Umum Dengan metode Heft 540 sebagai control Grafik dan perbandingan

59 BAB IV HASIL DAN PEMBAHASAN

4.1 Umum

Model matriks yang akan dianalisa : 3000 mm 3000 mm 500 mm 400 kN 500 mm 500 mm 500 mm 500 mm 500 mm 500 mm 500 mm 500 mm 500 mm 500 mm 500 mm 500 mm 400 kN Gambar 4.1 model balok tinggi Data-data yang diperlukan adalah : 5. Model struktur bangunan adalah balok tinggi h= L dengan panjang 3 meter, lebar 3 meter dan tebal 0,5 meter. 6. Beban yang bekerja adalah beban vertikal statis ekivalen sebesar 400 kN yang bekerja pada balok dengan perletakan sederhana sendi-rol. 7. Menganalisa tengangan yang terjadi akibat beban terpusat. Analisa struktur yang dilakukan adalah dengan finite element method untuk dua dimensi. Universitas Sumatera Utara 60 8. Sebagai perbandingan dari nilai tegangan yang diperoleh dengan metode elemen hingga akan dikontrol dengan metode Heft 540.

4.2 Elemen Segitiga

3000 mm 500 mm 400 kN 500 mm 500 mm 500 mm 500 mm 500 mm 500 mm 500 mm 500 mm 500 mm 500 mm 500 mm 1 2 3 4 5 7 8 9 10 11 6 12 13 14 400 kN 15 16 17 18 19 21 22 23 24 25 20 26 27 28 29 30 31 32 33 35 36 37 38 39 34 40 41 42 43 44 45 46 47 48 49 Gambar 4.2 Diskretisasi dan penomoran balok tinggi Universitas Sumatera Utara 61 Gambar 4.3 Penomoran bidang segitiga Data-data yang telah diketahui : Beban yang bekerja P = 400 kN Panjang L = 3 meter Lebar h = 3meter Mutu beton Rc = k350 Modulus Elastisitas beton E = 47000 �350�0,083 = 25332,0844 ��� 2 = 25332084.4 kNm 2 Poissons ratio v = 0.3 Perletakan = Sendi - rol tebal t = 0.5 meter 1 2 3 4 21 22 23 24 5 6 7 8 9 10 11 12 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 13 14 15 16 17 18 19 20 Universitas Sumatera Utara 62

4.2.1 Langkah Pertama : Diskretisasi Perjanjian Tanda dan Penefenisian Persamaan Matriks Global

3000 mm 500 mm 400 kN 500 mm 500 mm 500 mm 500 mm 500 mm 500 mm 500 mm 500 mm 500 mm 500 mm 500 mm 1 2 3 4 5 7 8 9 10 11 6 12 13 14 400 kN 15 16 17 18 19 21 22 23 24 25 20 26 27 28 29 30 31 32 33 35 36 37 38 39 34 40 41 42 43 44 45 46 47 48 49 1 2 3 4 21 22 23 24 5 6 7 8 9 10 11 12 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 13 14 15 16 17 18 19 20 Universitas Sumatera Utara 63 Persamaan matriks global : F1x R1x d1x d1x F1y R1y d1y d1y F2x R2x d2x d2x F2y R2y d2y d2y F3x R3x d3x d3x F3y R3y d3y d3y F4x R4x d4x d4x F4y R4y d4y d4y F5x R5x d5x d5x F5y R5y d5y d5y F6x R6x d6x d6x F6y R6y d6y d6y F7x R7x d7x d7x F7y R7y d7y d7y F8x R8x d8x d8x F8y R8y d8y d8y F9x R9x d9x d9x F9y R9y d9y d9y F10x R10x d10x d10x F10y R10y d10y d10y F11x R11x d11x d11x F11y R11y d11y d11y F12x R12x d12x d12x F12y R12y d12y d12y F13x R13x d13x d13x F13y R13y d13y d13y F14x R14x d14x d14x F14y R14y d14y d14y F15x R15x d15x d15x F15y R15y d15y d15y F16x R16x d16x d16x F16y R16y d16y d16y F17x R17x d17x d17x F17y R17y d17y d17y F18x R18x d18x d18x F18y R18y d18y d18y F19x R19x d19x d19x F19y R19y d19y d19y F20x R20x d20x d20x F20y R20y d20y d20y F21x R21x d21x d21x F21y R21y d21y d21y F22x R22x d22x d22x F22y R22y d22y d22y F23x R23x d23x d23x F23y R23y d23y d23y F24x R24x d24x d24x F24y R24y d24y d24y F25x R25x d25x d25x F25y = R25y K d25y = K d25y Universitas Sumatera Utara 64 F26x R26x d26x d26x F26y R26y d26y d26y F27x R27x d27x d27x F27y 400 kN d27y d27y F28x R28x d28x d28x F28y R28y d28y d28y F29x R29x d29x d29x F29y R29y d29y d29y F30x R30x d30x d30x F30y R30y d30y d30y F31x R31x d31x d31x F31y R31y d31y d31y F32x R32x d32x d32x F32y R32y d32y d32y F33x R33x d33x d33x F33y R33y d33y d33y F34x R34x d34x d34x F34y R34y d34y d34y F35x R35x d35x d35x F35y R35y d35y d35y F36x R36x d36x d36x F36y R36y d36y d36y F37x R37x d37x d37x F37y R37y d37y d37y F38x R38x d38x d38x F38y R38y d38y d38y F39x R39x d39x d39x F39y R39y d39y d39y F40x R40x d40x d40x F40y R40y d40y d40y F41x R41x d41x d41x F41y 400 kN d41y d41y F42x R42x d42x d42x F42y R42y d42y d42y F43x R43x d43x d43x F43y R43y d43y d43y F44x R44x d44x d44x F44y R44y d44y d44y F45x R45x d45x d45x F45y R45y d45y d45y F46x R46x d46x d46x F46y R46y d46y d46y F47x R47x d47x d47x F47y R47y d47y d47y F48x R48x d48x d48x F48y R48y d48y d48y F49x R49x d49x d49x F49y R49y d49y d49y K merupakan matriks 98x98 dengan dua derajat kebebasan dengan 49 titik. al pada titik 1 dan 43, Universitas Sumatera Utara 65 4.2.2Langkah kedua :Menentukan Matriks Kekakuan Lokal Pada jendela microsoft excel 2010: [k]= t A[B]T[D][B] A = 12bh = 120.53 = 0.75 m2 Elemen 1 0.5 m 0.5 m 0-0.5 = -0.5 0.5-0 = 0.5 0-0 = 0-0.5 = -0.5 0-0 = 0.5-0 = 0.5 -0.5 0.5 [ β ] = 1 -0.5 0.5 1 0.75 -0.5 -0.5 0.5 0.5 m 1 0.3 [ D ] = 25332084.4 0.3 1 Mpa 0.91 0.35 -0.5 -0.5 25332084.4 -0.5 -0.5 1 0.3 0.91 0.5 0.3 1 0.5 0.35 0.5 0.5 -0.5 -0.15 -0.175 -0.15 -0.5 -0.175 25332084.4 0.5 0.15 0.91 0.175 0.175 0.15 0.5 -0.5 -0.15 -0.175 -0.15 -0.5 -0.175 0.5 0.15 -0.5 0.5 [k ] = 0.5 0.75 25332084.4 0.175 1 -0.5 0.5 0.91 0.175 0.75 -0.5 -0.5 0.5 0.5 0.15 0.5 u1 v1 u2 v2 u3 v3 0.3375 0.1625 -0.25 -0.0875 -0.0875 -0.075 u1 0.1625 0.3375 -0.075 -0.0875 -0.0875 -0.25 v1 [k ] = 9499531.65 -0.25 -0.075 0.25 0.075 N u2 0.6825 -0.0875 -0.0875 0.0875 0.0875 m v2 -0.0875 -0.0875 0.0875 0.0875 u3 -0.075 -0.25 0.075 0.25 v3 m=3 i=1 j=2 y x � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � [D] = � � [D] = [D] = � 1−�2 1 � � 1 1−� 2 Universitas Sumatera Utara 66 Elemen 2 0.5 m 0.5 m 0.5-0.5 = 0.5-0 = 0.5 0-0.5 = -0.5 0-0.5 = -0.5 0.5 - 0 = 0.5 0.5- 0.5 = 0.5 -0.5 -0.5 [ β ] = 1 -0.5 0.5 1 0.75 -0.5 0.5 0.5 -0.5 m 1 0.3 [ D ] = 25332084.4 0.3 1 Mpa 0.75 0.35 -0.5 25332084.4 -0.5 1 0.3 0.75 0.5 0.5 0.3 1 -0.5 0.5 0.5 0.35 -0.5 -0.5 -0.175 -0.15 -0.5 25332084.4 0.5 0.15 0.175 0.75 -0.35 0.35 0.175 -0.5 -0.15 -0.175 -0.175 -0.15 -0.5 0.5 0.15 0.175 0.5 -0.5 -0.5 [k ] = 0.5 0.75 25332084.4 -0.35 0.35 0.175 1 -0.5 0.5 0.91 -0.5 -0.15 0.75 -0.5 0.5 0.5 -0.5 -0.175 u2 v2 u4 v4 u3 v3 0.0875 -0.0875 -0.0875 0.0875 u2 0.25 -0.075 -0.175 0.075 v2 [k ] = 9499531.65 -0.0875 -0.075 0.3375 -0.0875 -0.25 -0.0875 N u4 0.6825 -0.0875 -0.175 -0.0875 0.4375 0.175 -0.0875 m v4 0.075 -0.25 0.175 0.25 u3 0.0875 -0.0875 -0.0875 0.0875 v3 m=3 j = 4 i=2 y x � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � [D] = � � [D] = [D] = � 1−� 2 1 � � 1 1−� 2 Universitas Sumatera Utara 67

4.2.3 Langkah ketiga : Menggabungkan Matriks Kekakuan Lokal

Penggabungan matriks kekakuan global f = k11 k21 k31 d1 k12 k22 k32 d2 k13 k23 k33 d3 a b 1 2 3 1 3 2 f1 = ka11d1 ka12d2 ka13d3 f2 = ka21d1 ka22d2 ka23d3 kb22d2 kb13d3 kb12d4 km11d2 km13d4 km12d15 f3 = ka31d1 ka32d2 ka33d3 kb31d2 kb32d4 kb33d3 kc11d3 kc12d4 kc13d5 f4 = kb21d2 kb22d4 kb23d3 km33d4 km31d2 km32d15 kn33d4 kn31d15 kn32d17 k011d4 ko12d17 k013d6 kd11d4 kd12d6 kd13d5 kc22d4 kc21d3 kc23d5 f5 = kc33d5 kc31d3 kc32d4 kd33d5 kd31d4 kd32d6 ke11d5 ke12d6 ke13d7 f6 = kd22d6 kd21d4 kd23d5 ke22d6 ke21d5 ke23d7 kf11d6 kf12d8 kf13d7 kq11d6 kq12d19 kq13d8 kp33d6 kp31d17 kp32d19 ko33d6 ko31d4 ko32d17 f7 = ke31d5 ke32d6 ke33d7 kf31d6 kf32d8 kf33d7 kg11d7 kg12d8 kg13d9 f8 = kf21d6 kf22d8 kf23d7 kq31d6 kq32d19 kq33d8 kr31d19 kr32d21 kr33d8 ks11d8 ks12d21 ks13d10 kh11d8 kh12d10 kh13d9 kg21d7 kg22d8 kg23d9 f9 = kg 31d7 kg32d8 kg33d9 kh31d8 kh32d10 kh33d9 ki 11d9 ki 12d10 ki 13d11 f10 = kh21d8 kh22d10 kh23d9 ks31d8 ks32d21 ks33d10 kt31d21 kt32d23 kt33d10 ku11d10 ku12d23 ku13d12 kj11d10 kj12d12 kj13d11 ki21d9 ki22d10 ki23d11 f11 = ki31d9 ki 32d10 ki 33d11 kj31d10 kj32d12 kj33d11 kk11d11 kk12d12 kk13d13 f12 = kj21d10 kj22d12 kj23d11 ku31d10 ku32d23 ku33d12 kv31d23 kv32d25 kv33d12 kw11d12 kw12d25 kw13d14 kl11d12 kl12d14 kl13d13 kk21d11 kk22d12 kk23d13 f13 = kk32d11 kk31d12 kk33d13 kl31d12 kl32d14 kl33d13 f14 = kl21d12 kl22d14 kl23d13 kw31d12 kw32d25 kw33d14 kx31d25 kx32d27 kx33d14 f15 = km21d2 km2215 km23d4 ky11d15 ky12d16 kn11d15 kn12d17 kn13d4 f16 = ky21d15 ky22d16 ky23d17 kz11d16 kz12d18 kz13d17 kak11d16 kak12d29 kak13d18 f17 = kn218d15 kn22d17 kn23d4 ky31d15 ky32d16 ky33d17 kz31d16 kz32d18 kz33d17 kaa11d17 kaa12d18 kaa13d19 kp11d17 kp12d19 kp13d6 ko21D4 KO22D17 KO23D6 f18 = kz21d16 kz22d18 kz23d17 kak31d16 kak32d29 kak33d18 kal31d29 kal32d31 kal33d18 kam11d18 kam12d31 kam13d20 kab11d18 kab12d20 kab13d19 kaa21d17 kaa22d18 kaa23d19 f19 = kp21d17 kp22d19 kp23d18 kaa31d17 kaa32d18 kaa33d19 kab31d18 kab32d20 kab33d19 kac11d19 kac12d20 kac13d21 kr11d19 kr12d21 kr13d8 kq21d6 kq22d19 kq23d8 f20 = kab21d18 kab22d20 kab23d19 kam31d18 kam32d31 kam33d21 kan31d31 kan32d33 kan33d20 kao11d20 kao12d33 kao13d22 kad11d20 kad12d22 kad13d21 kac21d19 kac22d20 kac23d21 f21 = kr21d19 kr22d21 kr23d8 kac31d19 kac32d20 kac33d21 kad31d20 kad32d22 kad33d21 kae11d21 kae12d22 kae13d23 kt11d21 kt12d23 kt13d10 ks21d8 ks22d21 ks23d10 f22 = kad21d20 kad22d22 kad23d21 kao31d20 kao32d33 kao22d22 kap31d33 kap32d35 kap33d22 kaq11d22 kaq12d35 kaq13d24 kaf11d22 kaf12d24 kaf13d23 kae21d21 kae22d22 kae23d23 f23 = kt21d21 kt22d23 kt23d10 kae31d21 kae32d22 kae33d23 kaf31d22 kaf32d24 kaf33d23 kag11d23 kag12d24 kag13d25 kv11d23 kv12d25 kv13d12 ku21d10 ku22d23 ku23d12 f24 = kaf21d22 kaf22d24 kaf23d23 kaq31d22 kaq32d35 kaq33d24 kar31d35 kar32d37 kar33d24 kas11d24 kas12d37 kas13d26 kah11d24 kah12d26 kah138d25 kag21d23 kag22d24 kag23d25 f25 = kv21d23 kv22d25 kv23d12 kag31d23 kag32d24 kag33d25 kah31d24 kah32d26 kah33d25 kai11d25 kai12d26 kai13d27 kx11d25 kx12d27 kx13d14 kw21d12 kw22d25 kw23d14 f26 = kah21d24 kah22d26 kah23d25 kas31d24 kas32d37 kas33d26 kat31d37 kat32d39 kat33d26 kau11d26 kau12d39 kau13d28 kaj11d26 kaj12d28 kaj13d27 kai21d25 kai22d26 kai23d27 f27 = kx21d25 kx22d27 kx23d14 kai31d25 kai32d26 kai33d27 kaj31d26 kaj32d28 kaj33d27 f28 = kaj21d26 kaj22d28 kaj23d27 kau31d26 kau32d39 kau33d28 kav31d39 kav32d41 kav33d28 f29 = kak21d16 kak22d29 kak23d18 kal11d29 kal12d31 kal13d18 kaw11d29 kaw12d30 kaw13d31 f30 = kaw21d29 kaw22d30 kaw23d31 kax11d30 kax12d32 kax13d31 kbi11d30 kbi12d43 kbi13d32 f31 = kal21d29 kal22d31 kal23d18 kaw31d29 kaw32d30 kaw33d31 kax31d30 kax32d32 kax33d31 kay11d31 kay12d32 kay13d33 kan11d31 kan12d32 kan13d20 kam21d18 kam22d31 kam23d20 f32 = kax21d30 kax22d32 kax23d31 kbi11d30 kbi12d43 kbi13d32 kbj31d43 kbj32d44 kbj33d32 kbk11d32 kbk12d44 kbk13d34 kaz11d32 kaz12d34 kaz13d33 kay21d31 kay22d32 kay23d33 f33 = kan21d31 kan22d33 kan23d20 kay31d31 kay32d32 kay33d33 kaz31d32 kaz32d34 kaz33d33 kba11d33 kba12d34 kba13d35 kap11d33 kap12d35 kap13d22 kao21d20 kao22d33 kao23d22 f34 = kaz21d32 kaz2234 kaz23d33 kbk31d32 kbk32d44 kbk33d34 kbm31d44 kbm32d45 kbm33d34 kbn11d34 kbn12d45 kbn13d36 kbb11d34 kbb12d36 kbb13d35 kba21d33 kba22d34 kba23d35 f35 = kap21d33 kap22d35 kap23d22 kba31d33 kba32d34 kba33d35 kbb31d34 kbb32d36 kbb33d36 kbc11d35 kbc12d36 kbc13d37 kar11d35 kar12d37 kar13d24 kaq21s22 kaq22d35 kaq23d24 f36 = kbb21d34 kbb22d36 kbb23d35 kbn31d34 kbn32d45 kbn33d36 kbo31d45 kbo32d46 kbo33d36 kbp11d36 kbp12d46 kbp13d38 kbd11d36 kbd12d38 kbd13d37 kbc21d35 kbc22d36 kbc23d37 f37 = kar21d35 kar22d37 kar23d24 kbc31d35 kbc32d36 kbc33d37 kbd31d36 kbd32d38 kbd33d37 kbe11d37 kbe12d38 kbe13d39 kat11d37 kat12d39 kat13d26 kas21d24 kas22d37 kas23d26 f38 = kbd21d36 kbd22d38 kbd23d37 kbp31d36 kbp32d45 kbp33d38 kbq31d46 kbq32d47 kbq33d38 kbr11d38 kbr12d47 kbr13d40 kbf11d38 kbf12d40 kbf13d39 kbe21d37 kbe22d38 kbe23d39 f39 = kat21d37 kat22d39 kat23d26 kbe31d37 kbe32d38 kbe33d39 kbf31d38 kbf32d40 kbf33d39 kbg11d39 kbg12d40 kbg13d41 kav11d39 kav12d41 kav13d28 kau21d26 kau22d39 kau23d28 f40 = kbf21d38 kbf22d40 kbf23d39 kbr31d38 kbr32d47 kbr33d40 kbs31d47 kbs32d48 kbs33d40 kbt11d40 kbt12d48 kbt13d42 kbh11d40 kbh12d42 kbh13d41 kbg21d39 kbg2240 kbg23d41 f41 = kav21d39 kav22d41 kav23d28 kbg31d39 kbg32d40 kbg33d41 kbh31d40 kbh32d42 kbh33d41 f42 = kbh21d40 kbh22d42 kbh23d41 kbt31d40 bt32d48 kbt33d42 kbu31d48 kbu32d49 kbu33d42 f43 = kbi21d30 kbi22d43 kbi23d32 kbj11d43 kbj12d44 kbj13d32 f44 = kbj21d43 kbj22d44 kbj23d32 kbk21d32 kbk22d44 kbk23d34 kbm11d44 kbm12d45 kbm13d34 f45 = kbm21d44 kbm22d45 kbm23d34 kbn11d45 kbn12d34 kbn13d36 kbo11d45 kbo12d46 kbo13d36 f46 = kbo21d45 kbo22d46 kbo23d36 kbp21d36 kbp22d46 kbp23d38 kbq11d46 kbq12d47 kbq13d38 f47 = kbq21d46 kbq22d47 kbq23d38 kbr21d38 kbr22d47 kbr23d40 kbs11d47 kbs12d48 kbs13d40 f48 = kbs21d47 kbs22d48 kbs23d40 kbt21d40 kbt22d48 kbt23d42 kbu11d48 kbu12d49 kbu13d42 f49 = kbu21d48 kbu22d49 bu23d42 Universitas Sumatera Utara 68

4.2.4 Langkah Keempat : Mencari Perpindahan tiap-tiap titik

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d35y 2.24053E-06 d36x -15.11610691 d36x -1.31006E-06 d36y 0.812561477 d36y 7.0422E-08 d37x 10.35077057 d37x 8.97067E-07 d37y 83.98352171 d37y 7.27857E-06 d38x -42.25458325 d38x -3.66206E-06 d38y 12.55120542 d38y 1.08777E-06 d39x 27.84316596 d39x 2.41307E-06 d39y 218.8984413 d39y 1.89712E-05 d40x -92.57355538 d40x -8.02304E-06 d40y 32.99819072 d40y 2.85984E-06 d41x 46.09833922 d41x 3.99519E-06 d41y 565.577332 d41y 4.90167E-05 d42x -199.3042554 d42x -1.7273E-05 d42y 61.40299651 d42y 5.32159E-06 d43x -9.434495614 d43x -8.17656E-07 d43y -2.729377776 d43y -2.36546E-07 d44x -11.90613168 d44x -1.03186E-06 d44y -3.10273328 d44y -2.68904E-07 d45x -6.037773552 d45x -5.23274E-07 d45y -2.522491257 d45y -2.18616E-07 d46x -14.91508718 d46x -1.29264E-06 d46y 9.17569733 d46y 7.95227E-07 d47x -36.40229434 d47x -3.15487E-06 d47y 23.02010546 d47y 1.99508E-06 d48x -59.36092621 d48x -5.14461E-06 d48y 34.57565134 d48y 2.99656E-06 Universitas Sumatera Utara 70

4.2.5 Langkah Kelima : Menghitung Tegangan-Tegangan Yang Terjadi

{ �} = [�][�]{�} � � � � � � �� � = � 1 − � 2 � 1 � � 1 1 − � 2 � � � 1 2 �� � � 1 � 3 � 2 � 1 � 3 � 2 � 1 � 1 � 3 � 3 � 2 � 2 � ⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ � 1 � � 1 � � 3 � � 3 � � 2 � � 2 � ⎭ ⎪ ⎪ ⎬ ⎪ ⎪ ⎫ 3000 mm 500 mm 400 kN 500 mm 500 mm 500 mm 500 mm 500 mm 500 mm 500 mm 500 mm 500 mm 500 mm 500 mm 1 2 3 4 5 7 8 9 10 11 6 12 13 14 400 kN 15 16 17 18 19 21 22 23 24 25 20 26 27 28 29 30 31 32 33 35 36 37 38 39 34 40 41 42 43 44 45 46 47 48 49 A A B B C C D D E E F F Gambar 4.4 Pengambilan nilai tegangan pada potongan melintang A,B,C,D,E,F Universitas Sumatera Utara 71 4.2.5.1 Tegangan pada potongan A---A Elemen 1,3,5,7,9,11 Pada jendela microsoft excel 2010: Elemen 1 0.5 m 0.5 m 0-0.5 = -0.5 0.5-0 = 0.5 0-0 = 0-0.5 = -0.5 0-0 = 0.5-0 = 0.5 -0.5 0.5 [ β ] = 1 -0.5 0.5 1 0.75 -0.5 -0.5 0.5 0.5 m 1 0.3 [ D ] = 25310000 0.3 1 Mpa 0.91 0.35 27813186.8 8343956.044 -0.66667 0.666667 d1x -1.67319E-07 [ σ ] = 8343956.04 27813186.81 -0.66667 0.666667 d1y 7.03675E-08 0 9734615 -0.66667 -0.66667 0.666667 0.666667 d2x -1.28488E-07 d2y 2.41837E-07 d3x -1.23455E-07 d3y 7.06207E-08 σ x 2.58875E-08 0.721423 σ y 1.68752E-10 0.220698 kNm2 σ xy 1.43556E-07 1.397458 m=3 i=1 j=2 y x � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = [D] = � 1−� 2 1 � 0 � 1 0 0 0 1−� 2 Universitas Sumatera Utara 72 Elemen 3 0.5 m 0.5 m 0-0.5 = -0.5 0.5-0 = 0.5 0-0 = 0-0.5 = -0.5 0-0 = 0.5-0 = 0.5 -0.5 0.5 [ β ] = 1 -0.5 0.5 1 0.75 -0.5 -0.5 0.5 0.5 m 1 0.3 [ D ] = 25310000 0.3 1 Mpa 0.91 0.35 27813187 8343956 -0.66667 0.666667 d3x -1.28488E-07 [ σ ] = 8343956 27813187 -0.66667 0.666667 d3y 2.41837E-07 9734615 -0.66667 -0.66667 0.666667 0.666667 d4x -1.23455E-07 d4y 7.06207E-08 d5x -1.03921E-07 d5y 5.46972E-07 σ x 3.36E-09 1.790671 σ y 2.03E-07 5.685843 kNm2 σ xy -9.8E-08 -0.95172 m=5 i=3 j=4 y x � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = [D] = � 1−� 2 1 � � 1 0 0 1−� 2 Universitas Sumatera Utara 73 Elemen 5 0.5 m 0.5 m 0-0.5 = -0.5 0.5-0 = 0.5 0-0 = 0-0.5 = -0.5 0-0 = 0.5-0 = 0.5 -0.5 0.5 [ β ] = 1 -0.5 0.5 1 0.75 -0.5 -0.5 0.5 0.5 m 1 0.3 [ D ] = 25310000 0.3 1 Mpa 0.91 0.35 27813187 8343956 -0.66667 0.666667 d5x -1.0392E-07 [ σ ] = 8343956 27813187 -0.66667 0.666667 d5y 5.46972E-07 0 9734615 -0.66667 -0.66667 0.666667 0.666667 d6x -7.8364E-08 d6y 1.2231E-07 d7x -5.0937E-08 d7y 7.39768E-07 σ x 1.7E-08 1.546348 σ y 1.29E-07 3.717027 kNm2 σ xy -2.5E-07 -2.4121 m=7 i=5 j=6 y x � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = [D] = � 1−� 2 1 � 0 � 1 0 0 0 1−� 2 Universitas Sumatera Utara 74 Elemen 7 0.5 m 0.5 m 0-0.5 = -0.5 0.5-0 = 0.5 0-0 = 0-0.5 = -0.5 0-0 = 0.5-0 = 0.5 -0.5 0.5 [ β ] = 1 -0.5 0.5 1 0.75 -0.5 -0.5 0.5 0.5 m 1 0.3 [ D ] = 25310000 0.3 1 Mpa 0.91 0.35 27813187 8343956 -0.66667 0.666667 d7x -5.09369E-08 [ σ ] = 8343956 27813187 -0.66667 0.666667 d7y 7.39768E-07 0 9734615 -0.66667 -0.66667 0.666667 0.666667 d8x -1.01412E-07 d8y 1.25217E-07 d9x 3.01403E-07 d9y 1.36595E-06 σ x -3.4E-08 2.547297 σ y 4.17E-07 11.32994 kNm2 σ xy -1.7E-07 -1.70169 m=9 i=7 j=8 y x � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = [D] = � 1−� 2 1 � � 1 0 0 0 1−� 2 Universitas Sumatera Utara 75 Elemen 9 m = 11 0.5 m i = 9 0.5 m j = 10 0-0.5 = -0.5 0.5-0 = 0.5 0-0 = 0-0.5 = -0.5 0-0 = 0.5-0 = 0.5 -0.5 0.5 [ β ] = 1 -0.5 0.5 1 0.75 -0.5 -0.5 0.5 0.5 m 1 0.3 [ D ] = 25310000 0.3 1 Mpa 0.91 0.35 27813187 8343956 -0.66667 0.666667 d9x 3.01403E-07 [ σ ] = 8343956 27813187 -0.66667 0.666667 d9y 1.36595E-06 0 9734615 -0.66667 -0.66667 0.666667 0.666667 d10x 4.67537E-08 d10y 3.45144E-08 d11x 1.91169E-06 d11y 2.54742E-06 σ x -1.7E-07 1.850344 σ y 7.88E-07 20.49042 kNm2 σ xy 1.86E-07 1.809679 y x � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = [D] = � 1−� 2 1 � 0 � 1 0 0 0 1−� 2 Universitas Sumatera Utara 76 Elemen 11 m = 13 0.5 m i = 11 0.5 m j = 12 0-0.5 = -0.5 0.5-0 = 0.5 0-0 = 0-0.5 = -0.5 0-0 = 0.5-0 = 0.5 -0.5 0.5 [ β ] = 1 -0.5 0.5 1 0.75 -0.5 -0.5 0.5 0.5 m 1 0.3 [ D ] = 25310000 0.3 1 Mpa 0.91 0.35 27813187 8343956 -0.66667 0.666667 d11x 1.91169E-06 [ σ ] = 8343956 27813187 -0.66667 0.666667 d11y 2.54742E-06 0 9734615 -0.66667 -0.66667 0.666667 0.666667 d12x 1.05247E-06 d12y -2.97732E-07 d13x 8.15901E-06 d13y 3.63921E-06 σ x -5.7E-07 -9.85845 σ y 7.28E-07 15.46465 kNm2 σ xy 2.27E-06 22.07924 y x � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = [D] = � 1−� 2 1 � 0 � 1 0 0 0 1−� 2 Universitas Sumatera Utara 77

4.2.5.2 Tegangan σx pada potongan B—B Elemen

5,4,6,8,10,15 Pada jendela microsoft excel 2010: Elemen 2 0.5 m 0.5 m 0-0.5 = -0.5 0.5-0 = 0.5 0-0 = 0-0.5 = -0.5 0-0 = 0.5-0 = 0.5 -0.5 0.5 [ β ] = 1 -0.5 -0.5 0.5 1 0.75 -0.5 -0.5 0.5 0.5 m 1 0.3 [ D ] = 25332084.4 0.3 1 Mpa 0.91 0.35 27837455.4 8351237 -0.66667 0.666667 d2x -1.6732E-07 [ σ ] = 8351236.62 27837455 -0.66667 -0.66667 0.666667 d2y 7.03675E-08 9743109 -0.66667 -0.66667 0.666667 0.666667 d4x -1.2346E-07 d4y 7.06207E-08 d3x -1.2849E-07 d3y 2.41837E-07 σ x 2.92E-08 2.700244 σ y 2.26E-07 6.531554 kNm2 σ xy 2.61E-08 0.253869 m=3 i=1 j=2 y x � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = Universitas Sumatera Utara 78 Elemen 4 0.5 m 0.5 m 0-0.5 = -0.5 0.5-0 = 0.5 0-0 = 0-0.5 = -0.5 0-0 = 0.5-0 = 0.5 -0.5 0.5 [ β ] = 1 -0.5 0.5 1 0.75 -0.5 -0.5 0.5 0.5 m 1 0.3 [ D ] = 25332084.4 0.3 1 Mpa 0.91 0.35 27837455.38 8351237 -0.66667 0.666667 d4x -1.2346E-07 [ σ ] = 8351236.615 27837455 -0.66667 0.666667 d4y 7.06207E-08 9743109 -0.66667 -0.66667 0.666667 0.666667 d6x -7.8364E-08 d6y 1.2231E-07 d5x -1.0392E-07 d5y 5.46972E-07 σ x 3.01E-08 3.488905 σ y 3.18E-07 9.091315 kNm2 σ xy 4.75E-08 0.462622 m=5 i=3 j=4 y x � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = Universitas Sumatera Utara 79 Elemen 6 0.5 m 0.5 m 0-0.5 = -0.5 0.5-0 = 0.5 0-0 = 0-0.5 = -0.5 0-0 = 0.5-0 = 0.5 -0.5 0.5 [ β ] = 1 -0.5 0.5 1 0.75 -0.5 -0.5 0.5 0.5 m 1 0.3 [ D ] = 25332084.4 0.3 1 Mpa 0.91 0.35 27837455.38 8351237 -0.66667 0.666667 d6x -7.8364E-08 [ σ ] = 8351236.615 27837455 -0.66667 0.666667 d6y 1.2231E-07 9743109 -0.66667 -0.66667 0.666667 0.666667 d8x -1.0141E-07 d8y 1.25217E-07 d7x -5.0937E-08 d7y 7.39768E-07 σ x -1.5E-08 3.009954 σ y 4.12E-07 11.33067 kNm2 σ xy 2.02E-08 0.197032 m=7 i=5 j=6 y x � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = Universitas Sumatera Utara 80 Elemen 8 0.5 m 0.5 m 0-0.5 = -0.5 0.5-0 = 0.5 0-0 = 0-0.5 = -0.5 0-0 = 0.5-0 = 0.5 -0.5 0.5 [ β ] = 1 -0.5 0.5 1 0.75 -0.5 -0.5 0.5 0.5 m 1 0.3 [ D ] = 25332084.4 0.3 1 Mpa 0.91 0.35 27837455.38 8351237 -0.66667 0.666667 d8x -1.01412E-07 [ σ ] = 8351236.615 27837455 -0.66667 0.666667 d8y 1.25217E-07 9743109 -0.66667 -0.66667 0.666667 0.666667 d10x 4.67537E-08 d10y 3.45144E-08 d9x 3.01403E-07 d9y 1.36595E-06 σ x 9.88E-08 9.657472 σ y 8.27E-07 23.85079 kNm2 σ xy 2.08E-07 2.027298 m=9 i=7 j=8 y x � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = Universitas Sumatera Utara 81 Elemen 10 m = 11 0.5 m i = 9 0.5 m j = 10 0-0.5 = -0.5 0.5-0 = 0.5 0-0 = 0-0.5 = -0.5 0-0 = 0.5-0 = 0.5 -0.5 0.5 [ β ] = 1 -0.5 0.5 1 0.75 -0.5 -0.5 0.5 0.5 m 1 0.3 [ D ] = 25332084.4 0.3 1 Mpa 0.91 0.35 27837455.38 8351237 -0.66667 0.666667 d10x 4.67537E-08 [ σ ] = 8351236.615 27837455 -0.66667 0.666667 d10y 3.45144E-08 9743109 -0.66667 -0.66667 0.666667 0.666667 d12x 1.05247E-06 d12y -2.97732E-07 d11x 1.91169E-06 d11y 2.54742E-06 σ x 6.7E-07 32.65502 σ y 1.68E-06 52.23456 kNm2 σ xy 1.02E-06 9.955442 y x � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = Universitas Sumatera Utara 82 Elemen 12 m = 13 0.5 m i = 11 0.5 m j = 12 0-0.5 = -0.5 0.5-0 = 0.5 0-0 = 0-0.5 = -0.5 0-0 = 0.5-0 = 0.5 -0.5 0.5 [ β ] = 1 -0.5 0.5 1 0.75 -0.5 -0.5 0.5 0.5 m 1 0.3 [ D ] = 25332084.4 0.3 1 Mpa 0.91 0.35 27837455.38 8351237 -0.66667 0.666667 d12x 1.05247E-06 [ σ ] = 8351236.615 27837455 -0.66667 0.666667 d12y -2.97732E-07 9743109 -0.66667 -0.66667 0.666667 0.666667 d14x 1.23616E-05 d14y 7.19496E-07 d13x 8.15901E-06 d13y 3.63921E-06 σ x 7.54E-06 231.7971 σ y 2.62E-06 136.0265 kNm2 σ xy 5.42E-06 52.76718 y x � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = Universitas Sumatera Utara 83

4.2.5.3 Tegangan pada potongan C—C 14,16,18,50,55,54

Pada jendela microsoft excel 2010: Elemen 14 m = 4 j = 17 0.5 m 0.5 m i = 15 0.5-0.5 = 0.5-0 = 0.5 0-0.5 = -0.5 0-0.5 = -0.5 0.5 - 0 = 0.5 0.5- 0.5 = 0.5 -0.5 -0.5 [ β ] = 1 -0.5 0.5 1 0.75 -0.5 0.5 0.5 -0.5 m 1 0.3 [ D ] = 25310000 0.3 1 Mpa 0.75 0.35 27837455.4 8351236.615 0.666667 -0.66667 -0.66667 d15x -1.97538E-07 [ σ ] = 8351236.62 27837455.38 -0.66667 0.666667 d15y -1.89149E-07 9743109 -0.66667 0.666667 0.666667 -0.66667 d17x -6.88378E-07 d17y -7.2339E-07 d4x -1.23455E-07 d4y 7.06207E-08 σ x 1.05645E-07 -0.0335 σ y -3.56161E-07 -9.03234 kNm2 σ xy -8.56567E-07 -8.34563 y x � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = [D] = � 1−� 2 1 � � 1 1−� 2 Universitas Sumatera Utara 84 Elemen 16 m = 6 j = 19 0.5 m 0.5 m i = 17 0.5-0.5 = 0.5-0 = 0.5 0-0.5 = -0.5 0-0.5 = -0.5 0.5 - 0 = 0.5 0.5- 0.5 = 0.5 -0.5 -0.5 [ β ] = 1 -0.5 0.5 1 0.75 -0.5 0.5 0.5 -0.5 m 1 0.3 [ D ] = 25310000 0.3 1 Mpa 0.75 0.35 27837455 8351237 0.666667 -0.66667 -0.66667 d17x -6.88378E-07 [ σ ] = 8351237 27837455 -0.66667 0.666667 d17y -7.2339E-07 9743109 -0.66667 0.666667 0.666667 -0.66667 d19x -1.65079E-06 d19y -1.64978E-06 d6x -7.83636E-08 d6y 1.2231E-07 σ x 5.16E-08 -3.72219 σ y -6.2E-07 -16.7616 kNm2 σ xy -1.8E-06 -17.7617 y x � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = [D] = � 1−� 2 1 � � 1 1−� 2 Universitas Sumatera Utara 85 Elemen 18 m = 8 j = 21 0.5 m 0.5 m i = 19 0.5-0.5 = 0.5-0 = 0.5 0-0.5 = -0.5 0-0.5 = -0.5 0.5 - 0 = 0.5 0.5- 0.5 = 0.5 -0.5 -0.5 [ β ] = 1 -0.5 0.5 1 0.75 -0.5 0.5 0.5 -0.5 m 1 0.3 [ D ] = 25310000 0.3 1 Mpa 0.75 0.35 27837455 8351237 0.666667 -0.66667 -0.66667 d19x -1.6508E-06 [ σ ] = 8351237 27837455 -0.66667 0.666667 d19y -1.6498E-06 9743109 -0.66667 0.666667 0.666667 -0.66667 d21x -1.5442E-06 d21y -3.6332E-06 d8x -1.0141E-07 d8y 1.25217E-07 σ x 1.46E-06 29.60747 σ y -1.3E-06 -24.6142 kNm2 σ xy -2.4E-06 -23.7204 y x � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = [D] = � 1−�2 1 � � 1 1−� 2 Universitas Sumatera Utara 86 Elemen 20 m = 10 j = 23 0.5 m 0.5 m i = 21 0.5-0.5 = 0.5-0 = 0.5 0-0.5 = -0.5 0-0.5 = -0.5 0.5 - 0 = 0.5 0.5- 0.5 = 0.5 -0.5 -0.5 [ β ] = 1 -0.5 0.5 1 0.75 -0.5 0.5 0.5 -0.5 m 1 0.3 [ D ] = 25310000 0.3 1 Mpa 0.75 0.35 27837455 8351237 0.666667 -0.66667 -0.66667 d21x -1.54423E-06 [ σ ] = 8351237 27837455 -0.66667 0.666667 d21y -3.63322E-06 9743109 -0.66667 0.666667 0.666667 -0.66667 d23x -2.69777E-06 d23y -7.73805E-06 d10x 4.67537E-08 d10y 3.45144E-08 σ x 3.33E-06 69.81785 σ y -2.7E-06 -48.3772 kNm2 σ xy -6E-06 -57.9787 y x � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = [D] = � 1−� 2 1 � � 1 1−� 2 Universitas Sumatera Utara 87 Elemen 22 m = 12 j = 25 0.5 m 0.5 m i = 23 0.5-0.5 = 0.5-0 = 0.5 0-0.5 = -0.5 0-0.5 = -0.5 0.5 - 0 = 0.5 0.5- 0.5 = 0.5 -0.5 -0.5 [ β ] = 1 -0.5 0.5 1 0.75 -0.5 0.5 0.5 -0.5 m 1 0.3 [ D ] = 25310000 0.3 1 Mpa 0.75 0.35 27837455 8351237 0.666667 -0.66667 -0.66667 d23x -2.69777E-06 [ σ ] = 8351237 27837455 -0.66667 0.666667 d23y -7.73805E-06 9743109 -0.66667 0.666667 0.666667 -0.66667 d25x -4.14917E-06 d25y -1.91127E-05 d12x 1.05247E-06 d12y -2.97732E-07 σ x 9.27E-06 194.8368 σ y -7.6E-06 -133.644 kNm2 σ xy -1.4E-05 -131.638 y x � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = [D] = � 1−� 2 1 � � 1 1−� 2 Universitas Sumatera Utara 88 Elemen 24 m = 14 j = 27 0.5 m 0.5 m i = 25 0.5-0.5 = 0.5-0 = 0.5 0-0.5 = -0.5 0-0.5 = -0.5 0.5 - 0 = 0.5 0.5- 0.5 = 0.5 -0.5 -0.5 [ β ] = 1 -0.5 0.5 1 0.75 -0.5 0.5 0.5 -0.5 m 1 0.3 [ D ] = 25310000 0.3 1 Mpa 0.75 0.35 27837455 8351237 0.666667 -0.66667 -0.66667 d25x -4.14917E-06 [ σ ] = 8351237 27837455 -0.66667 0.666667 d25y -1.91127E-05 9743109 -0.66667 0.666667 0.666667 -0.66667 d27x 1.7642E-06 d27y -4.97496E-05 d14x 1.23616E-05 d14y 7.19496E-07 σ x 2.61E-05 556.0279 σ y -2E-05 -350.591 kNm2 σ xy -3E-05 -289.408 y x � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = [D] = � 1−� 2 1 � � 1 1−� 2 Universitas Sumatera Utara 89

4.2.5.4 Tegangan pada potongan D—D 55,57,59,31,33,35

Pada jendela microsoft excel 2010: Elemen 25 0.5 m 0.5 m 0-0.5 = -0.5 0.5-0 = 0.5 0-0 = 0-0.5 = -0.5 0-0 = 0.5-0 = 0.5 -0.5 0.5 [ β ] = 1 -0.5 0.5 1 0.75 -0.5 -0.5 0.5 0.5 m 1 0.3 [ D ] = 25310000 0.3 1 Mpa 0.91 0.35 27837455.38 8351236.62 -0.66667 0.666667 d15x -1.97538E-07 [ σ ] = 8351236.615 27837455.4 -0.66667 0.666667 d15y -1.89149E-07 9743109 -0.66667 -0.66667 0.666667 0.666667 d16x 2.31103E-07 d16y -6.20872E-08 d17x -6.88378E-07 d17y -7.2339E-07 σ x 2.8576E-07 4.980465 σ y -3.5616E-07 -7.52815 kNm2 σ xy -2.4252E-07 -2.36289 m=3 i=1 j=2 y x � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = [D] = � 1−� 2 1 � � 1 1−� 2 Universitas Sumatera Utara 90 Elemen 27 0.5 m 0.5 m 0-0.5 = -0.5 0.5-0 = 0.5 0-0 = 0-0.5 = -0.5 0-0 = 0.5-0 = 0.5 -0.5 0.5 [ β ] = 1 -0.5 0.5 1 0.75 -0.5 -0.5 0.5 0.5 m 1 0.3 [ D ] = 25310000 0.3 1 Mpa 0.91 0.35 27837455.38 8351236.615 -0.66667 0.666667 d17x -6.8838E-07 [ σ ] = 8351236.615 27837455.38 -0.66667 0.666667 d17y -7.2339E-07 9743109 -0.66667 -0.66667 0.666667 0.666667 d18x -4.616E-07 d18y -1.7628E-07 d19x -1.6508E-06 d19y -1.6498E-06 σ x 1.51183E-07 -0.94912 σ y -6.17593E-07 -15.9297 kNm2 σ xy -2.76871E-07 -2.69759 m=5 i=3 j=4 y x � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = [D] = � 1−� 2 1 � � 1 1−� 2 Universitas Sumatera Utara 91 Elemen 29 0.5 m 0.5 m 0-0.5 = -0.5 0.5-0 = 0.5 0-0 = 0-0.5 = -0.5 0-0 = 0.5-0 = 0.5 -0.5 0.5 [ β ] = 1 -0.5 0.5 1 0.75 -0.5 -0.5 0.5 0.5 m 1 0.3 [ D ] = 25310000 0.3 1 Mpa 0.91 0.35 27837455.4 8351236.615 -0.66667 0.666667 d19x -1.65079E-06 [ σ ] = 8351236.62 27837455.38 -0.66667 0.666667 d19y -1.64978E-06 9743109 -0.66667 -0.66667 0.666667 0.666667 d20x -2.1905E-06 d20y -6.32891E-07 d21x -1.54423E-06 d21y -3.63322E-06 σ x -3.59805E-07 -21.0589 σ y -1.3223E-06 -39.8141 kNm2 σ xy 7.48971E-07 7.297306 m=7 i=5 j=6 y x � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = [D] = � 1−� 2 1 � � 1 1−� 2 Universitas Sumatera Utara 92 Elemen 31 0.5 m 0.5 m 0-0.5 = -0.5 0.5-0 = 0.5 0-0 = 0-0.5 = -0.5 0-0 = 0.5-0 = 0.5 -0.5 0.5 [ β ] = 1 -0.5 0.5 1 0.75 -0.5 -0.5 0.5 0.5 m 1 0.3 [ D ] = 25310000 0.3 1 Mpa 0.91 0.35 27837455.38 8351236.615 -0.66667 0.666667 d21x -1.54423E-06 [ σ ] = 8351236.615 27837455.38 -0.66667 0.666667 d21y -3.63322E-06 9743109 -0.66667 -0.66667 0.666667 0.666667 d22x -1.08142E-06 d22y -2.48947E-06 d23x -2.69777E-06 d23y -7.73805E-06 σ x 3.08537E-07 -14.2647 σ y -2.73655E-06 -73.602 kNm2 σ xy -6.52291E-09 -0.06355 m=9 i=7 j=8 y x � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = [D] = � 1−� 2 1 � � 1 1−� 2 Universitas Sumatera Utara 93 Elemen 33 m = 11 0.5 m i = 9 0.5 m j = 10 0-0.5 = -0.5 0.5-0 = 0.5 0-0 = 0-0.5 = -0.5 0-0 = 0.5-0 = 0.5 -0.5 0.5 [ β ] = 1 -0.5 0.5 1 0.75 -0.5 -0.5 0.5 0.5 m 1 0.3 [ D ] = 25310000 0.3 1 Mpa 0.91 0.35 27837455.38 8351236.615 -0.66667 0.666667 d23x -2.69777E-06 [ σ ] = 8351236.615 27837455.38 -0.66667 0.666667 d23y -7.73805E-06 9743109 -0.66667 -0.66667 0.666667 0.666667 d24x 5.39637E-08 d24y -5.4225E-06 d25x -4.14917E-06 d25y -1.91127E-05 σ x 1.83449E-06 -12.2606 σ y -7.5831E-06 -195.773 kNm2 σ xy 5.76094E-07 5.612951 y x � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = [D] = � 1−� 2 1 � � 1 1−� 2 Universitas Sumatera Utara 94 Elemen 35 m = 13 0.5 m i = 11 0.5 m j = 12 0-0.5 = -0.5 0.5-0 = 0.5 0-0 = 0-0.5 = -0.5 0-0 = 0.5-0 = 0.5 -0.5 0.5 [ β ] = 1 -0.5 0.5 1 0.75 -0.5 -0.5 0.5 0.5 m 1 0.3 [ D ] = 25310000 0.3 1 Mpa 0.91 0.35 27837455.38 8351236.615 -0.66667 0.666667 d25x -4.14917E-06 [ σ ] = 8351236.615 27837455.38 -0.66667 0.666667 d25y -1.91127E-05 9743109 -0.66667 -0.66667 0.666667 0.666667 d26x 3.19857E-06 d26y -1.10742E-05 d27x 1.7642E-06 d27y -4.97496E-05 σ x 4.89849E-06 -34.2095 σ y -2.04246E-05 -527.662 kNm2 σ xy 9.30125E-06 90.62311 y x � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = [D] = � 1−� 2 1 � � 1 1−� 2 Universitas Sumatera Utara 95

4.2.5.5 Tegangan pada potongan E –E Elemen56,58,30,35,34,36

Pada jendela microsoft excel 2010: Elemen 26 m = 17 j = 18 0.5 m 0.5 m i = 16 0.5-0.5 = 0.5-0 = 0.5 0-0.5 = -0.5 0-0.5 = -0.5 0.5 - 0 = 0.5 0.5- 0.5 = 0.5 -0.5 -0.5 [ β ] = 1 -0.5 0.5 1 0.75 -0.5 0.5 0.5 -0.5 m 1 0.3 [ D ] = 25310000 0.3 1 Mpa 0.75 0.35 27837455.4 8351236.615 0.666667 -0.66667 -0.66667 d16x 2.31103E-07 [ σ ] = 8351236.62 27837455.38 -0.66667 0.666667 d16y -6.20872E-08 9743109 -0.66667 0.666667 0.666667 -0.66667 d18x -4.61603E-07 d18y -1.76281E-07 d17x -6.88378E-07 d17y -7.2339E-07 σ x 2.68704E-07 6.844261 σ y -7.61294E-08 0.124762 kNm2 σ xy -9.7065E-08 -0.94572 y x � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = [D] = � 1−� 2 1 � � 1 1−� 2 Universitas Sumatera Utara 96 Elemen 28 m = 19 j = 20 0.5 m 0.5 m i = 18 0.5-0.5 = 0.5-0 = 0.5 0-0.5 = -0.5 0-0.5 = -0.5 0.5 - 0 = 0.5 0.5- 0.5 = 0.5 -0.5 -0.5 [ β ] = 1 -0.5 0.5 1 0.75 -0.5 0.5 0.5 -0.5 m 1 0.3 [ D ] = 25310000 0.3 1 Mpa 0.75 0.35 27837455 8351237 0.666667 -0.66667 -0.66667 d18x -4.61603E-07 [ σ ] = 8351237 27837455 -0.66667 0.666667 d18y -1.76281E-07 9743109 -0.66667 0.666667 0.666667 -0.66667 d20x -2.1905E-06 d20y -6.32891E-07 d19x -1.65079E-06 d19y -1.64978E-06 σ x 6.21E-08 -0.81285 σ y -3E-07 -7.95511 kNm2 σ xy -4.7E-07 -4.62479 y x [D] = � 1−� 2 1 � � 1 1−� 2 � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = Universitas Sumatera Utara 97 Elemen 30 m = 21 j = 22 0.5 m 0.5 m i = 20 0.5-0.5 = 0.5-0 = 0.5 0-0.5 = -0.5 0-0.5 = -0.5 0.5 - 0 = 0.5 0.5- 0.5 = 0.5 -0.5 -0.5 [ β ] = 1 -0.5 0.5 1 0.75 -0.5 0.5 0.5 -0.5 m 1 0.3 [ D ] = 25310000 0.3 1 Mpa 0.75 0.35 27837455 8351237 0.666667 -0.66667 -0.66667 d20x -2.1905E-06 [ σ ] = 8351237 27837455 -0.66667 0.666667 d20y -6.3289E-07 9743109 -0.66667 0.666667 0.666667 -0.66667 d22x -1.0814E-06 d22y -2.4895E-06 d21x -1.5442E-06 d21y -3.6332E-06 σ x 1.97E-06 44.45269 σ y -1.2E-06 -18.0181 kNm2 σ xy 1.5E-06 14.63309 y x [D] = � 1−�2 1 � � 1 1−� 2 � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = Universitas Sumatera Utara 98 Elemen 32 m = 23 j = 24 0.5 m 0.5 m i = 22 0.5-0.5 = 0.5-0 = 0.5 0-0.5 = -0.5 0-0.5 = -0.5 0.5 - 0 = 0.5 0.5- 0.5 = 0.5 -0.5 -0.5 [ β ] = 1 -0.5 0.5 1 0.75 -0.5 0.5 0.5 -0.5 m 1 0.3 [ D ] = 25310000 0.3 1 Mpa 0.75 0.35 27837455 8351237 0.666667 -0.66667 -0.66667 d22x -1.08142E-06 [ σ ] = 8351237 27837455 -0.66667 0.666667 d22y -2.48947E-06 9743109 -0.66667 0.666667 0.666667 -0.66667 d24x 5.39637E-08 d24y -5.4225E-06 d23x -2.69777E-06 d23y -7.73805E-06 σ x 5.45E-06 135.3702 σ y -2E-06 -8.92218 kNm2 σ xy 2.3E-06 22.41521 y x [D] = � 1−� 2 1 � � 1 1−� 2 � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = Universitas Sumatera Utara 99 Elemen 34 m = 25 j = 26 0.5 m 0.5 m i = 24 0.5-0.5 = 0.5-0 = 0.5 0-0.5 = -0.5 0-0.5 = -0.5 0.5 - 0 = 0.5 0.5- 0.5 = 0.5 -0.5 -0.5 [ β ] = 1 -0.5 0.5 1 0.75 -0.5 0.5 0.5 -0.5 m 1 0.3 [ D ] = 25310000 0.3 1 Mpa 0.75 0.35 27837455 8351237 0.666667 -0.66667 -0.66667 d24x 5.39637E-08 [ σ ] = 8351237 27837455 -0.66667 0.666667 d24y -5.4225E-06 9743109 -0.66667 0.666667 0.666667 -0.66667 d26x 3.19857E-06 d26y -1.10742E-05 d25x -4.14917E-06 d25y -1.91127E-05 σ x 1.23E-05 310.4136 σ y -3.8E-06 -2.32139 kNm2 σ xy 7.46E-06 72.63883 y x [D] = � 1−� 2 1 � � 1 1−� 2 � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = Universitas Sumatera Utara 100 Elemen 36 m = 27 j = 28 0.5 m 0.5 m i = 26 0.5-0.5 = 0.5-0 = 0.5 0-0.5 = -0.5 0-0.5 = -0.5 0.5 - 0 = 0.5 0.5- 0.5 = 0.5 -0.5 -0.5 [ β ] = 1 -0.5 0.5 1 0.75 -0.5 0.5 0.5 -0.5 m 1 0.3 [ D ] = 25310000 0.3 1 Mpa 0.75 0.35 27837455 8351237 0.666667 -0.66667 -0.66667 d26x 3.19857E-06 [ σ ] = 8351237 27837455 -0.66667 0.666667 d26y -1.10742E-05 9743109 -0.66667 0.666667 0.666667 -0.66667 d27x 1.7642E-06 d27y -4.97496E-05 d28x 6.65225E-06 d28y -1.68695E-05 σ x 2.99E-05 617.2295 σ y -2.6E-05 -467.985 kNm2 σ xy -2.3E-05 -222.887 y x [D] = � 1−� 2 1 � � 1 1−� 2 � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = Universitas Sumatera Utara 101

4.2.5.6 Tegangan Pada potongan potongan F—F elemen37,39,41,43,45,47

Pada jendela microsoft excel 2010: Elemen 37 0.5 m 0.5 m 0-0.5 = -0.5 0.5-0 = 0.5 0-0 = 0-0.5 = -0.5 0-0 = 0.5-0 = 0.5 -0.5 0.5 [ β ] = 1 -0.5 0.5 1 0.75 -0.5 -0.5 0.5 0.5 m 1 0.3 [ D ] = 25332084.4 0.3 1 Mpa 0.91 0.35 27837455.38 8351236.62 -0.66667 0.666667 d16x -6.20872E-08 [ σ ] = 8351236.615 27837455.4 -0.66667 0.666667 d16y -6.88378E-07 9743109 -0.66667 -0.66667 0.666667 0.666667 d29x 6.23046E-07 d29y 5.83427E-07 d18x -4.61603E-07 d18y -1.76281E-07 σ x 4.5676E-07 15.566 σ y 3.414E-07 13.31812 kNm2 σ xy 5.8153E-07 5.665874 m=3 i=1 j=2 y x � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = [D] = � 1−� 2 1 � � 1 1−� 2 Universitas Sumatera Utara 102 Elemen 39 0.5 m 0.5 m 0-0.5 = -0.5 0.5-0 = 0.5 0-0 = 0-0.5 = -0.5 0-0 = 0.5-0 = 0.5 -0.5 0.5 [ β ] = 1 -0.5 0.5 1 0.75 -0.5 -0.5 0.5 0.5 m 1 0.3 [ D ] = 25332084.4 0.3 1 Mpa 0.91 0.35 27837455.38 8351236.615 -0.66667 0.666667 d18x -4.616E-07 [ σ ] = 8351236.615 27837455.38 -0.66667 0.666667 d18y -1.7628E-07 9743109 -0.66667 -0.66667 0.666667 0.666667 d31x 3.80907E-07 d31y 5.93875E-07 d20x -2.1905E-06 d20y -6.3289E-07 σ x 5.61674E-07 13.09339 σ y -3.04407E-07 -3.78324 kNm2 σ xy -6.39161E-07 -6.22742 m=5 i=3 j=4 y x � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = [D] = � 1−� 2 1 � � 1 1−� 2 Universitas Sumatera Utara 103 Elemen 41 0.5 m 0.5 m 0-0.5 = -0.5 0.5-0 = 0.5 0-0 = 0-0.5 = -0.5 0-0 = 0.5-0 = 0.5 -0.5 0.5 [ β ] = 1 -0.5 0.5 1 0.75 -0.5 -0.5 0.5 0.5 m 1 0.3 [ D ] = 25332084.4 0.3 1 Mpa 0.91 0.35 27837455.4 8351236.615 -0.66667 0.666667 d20x -2.1905E-06 [ σ ] = 8351236.62 27837455.38 -0.66667 0.666667 d20y -6.32891E-07 9743109 -0.66667 -0.66667 0.666667 0.666667 d33x -3.11473E-06 d33y 1.28862E-06 d22x -1.08142E-06 d22y -2.48947E-06 σ x -6.16153E-07 -27.4886 σ y -1.23772E-06 -39.6005 kNm2 σ xy 2.02039E-06 19.68492 m=7 i=5 j=6 y x � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = [D] = � 1−� 2 1 � � 1 1−� 2 Universitas Sumatera Utara 104 Elemen 43 0.5 m 0.5 m 0-0.5 = -0.5 0.5-0 = 0.5 0-0 = 0-0.5 = -0.5 0-0 = 0.5-0 = 0.5 -0.5 0.5 [ β ] = 1 -0.5 0.5 1 0.75 -0.5 -0.5 0.5 0.5 m 1 0.3 [ D ] = 25332084.4 0.3 1 Mpa 0.91 0.35 27837455.38 8351236.615 -0.66667 0.666667 d22x -1.08142E-06 [ σ ] = 8351236.615 27837455.38 -0.66667 0.666667 d22y -2.48947E-06 9743109 -0.66667 -0.66667 0.666667 0.666667 d35x -1.03281E-06 d35y -2.24053E-06 d24x 5.39637E-08 d24y -5.4225E-06 σ x 3.24098E-08 -15.4274 σ y -1.95536E-06 -54.1615 kNm2 σ xy 9.22878E-07 8.991696 m=9 i=7 j=8 y x � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = [D] = � 1−� 2 1 � � 1 1−� 2 Universitas Sumatera Utara 105 Elemen 45 m = 11 0.5 m i = 9 0.5 m j = 10 0-0.5 = -0.5 0.5-0 = 0.5 0-0 = 0-0.5 = -0.5 0-0 = 0.5-0 = 0.5 -0.5 0.5 [ β ] = 1 -0.5 0.5 1 0.75 -0.5 -0.5 0.5 0.5 m 1 0.3 [ D ] = 25332084.4 0.3 1 Mpa 0.91 0.35 27837455.38 8351236.615 -0.66667 0.666667 d24x 5.39637E-08 [ σ ] = 8351236.615 27837455.38 -0.66667 0.666667 d24y -5.4225E-06 9743109 -0.66667 -0.66667 0.666667 0.666667 d37x -8.97067E-07 d37y -7.27857E-06 d26x 3.19857E-06 d26y -1.10742E-05 σ x -6.3402E-07 -49.115 σ y -3.7678E-06 -110.18 kNm2 σ xy 8.59021E-07 8.369532 y x � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = [D] = � 1−� 2 1 � � 1 1−� 2 Universitas Sumatera Utara 106 Elemen 47 m = 13 0.5 m i = 11 0.5 m j = 12 0-0.5 = -0.5 0.5-0 = 0.5 0-0 = 0-0.5 = -0.5 0-0 = 0.5-0 = 0.5 -0.5 0.5 [ β ] = 1 -0.5 0.5 1 0.75 -0.5 -0.5 0.5 0.5 m 1 0.3 [ D ] = 25332084.4 0.3 1 Mpa 0.91 0.35 27837455.38 8351236.615 -0.66667 0.666667 d26x 3.19857E-06 [ σ ] = 8351236.615 27837455.38 -0.66667 0.666667 d26y -1.10742E-05 9743109 -0.66667 -0.66667 0.666667 0.666667 d39x -2.41307E-06 d39y -1.89712E-05 d28x 6.65225E-06 d28y -1.68695E-05 σ x -3.74109E-06 -136.408 σ y -3.86354E-06 -138.794 kNm2 σ xy -2.96224E-06 -28.8614 y x � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = [D] = � 1−� 2 1 � � 1 1−� 2 Universitas Sumatera Utara 107 0.72 1.79 1.55 2.55 1.85 -9.87 50 100 150 200 250 300 -15 -10 -5 5 tinggi lenga n cm tegangan kNm2 tegangan σx pada potongan A---A 2.70 3.49 3.01 9.66 32.66 231.80 500 1000 1500 2000 2500 3000 50 100 150 200 250 tegangan kNm2 tegangan σx potongan B--B -0.03 -3.72 29.61 69.82 194.84 556.03 50 100 150 200 250 300 -200 200 400 600 tegangan kNm2 tegangan σx pada potongan C--C 4.98 -0.95 -21.06 -14.26 -12.26 -34.21 50 100 150 200 250 300 -40 -20 20 tegangan kNm2 tegangan σx potongan D--D 6.84 -0.81 44.45 135.37 310.41 617.23 50 100 150 200 250 300 -200 200 400 600 800 tegangan kNm2 tegangan σx pada potongan E--E 15.57 13.09 -27.49 -15.43 -49.12 -136.41 50 100 150 200 250 300 -150 -100 -50 50 tegangan kNm2 tegangan σx pada potongan F--F -0.22 -5.69 -3.72 -11.33 -20.49 -15.46 50 100 150 200 250 300 -30 -20 -10 ti n g g i len g a n cm tegangan kNm2 tegangan σy pada potongan A---A -13.31 3.78 39.57 54.11 110.08 138.67 50 100 150 200 250 300 -50 50 100 150 tegangan kNm2 tegangan σy pada potongan B--B 9.02 16.75 24.59 48.34 133.53 350.28 50 100 150 200 250 300 100 200 300 400 tegangan kNm2 tegangan σy pada potongan C--C 7.52 15.92 39.78 73.54 195.60 527.20 50 100 150 200 250 300 200 400 600 tegangan kNm2 tegangan σy pada potongan D--D -0.12 7.95 18.00 8.91 2.32 467.58 50 100 150 200 250 300 -200 200 400 600 tegangan kNm2 tegangan σy pada potongan E--E tengah bentang -13.31 3.78 39.57 54.11 110.08 138.67 50 100 150 200 250 300 -50 50 100 150 ting g i leng a n c m tegangan kNm2 tegangan σy pada potongan F--F Grafik 4.1 Tegangan σ x pada potongan potongan melintang A,B,C,D,E,F Grafik 4.2 Tegangan σ y pada potongan potongan melintang A,B,C,D,E,F Universitas Sumatera Utara 108 -1.40 0.95 2.41 1.70 -1.81 -22.08 50 100 150 200 250 300 -30 -20 -10 10 ti n g g i len g a n cm tegangan kNm2 tegangan τxy pada potongan A---A -1.40 0.95 2.41 1.70 -1.81 -22.08 50 100 150 200 250 300 -30 -20 -10 10 tegangan kNm2 tegangan τxy pada potongan B--B 8.34 17.75 23.70 57.93 131.52 289.16 50 100 150 200 250 300 100 200 300 400 tegangan kNm2 tegangan τxy pada potongan C--C 2.36 2.70 -7.29 0.06 -5.61 -90.54 50 100 150 200 250 300 -100 -50 50 tegangan kNm2 tegangan Ϯxy pada potongan D--D 0.94 4.62 -14.62 -22.40 -72.58 222.69 50 100 150 200 250 300 -100 100 200 300 tegangan kNm2 tegangan τxy pada potongan E--E tengah bentang -5.66 6.22 -19.67 -8.98 -8.36 28.84 50 100 150 200 250 300 -40 -20 20 40 tegangan kNm2 tegangan τxy pada potongan F--F Grafik 4.3 Tegangan τ xy pada potongan potongan melintang A,B,C,D,E,F Universitas Sumatera Utara 109 1 1 2 2 3 3 Gambar 4.5 Pengambilan nilai tegangan pada potongan memanjang 1,5,3 Tegangan pada arah y diambil 3 potongan yang mewakili, Universitas Sumatera Utara 110

4.2.5.7 Tegangan Pada potongan potongan 1-- 1 elemen 15,54,36,48,60,75

Pada jendela microsoft excel 2010: Elemen 12 0.5 m 0.5 m 0-0.5 = -0.5 0.5-0 = 0.5 0-0 = 0-0.5 = -0.5 0-0 = 0.5-0 = 0.5 -0.5 0.5 [ β ] = 1 -0.5 0.5 1 0.75 -0.5 -0.5 0.5 0.5 m 1 0.3 [ D ] = 25310000 0.3 1 Mpa 0.91 0.35 27837455 8351237 -0.66667 0.666667 d12x 1.05247E-06 [ σ ] = 8351237 27837455 -0.66667 0.666667 d12y -2.97732E-07 9743109 -0.66667 -0.66667 0.666667 0.666667 d14x 1.23616E-05 d14y 7.19496E-07 d13x 8.15901E-06 d13y 3.63921E-06 σ x 7.54E-06 231.7971 σ y 2.62E-06 136.0265 kNm2 σ xy 5.42E-06 52.76718 m=3 i=1 j=2 y x � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = [D] = � 1−� 2 1 � � 1 1−� 2 Universitas Sumatera Utara 111 Elemen 24 0.5 m 0.5 m 0-0.5 = -0.5 0.5-0 = 0.5 0-0 = 0-0.5 = -0.5 0-0 = 0.5-0 = 0.5 -0.5 0.5 [ β ] = 1 -0.5 0.5 1 0.75 -0.5 -0.5 0.5 0.5 m 1 0.3 [ D ] = 25310000 0.3 1 Mpa 0.91 0.35 27837455 8351237 -0.66667 0.666667 d25x -4.14917E-06 [ σ ] = 8351237 27837455 -0.66667 0.666667 d25y -1.91127E-05 9743109 -0.66667 -0.66667 0.666667 0.666667 d27x 1.7642E-06 d27y -4.97496E-05 d14x 1.23616E-05 d14y 7.19496E-07 σ x 3.94E-06 220.1575 σ y 1.32E-05 400.9738 kNm2 σ xy -9.4E-06 -91.7554 m=5 i=3 j=4 y x � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = [D] = � 1−� 2 1 � � 1 1−� 2 Universitas Sumatera Utara 112 Elemen 36 0.5 m 0.5 m 0-0.5 = -0.5 0.5-0 = 0.5 0-0 = 0-0.5 = -0.5 0-0 = 0.5-0 = 0.5 -0.5 0.5 [ β ] = 1 -0.5 0.5 1 0.75 -0.5 -0.5 0.5 0.5 m 1 0.3 [ D ] = 25310000 0.3 1 Mpa 0.91 0.35 27837455 8351237 -0.66667 0.666667 d26x 3.19857E-06 [ σ ] = 8351237 27837455 -0.66667 0.666667 d26y -1.10742E-05 9743109 -0.66667 -0.66667 0.666667 0.666667 d28x 6.65225E-06 d28y -1.68695E-05 d27x 1.7642E-06 d27y -4.97496E-05 σ x 2.3E-06 -151.231 σ y -2.6E-05 -698.523 kNm2 σ xy -4.8E-06 -46.9596 m=7 i=5 j=6 y x � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = [D] = � 1−� 2 1 � � 1 1−� 2 Universitas Sumatera Utara 113 Elemen 48 0.5 m 0.5 m 0-0.5 = -0.5 0.5-0 = 0.5 0-0 = 0-0.5 = -0.5 0-0 = 0.5-0 = 0.5 -0.5 0.5 [ β ] = 1 -0.5 0.5 1 0.75 -0.5 -0.5 0.5 0.5 m 1 0.3 [ D ] = 25310000 0.3 1 Mpa 0.91 0.35 27837455 8351237 -0.66667 0.666667 d39x -2.41307E-06 [ σ ] = 8351237 27837455 -0.66667 0.666667 d39y -1.89712E-05 9743109 -0.66667 -0.66667 0.666667 0.666667 d41x -3.99519E-06 d41y -4.90167E-05 d28x 6.65225E-06 d28y -1.68695E-05 σ x -1.1E-06 -17.66 σ y 1.4E-06 30.19631 kNm2 σ xy -1.4E-05 -136.275 m=9 i=7 j=8 y x � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = [D] = � 1−� 2 1 � � 1 1−� 2 Universitas Sumatera Utara 114 Elemen 48 0.5 m 0.5 m 0-0.5 = -0.5 0.5-0 = 0.5 0-0 = 0-0.5 = -0.5 0-0 = 0.5-0 = 0.5 -0.5 0.5 [ β ] = 1 -0.5 0.5 1 0.75 -0.5 -0.5 0.5 0.5 m 1 0.3 [ D ] = 25310000 0.3 1 Mpa 0.91 0.35 27837455 8351237 -0.66667 0.666667 d39x -2.41307E-06 [ σ ] = 8351237 27837455 -0.66667 0.666667 d39y -1.89712E-05 9743109 -0.66667 -0.66667 0.666667 0.666667 d41x -3.99519E-06 d41y -4.90167E-05 d28x 6.65225E-06 d28y -1.68695E-05 σ x -1.1E-06 -17.66 σ y 1.4E-06 30.19631 kNm2 σ xy -1.4E-05 -136.275 m=9 i=7 j=8 y x � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = [D] = � 1−� 2 1 � � 1 1−� 2 Universitas Sumatera Utara 115 Elemen 60 m = 11 0.5 m i = 9 0.5 m j = 10 0-0.5 = -0.5 0.5-0 = 0.5 0-0 = 0-0.5 = -0.5 0-0 = 0.5-0 = 0.5 -0.5 0.5 [ β ] = 1 -0.5 0.5 1 0.75 -0.5 -0.5 0.5 0.5 m 1 0.3 [ D ] = 25310000 0.3 1 Mpa 0.91 0.35 27837455 8351237 -0.66667 0.666667 d40x 8.02304E-06 [ σ ] = 8351237 27837455 -0.66667 0.666667 d40y -2.85984E-06 9743109 -0.66667 -0.66667 0.666667 0.666667 d42x 1.7273E-05 d42y -5.32159E-06 d41x -3.99519E-06 d41y -4.90167E-05 σ x 6.17E-06 -85.3137 σ y -3.1E-05 -805.094 kNm2 σ xy -9.7E-06 -94.0534 y x � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = [D] = � 1−� 2 1 � � 1 1−� 2 Universitas Sumatera Utara 116 Elemen 72 m = 13 0.5 m i = 11 0.5 m j = 12 0-0.5 = -0.5 0.5-0 = 0.5 0-0 = 0-0.5 = -0.5 0-0 = 0.5-0 = 0.5 -0.5 0.5 [ β ] = 1 -0.5 0.5 1 0.75 -0.5 -0.5 0.5 0.5 m 1 0.3 [ D ] = 25310000 0.3 1 Mpa 0.91 0.35 27837455 8351237 -0.66667 0.666667 d48x 3.15487E-06 [ σ ] = 8351237 27837455 -0.66667 0.666667 d48y -1.9951E-06 9743109 -0.66667 -0.66667 0.666667 0.666667 d49x 5.14461E-06 d49y -2.9966E-06 d42x 1.7273E-05 d42y -5.3216E-06 σ x 1.33E-06 18.40599 σ y -2.2E-06 -50.6566 kNm2 σ xy 8.74E-06 85.19823 y x � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = [D] = � 1−� 2 1 � � 1 1−� 2 Universitas Sumatera Utara 117

4.2.5.8 Tegangan Pada potongan potongan 2-- 2 elemen 7,19,31,43,55,63

Pada jendela microsoft excel 2010: Elemen 7 0.5 m 0.5 m 0-0.5 = -0.5 0.5-0 = 0.5 0-0 = 0-0.5 = -0.5 0-0 = 0.5-0 = 0.5 -0.5 0.5 [ β ] = 1 -0.5 0.5 1 0.75 -0.5 -0.5 0.5 0.5 m 1 0.3 [ D ] = 25310000 0.3 1 Mpa 0.91 0.35 27837455 8351237 -0.66667 0.666667 d7x -5.09369E-08 [ σ ] = 8351237 27837455 -0.66667 0.666667 d7y 7.39768E-07 9743109 -0.66667 -0.66667 0.666667 0.666667 d8x -1.01412E-07 d8y 1.25217E-07 d9x 3.01403E-07 d9y 1.36595E-06 σ x -3.4E-08 2.54952 σ y 4.17E-07 11.33983 kNm2 σ xy -1.7E-07 -1.70317 m=3 i=1 j=2 y x � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = [D] = � 1−� 2 1 � � 1 1−� 2 Universitas Sumatera Utara 118 Elemen 19 0.5 m 0.5 m 0-0.5 = -0.5 0.5-0 = 0.5 0-0 = 0-0.5 = -0.5 0-0 = 0.5-0 = 0.5 -0.5 0.5 [ β ] = 1 -0.5 0.5 1 0.75 -0.5 -0.5 0.5 0.5 m 1 0.3 [ D ] = 25310000 0.3 1 Mpa 0.91 0.35 27837455 8351237 -0.66667 0.666667 d8x -1.01412E-07 [ σ ] = 8351237 27837455 -0.66667 0.666667 d8y 1.25217E-07 9743109 -0.66667 -0.66667 0.666667 0.666667 d21x -1.54423E-06 d21y -3.63322E-06 d10x 4.67537E-08 d10y 3.45144E-08 σ x -9.6E-07 -27.2812 σ y -6E-08 -9.71614 kNm2 σ xy -2.4E-06 -23.4502 m=5 i=3 j=4 y x � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = [D] = � 1−� 2 1 � � 1 1−� 2 Universitas Sumatera Utara 119 Elemen 31 0.5 m 0.5 m 0-0.5 = -0.5 0.5-0 = 0.5 0-0 = 0-0.5 = -0.5 0-0 = 0.5-0 = 0.5 -0.5 0.5 [ β ] = 1 -0.5 0.5 1 0.75 -0.5 -0.5 0.5 0.5 m 1 0.3 [ D ] = 25310000 0.3 1 Mpa 0.91 0.35 27837455 8351237 -0.66667 0.666667 d21x -1.54423E-06 [ σ ] = 8351237 27837455 -0.66667 0.666667 d21y -3.63322E-06 9743109 -0.66667 -0.66667 0.666667 0.666667 d22x -1.08142E-06 d22y -2.48947E-06 d23x -2.69777E-06 d23y -7.73805E-06 σ x 3.09E-07 -14.2647 σ y -2.7E-06 -73.602 kNm2 σ xy -6.5E-09 -0.06355 m=7 i=5 j=6 y x � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = [D] = � 1−� 2 1 � � 1 1−� 2 Universitas Sumatera Utara 120 Elemen 43 0.5 m 0.5 m 0-0.5 = -0.5 0.5-0 = 0.5 0-0 = 0-0.5 = -0.5 0-0 = 0.5-0 = 0.5 -0.5 0.5 [ β ] = 1 -0.5 0.5 1 0.75 -0.5 -0.5 0.5 0.5 m 1 0.3 [ D ] = 25310000 0.3 1 Mpa 0.91 0.35 27837455 8351237 -0.66667 0.666667 d22x -1.08142E-06 [ σ ] = 8351237 27837455 -0.66667 0.666667 d22y -2.48947E-06 9743109 -0.66667 -0.66667 0.666667 0.666667 d35x -1.03281E-06 d35y -2.24053E-06 d24x 5.39637E-08 d24y -5.4225E-06 σ x 3.24E-08 -15.4274 σ y -2E-06 -54.1615 kNm2 σ xy 9.23E-07 8.991696 m=9 i=7 j=8 y x � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = [D] = � 1−� 2 1 � � 1 1−� 2 Universitas Sumatera Utara 121 Elemen 55 m = 11 0.5 m i = 9 0.5 m j = 10 0-0.5 = -0.5 0.5-0 = 0.5 0-0 = 0-0.5 = -0.5 0-0 = 0.5-0 = 0.5 -0.5 0.5 [ β ] = 1 -0.5 0.5 1 0.75 -0.5 -0.5 0.5 0.5 m 1 0.3 [ D ] = 25310000 0.3 1 Mpa 0.91 0.35 27837455 8351237 -0.66667 0.666667 d35x -1.03281E-06 [ σ ] = 8351237 27837455 -0.66667 0.666667 d35y -2.24053E-06 9743109 -0.66667 -0.66667 0.666667 0.666667 d36x 1.31006E-06 d36y -7.0422E-08 d37x -8.97067E-07 d37y -7.27857E-06 σ x 1.56E-06 15.43044 σ y -3.4E-06 -80.4535 kNm2 σ xy 1.54E-06 14.97745 y x � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = [D] = � 1−� 2 1 � � 1 1−� 2 Universitas Sumatera Utara 122 Elemen 63 m = 13 0.5 m i = 11 0.5 m j = 12 0-0.5 = -0.5 0.5-0 = 0.5 0-0 = 0-0.5 = -0.5 0-0 = 0.5-0 = 0.5 -0.5 0.5 [ β ] = 1 -0.5 0.5 1 0.75 -0.5 -0.5 0.5 0.5 m 1 0.3 [ D ] = 25310000 0.3 1 Mpa 0.91 0.35 27837455 8351237 -0.66667 0.666667 d36x 1.31006E-06 [ σ ] = 8351237 27837455 -0.66667 0.666667 d36y -7.0422E-08 9743109 -0.66667 -0.66667 0.666667 0.666667 d46x 5.23274E-07 d46y 2.18616E-07 d38x 3.66206E-06 d38y -1.0878E-06 σ x -5.2E-07 -20.2655 σ y -6.8E-07 -23.2607 kNm2 σ xy 1.76E-06 17.15462 y x Universitas Sumatera Utara 123

4.2.5.9 Tegangan Pada potongan potongan 3-- 3 elemen 1,13,25,37,49,61

Pada jendela microsoft excel 2010: Elemen 1 0.5 m 0.5 m 0-0.5 = -0.5 0.5-0 = 0.5 0-0 = 0-0.5 = -0.5 0-0 = 0.5-0 = 0.5 -0.5 0.5 [ β ] = 1 -0.5 0.5 1 0.75 -0.5 -0.5 0.5 0.5 m 1 0.3 [ D ] = 25310000 0.3 1 Mpa 0.91 0.35 27837455 8351237 -0.66667 0.666667 d1x [ σ ] = 8351237 27837455 -0.66667 0.666667 d1y 9743109 -0.66667 -0.66667 0.666667 0.666667 d2x -1.67319E-07 d2y 7.03675E-08 d3x -1.28488E-07 d3y 2.41837E-07 σ x -1.1E-07 -1.75874 σ y 1.61E-07 3.556535 kNm2 σ xy -3.9E-08 -0.37752 m=3 i=1 j=2 y x � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = [D] = � 1−� 2 1 � � 1 1−� 2 Universitas Sumatera Utara 124 Elemen 13 0.5 m 0.5 m 0-0.5 = -0.5 0.5-0 = 0.5 0-0 = 0-0.5 = -0.5 0-0 = 0.5-0 = 0.5 -0.5 0.5 [ β ] = 1 -0.5 0.5 1 0.75 -0.5 -0.5 0.5 0.5 m 1 0.3 [ D ] = 25310000 0.3 1 Mpa 0.91 0.35 27837455 8351237 -0.66667 0.666667 d2x -1.67319E-07 [ σ ] = 8351237 27837455 -0.66667 0.666667 d2y 7.03675E-08 9743109 -0.66667 -0.66667 0.666667 0.666667 d15x -1.97538E-07 d15y -1.89149E-07 d4x -1.23455E-07 d4y 7.06207E-08 σ x -2E-08 -0.5594 σ y 1.69E-10 -0.16354 kNm2 σ xy -1.4E-07 -1.40075 m=5 i=3 j=4 y x � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = [D] = � 1−� 2 1 � � 1 1−� 2 Universitas Sumatera Utara 125 Elemen 25 0.5 m 0.5 m 0-0.5 = -0.5 0.5-0 = 0.5 0-0 = 0-0.5 = -0.5 0-0 = 0.5-0 = 0.5 -0.5 0.5 [ β ] = 1 -0.5 0.5 1 0.75 -0.5 -0.5 0.5 0.5 m 1 0.3 [ D ] = 25310000 0.3 1 Mpa 0.91 0.35 27837455 8351237 -0.66667 0.666667 d15x -1.97538E-07 [ σ ] = 8351237 27837455 -0.66667 0.666667 d15y -1.89149E-07 9743109 -0.66667 -0.66667 0.666667 0.666667 d16x 2.31103E-07 d16y -6.20872E-08 d17x -6.88378E-07 d17y -7.2339E-07 σ x 2.86E-07 4.980465 σ y -3.6E-07 -7.52815 kNm2 σ xy -2.4E-07 -2.36289 m=7 i=5 j=6 y x � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = [D] = � 1−� 2 1 � � 1 1−� 2 Universitas Sumatera Utara 126 Elemen 37 0.5 m 0.5 m 0-0.5 = -0.5 0.5-0 = 0.5 0-0 = 0-0.5 = -0.5 0-0 = 0.5-0 = 0.5 -0.5 0.5 [ β ] = 1 -0.5 0.5 1 0.75 -0.5 -0.5 0.5 0.5 m 1 0.3 [ D ] = 25310000 0.3 1 Mpa 0.91 0.35 27837455 8351237 -0.66667 0.666667 d16x 2.31103E-07 [ σ ] = 8351237 27837455 -0.66667 0.666667 d16y -6.20872E-08 9743109 -0.66667 -0.66667 0.666667 0.666667 d29x 6.23046E-07 d29y 5.83427E-07 d18x -4.61603E-07 d18y -1.76281E-07 σ x 2.61E-07 6.638015 σ y -7.6E-08 0.062889 kNm2 σ xy -3.1E-08 -0.30653 m=9 i=7 j=8 y x � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = [D] = � 1−� 2 1 � � 1 1−� 2 Universitas Sumatera Utara 127 Elemen 49 m = 11 0.5 m i = 9 0.5 m j = 10 0-0.5 = -0.5 0.5-0 = 0.5 0-0 = 0-0.5 = -0.5 0-0 = 0.5-0 = 0.5 -0.5 0.5 [ β ] = 1 -0.5 0.5 1 0.75 -0.5 -0.5 0.5 0.5 m 1 0.3 [ D ] = 25310000 0.3 1 Mpa 0.91 0.35 27837455 8351237 -0.66667 0.666667 d29x 6.23046E-07 [ σ ] = 8351237 27837455 -0.66667 0.666667 d29y 5.83427E-07 9743109 -0.66667 -0.66667 0.666667 0.666667 d30x 8.05836E-07 d30y 5.61995E-07 d31x 3.80907E-07 d31y 5.93875E-07 σ x 1.22E-07 3.450457 σ y 6.97E-09 1.21158 kNm2 σ xy -1.8E-07 -1.712 y x � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = [D] = � 1−� 2 1 � � 1 1−� 2 Universitas Sumatera Utara 128 Elemen 61 m = 13 0.5 m i = 11 0.5 m j = 12 0-0.5 = -0.5 0.5-0 = 0.5 0-0 = 0-0.5 = -0.5 0-0 = 0.5-0 = 0.5 -0.5 0.5 [ β ] = 1 -0.5 0.5 1 0.75 -0.5 -0.5 0.5 0.5 m 1 0.3 [ D ] = 25310000 0.3 1 Mpa 0.91 0.35 27837455 8351237 -0.66667 0.666667 d30x 8.05836E-07 [ σ ] = 8351237 27837455 -0.66667 0.666667 d30y 5.61995E-07 9743109 -0.66667 -0.66667 0.666667 0.666667 d43x d43y d32x 1.22374E-06 d32y 4.97918E-07 σ x -5.4E-07 -15.3117 σ y -4.3E-08 -5.67564 kNm2 σ xy -9.6E-08 -0.93596 y x � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = � � = � � − � � = [D] = � 1−� 2 1 � � 1 1−� 2 Universitas Sumatera Utara 129 Grafik 4.4 Tegangan τ xy pada potongan potongan memanjang 1,5,3 16.67 66.67 116.67 166.67 216.67 266.67 -1000 1000 50 100 150 200 250 300 ti n g g i l e n g a n c m tegangan kNm2 tegangan σy pada potongan 1--1 33.33 83.33 133.33 183.33 233.33 283.33 -100 -50 50 50 100 150 200 250 300 ti n g g i len g a n cm tegangan kNm2 tegangan σy pada potongan 2--2 33.33 83.33 133.33 183.33 233.33 283.33 -10 -5 5 50 100 150 200 250 300 ti n g g i len g a n cm tegangan kNm2 tegangan σy pada potongan 3--3 Universitas Sumatera Utara 130

4.3 Dengan metode Heft 540 sebagai control

Beban yang bekerja P = 400 kN Panjang L = 3 meter Lebar h = 3 meter Modulus Elastisitas beton E = 47000 �350�0,083 = 25332,0844 � �� 2 = 25332084.4 kNm 2 Poissons ratio v = 0.3 Perletakan = Sendi - rol tebal t = 0.5 meter panjang perletakan c = 0.3 meter ℎ � = 3 3 − 0,3 = 1,11 � � = 0,3 3 = 0,1 langkah 1 : dengan memakai tabel yang tersedia Tabel 4.1 ringkasan nama dan sistem perletakan berserta gambarnya M. R ὄsler, 5005 Universitas Sumatera Utara 131 Tabel 4.2 hasil kekuatan tarik pada suatu balok tinggi dengan dua tumpuan M. R ὄsler, 5005 Dari tabel : ZsP = 0,04 ZFP = 0.54 �� = 0,54�800 �� = 195 �� � ′ � = 0,04�800 �� = 35 �� Universitas Sumatera Utara 132 3000 mm 3000 mm 500 mm 400 kN 500 mm 500 mm 500 mm 500 mm 500 mm 500 mm 500 mm 500 mm 500 mm 500 mm 500 mm 500 mm 400 kN Gambar 4.6 Pemodelan tinggi lengan Z’s ZF Universitas Sumatera Utara 133 6.84 -0.81 44.41 135.25 310.14 616.69 50 100 150 200 250 300 -200 200 400 600 800 ti ng g i le ng a n c m tegangan kNm2

4.4 Grafik dan perbandingan

Nilai ZF dan Z’P dibandingkan dengan nilai tegangan maksimum yang terjadi pada tengah bentang, 3000 mm 3000 mm 500 mm 400 kN 500 mm 500 mm 500 mm 500 mm 500 mm 500 mm 500 mm 500 mm 500 mm 500 mm 500 mm 400 kN Ya = 245 cm Yb = 55 cm Gambar 4.7 Sketsa Pendekatan distribusi tegangan pada tengah bentang σ x pada tinggi lengan 545 cm � � = � � = 195 �� � 1 5 � 5,45 � 1 6 0,53 5 = 313,6 �� � 5 σ x pada tinggi lengan 55 cm � � = � � = 195 �� � 1 5 � 0,55 � 1 6 0,53 5 = 11,73 �� � 5 Untuk mengecek hasil dari tegangan-tegangan di tengah bentang secara analitis untuk tegangan-tegangan utama dapat dicari dengan rumus : Universitas Sumatera Utara 134 � 1 = � � + � � 2 + �� � � − � � 2 � 2 + � �� 2 = � ��� � 1 = � � + � � 2 − �� � � − � � 2 � 2 + � �� 2 = � ��� Hasil yang didapat dari metode elemen hingga untuk tegangan pada tengah bentang pada tiap ketinggian yang dibagi kedalam 6 titik adalah : 1 5 3 4 5 6 33,33 cm 83,33 cm 133,33 cm 183,33 cm 533,33 cm 583,33 cm σx kNm 5 -6.83859 0.815137 -44.4139 - 135.555 -310.143 -616.691 σykNm 5 -0.15465 7.948175 18.00543 8.9144 5.319364 467.5768 τxykNm 5 0.944891 4.650761 -14.6503 - 55.3957 -75.5755 555.6956 Tabel 4.3 tegangan yang terjadi pada 6 titik Perhitungan secara analitis : � 1 = � � + � � 2 + �� � � − � � 2 � 2 + � �� 2 = � ��� � 1 = −6.83859 + −0.15465 5 + �� −6.83859 − −0.15465 5 � 5 + 0.944891 5 � 1 = 0.005797454 �� � 5 Universitas Sumatera Utara 135 � 1 = −6.83859 + −0.15465 5 − �� −6.83859 − −0.15465 5 � 5 + 0.944891 5 � 1 = −6.968744904 �� � 5 Untuk seterusnya perhitungan dimasukkan kedalam tabel : 1 5 3 4 5 6 33,33 cm 83,33 cm 133,33 cm 183,33 cm 533,33 cm 583,33 cm σ5kNm 5 0.005797 10.51815 51.55734 15.31334 18.35364 511.5356 σ1kNm 5 -6.96874 -1.45784 -47.6688 -138.651 -356.177 -660.647 Tabel 4.4 tegangan yang didapat dengan rumus tegangan utama Perbandingan metode perhitungan : Elemen segitiga Metode heft 540 Secara analitis σxkNm 5 -310,143 -313,6 -356.177 Tabel 4.5 perbandingan tegangan dalam tiga metode Universitas Sumatera Utara 136 -0.00006 -0.00004 -0.00002 0.00002 50 100 150 200 250 300 350 len d u ta n m eter panjang bentang cm lendutan arah x pada potongan 1--1 -0.00000 -0.00000 0.000002 50 100 150 200 250 300 350 len d u ta n m eter panjang bentang cm lendutan arah x pada potongan 2--2 -4E-07 -2E-07 0.000000 0.000000 0.000000 0.000000 50 100 150 200 250 300 350 len d u ta n m eter panjang bentang cm lendutan arah x pada potongan 3--3 Grafik 4.6 Lendutan yang terjadi pada atas 1—1 , tengah 2 –2 , dan bawah 3 – 3 Universitas Sumatera Utara 137

4.5 Kontrol Lendutan