Penyelesaian Contoh Soal Latihan
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Analisa Struktur Portal dengan metode matrik E = 2100 t/cm2 = A1 = 100 cm2 α = I1 = 5000 cm4 sin36,87 = L1 = 500 cm cos36,87 = A2 = 40 cm2 sin90 = I2 = 1000 cm4 cos90 = L2 = 300 cm L = 400 cm P = 4 t M = 5 tm Elemen 1 Kekakuan elemen dalam koodinat global : f1 = (AE/L) 0 0 -(AE/L) g1 = 0 0 m1 = 0 (4EI/L) 0 f2 = -(AE/L) 0 0 (AE/L) g2 = 0 0 m2 = 0 (2EI/L) 0 Matrik kekakuan Elemen 1 pada koordinat global : = 420 0 0 -420 0 1.008 252 0 0 252 84000 0 sinα (12EI/L 3 ) (6EI/L 2 ) (6EI/L 2 ) -(12EI/L 3 ) -(6EI/L 2 ) (6EI/L 2 ) [ Kl (1) ]
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Transcript of Penyelesaian Contoh Soal Latihan
Contoh2Analisa Struktur Portal dengan metode matrikE=2100t/cm2sin=(3/5)A1=100cm2=36.87I1=5000cm4sin36,87=0.6L1=500cmcos36,87=0.8A2=40cm2sin90=1I2=1000cm4cos90=0L2=300cmL=400cmP=4tM=5tmElemen 1Kekakuan elemen dalam koodinat global :f1=(AE/L)00-(AE/L)00U1g1=0(12EI/L3)(6EI/L2)0-(12EI/L3)(6EI/L2)V1m1=0(6EI/L2)(4EI/L)0-(6EI/L2)(2EI/L)1f2=-(AE/L)00(AE/L)00U2g2=0-(12EI/L3)-(6EI/L2)0(12EI/L3)-(6EI/L2)V2m2=0(6EI/L2)(2EI/L)0-(6EI/L2)(4EI/L)2Matrik kekakuan Elemen 1 pada koordinat global :[ Kl(1) ]=42000-42000U101.0082520-1.008252V10252840000-252420001-4200042000U20-1.008-25201.008-252V20252420000-252840002Matrik Transformasi :[ T(1) ]=0.80.60000U1-0.60.80000V100100010000.80.60U2000-0.60.80V20000012[T(1)]T=0.8-0.60000U10.60.80000V100100010000.8-0.60U20000.60.80V20000012[T(1)]T[Kl(1)]=336-0.6048-151.2-3360.6048-151.2U12520.8064201.6-252-0.8064201.6V10252840000-252420001-3360.6048151.2336-0.6048151.2U2-252-0.8064-201.62520.8064-201.6V20252420000-252840002[T(1)]T [Kl(1)] [T(1)]U1V11U2V22[Kg(1)]=269.2201.1-151.2-269.2-201.1-151.2U1201.1151.8201.6-201.1-151.8201.6V1-151.2201.684000151.2-201.6420001-269.2-201.1151.2269.2201.1151.2U2-201.1-151.8-201.6201.1151.8-201.6V2-151.2201.642000151.2-201.6840002Elemen 2Kekakuan elemen dalam koodinat global :f2=(AE/L)00-(AE/L)00U2g20(12EI/L3)(6EI/L2)0-(12EI/L3)(6EI/L2)V2m20(6EI/L2)(4EI/L)0-(6EI/L2)(2EI/L)2f3-(AE/L)00(AE/L)00U3g30-(12EI/L3)-(6EI/L2)0(12EI/L3)-(6EI/L2)V3m30(6EI/L2)(2EI/L)0-(6EI/L2)(4EI/L)3Matrik kekakuan Elemen 2 pada koordinat global :[ Kl(2) ]=28000-28000U200.93331400-0.9333140V20140280000-140140002-2800028000U30-0.9333-14000.9333-140V30140140000-140280003Matrik Transformasi :[ T(2) ]=0-10000U2100000V200100020000-10U3000100V30000013[T(2)]T=010000U2-100000V20010002000010U3000-100V30000013[T(2)]T[Kl(2)]=00.93331400-0.9333140U2-2800028000V20140280000-1401400020-0.9333-14000.9333-140U328000-28000V30140140000-140280003[T(2)]T [Kl(2)] [T(2)]U2V22U3V33[Kg(2)]=0.93330140-0.93330140U2028000-2800V2140028000-1400140002-0.93330-1400.93330-140U30-280002800V3140014000-1400280003Overall/Structural Stiffness Matrik :U1V11U2V22U3V33F1=269.2201.1-151.2-269.2-201.1-151.2000U1G1201.1151.8201.6-201.1-151.8201.6000V1M1-151.2201.684000151.2-201.6420000001F2-269.2-201.1151.2270.1201.1291.2-0.93330140U2G2-201.1-151.8-201.6201.1431.8-201.60-2800V2M2-151.2201.642000291.2-201.6112000-1400140002F3000-0.93330-140.00.93330-140U3G30000-280002800V3M3000140014000-1400280003Batas Gaya dan Displacement:U1V11U2V22U3V33F1=269.2201.1-151.2-269.2-201.1-151.20000G1201.1151.8201.6-201.1-151.8201.60000M1-151.2201.684000151.2-201.642000.00000-4-269.2-201.1151.2270.1201.1291.2-0.93330140U20-201.1-151.8-201.6201.1431.8-201.60-2800V2500-151.2201.642000291.2-201.6112000-1400140002F3000-0.93330-1400.93330-1400G30000-2800028000M3000140014000-1400280000Matrik Kekakuan Struktur [K] diatur sesuai derajat kebebasan displacement:U2V22U1V11U3V33-4=270.1201.1291.2-269.2-201.1151.2-0.93330140U20201.1431.8-201.6-201.1-151.8-201.60-2800V2500291-201.6112000-151.2201.642000-1400140002F1-269.2-201.1-151.2269.2201.1-151.20000G1-201.1-151.8201.6201.1151.8201.60000M1151.2-201.642000-151.2201.6840000000F3-0.93330-1400000.93330-1400G30-2800000028000M3140014000000-1400280000Displacement :{ Fe }[ K ]{ U }U2V22-4=270.1201.1291.2U20201.1431.8-201.6V2500291-201.61120002{ U }[ K ]-1{ Fe }U2=-4V202500{ U }[ K ]-1{ Fe }U2=V22Gaya-gaya dalam setiap elemen :Elemen 1{ Fe }[T(1)] [Kg(1)]{U}f1=0U1g10V1m101f20.00000U2g20.00000V2m20.000002{ Fe }[T(1)] [Kg(1)] {U}f1=tg1tm1tmf2tg2tm2tmElemen 2{ Fe }[T(2)] [Kg(2)]{U}f2=0.00000U2g20.00000V2m20.000002f30U3g30V3m303{ Fe }[T(2)] [Kg(2)] {U}f2=tg2tm2tmf3tg3tm3tmHasil analisis:
Contoh1Penyelesaian contoh soal 1.Matrik kekakuan Elemen 1 pada koordinat global :[ Kl(1) ]=AE/20010-100000-10100000Matrik Transformasi :[ T(1) ]=0.8660.500-0.50.86600000.8660.500-0.50.866[T(1)]T=0.866-0.5000.50.86600000.866-0.5000.50.866[T(1)]T[Kl(1)]=AE/2000.8660-0.86600.50-0.50-0.86600.8660-0.500.50[T(1)]T [Kl(1)] [T(1)]u1v1u2v2[Kg(1)]=AE/2000.7500.433-0.750-0.433u10.4330.250-0.433-0.250v1-0.750-0.4330.7500.433u2-0.433-0.2500.4330.250v2x4/4[T(1)]T [Kl(1)] [T(1)]u1v1u2v2[Kg(1)]=AE/8003.0001.732-3.000-1.732u11.7321.000-1.732-1.000v1-3.000-1.7323.0001.732u2-1.732-1.0001.7321.000v2Matrik kekakuan Elemen 2 pada koordinat global :[ Kl(2) ]=AE/20010-100000-10100000Matrik Transformasi :[ T(2) ]=0.866-0.5000.50.86600000.866-0.5000.50.866[T(2)]T=0.8660.500-0.50.86600000.8660.500-0.50.866[T(2)]T[Kl(2)]=AE/2000.8660-0.8660-0.500.50-0.86600.86600.50-0.50[T(1)]T [Kl(1)] [T(1)]u2v2u3v3[Kg(2)]=AE/2000.750-0.433-0.7500.433u2-0.4330.2500.433-0.250v2-0.7500.4330.750-0.433u30.433-0.250-0.4330.250v3x4/4[T(1)]T [Kl(1)] [T(1)]u2v2u3v3[Kg(2)]=AE/8003.000-1.732-3.0001.732u2-1.7321.0001.732-1.000v2-3.0001.7323.000-1.732u31.732-1.000-1.7321.000v3Matrik kekakuan Elemen 3 pada koordinat global :[ Kl(3) ]=AE/200310-100000-10100000Matrik Transformasi :[ T(3) ]=1000010000100001[T(3)]T=1000010000100001[T(3)]T[Kl(3)]=AE/200310-100000-10100000[T(1)]T [Kl(1)] [T(1)]u1v1u3v3[Kg(3)]=AE/200310-10u10000v1-1010u30000v3x4/4[T(1)]T [Kl(1)] [T(1)]u1v1u3v3[Kg(3)]=AE/8002.3090.000-2.3090.000u10.0000.0000.0000.000v1-2.3090.0002.3090.000u30.0000.0000.0000.000v3Overall/Structural Stiffness Matrik :u1v1u2v2u3v3F1=AE/8005.3091.732-3.000-1.732-2.3090.000u1G11.7321.000-1.732-1.0000.0000.000v1F2-3.000-1.7326.0000.000-3.0001.732u2G2-1.732-1.0000.0002.0001.732-1.000v2F3-2.3090.000-3.0001.7325.309-1.732u3G30.0000.0001.732-1.000-1.7321.000v3Batas Gaya dan Displacement:u1v1u2v2u3v3F1=AE/8005.3091.732-3.000-1.732-2.3090.0000G11.7321.000-1.732-1.0000.0000.00005-3.000-1.7326.0000.000-3.0001.732u20-1.732-1.0000.0002.0001.732-1.000v20-2.3090.000-3.0001.7325.309-1.732u3G30.0000.0001.732-1.000-1.7321.0000Matrik Kekakuan Struktur [K] diatur sesuai derajat kebebasan displacement:u2v2u3u1v1v35=AE/8006.0000.000-3.000-3.000-1.7321.732u200.0002.0001.732-1.732-1.000-1.000v20-3.0001.7325.309-2.3090.000-1.732u3F1-3.000-1.732-2.3095.3091.7320.0000G1-1.732-1.0000.0001.7321.0000.0000G31.732-1.000-1.7320.0000.0001.0000Displacement{ Fe }=[ K ]{ U }U2v2u35AE/8006.0000.000-3.000U200.0002.0001.732V20-3.0001.7325.309U3{ U }=[ K ]{ Fe }U2800/AE6.0000.000-3.0005V20.0002.0001.7320U3-3.0001.7325.3090{ U }=[ K ]-1{ Fe }U2800/AE0.275-0.1870.2175V2-0.1870.825-0.3750U30.217-0.3750.4330U2=800/AE1.375=0.0001496198V2=-0.937=-0.0001020348U3=1.083=0.0001178231Reaksi Perletakan{ Fe }=[ K ]{ U }F1AE/800-3.000-1.732-2.309800/AE1.375G1-1.732-1.0000.000-0.937G31.732-1.000-1.7321.083F1=-5.000tG1=-1.443tG3=1.443tGaya-gaya dalam setiap elemen:Elemen 1{ Fe }=[ T(1) ][ Kg(1) ]{ U }{ U }f1AE/8003.4642.000-3.464-2.000800/AE00g10.0000.0000.0000.00000f2-3.464-2.0003.4642.0001.375=g20.0000.0000.0000.000-0.937={ Fe }=[T(1)][Kg(1)]{U(1)}f1-2.887tg10.000tf22.887tg20.000tElemen 2{ Fe }=[ T(2) ][ Kg(2) ]{ U }f2AE/8003.464-2.000-3.4642.000800/AE1.375g20.0000.0000.0000.000-0.937f3-3.4642.0003.464-2.0001.083g30.0000.0000.0000.0000.000{ Fe }=[T(2)][Kg(2)]{U(2)}f12.887tg10.000tf2-2.887tg20.000tElemen 3{ Fe }=[ T(3) ][ Kg(3) ]{ U }f2AE/8002.3090.000-2.3090.000800/AE0.000g20.0000.0000.0000.0000.000f3-2.3090.0002.3090.0001.083g30.0000.0000.0000.0000.000{ Fe }=[T(3)][Kg(3)]{U(3)}f1-2.500tg10.000tf32.500tg30.000tHasil analisis:
Contoh 1E =2,100t/cm2A =35.00cm21. GLOBLAL STIFFNESS MATRIXElemen Batang 1L =200cma =30o367.500.00-367.500.00[Kl] =0.000.000.000.00-367.500.00367.500.000.000.000.000.000.870.500.000.00[T] =-0.500.870.000.000.000.000.870.500.000.00-0.500.870.87-0.500.000.00[T]T =0.500.870.000.000.000.000.87-0.500.000.000.500.87[Kg]1 =[T]T [Kl] [T]0.87-0.500.000.00367.500.00-367.500.000.870.500.000.00[Kg]1 =0.500.870.000.000.000.000.000.00-0.500.870.000.000.000.000.87-0.50-367.500.00367.500.000.000.000.870.500.000.000.500.870.000.000.000.000.000.00-0.500.87318.260.00-318.260.000.870.500.000.00[Kg]1 =183.750.00-183.750.00-0.500.870.000.00-318.260.00318.260.000.000.000.870.50-183.750.00183.750.000.000.00-0.500.87u1v1u2v2275.63159.13-275.63-159.13u1[Kg]1 =159.1391.88-159.13-91.88v1-275.63-159.13275.63159.13u2-159.13-91.88159.1391.88v2Elemen Batang 2L =200cma =330o367.500.00-367.500.00[Kl] =0.000.000.000.00-367.500.00367.500.000.000.000.000.000.87-0.500.000.00[T] =0.500.870.000.000.000.000.87-0.500.000.000.500.870.870.500.000.00[T]T =-0.500.870.000.000.000.000.870.500.000.00-0.500.87[Kg]2 =[T]T [Kl] [T]0.870.500.000.00367.500.00-367.500.000.87-0.500.000.00[Kg]2 =-0.500.870.000.000.000.000.000.000.500.870.000.000.000.000.870.50-367.500.00367.500.000.000.000.87-0.500.000.00-0.500.870.000.000.000.000.000.000.500.87318.260.00-318.260.000.87-0.500.000.00[Kg]2 =-183.750.00183.750.000.500.870.000.00-318.260.00318.260.000.000.000.87-0.50183.750.00-183.750.000.000.000.500.87u2v2u3v3275.63-159.13-275.63159.13u2[Kg]2 =-159.1391.88159.13-91.88v2-275.63159.13275.63-159.13u3159.13-91.88-159.1391.88v3Elemen Batang 3L =346.4102cma =0o212.180.00-212.180.00[Kl] =0.000.000.000.00-212.180.00212.180.000.000.000.000.001000[T] =0100001000011000[T]T =010000100001[Kg]3 =[T]T [Kl] [T]1000212.180.00-212.180.001000[Kg]3 =01000.000.000.000.0001000010-212.180.00212.180.00001000010.000.000.000.000001212.180.00-212.180.001000[Kg]3 =0.000.000.000.000100-212.180.00212.180.0000100.000.000.000.000001u1v1u3v3212.180.00-212.180.00u1[Kg]3 =0.000.000.000.00v1-212.180.00212.180.00u30.000.000.000.00v32. OVERALL STIFFNESS MATRIXu1v1u2v2u3v3F1=487.80159.13-275.63-159.13-212.180.000u1G1159.1391.88-159.13-91.880.000.000v1F2 = 5-275.63-159.13551.250.00-275.63159.13u2u2G2 = 0-159.13-91.880.00183.75159.13-91.88v2v2F3 = 0-212.180.00-275.63159.13487.80-159.13u3u3G3 = 00.000.00159.13-91.88-159.1391.880v33. REARRAGEMENTu2v2u3u1v1v35=551.250.00-275.63-275.63-159.13159.13u2u200.00183.75159.13-159.13-91.88-91.88v2v20-275.63159.13487.80-212.180.00-159.13u3u3F1-275.63-159.13-212.18487.80159.130.000u1G1-159.13-91.880.00159.1391.880.000v1G3159.13-91.88-159.130.000.0091.880v34. UNKNOWN DISPLACEMENT AND REACTIONSDisplacement5=551.250.00-275.63u200.00183.75159.13v20-275.63159.13487.80u3551.250.00-275.6350.00183.75159.130Disusun seperti ini-275.63159.13487.8001.00000.0000-0.50000.009070.0000183.7500159.13220.00000Kolom pertama diusahakan menjadi satu1.0000-0.5774-1.76980.00000dapat dilakukan dengan cara membagi dengan kolom pertama1.00000.0000-0.50000.00907Usahakan kolom pertama menjadi nol0.0000183.7500159.13220.00000baris ke 2 dikurangi baris ke 10.0000-0.5774-1.2698-0.00907baris ke 3 dikurangi baris ke 11.00000.0000-0.50000.009070.00001.00000.86600.000000.00001.00002.19940.015711.00000.0000-0.50000.009070.00001.00000.86600.000000.00000.00001.33330.01571u3 =0.011783cmv2 =-0.0102040.000000=-0.010204cmu2 =0.0058910.0000000.009070=0.014962cmu2=0.014962v2-0.010204u30.011783Reaksi PerletakanF1=-275.63-159.13-212.180.01496=-5.000tonG1-159.13-91.880.00-0.01020-1.443tonG3159.13-91.88-159.130.011781.443ton5. MEMBER FORCESElemen Batang 1f1[T] [Kg] {u}g1f2g2f10.870.500.000.00275.63159.13-275.63-159.130g1-0.500.870.000.00159.1391.88-159.13-91.880f20.000.000.870.50-275.63-159.13275.63159.130.014962g20.000.00-0.500.87-159.13-91.88159.1391.88-0.010204f1318.26183.75-318.26-183.750g10.000.000.000.000f2-318.26-183.75318.26183.750.014962g20.000.000.000.00-0.010204f1-2.887tong10.000tonf22.887tong20.000tonElemen Batang 2f2[T] [Kg] {u}g2f3g3f20.87-0.500.000.00275.63-159.13-275.63159.130.014962g20.500.870.000.00-159.1391.88159.13-91.88-0.010204f30.000.000.87-0.50-275.63159.13275.63-159.130.011783g30.000.000.500.87159.13-91.88-159.1391.880f2318.26-183.75-318.26183.750.01496162g2-0.000.000.000.00-0.01020408f3-318.26183.75318.26-183.750.01178266g30.000.000.000.000f22.887tong2-0.000tonf3-2.887tong30.000tonElemen Batang 3f1[T] [Kg] {u}g1f3g3f11000212.180.00-212.180.000g101000.000.000.000.000f30010-212.180.00212.180.000.011783g300010.000.000.000.000f1212.180.00-212.180.000g10.000.000.000.000f3-212.180.00212.180.000.011783g30.000.000.000.000f1-2.500tong10.000tonf32.500tong30.000ton
1235 ton346,4102 cm200 cm200 cm100 cm30oYX123
Sheet3
