the matrix partition,QQ_kk==KK₁₁₁₁DD_uu++KK₁₂₁₂DD_kk,,

(1) (1) (2) (2) (3) (3) (4) (4) (5) (5) (6) (6) 0 0 = = --1208.331208.33DD11 ++ 1208.331208.33DD66 0 0 == 36253625DD11 ++ 145000145000DD33 ++ 290000290000DD55 0 0 == 906.25906.25DD22 ++ 7250072500DD33 ++ 145000145000DD44 --12001200 == 36253625DD₁₁ ++ 906.25906.25DD₂₂ ++ 435000435000DD₃₃ ++ 7250072500DD₄₄ ++ 145000145000DD₅₅ --2525 == 2424.222424.22DD22 ++ 906.25906.25DD33 ++ 906.25906.25DD44 0 0 == 1268.751268.75DD₁₁ ++ 36253625DD₃₃ ++ 36253625DD₅₅ -- 1208.331208.33DD₆₆ 4.937497862 4.937497862 = = --906.25906.25DD₃₃ -- 906.25906.25DD₄₄ 0 0 == 906.25(906.25(--8.2758)(108.2758)(10--33)) ++ 7250072500DD₃₃ ++ 145000145000DD₄₄ 5 5 = = --7.5521(7.5521(--8.2758)(108.2758)(10--33)) -- 906.25906.25DD₃₃ -- 906.25906.25DD₄₄ D D₂₂ = = --8.275862071(108.275862071(10--33)) 20 20 = = --2416.672416.67DD22

I

I

DD11 D D₂₂ D D33 D D₄₄ D D₅₅ D D₆₆ 0 0 0 0 0 0

Y

Y

Y

Y

I

I

00 --2525 --12001200 0 0 0 0 0 0 Q Q₇₇ Q Q88 Q Q99

Y

Y

==

I

I

1 1226688..7755 00 36362255 00 33662255 --11220088..3333 00 --6600..44116677 00 0 0 22442244..2222 909066..2255 990066..2255 00 00 --77..55552211 00 --2416.672416.67 3 3662255 990066..2255 443355000000 7722550000 114455000000 00 --906.25906.25 --33662255 00 0 0 990066..2255 7272550000 114455000000 00 00 --990066..2255 00 00 3 3662255 00 114455000000 00 229900000000 00 00 --33662255 00 --11220088..3333 00 00 00 00 11220088..3333 00 00 00 0 0 --7.55217.5521 --906.25906.25 --990066..2255 00 00 77..55552211 00 00 --6600..44116677 00 --33662255 00 --33662255 00 00 6600..44116677 00 0 0 --22441166..6677 00 00 00 00 00 00 22441166..3377 = =

C

C

0 0 0 0 0 0

S

S

7 7 8 8 9 9 = =

F

F

0 0 --2525 --12001200 0 0 0 0 0 0

V

V

1 1 2 2 3 3 4 4 5 5 6 6 16–10.

16–10. Determine Determine the the support support reactions reactions at at and and ..TTakeake , ,AA == 10 in10 in22ffoor r eeaacch h mememmbbeerr.. I I == 300 in300 in44 E E == 29291110103322 ksi, ksi, 3 3 1 1 10 ft 10 ft 20 ft 20 ft 2 2 1 1 3 3 2 2 ₇₇ 1 1 4 4 6 6 9 9 8 8 5 5 2 2 1 1 3 3 2 k 2 k_/ / ftft

5 54499 Solving Eqs

Solving Eqs..(1) to (6)(1) to (6)

Using these results and applying

Using these results and applyingQQkk==KK₂₁₂₁DDuu++KK₂₂₂₂DDkk,,

Superposition these results to

Superposition these results to those of FEM shown in those of FEM shown in Fig.Fig.aa_..

Ans. Ans. Ans. Ans. Ans. Ans. R R₉₉ == 2020 ++ 00 == 20 k20 k R R₈₈ == 00 ++ 00 == 00 R R₇₇ == 55 ++ 1515 == 20 k20 k Q Q99 = = --2416.67(2416.67(--0.008276)0.008276) == 2020 Q Q88 == 60.4167(1.32)60.4167(1.32)--3625(3625(--0.011)0.011)--3625(3625(--0.011)0.011) == 00 Q Q₇₇ = = --7.5521(7.5521(--0.008276)0.008276) -- 906.25(906.25(--0.011)0.011) -- 906.25(0.005552)906.25(0.005552) == 55 D D₆₆ == 1.321.32 D D₅₅ = = --0.0110.011 D D₄₄ == 0.0055520.005552 D D₃₃ = = --0.0110.011 D D₂₂ = = --0.0082760.008276 D D₁₁ == 1.321.32 16–

5 55500 Member Stiffness Matrices.

Member Stiffness Matrices.The origin of the global coordinate systemThe origin of the global coordinate system w wiilll l bbe e seset t aat t jjooiinnt t . F. Foor r mmeemmbbeer r ,, ,, and and k k₁₁ F Foor r mmeemmbbeer r ,, aannd d .. 2 2EIEI L L == 2[29(10 2[29(1033)](700))](700) [16(12)] [16(12)] == 211458 k211458 k

#

#

inin 4 4EIEI L L == 4[29(10 4[29(1033)](700))](700) [16(12)] [16(12)] == 422917 k422917 k

#

#

inin 6 6EIEI L L22 == 6[29(10 6[29(1033)](700))](700) [16(12)] [16(12)]22 == 3304.04 k3304.04 k 12 12EIEI L L33 == 12[29(10 12[29(1033)](700))](700) [16(12)] [16(12)]33 == 34.4170 k/in34.4170 k/in AE AE L L == 20[29(10 20[29(1033)])] 16(12) 16(12) == 3020.83 k/in.3020.83 k/in. l l_{yy ==} 1616 -- 00 16 16 == 11 L L == 16 ft,16 ft,ll_{xx ==} 2424 -- 2424 16 16 == 00

ƒƒ

22

ƒƒ

8 8 9 9 5 5 1 1 2 2 3 3 = =

F

F

8 9 5 1 2 3 8 9 5 1 2 3 2 2001133..8899 00 00 -2-2001133..8899 00 00 0 0 1100..11997766 11446688..4466 00 --1100..11997766 11446688..4466 0 0 11446688..4466 228811994444 00 --11446688..4466 114400997722 --22001133..8899 00 00 22001133..8899 00 00 0 0 --10.197610.1976 --11446688..4466 00 1100..11997766 --1468.461468.46 0 0 11446688..4466 114400997722 00 --11446688..4466 228811994444

V

V

2 2EIEI L L == 2[29(10 2[29(1033)](700))](700) [24(12)] [24(12)] == 140972 k140972 k

#

#

inin 4 4EIEI L L == 4[29(10 4[29(1033)](700))](700) [24(12)] [24(12)] == 281944 k281944 k

#

#

inin 6 6EIEI L L22 == 6[29(10 6[29(1033)](700))](700) [24(12)] [24(12)]22 == 1468.46 k1468.46 k 12 12EIEI L L33 == 12[29(10 12[29(1033)](700))](700) [24(12)] [24(12)]33 == 10.1976 k/in10.1976 k/in AE AE L L == 20[29(10 20[29(1033)])] 24(12) 24(12) == 2013.89 k/in2013.89 k/in l l_{yy ==} 00 -- 00 24 24 == 00 l l_{xx ==} 2424 -- 00 24 24 == 11 L L == 24 ft24 ft

ƒƒ

11

ƒƒ

1 1 16–11.

16–11. Determine the structure stiffness matrixDetermine the structure stiffness matrixKKfor thefor the f frraammee.. TTaakke e ,, ,, ffoorr each member. each member. A A == 20 in20 in22 I I == 700 in700 in44 E E == 29291110103322 ksi ksi 1 1 2 2 9 9 8 8 3 3 2 2 1 1 3 3 16 ft 16 ft 20 k 20 k 12 ft 12 ft 12 ft 12 ft 1 1 2 2 4 4 7 7 6 6 5 5

5 55511 k

k₂₂

 Structure Stiffness Matrix.

 Structure Stiffness Matrix.It iIt is a s a 99 9 ma9 matritrix sx sincince the the he higheighest cst code ode numnumber ber is 9is 9.. Thus, Thus, K K ==

I

I

2 2004488..3311 00 --3304.043304.04 --33330044..0044 00 --3344..44117700 00 --22001133..8899 00 0 0 33003311..0033 --11446688..4466 00 --11446688..4466 00 --33002200..8833 00 --10.197610.1976 --3304.043304.04 --11446688..4466 707044886611 221111445588 114400997722 33330044..0044 00 00 11446688..4466 --33330044..0044 00 221111445588 442222991177 00 33330044..0044 00 00 00 0 0 --11446688..4466 114400997722 00 282811994444 00 00 00 11446688..4466 --3344..44117700 00 33330044..0044 33330044..0044 00 3344..44117700 00 00 00 0 0 --33002200..8833 00 00 00 00 33002200..8833 00 00 --22001133..8899 00 00 00 00 00 00 22001133..8899 00 0 0 --1100..11997766 14146688..4466 00 11446688..4466 00 00 00 1100..11997766

Y

Y

* * 1 1 2 2 3 3 6 6 7 7 4 4 = =

F

F

1 2 3 6 7 4 1 2 3 6 7 4 3 344..44117700 00 --3304.043304.04 -3-344..44117700 00 --3304.043304.04 0 0 33002200..8833 00 00 --33002200..8833 00 --33330044..0044 00 442222991177 33330044..0044 00 221111445588 --3344..44117700 00 33330044..0044 3344..44117700 00 33330044..0044 0 0 --33002200..8833 00 00 33002200..8833 00 --33330044..0044 00 221111445588 33330044..0044 00 442222991177

V

V

16–

16–11. 11. ConContintinuedued

 Known Nodal Loads and Deflections.

 Known Nodal Loads and Deflections.The nodal loads acting on theThe nodal loads acting on the uncon

unconstrainstrained degree oed degree of freedom (f freedom (code numcode number 1,ber 1,2,2,3,3,4,4,and 5) areand 5) are shown in Fig.

shown in Fig.aa_andandbb_..

Q Q_kk andandDD_kk ==

D

D

0 0 0 0 0 0 0 0

T

T

 6  6  7  7  8  8  9  9 = =

E

E

0 0 --13.7513.75 1080 1080 0 0 0 0

U

U

 1  1  2  2  3  3  4  4  5  5 *16–12.

*16–12. Determine the support reactions at the pinsDetermine the support reactions at the pins a annd d ..TTaakke e ,, ,, ffoorr each member. each member. A A == 20 in20 in22 I I == 700 in700 in44 E E == 29291110103322 ksi ksi 3 3 1 1 1 1 2 2 9 9 8 8 3 3 2 2 1 1 3 3 16 ft 16 ft 20 k 20 k 12 ft 12 ft 12 ft 12 ft 1 1 2 2 4 4 7 7 6 6 5 5

5 55522  Loads-Displacement Relation.

 Loads-Displacement Relation.ApplyingApplyingQQ==KDKD,,

= =

From the matrix partition,

From the matrix partition,QQ_kk==KK₁₁₁₁DD_uu++KK₁₂₁₂DD_kk,,

(1) (1) (2) (2) (3) (3) (4) (4) (5) (5) Solving Eqs Solving Eqs..(1) to (5),(1) to (5),

Using these results and applying Using these results and applying

Superposition these results to those of FEM shown in Fig. Superposition these results to those of FEM shown in Fig.aa_..

Ans. Ans. Ans. Ans. Ans. Ans. Ans. Ans. R R₉₉ == 1.5101.510 ++ 6.256.25 == 7.76 k7.76 k R R88 = = --3.3603.360 ++ 00 = = --3.36 k3.36 k R R77 == 12.2412.24 ++ 00 == 12.2 k12.2 k R R66 == 3.3603.360 ++ 00 == 3.36 k3.36 k Q Q₉₉ = = --10.1976(10.1976(--0.004052)0.004052) ++ 1468.46(0.002043)1468.46(0.002043) ++ 1468.46(1468.46(--0.001008)0.001008) == 1.5101.510 Q Q₈₈ = = --2013.89(0.001668)2013.89(0.001668) = = --3.3603.360 Q Q₇₇ = = --3020.83(3020.83(--0.004052)0.004052) == 12.2412.24 Q Q₆₆ = = --34.4170(0.001668)34.4170(0.001668) ++ 3304.04(0.002043)3304.04(0.002043) ++ 3304.04(3304.04(--0.001008)0.001008) == 3.3603.360 Q Q_uu == KK₂₁₂₁DD_uu ++ KK₂₂₂₂DD_kk,, D D₅₅ = = --0.0010420.001042 D D₄₄ = = --0.0010080.001008 D D₃₃ == 0.0020430.002043 D D₂₂ = = --0.0040520.004052 D D₁₁ == 0.0016680.001668 0 0 = = --1468.461468.46DD22 ++ 140972140972DD33 ++ 281944281944DD55 0 0 = = --3304.043304.04DD11 ++ 211458211458DD33 ++ 422917422917DD44 90 90 = = --3304.043304.04DD11 -- 1468.461468.46DD22 ++ 704861704861DD33 ++ 211458211458DD44 ++ 140972140972DD55 --13.7513.75 == 3031.033031.03DD₂₂ -- 1468.461468.46DD₃₃ -- 1468.461468.46DD₅₅ 0 0 == 2048.312048.31DD11 -- 3304.043304.04DD33 -- 3304.043304.04DD44

I

I

DD₁₁ D D₂₂ D D33 D D44 D D₅₅ 0 0 0 0 0 0 0 0

Y

Y

I

I

22004488..3311 00 --3304.043304.04 --33330044..0044 00 --3344..44117700 00 --22001133..8899 00 0 0 33003311..0033 --11446688..4466 00 --11446688..4466 00 --33002200..8833 00 --10.197610.1976 --3304.043304.04 --11446688..4466 707044886611 221111445588 114400997722 33330044..0044 00 00 11446688..4466 --33330044..0044 00 221111445588 442222991177 00 33330044..0044 00 00 00 0 0 --11446688..4466 114400997722 00 228811994444 00 00 00 11446688..4466 --3344..44117700 00 33330044..0044 33330044..0044 00 3344..44117700 00 00 00 0 0 --33002200..8833 00 00 00 00 33002200..8833 00 00 --22001133..8899 00 00 00 00 00 00 22001133..8899 00 0 0 --1100..11997766 11446688..4466 00 14146688..4466 00 00 00 1100..11997766

Y

Y

I

I

00 --13.7513.75 90 90 0 0 0 0 Q Q₆₆ Q Q₇₇ Q Q₈₈ Q Q₉₉

Y

Y

16–