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.
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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
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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
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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
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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
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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,
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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
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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
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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
d2x d2y
0.5875 0.5375 0.2 -0.2125
-0.2 d3x
-0.0875 0.3
0.0875 0 -0.0875
-0.5 d3y
0 0.0875 0.675 0.1625
-0.5 0.1625 -0.0875 -0.075 d4x
0.2 0.1625 0.7125
-0.2 -0.2125 -0.0875 -0.25
d4y -0.2125 -0.0875
-0.5 -0.2
1.425 0.0375 0.075
-0.125 -0.125
0.125 d5x
-0.2 -0.5
0.1625 -0.2125 0.0375
1.675 0.0875 -0.25
d5y 0.1625 -0.0875
0 0.0875 0.5875 0.1625 -0.5
0.1625 -0.0875 -0.075
d6x -0.075 -0.125
0.2 0 0.1625 0.4625
-0.2 -0.2125 -0.0875
-0.25 d6y
0 -0.2125 -0.0875 -0.5
-0.2 1.425
0.0375 0.2 -0.2125
-0.2 d7x
-0.2 -0.5 0.1625 -0.2125
0.0375 1.675
0.0875 0 -0.0875
-0.5 d7y
0 -0.0875 -0.0875 0.0875
0.675 0.1625 -0.5 0.1625
-0.0875 -0.075 d8x
0 -0.075 -0.25
0.2 0.1625 0.7125
-0.2 -0.2125 -0.0875
-0.25 d8y
0 -0.2125 -0.0875 -0.5
-0.2 1.425 0.0375
0.2 -0.2125
-0.2 d9x
-0.2 -0.5
0.1625 -0.2125 0.0375 1.675
0.0875 -0.0875
-0.5 d9y
-0.0875 -0.0875 0 0.0875
0.675 0.1625 -0.5 0.1625 -0.0875 -0.075
d10x -0.075
-0.25 0.2
0.1625 0.7125 -0.2 -0.2125 -0.0875
-0.25 d10y
0 -0.2125 -0.0875 -0.5
-0.2 1.425 0.0375
0.2 -0.2125 -0.2
d11x -0.2
-0.5 0.1625 -0.2125
0.0375 1.675 0.0875
0 -0.0875 -0.5
d11y -0.0875 -0.0875
0 0.0875 0.675 0.1625
-0.5 0.1625 -0.0875 -0.075
d12x -0.075
-0.25 0.2
0 0.1625 0.7125 -0.2 -0.2125 -0.0875
-0.25 d12y
-0.2125 -0.0875 -0.5
-0.2 1.425
0.0375 0.2 -0.2125
-0.2 0 0.0875
-0.25 -0.0875 d13x
-0.2 -0.5 0.1625 -0.2125 0.0375
1.675 0.0875
0 -0.0875 -0.5
0.2 0 -0.075 -0.0875
d13y 0 0.0875 -0.0875 -0.0875
0.3375 -0.25
0.25 d14x
0.075 0.05
-0.25 0.375
-0.125 -0.125
d14y 0 -0.2125 -0.0875
-0.25 -0.125 0.7125
-0.125 0 0.0875
0.0875 d15x
-0.2 -0.5
0.25 -0.125 -0.125
1 0.2
0.075 d15y
-0.25 -0.075 0 -0.0875
0.7125 0.1625
-0.25 -0.0875
-0.125 -0.125
d16x -0.0875 -0.0875
-0.25 0.1625
0.675 -0.075
-0.0875 -0.25
d16y -0.25
-0.075 0.7125
0.1625 0.2
-0.2125 -0.2
-0.25 -0.0875
d17x -0.0875
-0.0875 0.1625
0.675 0.0875
-0.0875 -0.5
-0.075 -0.0875
d17y -0.5
-0.2 0.2
-0.2125 -0.0875
0.0875 1.425
0.0375 -0.5
0.1625 -0.2125
-0.2 d18x
0.1625 -0.2125 0.0875
-0.2 -0.5
0.2 0.0375
1.675 -0.2
-0.2125 -0.0875
-0.5 d18y
-0.2125 -0.0875
-0.5 -0.2
1.425 0.0375
0.2 -0.2125
-0.2 0.0875
-0.5 0.1625
d19x -0.2
-0.5 0.1625
-0.2125 0.0375
1.675 0.0875
-0.0875 -0.5
0.2 -0.2
-0.2125 d19y
-0.25 -0.075
0.2 -0.2125
-0.0875 -0.25
-0.0375 1.425
0.0375 -0.5
0.1625 -0.2125 -0.2
d20x 0 -0.0875 -0.0875
0 0.0875 -0.2
-0.5 0.45
-0.125 0.0375
1.675 -0.2
-0.2125 -0.0875 -0.5
d20y -0.2125
-0.0875 -0.5
-0.2 1.3375
0.0375 0.0875
0.2 -0.2125 -0.2
0.0875 -0.5
0.1625 d21x
-0.2 -0.5
0.1625 -0.2125
0.0375 1.425
0.0875 0.25 -0.0875
-0.5 0.2
-0.2 -0.2125
d21y -0.5
-0.2 0.2
-0.2125 -0.0875
0.0875 1.175
0.0375 -0.5
0.1625 -0.2125 -0.2
0.25 d22x
0 0.1625 -0.2125 0.0875
-0.2 -0.5
0.2 0.0375
1.55 -0.2
-0.2125 -0.0875 -0.5
0.125 d22y
-0.2125 -0.0875
-0.5 -0.2 1.5875
0.0375 0.2 -0.2125
-0.2 0.0875
-0.5 0.1625
d23x -0.2
-0.5 0.1625
-0.2125 0.0375 1.5125
0.0875 0 -0.0875
-0.5 0.2
-0.2 -0.2125
d23y -0.5
-0.2 0.2
0 -0.2125 -0.0875
0.0875 1.425 0.0375
-0.5 0.1625 -0.2125
-0.2 d24x
0.1625 -0.2125 0 0.0875
-0.2 -0.5
0.2 0.0375
1.675 -0.2
-0.2125 -0.0875 -0.5
d24y 0 -0.2125
-0.0875 -0.5
-0.2 1.425
0.0375 0.2 -0.2125
-0.2 0.0875
-0.5 0.1625
d25x
7875000
-0.2 -0.5
0.1625 -0.2125 0.0375 1.675 0.0875
0 -0.0875 -0.5
0.2 -0.2 -0.2125
d25y
0.6825
-0.5 -0.2
0.2 0 -0.2125 -0.0875
0.0875 1.425 0.0375
-0.5 0.1625 -0.2125
-0.2 d26x
0 0.1625 -0.2125 0.0875
-0.2 -0.5
0.2 0 0.0375
1.675 -0.2 -0.2125 -0.0875
-0.5 d26y
0 -0.2125 -0.0875
-0.5 -0.2
1.425 0.0375
0.2 -0.2125 -0.2
0.0875 -0.5
0.1625 d27x
-0.2 -0.5 0.1625 -0.2125 0.0375
1.675 0.0875 0 -0.0875
-0.5 0.2
-0.2 -0.2125 d27y
-0.25 -0.125
0 -0.2125 -0.0875 0.0875 0.7125
-0.125 -0.25
0.25 d28x
0.25 -0.125
-0.2 -0.5
0.2 0 -0.125
1 -0.125 -0.125
-400 d28y
0 -0.2125 -0.0875 -0.25
-0.125 0.7125 -0.125
0.0875 -0.25
0.25 d29x
-0.2 -0.5
0.25 -0.125 -0.125
1 0.2
0 -0.125 -0.125 d29y
-0.25 -0.075
0.2 0.7125
0.1625 -0.25
-0.0875 -0.2125
-0.2 d30x
-0.0875 -0.0875
0.0875 0.1625
0.675 -0.075
-0.0875 -0.0875
-0.5 d30y
-0.25 -0.075
0.7125 0.1625
0.2 -0.2125
-0.2 -0.25
-0.0875 d31x
-0.0875 -0.0875
0.1625 0.675
0.0875 -0.0875
-0.5 -0.075
-0.0875 d31y
-0.5 -0.2
0.2 -0.2125
-0.0875 0.0875
1.425 0.0375
-0.625 0.0375
-0.0875 -0.075
d32x 0.1625
-0.2125 0.0875
-0.2 -0.5
0.2 0.0375
1.675 -0.2
-0.4625 -0.0875
-0.25 d32y
0.2125 0.1625
-0.5 -0.2
1.25 -0.0375
0.25 0.125
-0.2125 -0.2
-0.25 -0.0875
-0.5 0.1625
d33x 0.0375
0.0875 0.1625
-0.2125 -0.05
1.175 0.0875
-0.0875 -0.5
0.05 -0.0875
-0.2 -0.2125
d33y -0.5
-0.2 0.2
-0.2125 -0.0875
0.0875 1.05
1.425 0.0375
0.1625 -0.2125 -0.2
d34x 0.1625
-0.2125 0 0.0875
-0.2 -0.5
0.2 0.1625
0.0375 1.675
-0.2125 -0.0875 -0.5
d34y -0.2125
-0.0875 -0.5
-0.2 1.425
0.0375 0.2 -0.2125
-0.2 0.0875
-0.5 0.1625
d35x -0.2
-0.5 0.1625
-0.2125 0.0375
1.675 0.0875 0 -0.0875
-0.5 0.2
-0.2 -0.2125
d35y -0.5
-0.2 0.2
-0.2125 -0.0875
0.0875 1.425
0.0375 -0.5
0.1625 -0.2125 -0.2
d36x 0 0.1625
-0.2125 0 0.0875
-0.2 -0.5
0.2 0 0.0375
1.675 -0.2 -0.2125 -0.0875
-0.5 d36y
-0.2125 -0.0875
-0.5 -0.2
1.425 0.0375
0.2 -0.2125 -0.2
0.0875 -0.5
0.1625 d37x
-0.2 -0.5 0.1625
-0.2125 0.0375 1.675 0.0875
0 -0.0875 -0.5
0.2 -0.2 -0.2125
d37y -0.5
-0.2 0.2
0 -0.2125 -0.0875
0.0875 1.425
0.0375 -0.5
0.1625 -0.2125 -0.2
d38x 0 0.1625
-0.2125 0 0.0875
-0.2 -0.5
0.2 0 0.0375
1.675 -0.2 -0.2125 -0.0875
-0.5 d38y
0 -0.2125 -0.0875 -0.5
-0.2 1.425
0.0375 0.2 -0.2125
-0.2 0.0875
-0.5 0.1625
d39x -0.2
-0.5 0.1625 -0.2125 0.0375 1.675 0.0875
0 -0.0875 -0.5
0.075 0.125
-0.2 -0.2125 d39y
-0.5 -0.2
0.2 0 -0.2125 -0.0875
0.0875 1.425
0.0375 -0.5
0.1625 -0.2125 -0.2
d40x 0 0.1625 -0.2125
0 0.0875 -0.2
-0.5 0.2
0 0.0375 1.675
-0.2 -0.2125 -0.0875 -0.5
d40y 0 -0.2125 -0.0875
-0.5 -0.2
1.425 0.0375
0.2 -0.2125 -0.2
0.0875 -0.5
0.1625 d41x
-0.2 -0.5 0.1625 -0.2125 0.0375
1.675 0.0875 0 -0.0875
-0.5 0.2
-0.2 -0.2125 d41y
-0.25 -0.125
0 -0.2125 -0.0875 0.0875 0.7125 -0.125
-0.25 0.25
d42x 0.25
-0.125 -0.2
-0.5 0.2
0 -0.125 1 -0.125
-0.125 -400
d42y 0 -0.2125 -0.0875
-0.25 0.25 0.7125
-0.125 0.0875
d43x -0.2
-0.5 -0.125 -0.125 -0.125 1
0.2 d43y
-0.25 -0.075
0.2 0.375
-0.125 -0.125
d44x -0.0875
-0.0875 0.0875
0.3375 -0.25
d44y -0.5
-0.2 0.2
-0.125 0.75
-0.125 -0.125
-0.125 d45x
0.1625 -0.2125
0.0875 -0.125
-0.25 -0.125
0.9625 -0.25
d45y -0.5
-0.2125 0 -0.0875
0.05 -0.125
0.8375 0.0375 -0.125
-0.125 d46x
0.175 -0.2125
0 -0.0875 -0.25
-0.125 -0.25
0.0375 1.2125
-0.25 d46y
-0.5 -0.2
0.2 -0.125
0.75 -0.125 -0.125
-0.125 d47x
0 0.1625 -0.2125 0 0.0875
-0.125 -0.25 -0.125
0.9625 -0.25
d47y -0.5
-0.2 0.2
0 -0.125 0.75
-0.125 -0.125 -0.125
d48x 0 0.1625 -0.2125
0 0.0875 0 -0.125
-0.25 -0.125 0.9625
-0.25 d48y
-0.5 -0.2
0.075 0 -0.125
0.75 -0.125
0 0.1625 -0.2125 0 0.0875
0 -0.125 -0.25 -0.125
0.9625
- 1
Universitas Sumatera Utara
69
d2x 1.930606435
d2x 1.67319E-07
d2y -0.811933141
d2y -7.03675E-08
d3x 1.482553079
d3x 1.28488E-07
d3y -2.790426168
d3y -2.41837E-07
d4x 1.424483536
d4x 1.23455E-07
d4y -0.814853841
d4y -7.06207E-08
d5x 1.199089525
d5x 1.03921E-07
d5y -6.311212328
d5y -5.46972E-07
d6x 0.904195257
d6x 7.83636E-08
d6y -1.411263991
d6y -1.2231E-07
d7x 0.587733918
d7x 5.09369E-08
d7y -8.535789311
d7y -7.39768E-07
d8x 1.170140552
d8x 1.01412E-07
d8y -1.444811124
d8y -1.25217E-07
d9x -3.477726878
d9x -3.01403E-07
d9y -15.76094939
d9y -1.36595E-06
d10x -0.539465838
d10x -4.67537E-08
d10y -0.398242539
d10y -3.45144E-08
d11x -22.05795839
d11x -1.91169E-06
d11y -29.3932824
d11y -2.54742E-06
d12x -12.14392321
d12x -1.05247E-06
d12y 3.43536856
d12y 2.97732E-07
d13x -94.14247393
d13x -8.15901E-06
d13y -41.99089183
d13y -3.63921E-06
d14x -142.6338757
d14x -1.23616E-05
d14y -8.301881206
d14y -7.19496E-07
d15x 2.27928342
d15x 1.97538E-07
d15y 2.182489056
d15y 1.89149E-07
d16x -2.666572209
d16x -2.31103E-07
d16y 0.716391339
d16y 6.20872E-08
d17x 7.942822728
d17x 6.88378E-07
d17y 8.346807556
d17y 7.2339E-07
d18x 5.326192069
d18x 4.61603E-07
d18y 2.034015189
d18y 1.76281E-07
d19x 19.04762087
d19x 1.65079E-06
d19y 19.03592298
d19y 1.64978E-06
d20x 25.27501966
d20x 2.1905E-06
d20y 7.302592777
d20y 6.32891E-07
d21x 17.81799135
d21x 1.54423E-06
d21y 41.9217993
d21y 3.63322E-06
d22x 12.47793091
d22x 1.08142E-06
d22y 28.72461596
d22y 2.48947E-06
d23x 31.12807125
d23x 2.69777E-06
d23y 89.28518176
d23y 7.73805E-06
d24x -0.622658543
d24x -5.39637E-08
d24y 62.56731542
d24y 5.4225E-06
d25x
7875000
47.87507183 d25x
4.14917E-06 d25y
0.6825
220.5306977 d25y
1.91127E-05 d26x
-36.90653047 d26x
-3.19857E-06 d26y
127.7787123 d26y
1.10742E-05 d27x
-20.35614645 d27x
-1.7642E-06 d27y
574.0342441 d27y
4.97496E-05 d28x
-76.7567271 d28x
-6.65225E-06 d28y
194.647602 d28y
1.68695E-05 d29x
-7.188986558 d29x
-6.23046E-07 d29y
-6.731855335 d29y
-5.83427E-07 d30x
-9.298112022 d30x
-8.05836E-07 d30y
-6.484556068 d30y
-5.61995E-07 d31x
-4.395080692 d31x
-3.80907E-07 d31y
-6.852406834 d31y
-5.93875E-07 d32x
-14.12002033 d32x
-1.22374E-06 d32y
-5.745209879 d32y
-4.97918E-07 d33x
35.9392077 d33x
3.11473E-06 d33y
-14.86867693 d33y
-1.28862E-06 d34x
0.908357885 d34x
7.87244E-08 d34y
-9.583200444 d34y
-8.30544E-07 d35x
11.91699241 d35x
1.03281E-06 d35y
25.85232551 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
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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 �
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− �
�
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[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 �
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− �
�
=
[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
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− �
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[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
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[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
�
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− �
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=
[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
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=
[D] =
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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
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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
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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
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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 �
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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
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=
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
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=
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
�
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= �
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= �
�
− �
�
=
[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 �
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= �
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= �
�
− �
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=
[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 �
�
= �
�
− �
�
= �
�
= �
�
− �
�
= �
�
= �
�
− �
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= �
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= �
�
− �
�
= �
�
= �
�
− �
�
= �
�
= �
�
− �
�
=
[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 �
�
= �
�
− �
�
= �
�
= �
�
− �
�
= �
�
= �
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− �
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= �
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= �
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− �
�
= �
�
= �
�
− �
�
= �
�
= �
�
− �
�
=
[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 �
�
= �
�
− �
�
= �
�
= �
�
− �
�
= �
�
= �
�
− �
�
= �
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= �
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− �
�
= �
�
= �
�
− �
�
= �
�
= �
�
− �
�
=
[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 �
�
= �
�
− �
�
= �
�
= �
�
− �
�
= �
�
= �
�
− �
�
= �
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= �
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− �
�
= �
�
= �
�
− �
�
= �
�
= �
�
− �
�
=
[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
�
�
= �
�
− �
�
= �
�
= �
�
− �
�
= �
�
= �
�
− �
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= �
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= �
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− �
�
= �
�
= �
�
− �
�
= �
�
= �
�
− �
�
=
[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
�
�
= �
�
− �
�
= �
�
= �
�
− �
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= �
�
= �
�
− �
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= �
�
= �
�
− �
�
= �
�
= �
�
− �
�
= �
�
= �
�
− �
�
=
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
�
�
= �
�
− �
�
= �
�
= �
�
− �
�
= �
�
= �
�
− �
�
= �
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= �
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− �
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= �
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= �
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− �
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= �
�
= �
�
− �
�
=
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
�
�
= �
�
− �
�
= �
�
= �
�
− �
�
= �
�
= �
�
− �
�
= �
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= �
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− �
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= �
�
= �
�
− �
�
= �
�
= �
�
− �
�
=
[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 �
�
= �
�
− �
�
= �
�
= �
�
− �
�
= �
�
= �
�
− �
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= �
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= �
�
= �
�
− �
�
= �
�
= �
�
− �
�
=
[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 �
�
= �
�
− �
�
= �
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= �
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− �
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− �
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�
− �
�
= �
�
= �
�
− �
�
=
[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 �
�
= �
�
− �
�
= �
�
= �
�
− �
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= �
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− �
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= �
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− �
�
= �
�
= �
�
− �
�
=
[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 �
�
= �
�
− �
�
= �
�
= �
�
− �
�
= �
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− �
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= �
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= �
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− �
�
= �
�
= �
�
− �
�
=
[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 �
�
= �
�
− �
�
= �
�
= �
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− �
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= �
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= �
�
− �
�
=
[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
�
�
= �
�
− �
�
= �
�
= �
�
− �
�
= �
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= �
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− �
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= �
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− �
�
= �
�
= �
�
− �
�
= �
�
= �
�
− �
�
=
[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 �
�
= �
�
− �
�
= �
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= �
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− �
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− �
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= �
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= �
�
− �
�
=
[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
�
�
= �
�
− �
�
= �
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= �
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− �
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= �
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= �
�
− �
�
=
[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 �
�
= �
�
− �
�
= �
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= �
�
− �
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= �
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− �
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= �
�
= �
�
− �
�
=
[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 �
�
= �
�
− �
�
= �
�
= �
�
− �
�
= �
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= �
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− �
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= �
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− �
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= �
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= �
�
− �
�
= �
�
= �
�
− �
�
=
[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
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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
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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
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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
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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 ��
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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
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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 :
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�
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
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�
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
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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
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4.5 Kontrol Lendutan