Soal-soal dan penyelesaian METODE Distribusi Momen { Metode Cross }

(1)

Soal-soal dan penyelesaian

METODE Distribusi Momen {

Metode Cross

}

No. 01. Analisislah Portal berikut dengan Metode Cross

2

a. Perhitungan MOMEN PRIMER (MO)

MoAB = - Pab2/L2 = - 600.3.22/52 = -288 kg.m MoBA = Pa2b/L2 = + 600.32.2/52 = 432 kg.m MoBC = - qL2/12 = - 300.62/12 = -625 kg.m MoCB = qL2/12 = + 300.62/12 = 625 kg.m MoCD = - Pa2b/L2 = - 600.32.2/52 = -432 kg.m MoDC = Pab2/L2 = + 600.3.22/52 = 288 kg.m

b. Perhitungan Koefisien Distribusi Titik Simpul A

m

AB = 0

Titik Simpul D

(2)

5 m Titik Simpul B B C 2.EI 5 m EI A 4.EIBA 4.EI LBA 5 4.EIBC 4.2EI LBC 5 maka Titik Simpul C 5 m B C 2.EI EI 5 m D 4.EICB 4.2EI LCB 5 4.EICD 4.EI LCD 5 maka

c. Proses Distribusi Momen

Lihat halaman selanjutnya….

1.00000 0.66667 2.40

m

CD = 0.80 = 0.33333 2.40

m

CB = 1.60 = = = = 0.80

m

CD = kCD kCD kCB + kCD = = = 1.60

m

CB = kCB kCB kCB + kCD 2.40 0.33333 0.66667 1.00000 1.60

m

BA =

m

BC = 0.80 2.40 = = 1.60 = = kBA = = =

m

BA =

m

BC = kBA kBA + kBC kBC kBA + kBC 0.80 kBC =

(3)

Tabel Distribusi Momen A D AB BA BC CB CD DC 0.80 0.80 1.60 1.60 0.80 0.80 No. DF 0 0.3333333 0.66666667 0.6666667 0.33333333 0 FEM -288.00000 432.00000 -625.00000 625.00000 -432.00000 288.00000 Bal 0.00000 64.33333 128.66667 -128.66667 -64.33333 0.00000 CO 32.16667 0.00000 -64.33333 64.33333 0.00000 -32.16667 Bal 0.00000 21.44444 42.88889 -42.88889 -21.44444 0.00000 CO 10.72222 0.00000 -21.44444 21.44444 0.00000 -10.72222 Bal 0.00000 7.14815 14.29630 -14.29630 -7.14815 0.00000 CO 3.57407 0.00000 -7.14815 7.14815 0.00000 -3.57407 Bal 0.00000 2.38272 4.76543 -4.76543 -2.38272 0.00000 CO 1.19136 0.00000 -2.38272 2.38272 0.00000 -1.19136 Bal 0.00000 0.79424 1.58848 -1.58848 -0.79424 0.00000 CO 0.39712 0.00000 -0.79424 0.79424 0.00000 -0.39712 Bal 0.00000 0.26475 0.52949 -0.52949 -0.26475 0.00000 CO 0.13237 0.00000 -0.26475 0.26475 0.00000 -0.13237 Bal 0.00000 0.08825 0.17650 -0.17650 -0.08825 0.00000 CO 0.04412 0.00000 -0.08825 0.08825 0.00000 -0.04412 Bal 0.00000 0.02942 0.05883 -0.05883 -0.02942 0.00000 CO 0.01471 0.00000 -0.02942 0.02942 0.00000 -0.01471 Bal 0.00000 0.00981 0.01961 -0.01961 -0.00981 0.00000 CO 0.00490 0.00000 -0.00981 0.00981 0.00000 -0.00490 Bal 0.00000 0.00327 0.00654 -0.00654 -0.00327 0.00000 CO 0.00163 0.00000 -0.00327 0.00327 0.00000 -0.00163 Bal 0.00000 0.00109 0.00218 -0.00218 -0.00109 0.00000 CO 0.00054 0.00000 -0.00109 0.00109 0.00000 -0.00054 Bal 0.00000 0.00036 0.00073 -0.00073 -0.00036 0.00000 CO 0.00018 0.00000 -0.00036 0.00036 0.00000 -0.00018 Bal 0.00000 0.00012 0.00024 -0.00024 -0.00012 0.00000 CO 0.00006 0.00000 -0.00012 0.00012 0.00000 -0.00006 Bal 0.00000 0.00004 0.00008 -0.00008 -0.00004 0.00000 CO 0.00002 0.00000 -0.00004 0.00004 0.00000 -0.00002 Bal 0.00000 0.00001 0.00003 -0.00003 -0.00001 0.00000 CO 0.00001 0.00000 -0.00001 0.00001 0.00000 -0.00001 Bal 0.00000 0.00000 0.00001 -0.00001 0.00000 0.00000 CO 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 Bal 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 CO 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 Bal 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 -239.75000 528.50000 -528.50000 528.50000 -528.50000 239.75000 Check Statika 0.00000 0.00000 16 17 18 15 Total 11 12 13 14 7 8 9 10 3 4 5 6 Joint B C 2 Batang K 1

(4)

Proses Distribusi Momen A D AB BA BC CB CD DC kAB kAB kBC kCB kCD kDC No. DF (m) 0 mBA mBC mCB mCD ₀ FEM MoAB MoBA MoBC MoCB MoCD MoDC

Bal 0 (MoBA+MoBC).(-mBA) (MoBA+MoBC).(-mBC) (MoCB+MoCD).(-mCB) (MoCB+MoCD).(-mCD) 0

CO A = Bal.1 BA / 2 B = Bal.1 AB / 2 C = Bal. 1 CB / 2 D = Bal. 1 BC / 2 E = Bal. 1 DC / 2 F = Bal. 1 CD / 2

Bal 0 (B + C).(-mBA) (B + C).(-mBC) (D + E).(-mCB) (D + E).(-mCD)

CO G = Bal.2 BA / 2 H = Bal.2 AB / 2 I = Bal. 2 CB / 2 J = Bal. 2 BC / 2 K = Bal. K DC / 2 L = Bal. 2 CD / 2

Bal 0 (H + I).(-mBA) (H + I).(-mBC) (J + K).(-mCB) (J + K).(-mCD) 0

CO Proses tersebut diatas diulang hingga hasil bagi ( CO ) sudah mendekati nilai 0 , ( 0,00000 ) Bal

CO 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

Bal 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

MAB MBA MBC MCB MCD MDC

Check Statika

MAB Diperoleh dari hasil penjumlahan kebawah (menurut kolom) dimulai dari Momen Primer MOAB sampai pada kondisi balance (bal) sudah bernilai 0,00000.

Untuk Momen Total pada Titik simpul yang lain dilakukan dengan cara yang sama.

jumlah satu titik simpul harus = 0 jumlah satu titik simpul harus = 0 5 Total 4 2 3 C Batang K 1 Joint B

(5)

d. Perhitungan Momen Desain

Karena kondisi beban dan rangka simetris, maka dianalisis Portal tanpa Pergoyangan.

untuk itulah proses distribusi momen (cross) hanya satu kali yaitu akibat beban luar (tanpa pergoyangan). Harga momen desain sama dengan harga cross dari beban luar (lihat jumlah momen dari proses cross).

MAB = -239.8 kg.m MBA = 528.5 kg.m MBC = -528.5 kg.m MCB = 528.5 kg.m MCD = -528.5 kg.m MDC = 239.8 kg.m

l. Free Body untuk Perhitungan Reaksi Perletakan

Batang AB Batang CD SMB = 0 RCH = - RBH = -182.3kg g RAH = {(600.2)+(239,75)-(528,5)}/(5) = 182.25 kg f RDH = - RAH = -417.7kg g SH = 0 RAH + RBH = P RBH = P - RAH = 417.75 kg f Batang BC SMC = 0 RBV = {(300.5.5/2)+(528,5)-(528,5)}/(5) = 750.00 kg h SH = 0 RBV + RCV = Q RCV = Q - RBV = 750.00 kg h

(6)

m. Perhitungan Momen, Gaya Lintang dan Gaya Normal

Potongan AB interval Potongan BC interval

Mx1 = RAH.X - MAB 0 < X < 2 Mx = RBV.x - 1/2.q.x2- MBC 0 < X < 5 Dx1 = RAH Dx = RBV - q.x Nx1 = - RAV = RBV Nx = - RBH Mx2 = RAH.(3+X) - P.X - MAB 0 < X < 3 Dx2 = RAH - P Nx2 = - RAV Pot. AB Pot. BC x Mx Dx Nx x Mx Dx Nx 0 -239.8 182.3 -750 0 -528.5 750 -418 1 -57.5 182.3 -750 0.5 -191 600 -418 2 124.7 182.3 -750 1 71.5 450 -418 3 307.0 182.3 -750 1.5 259 300 -418 0 307.0 -417.7 -750 2 371.5 150 -418 1 -110.7 -417.7 -750 2.5 409.0 0 -418 2 -528.5 -417.7 -750 3 371.5 -150 -418 3.5 259 -300 -418 4 71.5 -450 -418 4.5 -191 -600 -418 5 -528.5 -750 -418

Gbr. Bidang Momen Gbr. Bidang Lintang

Gbr. Bidang Normal Interval 0 < X < 5 Interval 0 < X < 3 0 < X < 2

(7)

No. 02. Analisislah Portal berikut dengan Metode Cross

a

36.87o

a. Perhitungan MOMEN PRIMER (MO)

MoBE = PL = + 400.1 = 400 kg.m

MoBC = - PL/8 = - 1000.4/8 = -500 kg.m

MoCB = + PL/8 = + 1000.4/8 = 500 kg.m

Momen primer akibat pergoyangan

MAB=MBA MXAB = MXBA = (6.EI_AB.Dh)/LAB 2

= -0.375 EI.DH misal EI.DH = X, maka

MXAB = MXBA = -37.5 X MBC=MCB MXBC = MXCB = (6.EIBC.Dv)/LBC 2 _{Dv = Dh / Tan a} = 1.000 X MCD=MDC MXCD = MXDC = (6.EI_CD.Dr)/L_CD2 Dr = Dh / Sin a = -0.400 X MAB=MBA = -37.5 .X MBC=MCB = 100 .X MCD=MDC = -40 .X

b. Perhitungan Koefisien Distribusi Titik Simpul A

m

AB = 0

Titik Simpul D

m

DC = 0

(8)

4 m Titik Simpul B B C 2.EI 4 m EI A 4.EI_BA 4.EI L_BA 4 4.EI_BC 4.2EI L_BC 4 maka Titik Simpul C 4 m B C 2.EI EI 5 m D 4.EI_CB 4.2EI L_CB 4 4.EI_CD 4.EI L_CD 5 maka

c. Proses Distribusi Momen

Lihat halaman selanjutnya….

0.80 = 1.00000 0.71429 2.80 0.28571 = = =

m

CD

= 0.80 = 2.80

m

CB

= 2.00 =

m

CD

= kCD kCD kCB + kCD = = 2.00

m

CB

= kCB kCB kCB + kCD 3.00 0.33333 0.66667 1.00000 2.00

m

BA

=

m

BC

= 1.00 3.00 = = 2.00 = = kBA = = =

m

BA

=

m

BC

= kBA kBA + kBC kBC kBA + kBC 1.00 kBC =

(9)

Tabel Distribusi Momen, akibat Beban Luar A D AB BA BE BC CB CD DC 1.00 1.00 0.00 2.00 2.00 0.80 0.80 No. DF 0 0.33333333 0 0.66666667 0.71428571 0.28571429 0 FEM 0.00000 0.00000 400.00000 -500.00000 500.00000 0.00000 0.00000 Bal 0.00000 33.33333 0.00000 66.66667 -357.14286 -142.85714 0.00000 CO 16.66667 0.00000 0.00000 -178.57143 33.33333 0.00000 -71.42857 Bal 0.00000 59.52381 0.00000 119.04762 -23.80952 -9.52381 0.00000 CO 29.76190 0.00000 0.00000 -11.90476 59.52381 0.00000 -4.76190 Bal 0.00000 3.96825 0.00000 7.93651 -42.51701 -17.00680 0.00000 CO 1.98413 0.00000 0.00000 -21.25850 3.96825 0.00000 -8.50340 Bal 0.00000 7.08617 0.00000 14.17234 -2.83447 -1.13379 0.00000 CO 3.54308 0.00000 0.00000 -1.41723 7.08617 0.00000 -0.56689 Bal 0.00000 0.47241 0.00000 0.94482 -5.06155 -2.02462 0.00000 CO 0.23621 0.00000 0.00000 -2.53077 0.47241 0.00000 -1.01231 Bal 0.00000 0.84359 0.00000 1.68718 -0.33744 -0.13497 0.00000 CO 0.42180 0.00000 0.00000 -0.16872 0.84359 0.00000 -0.06749 Bal 0.00000 0.05624 0.00000 0.11248 -0.60257 -0.24103 0.00000 CO 0.02812 0.00000 0.00000 -0.30128 0.05624 0.00000 -0.12051 Bal 0.00000 0.10043 0.00000 0.20086 -0.04017 -0.01607 0.00000 CO 0.05021 0.00000 0.00000 -0.02009 0.10043 0.00000 -0.00803 Bal 0.00000 0.00670 0.00000 0.01339 -0.07173 -0.02869 0.00000 CO 0.00335 0.00000 0.00000 -0.03587 0.00670 0.00000 -0.01435 Bal 0.00000 0.01196 0.00000 0.02391 -0.00478 -0.00191 0.00000 CO 0.00598 0.00000 0.00000 -0.00239 0.01196 0.00000 -0.00096 Bal 0.00000 0.00080 0.00000 0.00159 -0.00854 -0.00342 0.00000 CO 0.00040 0.00000 0.00000 -0.00427 0.00080 0.00000 -0.00171 Bal 0.00000 0.00142 0.00000 0.00285 -0.00057 -0.00023 0.00000 CO 0.00071 0.00000 0.00000 -0.00028 0.00142 0.00000 -0.00011 Bal 0.00000 0.00009 0.00000 0.00019 -0.00102 -0.00041 0.00000 CO 0.00005 0.00000 0.00000 -0.00051 0.00009 0.00000 -0.00020 Bal 0.00000 0.00017 0.00000 0.00034 -0.00007 -0.00003 0.00000 CO 0.00008 0.00000 0.00000 -0.00003 0.00017 0.00000 -0.00001 Bal 0.00000 0.00001 0.00000 0.00002 -0.00012 -0.00005 0.00000 CO 0.00001 0.00000 0.00000 -0.00006 0.00001 0.00000 -0.00002 Bal 0.00000 0.00002 0.00000 0.00004 -0.00001 0.00000 0.00000 CO 0.00001 0.00000 0.00000 0.00000 0.00002 0.00000 0.00000 Bal 0.00000 0.00000 0.00000 0.00000 -0.00001 -0.00001 0.00000 CO 0.00000 0.00000 0.00000 -0.00001 0.00000 0.00000 0.00000 Bal 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 52.70270 105.40540 400.00000 -505.40540 172.97297 -172.97297 -86.48649 Check Statika 0.00000 0.00000 16 17 18 15 Total 11 12 13 14 7 8 9 10 3 4 5 6 Joint B C 2 Batang K 1

(10)

Tabel Distribusi Momen, akibat Pelepasan Pendel 1 A D AB BA BE BC CB CD DC 1.00 1.00 0.00 2.00 2.00 0.80 0.80 No. DF 0 0.333333 0 0.66666667 0.71428571 0.285714 0 FEM -37.50000 -37.50000 0.00000 100.00000 100.00000 -40.00000 -40.00000 Bal 0.00000 -20.83333 0.00000 -41.66667 -42.85714 -17.14286 0.00000 CO -10.41667 0.00000 0.00000 -21.42857 -20.83333 0.00000 -8.57143 Bal 0.00000 7.14286 0.00000 14.28571 14.88095 5.95238 0.00000 CO 3.57143 0.00000 0.00000 7.44048 7.14286 0.00000 2.97619 Bal 0.00000 -2.48016 0.00000 -4.96032 -5.10204 -2.04082 0.00000 CO -1.24008 0.00000 0.00000 -2.55102 -2.48016 0.00000 -1.02041 Bal 0.00000 0.85034 0.00000 1.70068 1.77154 0.70862 0.00000 CO 0.42517 0.00000 0.00000 0.88577 0.85034 0.00000 0.35431 Bal 0.00000 -0.29526 0.00000 -0.59051 -0.60739 -0.24295 0.00000 CO -0.14763 0.00000 0.00000 -0.30369 -0.29526 0.00000 -0.12148 Bal 0.00000 0.10123 0.00000 0.20246 0.21090 0.08436 0.00000 CO 0.05062 0.00000 0.00000 0.10545 0.10123 0.00000 0.04218 Bal 0.00000 -0.03515 0.00000 -0.07030 -0.07231 -0.02892 0.00000 CO -0.01757 0.00000 0.00000 -0.03615 -0.03515 0.00000 -0.01446 Bal 0.00000 0.01205 0.00000 0.02410 0.02511 0.01004 0.00000 CO 0.00603 0.00000 0.00000 0.01255 0.01205 0.00000 0.00502 Bal 0.00000 -0.00418 0.00000 -0.00837 -0.00861 -0.00344 0.00000 CO -0.00209 0.00000 0.00000 -0.00430 -0.00418 0.00000 -0.00172 Bal 0.00000 0.00143 0.00000 0.00287 0.00299 0.00120 0.00000 CO 0.00072 0.00000 0.00000 0.00149 0.00143 0.00000 0.00060 Bal 0.00000 -0.00050 0.00000 -0.00100 -0.00102 -0.00041 0.00000 CO -0.00025 0.00000 0.00000 -0.00051 -0.00050 0.00000 -0.00020 Bal 0.00000 0.00017 0.00000 0.00034 0.00036 0.00014 0.00000 CO 0.00009 0.00000 0.00000 0.00018 0.00017 0.00000 0.00007 Bal 0.00000 -0.00006 0.00000 -0.00012 -0.00012 -0.00005 0.00000 CO -0.00003 0.00000 0.00000 -0.00006 -0.00006 0.00000 -0.00002 Bal 0.00000 0.00002 0.00000 0.00004 0.00004 0.00002 0.00000 CO 0.00001 0.00000 0.00000 0.00002 0.00002 0.00000 0.00001 Bal 0.00000 -0.00001 0.00000 -0.00001 -0.00001 -0.00001 0.00000 CO 0.00000 0.00000 0.00000 -0.00001 -0.00001 0.00000 0.00000 Bal 0.00000 0.00000 0.00000 0.00000 0.00001 0.00000 0.00000 CO 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 Bal 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 CO 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 Bal 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 -45.27027 -53.04054 0.00000 53.04054 52.70270 -52.70270 -46.35135 Check Statika Joint B C 2 Batang K 1 3 4 5 6 7 8 9 10 15 Total 11 12 13 14 0.00000 0.00000 16 17 18