1). Struktur Truss P1 10 ton P2 15 ton 6 m P3 5 ton 4 m 4.5 m 4.5 m 4 m Diketahui A = 250 = 0.00025 E = 215000 = 21500000 3 Perputa Searah 1 5 2 3 4 1 2 4 5 6 7 8 9 Mencari Panjang Batang Batang 1 = √ 8.5 ² + 6² = 10.4 m Batang 6 = 4 m Batang 2 = √ 4.5 ² + 6² = 7.5 m Batang 7 = 4.5 m Batang 3 = 6 m Batang 8 = 4.5 m Batang 4 = 7.5 m Batang 9 = 4 m Batang 5 = 10.4 m = 6 = 1.3 = 53.13 = 6 = 0.7 = 35. 4.5 8.5 mm 2 m 2 N/mm 2 t/m 2 α5 α4 α1 α2 β1 Mencari Nilai β1 dan β2 β1 tg -1 tg -1 β2 tg -1 tg -1
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No 11). Struktur TrussP110tonP215ton
6mP35ton4m4.5m4.5m4mDiketahuiA=250mm2=0.00025m2E=215000N/mm2=21500000t/m2354Perputaran sudutSearah jarum jam152341122415266789Mencari Panjang BatangBatang 1=8.5+6=10.4mBatang 6=4mBatang 2=4.5+6=7.5mBatang 7=4.5mBatang 3=6mBatang 8=4.5mBatang 4=7.5mBatang 9=4mBatang 5=10.4mMencari Nilai 1 dan 21=tg-16=tg-11.3=53.13010235422=tg-16=tg-10.7=35.21759296824.58.5Penentuan SudutElemenSudut (o)CSC2S2CSPanjang Batang (m)Ket :135.20.80.60.70.30.510.4C=Cos253.10.60.80.40.60.57.5S=Sin390.00.01.00.01.00.06.01=24306.90.6-0.80.40.6-0.57.52=15324.80.8-0.60.70.3-0.510.460.01.00.01.00.00.04.070.01.00.01.00.00.04.580.01.00.01.00.00.04.590.01.00.01.00.00.04.0Dearajat KebebasanNx=2JTK-2NS-NR=12-4-0=8Derajat KebebasanMencari Kekakuan Lokal elemenPersamaan dasar Kekakuan Lokal ElemenC2CS-'C2-'CS[K]=AECSS2-'CS-'S2L-'C2-'CSC2CS-'CS-'S2CSS2
Kekakuan Lokal Elemen I (Nodal 1-Nodal 3)u1v1u3v30.70.5-0.7-0.50.0640.045-0.064-0.045u1[Ke 1]=53750.50.3-0.5-0.3[Ke 1]=53750.0450.032-0.045-0.032v1s10.4-0.7-0.50.70.5-0.064-0.0450.0640.045u3-0.5-0.30.50.3-0.045-0.0320.0450.032v3
Kekakuan Lokal Elemen 2 (Nodal 2-Nodal 3)u2v2u3v30.40.5-0.4-0.50.0480.064-0.048-0.064u2[Ke 2]=53750.50.6-0.5-0.6[Ke 2]=53750.0640.085-0.064-0.085v2s7.5-0.4-0.50.40.5-0.048-0.0640.0480.064u3-0.5-0.60.50.6-0.064-0.0850.0640.085v3Kekakuan Lokal Elemen 3 (Nodal 4-Nodal 3)u4v4u3v30.00.0-0.0-0.00.0000.000-0.000-0.000u4[Ke 3]=53750.01.0-0.0-1.0[Ke 3]=53750.0000.167-0.000-0.167v4s6.0-0.0-0.00.00.0-0.000-0.0000.0000.000u3-0.0-1.00.01.0-0.000-0.1670.0000.167v3Kekakuan Lokal Elemen 4 (Nodal 3-Nodal 5)u3v3u5v50.4-0.5-0.40.50.048-0.064-0.0480.064u30.048-0.064[Ke 4]=5375-0.50.60.5-0.6[Ke 4]=5375-0.0640.0850.064-0.085v3S7.5-0.40.50.4-0.5-0.0480.0640.048-0.064u50.5-0.6-0.50.60.064-0.085-0.0640.085v5Kekakuan Lokal Elemen 5 (Nodal 3-Nodal 6)u3v3u6v60.7-0.5-0.70.50.064-0.045-0.0640.045u3[Ke 5]=5375-0.50.30.5-0.3[Ke 5]=5375-0.0450.0320.045-0.032v3S 10.4-0.70.50.7-0.5-0.0640.0450.064-0.045u60.5-0.3-0.50.30.045-0.032-0.0450.032v6Kekakuan Lokal Elemen 6 (Nodal 1-Nodal 2)u1v1u2v21.00.0-1.00.00.2500.000-0.2500.000u1[Ke 6]=53750.00.00.00.0[Ke 6]=53750.0000.0000.0000.000v1s4.0-1.00.01.00.0-0.2500.0000.2500.000u20.00.00.00.00.0000.0000.0000.000v2Kekakuan Lokal Elemen 7 (Nodal 2-Nodal 4)u2v2u4v40.2220.0000.0000.0000.0000.0000.0001.00.0-1.00.00.2220.000-0.2220.000u2[Ke 7]=53750.00.00.00.0[Ke 7]=53750.0000.0000.0000.000v2s4.5-1.00.01.00.0-0.2220.0000.2220.000u40.00.00.00.00.0000.0000.0000.000v4Kekakuan Lokal Elemen 8 (Nodal 4-Nodal 5)u4v4u5v51.00.0-1.00.00.2220.000-0.2220.000u4[Ke 8]=53750.00.00.00.0[Ke 8]=53750.0000.0000.0000.000v4s4.5-1.00.01.00.0-0.2220.0000.2220.000u50.00.00.00.00.0000.0000.0000.000v5Kekakuan Lokal Elemen 9 (Nodal 5-Nodal 6)u5v5u6v61.00.0-1.00.00.2500.000-0.2500.000u5[Ke 9]=53750.00.00.00.0[Ke 9]=53750.0000.0000.0000.000v54.0-1.00.01.00.0-0.2500.0000.2500.000u60.00.00.00.00.0000.0000.0000.000v6Mencari Matriks kekakuan Global Struktur[F]=[Kg][U]
u1v1u2v2u3v3u4v4u5v5u6v6u1v1u2v2u3v3u4v4u5v5u6v6F1=0.3140.045-0.2500.000-0.064-0.045000000u1U100.3140.045-0.2500.000-0.064-0.045000000u1G10.0450.0320.0000.000-0.045-0.032000000v1V100.0450.0320.0000.000-0.045-0.032000000v1F2-0.2500.0000.5200.064-0.048-0.064-0.2220.0000000u2U2-0.2500.0000.5200.064-0.048-0.064-0.2220.0000000u20.5200.064-0.048-0.064-0.2220.00000G20.0000.0000.0640.085-0.064-0.0850.0000.0000000v2V20.0000.0000.0640.085-0.064-0.0850.0000.0000000v20.0640.085-0.064-0.0850.0000.00000F3-0.064-0.045-0.048-0.0640.2240.000-0.000-0.000-0.0480.064-0.0640.045u3U3-0.064-0.045-0.048-0.0640.2240.000-0.000-0.000-0.0480.064-0.0640.045u3-0.048-0.0640.2240.000-0.000-0.000-0.0480.064G3-0.045-0.032-0.064-0.0850.0000.401-0.000-0.1670.064-0.0850.045-0.032v3V3-0.045-0.032-0.064-0.0850.0000.401-0.000-0.1670.064-0.0850.045-0.032v3-0.064-0.0850.0000.401-0.000-0.1670.064-0.0853333333F400-0.2220.000-0.000-0.0000.4440.000-0.2220.00000u4U400-0.2220.000-0.000-0.0000.4440.000-0.2220.00000u4-0.2220.000-0.000-0.0000.4440.000-0.22222222220G40000-0.000-0.1670.0000.1670.0000.00000v4V4000.0000.000-0.000-0.1670.0000.1670.0000.00000v40.0000.000-0.000-0.1670.0000.16700F50000-0.0480.064-0.2220.0000.520-0.064-0.2500.000u5U50000-0.0480.064-0.2220.0000.520-0.064-0.2500.000u50.0000.000-0.0480.064-0.2220.0000.5202222222-0.064G500000.064-0.0850.0000.000-0.0640.0850.0000.000v5V500000.064-0.0850.0000.000-0.0640.0850.0000.000v50.0000.0000.064-0.0850.0000.000-0.0640.0853333333F60000-0.0640.04500-0.2500.0000.314-0.045u6U600000-0.0640.04500-0.2500.0000.314-0.045u6G600000.045-0.032000.0000.000-0.0450.032v6V6000000.045-0.032000.0000.000-0.0450.032v6Karena pada nodal 1 dan 6 perletakan sendi, nilai U1;V1 dan U6;V6 = 0, maka matriks yang 3.06-2.29-0.000.002.00-0.000.940.71tersisa yaitu matriks 8 x 8-2.2933.475.8515.64-1.5015.64-0.7110.73-0.005.857.79-0.00-0.00-0.00-0.00-5.850=0.5200.064-0.048-0.064-0.2220.0000.0000.000U20.0015.64-0.0015.640.0015.640.0015.64-50.0640.085-0.064-0.0850.0000.0000.0000.000V22.00-1.50-0.000.004.25-0.002.001.5015-0.048-0.0640.2240.000-0.000-0.000-0.0480.064U3-0.0015.640.0015.64-0.0021.640.0015.6410-0.064-0.0850.0000.401-0.000-0.1670.064-0.085V30.94-0.71-0.000.002.000.003.062.290-0.2220.000-0.000-0.0000.4440.000-0.2220.000U40.7110.73-5.8515.641.5015.642.2933.4700.0000.000-0.000-0.1670.0000.1670.0000.000V400.0000.000-0.0480.064-0.2220.0000.520-0.064U500.0000.0000.064-0.0850.0000.000-0.0640.085V50-5Matriks di atas diubah menjadi matriks perpindahan1510[F]=[K][U][U]=[K]-1[F]000U2=3.06-2.29-0.000.002.00-0.000.940.7100V2-2.2933.475.8515.64-1.5015.64-0.7110.73-5U3-0.005.857.79-0.00-0.00-0.00-0.00-5.8515V310.0015.64-0.0015.640.0015.640.0015.6410U453752.00-1.50-0.000.004.25-0.002.001.500V4-0.0015.640.0015.64-0.0021.640.0015.640U50.94-0.71-0.000.002.000.003.062.290V50.7110.73-5.8515.641.5015.642.2933.470DAFTAR SUDUT-'C-'SCSU2=11.4710.0021m11.4711-0.8-0.60.80.6V276.7800.014376.7802-0.6-0.80.60.8U387.6850.016387.6853-0.0-1.00.01.0V3178.2130.014678.2134-0.60.80.6-0.8U453757.5000.00147.5005-0.80.60.8-0.6V478.2130.014678.2136-1.00.01.00.0U53.5290.00073.5297-1.00.01.00.0V515.0960.002815.0968-1.00.01.00.09-1.00.01.00.0A). Perhitungan Gaya - gaya Batang(S)S=AE[-C -S C S ] x ULElemen 1U1=00S1=5375-0.8-0.60.80.6V1=0010.4U3=0.01630.0163V3=0.01460.0146=516.6120311821x0.0217191062=11.220t0.0217191062Elemen 2U2=0.00210.0021S2=5375-0.6-0.80.60.8V2=0.01430.01437.5U3=0.01630.0163V3=0.01460.0146=716.6666666667x0.0087209302=6.250t0.0087209302Elemen 3U4=0.00140.0014S3=5375-0.0-1.00.01.0V4=0.01460.01466.0U3=0.01630.0163V3=0.01460.0146=895.8333333333x0=0.000t0Elemen 4U3=0.01630.0163S4=5375-0.60.80.6-0.8V3=0.01460.01467.5U5=0.00070.0007V5=0.00280.0028=716.6666666667x-0=-0.000t-0Elemen 5U3=0.01630.0163S5=5375-0.80.60.8-0.6V3=0.01460.014610.4U6=0.00000.0000U6=0.00000.0000=516.6120311821x-0.0049361605=-2.550t-0.0049361605Elemen 6U1=0.00000.0000S6=5375-1.00.01.00.0V1=0.00000.00004.0U2=0.00210.0021V2=0.01430.0143=1343.75x0.0021340629=2.868t0.0021340629Elemen 7U2=0.00210.0021S7=5375-1.00.01.00.0V2=0.01430.01434.5U4=0.00140.0014V4=0.01460.0146=1194.4444444444x-0.0007387141=-0.882t-0.0007387141Elemen 8U4=0.00140.0014S8=5375-1.00.01.00.0V4=0.01460.01464.5U5=0.00070.0007V5=0.00280.0028=1194.4444444444x-0.0007387141=-0.882t-0.0007387141Elemen 9U5=0.00070.0007S9=5375-1.00.01.00.0V5=0.00280.00284.0U6=0.00000.0000V6=0.00000.0000=1343.75x-0.0006566347=-0.882t-0.0006566347B). Reaksi Pada Tumpuan[F]=[K][U]
Matriks Tumpuan yaitu :U2u2v2u3v3u4v4u5v5V2F1=-0.2500.000-0.064-0.0450.0000.0000.0000.000u1U3G10.0000.000-0.045-0.0320.0000.0000.0000.000v1V3F60.0000.000-0.0640.0450.0000.000-0.2500.000u6U4G60.0000.0000.045-0.0320.0000.0000.0000.000v6V4U5V5
0.00210.01430.0021F1=-0.2500.000-0.064-0.0450.0000.0000.0000.0000.01630.0143G153750.0000.000-0.045-0.0320.0000.0000.0000.0000.01460.0163F610.0000.000-0.0640.0450.0000.000-0.2500.0000.00140.0146G60.0000.0000.045-0.0320.0000.0000.0000.0000.01460.0014-0.2500.000-0.064-0.04500000.00070.01460.0000.000-0.045-0.03200000.00280.00070.0000.000-0.0640.04500-0.2500.0000.00280.0000.0000.045-0.032000.0000.000
0.00210.0143F1=-1343.750-344.8057205811-243.392273351400000.0163G100-243.3922733514-171.80631060100000.0146F600-344.8057205811243.392273351400-1343.7500.0014G600243.3922733514-171.80631060100000.01460.00070.0028Reaksi yang terjadi pada TumpuanF1=-12.034ton-12.0343137255G1-6.471-6.4705882353-1343.80.0-344.8-243.40.00.00.00.0F6-2.966-2.96568627450.00.0-243.4-171.80.00.00.00.0G61.4711.47058823530.00.0-344.8243.40.00.0-1343.80.00.00.0243.4-171.80.00.00.00.0Kontrolv=0P1-P3+G1+G6=010-5+-6.471+1.471=05+-5=00.000=0OKC). Diagram Freebody1). Akibat Gaya Batang11.220t-2.550t
+11.220t+-6.25t-2.550t+2.9t--0.882t-0.882t
No 22). Struktur Portalq=3t/m
34m2342A=120cm28.0mI=500cm41E=2100T/cm24Ket:A,I,E sama untuk semua elemen15
6m6m
Portal 2-dimensi dengan pembebanan tergambar di atas, akan dianalisis dengan menggunakan5 titik nodal dan 4 elemen. Struktur berada pada bidang X-Y dan pusat sumbu global dipilih di nodal1. Untuk membedakan antara nomor titik nodal dan nomor elemen, maka nomor titik nodal ditulis da-lam lingkaran, sedangkan semua nomor elemen ditulis dalam segi empat.Beban Nodal Ekivalenq=0.025t/cm
34721.1102550928cmMomen PrimerM34=q x L2=0.0249615088x721.1102550928=-1081.7t.cm1212M43=1081.7t.cm
P12=q x L=0.03x721.1102550928=10.8ton22P21=10.8ton
Selanjutnya, masing - masing momen (M) dan reaksi (P) disebut M1 dan M2; P1 dan P2, sehingganilai momen dan reaksi tersebut diekivalenkan & dapat digambarkan ulang sebagai berikut.Mencari Nilai a M1-1081.7T.cm33.69P1-10.8T
M21081.7T.cmP2-10.8T
Mencari Panjang Batang Elemen 2 dan Elemen 3.Elemen 2=600+400=721.1cm7.2111025509mElemen 3=600+400=721.1cm7.211m1). Matriks Kekakuan elemenfi=AE/L00 AE/L00gi012EI/L36EI/L20 12EI/L36EI/L2mi06EI/L24EI/L0 6EI/L22EI/Lfj AE/L00AE/L00gj0 12EI/L3 6EI/L2012EI/L3 6EI/L2mj06EI/L22EI/L0 6EI/L24EI/L2). Matriks Transformasi[T]=cos asin a0000 sin acos a0000001000000cos asin a0000 sin acos a00000013). Penentuan SudutElemenSudut (o)CSLAEI190018001202100500233.690.830.55721.112021005003326.30.83-0.55721.112021005004270-0-18001202100500Elemen 1 dari Nodal 1 ke 2[Ke1]=315.000-315.000315.00.00.0-315.00.00.000.029.843750-0.029.843750.00.09.80.0-0.09.809.8437552500-9.8437526250.09.85250.00.0-9.82625.0-315.000315.000-315.00.00.0315.00.00.00-0.02-9.8437500.02-9.843750.0-0.025-9.80.00.0-9.809.8437526250-9.8437552500.09.82625.00.0-9.85250.0
Transformasi Koordinat Elemen 1(Untuk Sudut, Lih. Tabel Penentuan Sudut)[T]=010000010000-100000-100000001000001000000010000010000-100000-100000001000001
Matriks Transformasi Koordinat Elemen 1[T]T=0-100000-100001000001000000010000010000000-100000-10000100000100000001000001
Matriks Kekakuan Global Elemen 1[Kg]=[T]T.[Ke1].[T]
0-0-10-00-100-0-10-00-1031500-315-0031500-315-0001052500-10262501052500-102625-00100-010-00100-010-315-0-03150-0-315-0-03150-001026250-10525001026250-105250
U1V11U2V2200-10-0-0-10U10.0246093750-9.84375-0.024609375-0-9.843750315.000-0-315.000V103150-0-3150[Kg1]=-100525010-026251-9.84375052509.84375-02625-0-0100010U2-0.024609375-09.843750.02460937509.84375-0-315.00-00315.00-0V2-0-315-00315-0-100262510-052502-9.84375026259.84375-05250
Elemen 2 dari Nodal 2 ke 3[Ke2]=349.500-349.500349.50.00.0-349.50.00.000.0312.11538461540-0.0312.11538461540.00.012.10.0-0.012.1012.11538461545824.35206036490-12.11538461542912.17603018240.012.15824.40.0-12.12912.2-349.500349.500-349.50.00.0349.50.00.00-0.03-12.115384615400.03-12.11538461540.0-0.0-12.10.00.0-12.1012.11538461542912.17603018240-12.11538461545824.35206036490.012.12912.20.0-12.15824.4
Transformasi Koordinat Elemen 2(Untuk Sudut, Lih. Tabel Penentuan Sudut)[T]=0.80.60.00.00.00.00.80.60.00.00.00.0-0.60.80.00.00.00.0-0.60.80.00.00.00.00.00.01.00.00.00.00.00.01.00.00.00.00.00.00.00.80.60.00.00.00.00.80.60.00.00.00.0-0.60.80.00.00.00.0-0.60.80.00.00.00.00.00.01.00.00.00.00.00.01.0
Matriks Transformasi Koordinat Elemen 2[T]T=0.83-0.550.000.000.000.000.83-0.550.000.000.000.000.550.830.000.000.000.000.550.830.000.000.000.000.000.001.000.000.000.000.000.001.000.000.000.000.000.000.000.83-0.550.000.000.000.000.83-0.550.000.000.000.000.550.830.000.000.000.000.550.830.000.000.000.000.000.001.000.000.000.000.000.001.00
Matriks Kekakuan Global Elemen 2[Kg]=[T]T.[Ke2].[T]
291-0-7-2910-7291-0-7-2910-7194010-194-010194010-194-01001258240-12291201258240-122912-29107291-07-29107291-07-194-0-101940-10-194-0-101940-1001229120-12582401229120-125824
U2V22U3V33241.9161.3-6.7-241.9-161.3-6.7U2241.94161.27-6.72-241.94-161.27-6.72161.3107.510.1-161.3-107.510.1V2161.27107.5510.08-161.27-107.5510.08[Kg2]=-6.710.15824.46.7-10.12912.22-6.7210.085824.356.72-10.082912.18-241.9-161.36.7241.9161.36.7U3-241.94-161.276.72241.94161.276.72-161.3-107.5-10.1161.3107.5-10.1V3-161.27-107.55-10.08161.27107.55-10.08-6.710.12912.26.7-10.15824.43-6.7210.082912.186.72-10.085824.35
Elemen 3 dari Nodal 3 ke 4[Ke3]=349.500-349.500349.50.00.0-349.50.00.000.0312.11538461540-0.0312.11538461540.00.012.10.0-0.012.1012.11538461545824.35206036490-12.11538461542912.17603018240.012.15824.40.0-12.12912.2-349.500349.500-349.50.00.0349.50.00.00-0.03-12.115384615400.03-12.11538461540.0-0.0-12.10.00.0-12.1012.11538461542912.17603018240-12.11538461545824.35206036490.012.12912.20.0-12.15824.4
Transformasi Koordinat Elemen 3(Untuk Sudut, Lih. Tabel Penentuan Sudut)[T]=0.83-0.550.000.000.000.000.83-0.550.000.000.000.000.550.830.000.000.000.000.550.830.000.000.000.000.000.001.000.000.000.000.000.001.000.000.000.000.000.000.000.83-0.550.000.000.000.000.83-0.550.000.000.000.000.550.830.000.000.000.000.550.830.000.000.000.000.000.001.000.000.000.000.000.001.00
Matriks Transformasi Koordinat Elemen 3[T]T=0.830.550.000.000.000.000.830.550.000.000.000.00-0.550.830.000.000.000.00-0.550.830.000.000.000.000.000.001.000.000.000.000.000.001.000.000.000.000.000.000.000.830.550.000.000.000.000.830.550.000.000.000.00-0.550.830.000.000.000.00-0.550.830.000.000.000.000.000.001.000.000.000.000.000.001.00
Matriks Kekakuan Global Elemen 3[Kg]=[T]T.[Ke3].[T]
29107-291-0729107-291-07-194010194-010-194010194-01001258240-12291201258240-122912-291-0-72910-7-291-0-72910-7194-0-10-1940-10194-0-10-1940-1001229120-12582401229120-125824
U3V33U4V44242-1617-2421617U3241.94-161.276.72-241.94161.276.72-16110810161-10810V3-161.27107.5510.08161.27-107.5510.08[Kg3]=7105824-7-10291236.7210.085824.35-6.72-10.082912.18-242161-7242-161-7U4-241.94161.27-6.72241.94-161.27-6.72161-108-10-161108-10V4161.27-107.55-10.08-161.27107.55-10.087102912-7-10582446.7210.082912.18-6.72-10.085824.35
Elemen 4 dari Nodal 4 ke 5[Ke4]=315.000-315.000315.00.00.0-315.00.00.000.029.843750-0.029.843750.00.09.80.0-0.09.809.8437552500-9.8437526250.09.85250.00.0-9.82625.0-315.000315.000-315.00.00.0315.00.00.00-0.02-9.8437500.02-9.843750.0-0.0-9.80.00.0-9.809.8437526250-9.8437552500.09.82625.00.0-9.85250.0
Transformasi Koordinat Elemen 4(Untuk Sudut, Lih. Tabel Penentuan Sudut)[T]=-0.00-1.000.000.000.000.00-0.00-1.000.000.000.000.001.00-0.000.000.000.000.001.00-0.000.000.000.000.000.000.001.000.000.000.000.000.001.000.000.000.000.000.000.00-0.00-1.000.000.000.000.00-0.00-1.000.000.000.000.001.00-0.000.000.000.000.001.00-0.000.000.000.000.000.000.001.000.000.000.000.000.001.00
Matriks Transformasi Koordinat Elemen 4[T]T=-0.001.000.000.000.000.00-0.001.000.000.000.000.00-1.00-0.000.000.000.000.00-1.00-0.000.000.000.000.000.000.001.000.000.000.000.000.001.000.000.000.000.000.000.00-0.001.000.000.000.000.00-0.001.000.000.000.000.00-1.00-0.000.000.000.000.00-1.00-0.000.000.000.000.000.000.001.000.000.000.000.000.001.00
Matriks Kekakuan Global Elemen 4[Kg]=[T]T.[Ke4].[T]
-00100-010-00100-010-315-0-03150-0-315-0-03150-001052500-10262501052500-1026250-0-10-00-100-0-10-00-1031500-315-0031500-315-0001026250-10525001026250-105250
U4V44U5V550010-0-010U40.020.009.84-0.02-0.009.840315-0-0-315-0V40.00315.00-0.00-0.00-315.00-0.00[Kg4]=10-05250-100262549.84-0.005250.00-9.840.002625.00-0-0-1000-10U5-0.02-0.00-9.840.020.00-9.84-0-315003150V5-0.00-315.000.000.00315.000.0010-02625-100525059.84-0.002625.00-9.840.005250.00
4). Overall Stiffness MatriksU1V11U2V22U3V33U4V44U5V55u1v1o1u2v2o2u3v3o3u4v4o4u5v5o5F1=00-10-0-0-10000000000000-10-0-0-10000000000u1G103150-0-3150000000000003150-0-3150000000000v1M1-100525010-026250000000000-100525010-02625000000000o1F2-0-0102421613-241.9449631324-161.2742407342-6.7204062235000000U2-0-0102421613-242-161-7000000u2G2-0-315-016142310-161.2742407342-107.549762520610.0806093352000000V2-0-315-016142310-161-10810000000v2M2-1002625310110746.7204062235-10.08060933522912.17603018240000002-1002625310110747-102912000000o20.00000-242-1617483.8899262648013.440812447-241.9449631324161.27424073426.7000U3000-242-1617484013-2421617000u3-10.82000-161-108-100215.09952504120161.2742407342-107.549762520610.1000V3000-161-108-1002150161-10810000v3-1082000-710291213.440812447011648.7041207298-6.7204062235-10.08060933522912.20003000-710291213011649-7-102912000o30.00000000-241.9449631324161.2742407342-6.7204062235241.9695725074-161.27424073423.1-0.024609375-09.84375U4000000-242161-7242-1613-0-010u4-10.82000000161.2742407342-107.5497625206-10.0806093352-161.2742407342422.5497625206-10.1-0-315-0V4000000161-108-10-161423-10-0-315-0v410820000006.720406223510.08060933522912.17603018243.1233437765-10.080609335211074.4-9.8437502625400000071029123-1011074-1002625o4F5000000000-0.024609375-0-9.80.0246093750-9.843750000000000-0-0-1000-10u5G5000000000-0-3150.0031500000000000-0-315003150v5M50000000009.84375-02625.0-9.8437505250000000000010-02625-1005250o5RearrangementU2V22U3V33U4V44F2=2421613-242-161-7000U22421613-242-161-7000G216142310-161-10810000V216142310-161-10810000M2310110747-1029120002310110747-1029120000-242-1617484013-2421617U3-242-1617484013-2421617-10.8-161-108-1002150161-10810V3-161-108-1002150161-10810-1082-710291213011649-7-1029123-710291213011649-7-1029120000-242161-7242-1613U4000-242161-7242-1613-10.8000161-108-10-161423-10V4000161-108-10-161423-10108200071029123-1011074400071029123-10110745).Mencari Nilai Perpindahan[F]=[K][U][U]=[K]-1[F]
U2V22U3V33U4V44 1U2=2421613-242-161-7000F238.540.00-0.0334.406.210.0230.26-0.00-0.04V216142310-161-10810000G20.000.00-0.000.000.00-0.000.000.00-0.002310110747-102912000M2-0.03-0.000.00-0.040.01-0.00-0.040.000.00U3-242-1617484013-2421617034.400.00-0.0434.410.000.0234.40-0.00-0.04V3-161-108-1002150161-10810-10.826.210.000.010.009.310.00-6.210.00-0.013-710291213011649-7-102912-1081.70.02-0.00-0.000.020.000.000.020.00-0.00U4000-242161-7242-1613030.260.00-0.0434.40-6.210.0238.54-0.00-0.03V4000161-108-10-161423-10-10.82-0.000.000.00-0.000.000.00-0.000.000.00400071029123-10110741081.7-0.04-0.000.00-0.04-0.01-0.00-0.030.000.00U2V22U3V33U4V44U2=38.540.00-0.0334.406.210.0230.26-0.00-0.04F20V20.000.00-0.000.000.00-0.000.000.00-0.00G202-0.03-0.000.00-0.040.01-0.00-0.040.000.00M20U334.400.00-0.0434.410.000.0234.40-0.00-0.0400V36.210.000.010.009.310.00-6.210.00-0.01-11-1130.02-0.00-0.000.020.000.000.020.00-0.00-1081.7-1082U430.260.00-0.0434.40-6.210.0238.54-0.00-0.0300V4-0.000.000.00-0.000.000.00-0.000.000.00-11-114-0.04-0.000.00-0.04-0.01-0.00-0.030.000.0010821082U2=-129.100cm-129.0996280841V2-0.016-0.016177745320.0160.0159325622U3-57.100-57.1003760474V3-108.043-108.04260200393-0.164-0.1642945814U414.87314.8734698476V4-0.052-0.052499421940.2700.26963283346). Reaksi Pada tumpuan[F]=[K][U]
-129.100-129.0996280841-0.016-0.0161777453F1=-0-0-100000000.0160.0159325622-0-0-10000000G1-0-3150000000-57.100-57.1003760474-0-3150000000M110-02625000000-108.043-108.042602003910-02625000000F5000000-0-0-10-0.164-0.1642945814000000-0.024609375-0-9.84375G5000000-0-315014.87314.8734698476000000-0-3150M500000010-02625-0.052-0.05249942190000009.84375-026250.2700.2696328334F1=3.023.0202250004G15.105.0959897618M1-1229.00121.633-1229.0014880937F5-3.02-374.805-3.0202250004G516.5416.537317891M5854.20854.1969063798
7). Gaya BatangElemen 1f1=g1m1[T].[Kg1].[U]f2g2m2
01000000-10-0-0-10001000000-10-0-0-100-10000003150-0-31500-10000003150-0-31500001000-100525010-026250001000-100525010-026250000010-0-0100010-129.100000010-0-0100010-129.0996280841000-100-0-315-00315-0-0.016000-100-0-315-00315-0-0.0161777453000001-100262510-052500.016000001-100262510-052500.0159325622f1=5.0959897618t03150-0-31505.0959897618g1-3.0202250004t-0.02460937509.843750.024609375-09.84375-3.0202250004m1-1229.0014880937t.cm-9.84375052509.84375-02625-1229.0014880937f2-5.0959897618t-0-315003150-5.0959897618g23.0202250004t0.024609375-0-9.84375-0.0246093750-9.843753.0202250004m2-1187.1785122346t.cm-9.84375026259.84375-05250-1187.1785122346Elemen 2f2=g2m2[T].[Kg2].[U]f3g3m30.80.60000242161-7-242-161-7-129.1000.80.60.00.00.00.0242161-7-242-161-7-129.0996280841-0.60.8000016110810-161-10810-0.016-0.60.80.00.00.00.016110810-161-10810-0.0161777453001000-71058247-1029120.0160.00.01.00.00.00.0-71058247-1029120.01593256220000.80.60-242-16172421617-57.1000.00.00.00.80.60.0-242-16172421617-57.1003760474000-0.60.80-161-108-10161108-10-108.0430.00.00.0-0.60.80.0-161-108-10161108-10-108.0426020039000001-71029127-105824-0.1640.00.00.00.00.01.0-71029127-105824-0.1642945814f2=5.3397256214t290.7692307692193.84615384620-290.7692307692-193.846153846205.3397256214g22.5648003809t-0.01863905330.027958579912.11538461540.0186390533-0.027958579912.11538461542.5648003809m21187.1785122346t.cm-6.720406223510.08060933525824.35206036496.7204062235-10.08060933522912.17603018241187.1785122346f3-5.3397256214t-290.7692307692-193.84615384620290.7692307692193.84615384620-5.3397256214g3-2.5648003809t0.0186390533-0.0279585799-12.1153846154-0.01863905330.0279585799-12.1153846154-2.5648003809m3662.3253446647t.cm-6.720406223510.08060933522912.17603018246.7204062235-10.08060933525824.3520603649662.3253446647Elemen 3f3=g3m3[T].[Kg3].[U]f4g4m4
0.83-0.60000242-1617-2421617-57.1000.8-0.60.00.00.00.0242-1617-2421617-57.10037604740.60.830000-16110810161-10810-108.0430.60.80.00.00.00.0-16110810161-10810-108.04260200390010007105824-7-102912-0.1640.00.01.00.00.00.07105824-7-102912-0.16429458140000.83-0.60-242161-7242-161-714.8730.00.00.00.8-0.60.0-242161-7242-161-714.87346984760000.60.830161-108-10-161108-10-0.0520.00.00.00.60.80.0161-108-10-161108-10-0.05249942190000017102912-7-1058240.2700.00.00.00.00.01.07102912-7-1058240.2696328334
f3=5.6862325797t290.7692307692-193.8461538462-0-290.7692307692193.8461538462-05.6862325797g3-3.0845608184t0.01863905330.027958579912.1153846154-0.0186390533-0.027958579912.1153846154-3.0845608184m3-1744.0t.cm6.720406223510.08060933525824.3520603649-6.7204062235-10.08060933522912.1760301824-1743.9907273039f4-5.6862325797t-290.7692307692193.84615384620290.7692307692-193.84615384620-5.6862325797g43.0845608184t-0.0186390533-0.0279585799-12.11538461540.01863905330.0279585799-12.11538461543.0845608184m4-480.3177113077t.cm6.720406223510.08060933522912.1760301824-6.7204062235-10.08060933525824.3520603649-480.3177113077Elemen 4f4=g4m4[T].[Kg4].[U]f5g5m5
-0.00-1.000000010-0-01014.873-0.0-1.00.00.00.00.00010-0-01014.87346984761.0-0.0000000315-0-0-315-0-0.0521.0-0.00.00.00.00.00315-0-0-315-0-0.052499421900100010-05250-10026250.2700.00.01.00.00.00.010-05250-10026250.2696328334000-0.00-1.00-0-0-1000-1000.00.00.0-0.0-1.00.0-0-0-1000-1000001.0-0.000-0-31500315000.00.00.01.0-0.00.0-0-315003150000000110-02625-100525000.00.00.00.00.01.010-02625-10052500
f4=16.537317891t-0-31500315016.537317891g43.0202250004t0.024609375-09.84375-0.02460937509.843753.0202250004m41562.0t.cm9.84375-05250-9.84375026251561.9830939468f5-16.537317891t03150-0-3150-16.537317891g5-3.0202250004t-0.0246093750-9.843750.024609375-0-9.84375-3.0202250004m5854.1969063798t.cm9.84375-02625-9.8437505250854.1969063798
Percobaan2). Struktur Portal32m23P=10ton42q=2t/mA=100cm26mI=5000cm41E=2100T/cm24Ket:A,I,E sama untuk semua elemen15
6m10m
Portal 2-dimensi dengan pembebanan tergambar di atas, akan dianalisis dengan menggunakan5 titik nodal dan 4 elemen. Struktur berada pada bidang X-Y dan pusat sumbu global dipilih di nodal1. Untuk membedakan antara nomor titik nodal dan nomor elemen, maka nomor titik nodal ditulis da-lam lingkaran, sedangkan semua nomor elemen ditulis dalam segi empat.Beban Nodal Ekivalenq=0.02t/cm
12600cmMomen PrimerM12=q x L2=0.02x600=-600.0t.cm1212M21=600.0t.cm
P12=q x L=0.02x600=6.0ton5022P21=6.0ton
Selanjutnya, masing - masing momen (M) dan reaksi (P) disebut M1 dan M2; P1 dan P2, sehingganilai momen dan reaksi tersebut diekivalenkan & dapat digambarkan ulang sebagai berikut.M2600T.cmP26TP310TP16TM1-600T.cmMencari Panjang Batang Elemen 2 dan Elemen 3.Elemen 2=600+200=632.5cmElemen 3=1000+200=1019.8cm1). Matriks Kekakuan elemenfi=AE/L00 AE/L00gi012EI/L36EI/L20 12EI/L36EI/L2mi06EI/L24EI/L0 6EI/L22EI/Lfj AE/L00AE/L00gj0 12EI/L3 6EI/L2012EI/L3 6EI/L2mj06EI/L22EI/L0 6EI/L24EI/L2). Matriks Transformasi[T]=cos asin a0000 sin acos a0000001000000cos asin a0000 sin acos a00000013). Penentuan SudutElemenSudut (o)CSLAEI1900160010021005000218.40.950.32632.5100210050003348.70.98-0.201019.8100210050004270-0-160010021005000Elemen 1 dari Nodal 1 ke 2[Ke1]=420.000-420.000420.00.00.0-420.00.00.001.0082520-1.0082520.01.0252.00.0-1.0252.00252-840000-252420000.0252.0-84000.00.0-252.042000.0-420.000420.000-420.00.00.0420.00.00.00-1.01-25201.01-2520.0-1.008-252.00.01.0-252.00252420000-252840000.0252.042000.00.0-252.084000.0
Transformasi Koordinat Elemen 1(Untuk Sudut, Lih. Tabel Penentuan Sudut)[T]=0.80.600000.80.60.00.00.00.0-0.60.80000-0.60.80.00.00.00.00010000.00.01.00.00.00.00000.80.600.00.00.00.80.60.0000-0.60.800.00.00.0-0.60.80.00000010.00.00.00.00.01.0
Matriks Transformasi Koordinat Elemen 1[T]T=0.8-0.60.00.00.00.00.8-0.60.00.00.00.00.60.80.00.00.00.00.60.80.00.00.00.00.00.01.00.00.00.00.00.01.00.00.00.00.00.00.00.8-0.60.00.00.00.00.8-0.60.00.00.00.00.60.80.00.00.00.00.60.80.00.00.00.00.00.01.00.00.00.00.00.01.0
Matriks Kekakuan Global Elemen 1[Kg]=[T]T.[Ke1].[T]
336-1-151-3361-151336-1-151-3361-1512521202-252-12022521202-252-12020252-840000-252420000252-840000-25242000-3361151336-1151-3361151336-1151-252-1-2022521-202-252-1-2022521-2020252420000-252840000252420000-25284000
U1V11U2V22269201-151-269-201-151U1269.16288201.11616-151.2-269.16288-201.11616-151.2201151.85202-201-151.85202V1201.11616151.84512201.6-201.11616-151.84512201.6[Kg1]=-151202-84000151-202420001-151.2201.6-84000151.2-201.642000-269-201151269201151U2-269.16288-201.11616151.2269.16288201.11616151.2-201-151.85-202201151.85-202V2-201.11616-151.84512-201.6201.11616151.84512-201.6-15120242000151-202840002-151.2201.642000151.2-201.684000
Elemen 2 dari Nodal 2 ke 3[Ke2]=332.000-332.000332.00.00.0-332.00.00.000.50157.50-0.50157.50.00.5157.50.0-0.5157.50157.566407.8308635360-157.533203.9154317680.0157.566407.80.0-157.533203.9-332.000332.000-332.00.00.0332.00.00.00-0.50-157.500.50-157.50.0-0.5-157.50.00.5-157.50157.533203.9154317680-157.566407.8308635360.0157.533203.90.0-157.566407.8
Transformasi Koordinat Elemen 2(Untuk Sudut, Lih. Tabel Penentuan Sudut)[T]=0.90.30.00.00.00.00.90.30.00.00.00.0-0.30.90.00.00.00.0-0.30.90.00.00.00.00.00.01.00.00.00.00.00.01.00.00.00.00.00.00.00.90.30.00.00.00.00.90.30.00.00.00.0-0.30.90.00.00.00.0-0.30.90.00.00.00.00.00.01.00.00.00.00.00.01.0
Matriks Transformasi Koordinat Elemen 2[T]T=0.95-0.320.000.000.000.000.95-0.320.000.000.000.000.320.950.000.000.000.000.320.950.000.000.000.000.000.001.000.000.000.000.000.001.000.000.000.000.000.000.000.95-0.320.000.000.000.000.95-0.320.000.000.000.000.320.950.000.000.000.000.320.950.000.000.000.000.000.001.000.000.000.000.000.001.00
Matriks Kekakuan Global Elemen 2[Kg]=[T]T.[Ke2].[T]
315-1050-3151050315-1050-315105000158-0-015800158-0-01585014966408-50-149332045014966408-50-14933204-3151050315-1050-3151050315-1050-0-0-15800-158-0-0-15800-1585014933204-50-149664085014933204-50-14966408
U2V22U3V3333200-33200U2332.040.000.00-332.040.000.00001580-0158V20.000.50157.500.00-0.50157.50[Kg2]=0158664080-1583320420.00157.5066407.830.00-157.5033203.92-3320033200U3-332.040.000.00332.040.000.000-0-15800-158V30.00-0.50-157.500.000.50-157.500158332040-1586640830.00157.5033203.920.00-157.5066407.83
Elemen 3 dari Nodal 3 ke 4[Ke3]=205.900-205.900205.90.00.0-205.90.00.000.1260.57692307690-0.1260.57692307690.00.160.60.0-0.160.6060.576923076941184.38837901860-60.576923076920592.19418950930.060.641184.40.0-60.620592.2-205.900205.900-205.90.00.0205.90.00.00-0.12-60.576923076900.12-60.57692307690.0-0.1-60.60.00.1-60.6060.576923076920592.19418950930-60.576923076941184.38837901860.060.620592.20.0-60.641184.4
Transformasi Koordinat Elemen 3(Untuk Sudut, Lih. Tabel Penentuan Sudut)[T]=0.98-0.200.000.000.000.000.98-0.200.000.000.000.000.200.980.000.000.000.000.200.980.000.000.000.000.000.001.000.000.000.000.000.001.000.000.000.000.000.000.000.98-0.200.000.000.000.000.98-0.200.000.000.000.000.200.980.000.000.000.000.200.980.000.000.000.000.000.001.000.000.000.000.000.001.00
Matriks Transformasi Koordinat Elemen 3[T]T=0.980.200.000.000.000.000.980.200.000.000.000.00-0.200.980.000.000.000.00-0.200.980.000.000.000.000.000.001.000.000.000.000.000.001.000.000.000.000.000.000.000.980.200.000.000.000.000.980.200.000.000.000.00-0.200.980.000.000.000.00-0.200.980.000.000.000.000.000.001.000.000.000.000.000.001.00
Matriks Kekakuan Global Elemen 3[Kg]=[T]T.[Ke3].[T]
202400-202-400202400-202-400-00610-061-00610-061-12594118412-5920592-12594118412-5920592-202-400202400-202-4002024000-0-61-00-610-0-61-00-61-12592059212-5941184-12592059212-5941184
U3V33U4V4420600-20600U3205.920.000.00-205.920.000.0000610-061V30.000.1260.580.00-0.1260.58[Kg3]=061411840-612059230.0060.5841184.390.00-60.5820592.19-2060020600U4-205.920.000.00205.920.000.000-0-6100-61V40.00-0.12-60.580.000.12-60.58061205920-614118440.0060.5820592.190.00-60.5841184.39
Elemen 4 dari Nodal 4 ke 5[Ke4]=350.000-350.000350.00.00.0-350.00.00.000.581750-0.581750.00.6175.00.0-0.6175.00175700000-175350000.0175.070000.00.0-175.035000.0-350.000350.000-350.00.00.0350.00.00.00-0.58-17500.58-1750.0-0.6-175.00.00.6-175.00175350000-175700000.0175.035000.00.0-175.070000.0
Transformasi Koordinat Elemen 4(Untuk Sudut, Lih. Tabel Penentuan Sudut)[T]=-0.00-1.000.000.000.000.00-0.00-1.000.000.000.000.001.00-0.000.000.000.000.001.00-0.000.000.000.000.000.000.001.000.000.000.000.000.001.000.000.000.000.000.000.00-0.00-1.000.000.000.000.00-0.00-1.000.000.000.000.001.00-0.000.000.000.000.001.00-0.000.000.000.000.000.000.001.000.000.000.000.000.001.00
Matriks Transformasi Koordinat Elemen 4[T]T=-0.001.000.000.000.000.00-0.001.000.000.000.000.00-1.00-0.000.000.000.000.00-1.00-0.000.000.000.000.000.000.001.000.000.000.000.000.001.000.000.000.000.000.000.00-0.001.000.000.000.000.00-0.001.000.000.000.000.00-1.00-0.000.000.000.000.00-1.00-0.000.000.000.000.000.000.001.000.000.000.000.000.001.00
Matriks Kekakuan Global Elemen 4[Kg]=[T]T.[Ke4].[T]
-035000-3500-035000-3500-1-017510175-1-017510175-175-070000175035000-175-0700001750350000-3500-035000-3500-0350010-175-1-0-17510-175-1-0-175-175-035000175070000-175-035000175070000
U4V44U5V5535000-35000U4350.000.000.00-350.000.000.00011750-1175V40.000.58175.000.00-0.58175.00[Kg4]=0175700000-1753500040.00175.0070000.000.00-175.0035000.00-3500035000U5-350.000.000.00350.000.000.000-1-17501-175V50.00-0.58-175.000.000.58-175.000175350000-1757000050.00175.0035000.000.00-175.0070000.004). Overall Stiffness MatriksU1V11U2V22U3V33U4V44U5V55u1v1o1u2v2o2u3v3o3u4v4o4u5v5o56.0=269201-151-269-201-1510000000000269201-151-269-201-151000000000u10201152202-201-1522020000000000201152202-201-152202000000000v1-600-151202-84000151-202420000000000000-151202-84000151-20242000000000000o16-269-201151601201151-332.039154317700000000U2-269-201151601201151-33200000000u20-201-152-202201152-440-0.4980587315157.5000000V2-201-152-202201152-440-0158000000v2600-15120242000151-441504080-157.533203.9154317680000002-15120242000151-441504080-15833204000000o20000-33200537.961096212800-205.921941895100000U3000-3320053800-20600000u3-100000-0-15800.6168598518-96.92307692310-0.118801120360.5769230769000V30000-0-15801-970-061000v300000158332040-96.9230769231107592.2192425550-60.576923076920592.194189509300030000158332040-971075920-6120592000o3F4000000-205.921941895100555.921941895100-35000U4000000-2060055600-35000u4G40000000-0.1188011203-60.576923076900.7021344537114.42307692310-0.5833333333175V40000000-0-61011140-1175v4M4000000060.576923076920592.19418950930114.4230769231111184.3883790190-1753500040000000612059201141111840-17535000o4F5000000000-35000350000000000000-3500035000u5G50000000000-0.5833333333-17500.5833333333-17500000000000-1-17501-175v5M50000000000175350000-1757000000000000000175350000-17570000o5RearrangementU2V22U3V33U4V446=601201151-33200000U2601201151-332000000201152-440-0158000V2201152-440-0158000600151-441504080-158332040002151-441504080-158332040000-3320053800-20600U3-3320053800-20600-100-0-15801-970-061V30-0-15801-970-06100158332040-971075920-612059230158332040-971075920-6120592F4000-2060055600U4000-2060055600G40000-0-6101114V40000-0-6101114M400006120592011411118440000612059201141111845).Mencari Nilai Perpindahan[F]=[K][U][U]=[K]-1[F]
U2V22U3V33U4V44 1U2=601201151-3320000060.01-0.01-0.000.01-0.020.000.00-0.010.00V2201152-440-01580000-0.010.030.00-0.010.04-0.00-0.000.01-0.002151-441504080-15833204000600-0.000.000.00-0.000.000.00-0.000.00-0.00U3-3320053800-2060000.01-0.01-0.000.01-0.020.000.00-0.010.00V30-0-15801-970-061-10-0.020.040.00-0.024.100.00-0.011.93-0.0130158332040-971075920-612059200.00-0.000.000.000.000.000.000.00-0.00U4000-2060055600F40.00-0.00-0.000.00-0.010.000.00-0.000.00V40000-0-6101114G4-0.010.010.00-0.011.930.00-0.002.85-0.004000061205920114111184M40.00-0.00-0.000.00-0.01-0.000.00-0.000.00U2V22U3V33U4V44U2=0.01-0.01-0.000.01-0.020.000.00-0.010.0066V2-0.010.030.00-0.010.04-0.00-0.000.01-0.00002-0.000.000.00-0.000.000.00-0.000.00-0.00600600U30.01-0.01-0.000.01-0.020.000.00-0.010.0000V3-0.020.040.00-0.024.100.00-0.011.93-0.01-10-1030.00-0.000.000.000.000.000.000.00-0.0000U40.00-0.00-0.000.00-0.010.000.00-0.000.00F40V4-0.010.010.00-0.011.930.00-0.002.85-0.00G4040.00-0.00-0.000.00-0.01-0.000.00-0.000.00M40U2=0.288086899cm0.288086899V2-0.4687532933-0.46875329332-0.0273119967-0.0273119967U30.20718940110.2071894011V3-39.1883964544-39.18839645443-0.0460326744-0.0460326744U40.07674610510.0767461051V4-18.5885238299-18.588523829940.04900672130.04900672136). Reaksi Pada tumpuan[F]=[K][U]
0.2880868990.288086899-0.4687532933-0.4687532933F1=-269-201-151000000-0.0273119967-0.0273119967-269-201-151000000G1-201-1522020000000.20718940110.2071894011-201-152202000000M1151-20242000000000-39.1883964544-39.1883964544151-20242000000000F5000000-35000-0.0460326744-0.0460326744000000-35000G50000000-1-1750.07674610510.07674610510000000-0.5833333333-175M5000000017535000-18.5885238299-18.58852382990000000175350000.04900672130.0490067213F1=20.8611368003G17.7328706683M1-1009.0444565821F5-26.8611368003G52.2671293317M5-1537.7564230943
Percobaan 22). Struktur Portal32m23P=10ton42q=2t/mA=100cm26mI=5000cm41E=2100T/cm24Ket:A,I,E sama untuk semua elemen15
6m10m
Portal 2-dimensi dengan pembebanan tergambar di atas, akan dianalisis dengan menggunakan5 titik nodal dan 4 elemen. Struktur berada pada bidang X-Y dan pusat sumbu global dipilih di nodal1. Untuk membedakan antara nomor titik nodal dan nomor elemen, maka nomor titik nodal ditulis da-lam lingkaran, sedangkan semua nomor elemen ditulis dalam segi empat.Beban Nodal Ekivalenq=0.02t/cm
12600cmMomen PrimerM12=q x L2=0.02x600=-600.0t.cm1212M21=600.0t.cm
P12=q x L=0.02x600=6.0ton22P21=6.0ton
Selanjutnya, masing - masing momen (M) dan reaksi (P) disebut M1 dan M2; P1 dan P2, sehingganilai momen dan reaksi tersebut diekivalenkan & dapat digambarkan ulang sebagai berikut.M2600T.cmP26TP310TP16TM1-600T.cmMencari Panjang Batang Elemen 2 dan Elemen 3.Elemen 2=600+200=632.5cmElemen 3=1000+200=1019.8cm1). Matriks Kekakuan elemenfi=AE/L00 AE/L00gi012EI/L36EI/L20 12EI/L36EI/L2mi06EI/L24EI/L0 6EI/L22EI/Lfj AE/L00AE/L00gj0 12EI/L3 6EI/L2012EI/L3 6EI/L2mj06EI/L22EI/L0 6EI/L24EI/L2). Matriks Transformasi[T]=cos asin a0000 sin acos a0000001000000cos asin a0000 sin acos a00000013). Penentuan SudutElemenSudut (o)CSLAEI1900160010021005000218.40.950.32632.5100210050003348.70.98-0.201019.8100210050004270-0-160010021005000Elemen 1 dari Nodal 1 ke 2[Ke1]=350.000-350.000350.00.00.0-350.00.00.000.581750-0.581750.00.6175.00.0-0.6175.00175-700000-175350000.0175.0-70000.00.0-175.035000.0-350.000350.000-350.00.00.0350.00.00.00-0.58-17500.58-1750.0-0.583-175.00.00.6-175.00175350000-175-700000.0175.035000.00.0-175.0-70000.0
Transformasi Koordinat Elemen 1(Untuk Sudut, Lih. Tabel Penentuan Sudut)[T]=010000010000-100000-100000001000001000000010000010000-100000-100000001000001
Matriks Transformasi Koordinat Elemen 1[T]T=0-100000-100001000001000000010000010000000-100000-10000100000100000001000001
Matriks Kekakuan Global Elemen 1[Kg]=[T]T.[Ke1].[T]
0-1-175-01-1750-1-175-01-17535000-350-0035000-350-000175-700000-175350000175-700000-17535000-011750-1175-011750-1175-350-0-03500-0-350-0-03500-00175350000-175-700000175350000-175-70000
U1V11U2V2210-175-1-0-175U10.58333333330-175-0.5833333333-0-1750350.000-0-350.000V103500-0-3500[Kg1]=-1750-70000175-0350001-1750-70000175-035000-1-017510175U2-0.5833333333-01750.58333333330175-0-350.00-00350.00-0V2-0-350-00350-0-175035000175-0-700002-175035000175-0-70000
Elemen 2 dari Nodal 2 ke 3[Ke2]=332.000-332.000332.00.00.0-332.00.00.000.50157.50-0.50157.50.00.5157.50.0-0.5157.50157.5-66407.8308635360-157.533203.9154317680.0157.5-66407.80.0-157.533203.9-332.000332.000-332.00.00.0332.00.00.00-0.50-157.500.50-157.50.0-0.5-157.50.00.5-157.50157.533203.9154317680-157.5-66407.8308635360.0157.533203.90.0-157.5-66407.8
Transformasi Koordinat Elemen 2(Untuk Sudut, Lih. Tabel Penentuan Sudut)[T]=0.90.30.00.00.00.00.90.30.00.00.00.0-0.30.90.00.00.00.0-0.30.90.00.00.00.00.00.01.00.00.00.00.00.01.00.00.00.00.00.00.00.90.30.00.00.00.00.90.30.00.00.00.0-0.30.90.00.00.00.0-0.30.90.00.00.00.00.00.01.00.00.00.00.00.01.0
Matriks Transformasi Koordinat Elemen 2[T]T=0.95-0.320.000.000.000.000.95-0.320.000.000.000.000.320.950.000.000.000.000.320.950.000.000.000.000.000.001.000.000.000.000.000.001.000.000.000.000.000.000.000.95-0.320.000.000.000.000.95-0.320.000.000.000.000.320.950.000.000.000.000.320.950.000.000.000.000.000.001.000.000.000.000.000.001.00
Matriks Kekakuan Global Elemen 2[Kg]=[T]T.[Ke2].[T]
315-0-50-3150-50315-0-50-3150-501050149-105-01491050149-105-01490158-664080-158332040158-664080-15833204-315050315-050-315050315-050-105-0-1491050-149-105-0-1491050-1490158332040-158-664080158332040-158-66408
U2V22U3V33298.999.5-49.8-298.9-99.5-49.8U2298.8999.46-49.81-298.89-99.46-49.8199.533.7149.4-99.5-33.7149.4V299.4633.65149.42-99.46-33.65149.42[Kg2]=-49.8149.4-66407.849.8-149.433203.92-49.81149.42-66407.8349.81-149.4233203.92-298.9-99.549.8298.999.549.8U3-298.89-99.4649.81298.8999.4649.81-99.5-33.7-149.499.533.7-149.4V3-99.46-33.65-149.4299.4633.65-149.42-49.8149.433203.949.8-149.4-66407.83-49.81149.4233203.9249.81-149.42-66407.83
Elemen 3 dari Nodal 3 ke 4[Ke3]=205.900-205.900205.90.00.0-205.90.00.000.1260.57692307690-0.1260.57692307690.00.160.60.0-0.160.6060.5769230769-41184.38837901860-60.576923076920592.19418950930.060.6-41184.40.0-60.620592.2-205.900205.900-205.90.00.0205.90.00.00-0.12-60.576923076900.12-60.57692307690.0-0.1-60.60.00.1-60.6060.576923076920592.19418950930-60.5769230769-41184.38837901860.060.620592.20.0-60.6-41184.4
Transformasi Koordinat Elemen 3(Untuk Sudut, Lih. Tabel Penentuan Sudut)[T]=0.98-0.200.000.000.000.000.98-0.200.000.000.000.000.200.980.000.000.000.000.200.980.000.000.000.000.000.001.000.000.000.000.000.001.000.000.000.000.000.000.000.98-0.200.000.000.000.000.98-0.200.000.000.000.000.200.980.000.000.000.000.200.980.000.000.000.000.000.001.000.000.000.000.000.001.00
Matriks Transformasi Koordinat Elemen 3[T]T=0.980.200.000.000.000.000.980.200.000.000.000.00-0.200.980.000.000.000.00-0.200.980.000.000.000.000.000.001.000.000.000.000.000.001.000.000.000.000.000.000.000.980.200.000.000.000.000.980.200.000.000.000.00-0.200.980.000.000.000.00-0.200.980.000.000.000.000.000.001.000.000.000.000.000.001.00
Matriks Kekakuan Global Elemen 3[Kg]=[T]T.[Ke3].[T]
202012-202-012202012-202-012-4005940-059-4005940-059061-411840-6120592061-411840-6120592-202-0-122020-12-202-0-122020-1240-0-59-400-5940-0-59-400-59061205920-61-41184061205920-61-41184
U3V33U4V44198-4012-1984012U3198.01-39.5811.88-198.0139.5811.88-4085940-859V3-39.588.0359.4039.58-8.0359.40[Kg3]=1259-41184-12-5920592311.8859.40-41184.39-11.88-59.4020592.19-19840-12198-40-12U4-198.0139.58-11.88198.01-39.58-11.8840-8-59-408-59V439.58-8.03-59.40-39.588.03-59.40125920592-12-59-41184411.8859.4020592.19-11.88-59.40-41184.39
Elemen 4 dari Nodal 4 ke 5[Ke4]=350.000-350.000350.00.00.0-350.00.00.000.581750-0.581750.00.6175.00.0-0.6175.00175-700000-175350000.0175.0-70000.00.0-175.035000.0-350.000350.000-350.00.00.0350.00.00.00-0.58-17500.58-1750.0-0.6-175.00.00.6-175.00175350000-175-700000.0175.035000.00.0-175.0-70000.0
Transformasi Koordinat Elemen 4(Untuk Sudut, Lih. Tabel Penentuan Sudut)[T]=-0.00-1.000.000.000.000.00-0.00-1.000.000.000.000.001.00-0.000.000.000.000.001.00-0.000.000.000.000.000.000.001.000.000.000.000.000.001.000.000.000.000.000.000.00-0.00-1.000.000.000.000.00-0.00-1.000.000.000.000.001.00-0.000.000.000.000.001.00-0.000.000.000.000.000.000.001.000.000.000.000.000.001.00
Matriks Transformasi Koordinat Elemen 4[T]T=-0.001.000.000.000.000.00-0.001.000.000.000.000.00-1.00-0.000.000.000.000.00-1.00-0.000.000.000.000.000.000.001.000.000.000.000.000.001.000.000.000.000.000.000.00-0.001.000.000.000.000.00-0.001.000.000.000.000.00-1.00-0.000.000.000.000.00-1.00-0.000.000.000.000.000.000.001.000.000.000.000.000.001.00
Matriks Kekakuan Global Elemen 4[Kg]=[T]T.[Ke4].[T]
-011750-1175-011750-1175-350-0-03500-0-350-0-03500-00175-700000-175350000175-700000-175350000-1-175-01-1750-1-175-01-17535000-350-0035000-350-000175350000-175-700000175350000-175-70000
U4V44U5V5510175-1-0175U40.580.00175.00-0.58-0.00175.000350-0-0-350-0V40.00350.00-0.00-0.00-350.00-0.00[Kg4]=175-0-70000-1750350004175.00-0.00-70000.00-175.000.0035000.00-1-0-17510-175U5-0.58-0.00-175.000.580.00-175.00-0-350003500V5-0.00-350.000.000.00350.000.00175-035000-1750-700005175.00-0.0035000.00-175.000.00-70000.004). Overall Stiffness MatriksU1V11U2V22U3V33U4V44U5V55u1v1o1u2v2o2u3v3o3u4v4o4u5v5o56.0=10-175-1-0-175000000000010-175-1-0-175000000000u1003500-0-3500000000000003500-0-3500000000000v1-600-1750-70000175-0350000000000000-1750-70000175-035000000000000o16-1-017529999125-298.8850447591-99.4623286759-49.8058731477000000U2-1-017529999125-299-99-50000000u20-0-350-099384149-99.4623286759-33.6521682901149.417619443000000V2-0-350-099384149-99-34149000000v2600-175035000125149-13640849.8058731477-149.41761944333203.9154317680000002-175035000125149-13640850-14933204000000o20000-299-9950496.891481239759.884801603861.6859851801-198.006436480739.577527072111.8801120324000U3000-299-99504976062-1984012000u3-10000-99-34-14959.884801603841.6864748248-90.017059280939.5775270721-8.034306534759.400560162000V3000-99-34-1496042-9040-859000v30000-501493320461.6859851801-90.0170592809-107592.219242555-11.8801120324-59.40056016220592.19418950930003000-501493320462-90-107592-12-5920592000o3F4000000-198.006436480739.5775270721-11.8801120324198.589769814-39.5775270721163.1198879676-0.5833333333-0175U4000000-19840-12199-40163-1-0175u4G400000039.5775270721-8.0343065347-59.400560162-39.5775270721358.0343065347-59.400560162-0-350-0V400000040-8-59-40358-59-0-350-0v4M400000011.880112032459.40056016220592.1941895093163.1198879676-59.400560162-111184.388379019-1750350004000000125920592163-59-111184-175035000o4F5000000000-0.5833333333-0-1750.58333333330-1750000000000-1-0-17510-175u5G5000000000-0-3500035000000000000-0-350003500v5M5000000000175-035000-1750-700000000000000175-035000-1750-70000o5RearrangementU2V22U3V33U4V446=29999125-299-99-50000U229999125-299-99-50000099384149-99-34149000V299384149-99-34149000600125149-13640850-149332040002125149-13640850-149332040000-299-99504976062-1984012U3-299-99504976062-1984012-10-99-34-1496042-9040-859V3-99-34-1496042-9040-8590-501493320462-90-107592-12-59205923-501493320462-90-107592-12-5920592F4000-19840-12199-40163U4000-19840-12199-40163G400040-8-59-40358-59V400040-8-59-40358-59M4000125920592163-59-1111844000125920592163-59-1111845).Mencari Nilai Perpindahan[F]=[K][U][U]=[K]-1[F]
U2V22U3V33U4V44 1U2=29999125-299-99-5000060.74-0.000.000.610.39-0.000.530.000.00V299384149-99-341490000-0.000.000.000.000.000.00-0.000.000.002125149-13640850-149332040006000.000.00-0.000.00-0.00-0.000.00-0.000.00U3-299-99504976062-198401200.610.000.000.590.070.000.57-0.000.00V3-99-34-1496042-9040-859-100.390.00-0.000.070.99-0.00-0.130.000.003-501493320462-90-107592-12-59205920-0.000.00-0.000.00-0.00-0.000.00-0.00-0.00U4000-19840-12199-40163F40.53-0.000.000.57-0.130.000.600.000.00V400040-8-59-40358-59G40.000.00-0.00-0.000.00-0.000.000.00-0.004000125920592163-59-111184M40.000.000.000.000.00-0.000.00-0.00-0.00U2V22U3V33U4V44U2=0.74-0.000.000.610.39-0.000.530.000.0066V2-0.000.000.000.000.000.00-0.000.000.000020.000.00-0.000.00-0.00-0.000.00-0.000.00600600U30.610.000.000.590.070.000.57-0.000.0000V30.390.00-0.000.070.99-0.00-0.130.000.00-10-103-0.000.00-0.000.00-0.00-0.000.00-0.00-0.0000U40.53-0.000.000.57-0.130.000.600.000.00F40V40.000.00-0.00-0.000.00-0.000.000.00-0.00G4040.000.000.000.000.00-0.000.00-0.00-0.00M40U2=0.7644306676cm0.7644306676V2-0.0265290063-0.026529006320.00929694110.0092969411U33.44310549813.4431054981V3-8.1259993422-8.125999342230.0117691810.011769181U45.04813168895.0481316889V4-0.0020424222-0.002042422240.00561356530.00561356536). Reaksi Pada tumpuan[F]=[K][U]
0.76443066760.7644306676-0.0265290063-0.0265290063F1=-1-0-1750000000.00929694110.0092969411-1-0-175000000G1-0-35000000003.44310549813.4431054981-0-3500000000M1175-035000000000-8.1259993422-8.1259993422175-035000000000F5000000-1-0-1750.0117691810.011769181000000-0.5833333333-0-175G5000000-0-35005.04813168895.0481316889000000-0-3500M5000000175-035000-0.0020424222-0.0020424222000000175-0350000.00561356530.0056135653F1=-2.0729ton-2.0728825879G19.28529.2851522162M1459.1683459.1683065173F5-3.9271-3.9271174121G50.71480.7148477838M51079.8981079.89783094137). Gaya BatangElemen 1f1=g1m1[T].[Kg1].[U]f2g2m201000010-175-1-0-175001000010-175-1-0-1750-10000003500-0-35000-10000003500-0-35000001000-1750-70000175-0350000001000-1750-70000175-0350000000010-1-0175101750.7644306676000010-1-0175101750.7644306676000-100-0-350-00350-0-0.0265290063000-100-0-350-00350-0-0.0265290063000001-175035000175-0-700000.0092969411000001-175035000175-0-700000.0092969411f1=9.2851522162t03500-0-35009.2851522162g12.0728825879t-0.583333333301750.5833333333-01752.0728825879m1459.1683065173t.cm-1750-70000175-035000459.1683065173f2-9.2851522162t-0-350003500-9.2851522162g2-2.0728825879t0.5833333333-0-175-0.58333333330-175-2.0728825879m2-517.0105125659t.cm-175035000175-0-70000-517.0105125659Elemen 2f2=g2m2[T].[Kg2].[U]f3g3m30.90.3000029999-50-299-99-500.76443066760.90.30.00.00.00.029999-50-299-99-500.7644306676-0.30.900009934149-99-34149-0.0265290063-0.30.90.00.00.00.09934149-99-34149-0.0265290063001000-50149-6640850-149332040.00929694110.00.01.00.00.00.0-50149-6640850-149332040.00929694110000.90.30-299-995029999503.44310549810.00.00.00.90.30.0-299-995029999503.4431054981000-0.30.90-99-34-1499934-149-8.12599934220.00.00.0-0.30.90.0-99-34-1499934-149-8.1259993422000001-501493320450-149-664080.0117691810.00.00.00.00.01.0-501493320450-149-664080.011769181f2=6.6618136408t3151050-315-10506.6618136408g27.5668052612t-0.15750.4725157.50.1575-0.4725157.57.5668052612m21117.0105125659t.cm-49.8058731477149.417619443-66407.83086353649.8058731477-149.41761944333203.9154317681117.0105125659f3-6.6618136408t-315-10503151050-6.6618136408g3-7.5668052612t0.1575-0.4725-157.5-0.15750.4725-157.5-7.5668052612m3870.7463787218t.cm-49.8058731477149.41761944333203.91543176849.8058731477-149.417619443-66407.830863536870.7463787218Elemen 3f3=g3m3[T].[Kg3].[U]f4g4m40.98-0.20000198-4012-19840123.44310549811.0-0.20.00.00.00.0198-4012-19840123.44310549810.20.980000-4085940-859-8.12599934220.21.00.00.00.00.0-4085940-859-8.12599934220010001259-41184-12-59205920.0117691810.00.01.00.00.00.01259-41184-12-59205920.0117691810000.98-0.20-19840-12198-40-125.04813168890.00.00.01.0-0.20.0-19840-12198-40-125.04813168890000.20.98040-8-59-408-59-0.00204242220.00.00.00.21.00.040-8-59-408-59-0.0020424222000001125920592-12-59-411840.00561356530.00.00.00.00.01.0125920592-12-59-411840.0056135653f3=3.9910486301t201.9230769231-40.38461538460-201.923076923140.384615384603.9910486301g30.0692051662t0.02329881660.116494082860.5769230769-0.0232988166-0.116494082860.57692307690.0692051662m3-870.7463787218t.cm11.880112032459.400560162-41184.3883790186-11.8801120324-59.40056016220592.1941895093-870.7463787218f4-3.9910486301t-201.923076923140.38461538460201.9230769231-40.38461538460-3.9910486301g4-0.0692051662t-0.0232988166-0.1164940828-60.57692307690.02329881660.1164940828-60.5769230769-0.0692051662m4-490.473474776t.cm11.880112032459.40056016220592.1941895093-11.8801120324-59.400560162-41184.3883790186-490.473474776Elemen 4f4=g4m4[T].[Kg4].[U]f5g5m5-0.00-1.0000010175-1-01755.0481316889-0.0-1.00.00.00.00.010175-1-01755.04813168891.0-0.0000000350-0-0-350-0-0.00204242221.0-0.00.00.00.00.00350-0-0-350-0-0.0020424222001000175-0-70000-1750350000.00561356530.00.01.00.00.00.0175-0-70000-1750350000.0056135653000-0.00-1.00-1-0-17510-17500.00.00.0-0.0-1.00.0-1-0-17510-17500001.0-0.000-0-35000350000.00.00.01.0-0.00.0-0-3500035000000001175-035000-1750-7000000.00.00.00.00.01.0175-035000-1750-700000f4=0.7148477838t-0-3500035000.7148477838g43.9271174121t0.5833333333-0175-0.583333333301753.9271174121m4490.473474776t.cm175-0-70000-175035000490.473474776f5-0.7148477838t03500-0-3500-0.7148477838g5-3.9271174121t-0.58333333330-1750.5833333333-0-175-3.9271174121m51079.8978309413t.cm175-035000-1750-700001079.8978309413
Sheet3
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