SOAL :B ton

A ton

a Elemen 1A = 400.00 cm² a = 0.00 cmI = 5000.00 b = 400.00 cmL = 500.00 cm c = 250.00 cm

d = 250.00 cmElemen 2 A = 2.00 Ton

b A = 425.00 cm² B = 4.00 TonI = 5500.00L = 400.00 cm

c d

E = 2100.00 t/cm²360.00 derajad270.00 derajad

MENGHITUNG MOMEN PRIMER

Batang 2

V2 = 0.00 V3 = 1.02

M2 = 0.00 M3 = 0.00L² L²

0.00 0.00

0.00 1.02

Batang 1

V1 = 3.91 V2 = 3.91

M1 = 390.63 M2 = 390.63L² L²

390.63 390.63

3.91 3.91

cm^4

cm^4

α btg 1 =α btg 2 =

A.a² ( 3b + a ) = A.b² ( b + 3a ) =L³ L³

A.b .a² = A.b² .a =

B.d² ( 3c + d ) = B.c² ( c + 3d ) =L³ L³

B.c .d² = B.c² .d =

2 1

3

1

2

V2 (ton) V1 (ton)

H2 (ton)390.63 390.63

V1 = 3.91V2 = 3.91H2 = 1.02H3 = 0.00

0.00H1 (ton)

PENYELESAIAN :

PERSAMAAN MATRIX KEKAKUAN BATANG/ELEMEN

A E0.00 0.00

- A E0.00 0.00L L

0.0012 EI 6 EI

0.00 - 12 EI 6 EI

fi L² L² uigi

0.006 EI 4 EI

0.00 - 6 EI 2 EI vi

mi =

L² L L² Lfj - A E

0.00 0.00A E

0.00 0.00uj

gj L L vjmj

0.00 - 12 EI - 6 EI

0.0012 EI - 6 EI θj

L² L²

0.006 EI 2 EI

0.00 - 6 EI 4 EI

L² L L² L

PERSAMAAN MATRIX TRANSFORMASI KOORDINAT

ui 0.00 0.00 0.00 0.00 Uivi 0.00 0.00 0.00 0.00 Vi

=0.00 0.00 1.00 0.00 0.00 0.00

uj 0.00 0.00 0.00 0.00 Ujvj 0.00 0.00 0.00 0.00 Vjθj 0.00 0.00 0.00 0.00 0.00 1.00 θj

L³ L³

θi

L³ L³

Cos. α Sin. α -Sin. α Cos. α

θi θiCos. α Sin. α -Sin. α Cos. α

2

3

1

1

2

MATRIK KEKAKUAN ELEMEN

BATANG 2 (ELEMEN 2)

[ T ]

1.00 0.00 0.00 0.00 0.00 0.000.00 1.00 0.00 0.00 0.00 0.000.00 0.00 1.00 0.00 0.00 0.000.00 0.00 0.00 1.00 0.00 0.000.00 0.00 0.00 0.00 1.00 0.000.00 0.00 0.00 0.00 0.00 1.00

1.00 0.00 0.00 0.00 0.00 0.000.00 1.00 0.00 0.00 0.00 0.000.00 0.00 1.00 0.00 0.00 0.000.00 0.00 0.00 1.00 0.00 0.000.00 0.00 0.00 0.00 1.00 0.000.00 0.00 0.00 0.00 0.00 1.00

1680.00 0.00 0.00 -1680.00 0.00 0.000.00 1.01 252.00 0.00 -1.01 252.000.00 252.00 84000.00 0.00 -252.00 42000.00

-1680.00 0.00 0.00 1680.00 0.00 0.000.00 -1.01 -252.00 0.00 1.01 -252.000.00 252.00 42000.00 0.00 -252.00 84000.00

=

1680.00 0.00 0.00 -1680.00 0.00 0.000.00 1.01 252.00 0.00 -1.01 252.000.00 252.00 84000.00 0.00 -252.00 42000.00

-1680.00 0.00 0.00 1680.00 0.00 0.000.00 -1.01 -252.00 0.00 1.01 -252.000.00 252.00 42000.00 0.00 -252.00 84000.00

[ T ] =

1680.00 0.00 0.00 -1680.00 0.00 0.000.00 1.01 252.00 0.00 -1.01 252.00

= 0.00 252.00 84000.00 0.00 -252.00 42000.00

-1680.00 0.00 0.00 1680.00 0.00 0.000.00 -1.01 -252.00 0.00 1.01 -252.000.00 252.00 42000.00 0.00 -252.00 84000.00

BATANG 1 (ELEMEN 1)

[ T ]

0.00 -1.00 0.00 0.00 0.00 0.001.00 0.00 0.00 0.00 0.00 0.000.00 0.00 1.00 0.00 0.00 0.000.00 0.00 0.00 0.00 -1.00 0.000.00 0.00 0.00 1.00 0.00 0.000.00 0.00 0.00 0.00 0.00 1.00

[ T ] T

[ Kl ]

[ T ] T [ Kl(2) ]

[ Kg(2) ] [ T ] T [ Kl(2) ]

[Kg(2)]

0.00 1.00 0.00 0.00 0.00 0.00-1.00 0.00 0.00 0.00 0.00 0.000.00 0.00 1.00 0.00 0.00 0.000.00 0.00 0.00 0.00 1.00 0.000.00 0.00 0.00 -1.00 0.00 0.000.00 0.00 0.00 0.00 0.00 1.00

2231.25 0.00 0.00 -2231.25 0.00 0.000.00 2.17 433.13 0.00 -2.17 433.130.00 433.13 115500.00 0.00 -433.13 57750.00

-2231.25 0.00 0.00 2231.25 0.00 0.000.00 -2.17 -433.13 0.00 2.17 -433.130.00 433.13 57750.00 0.00 -433.13 115500.00

=

0.00 2.17 433.13 0.00 -2.17 433.13-2231.25 0.00 0.00 2231.25 0.00 0.00

0.00 433.13 115500.00 0.00 -433.13 57750.000.00 -2.17 -433.13 0.00 2.17 -433.13

2231.25 0.00 0.00 -2231.25 0.00 0.000.00 433.13 57750.00 0.00 -433.13 115500.00

[ T ] =

2.17 0.00 433.13 -2.17 0.00 433.130.00 2231.25 0.00 0.00 -2231.25 0.00

= 433.13 0.00 115500.00 -433.13 0.00 57750.00

-2.17 0.00 -433.13 2.17 0.00 -433.130.00 -2231.25 0.00 0.00 2231.25 0.00

433.13 0.00 57750.00 -433.13 0.00 115500.00

MENYUSUN OVERALL STIFFNESS MATRIX

=

1680.00 0.00 0.00 -1680.00 0.00 0.00 0.00 0.00 0.000.00 1.01 252.00 0.00 -1.01 252.00 0.00 0.00 0.000.00 252.00 84000.00 0.00 -252.00 42000.00 0.00 0.00 0.00

-1680.00 0.00 0.00 1680.00 0.00 0.00 0.00 0.00 0.000.00 -1.01 -252.00 0.00 1.01 -252.00 0.00 0.00 0.000.00 252.00 42000.00 0.00 -252.00 84000.00 0.00 0.00 0.000.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.000.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.000.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

[ T ] T

[ Kl ]

[ T ] T [ Kl(1) ]

[ Kg(1) ] [ T ] T [ Kl(1) ]

[Kg(1)]

[Kg(2)]

=

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.000.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.000.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.000.00 0.00 0.00 2.17 0.00 433.13 -2.17 0.00 433.130.00 0.00 0.00 0.00 2231.25 0.00 0.00 -2231.25 0.000.00 0.00 0.00 433.13 0.00 115500.00 -433.13 0.00 57750.000.00 0.00 0.00 -2.17 0.00 -433.13 2.17 0.00 -433.130.00 0.00 0.00 0.00 -2231.25 0.00 0.00 2231.25 0.000.00 0.00 0.00 433.13 0.00 57750.00 -433.13 0.00 115500.00

1680.00 0.00 0.00 -1680.00 0.00 0.00 0.00 0.00 0.000.00 1.01 252.00 0.00 -1.01 252.00 0.00 0.00 0.000.00 252.00 84000.00 0.00 -252.00 42000.00 0.00 0.00 0.00

-1680.00 0.00 0.00 1682.17 0.00 433.12 -2.17 0.00 433.130.00 -1.01 -252.00 0.00 2232.26 -252.00 0.00 -2231.25 0.000.00 252.00 42000.00 433.12 -252.00 199500.00 -433.13 0.00 57750.000.00 0.00 0.00 -2.17 0.00 -433.13 2.17 0.00 -433.130.00 0.00 0.00 0.00 -2231.25 0.00 0.00 2231.25 0.000.00 0.00 0.00 433.13 0.00 57750.00 -433.13 0.00 115500.00

UNKNOWN DISPLACEMENT AND REACTIONS

H2 1.02 1682.17 0.00 433.12V2 -3.91 = 0.00 2232.26 -252.00M2 -390.63 433.12 -252.00 199500.00

-1

1682.17 0.00 433.12 1.02 = 0.00 2232.26 -252.00 -3.91

433.12 -252.00 199500.00 -390.63

0.000595 0.000000 -0.000001 1.02 = 0.000000 0.000448 0.000001 -3.91

-0.000001 0.000001 0.000005 -390.63

U2 0.0011V2 = -0.0020θ2 -0.0020

F1 -1680.00 0.00 0.00 -1.87G1 0.00 -1.01 252.00 0.0011 -0.49M1

=0.00 -252.00 42000.00 -0.0020

=-81.95

F3 -2.17 0.00 -433.13 -0.0020 0.85G3 0.00 -2231.25 0.00 4.40M3 433.13 0.00 57750.00 -112.88

[Kg(1)]

U2

V2

θ2

U2

V2

θ2

Reaksi Tumpuan Global

F1 -1.87 0.00 -1.87G1 -0.49 0.00 -0.49M1

=-81.95 + 0.00 = -81.95

F3 0.85 0.00 0.85G3 4.40 3.91 8.31M3 -112.88 -390.63 -503.50

MEMBER FORCES

ELEMEN 2

0.00000.0000

[ U ] =0.00000.0011

-0.0020-0.0020

[ T ]

1680.00 0.00 0.00 -1680.00 0.00 0.000.00 1.01 252.00 0.00 -1.01 252.00

=0.00 252.00 84000.00 0.00 -252.00 42000.00

-1680.00 0.00 0.00 1680.00 0.00 0.000.00 -1.01 -252.00 0.00 1.01 -252.000.00 252.00 42000.00 0.00 -252.00 84000.00

f1 -1.87g1 -0.49m1

= [ T ] [ U ] =-81.95

f2 1.87g2 0.49m2 -164.39

Reaksi Tumpuan Lokal

f1 -1.87 0.00 -1.87g1 -0.49 0.00 -0.49m1

=-81.95

+0.00

=-81.95

f2 1.87 0.00 1.87g2 0.49 1.02 1.52m2 -164.39 0.00 -164.39

ELEMEN 10.0011

-0.0020

[ U ] =-0.00200.00000.00000.0000

[ Kg(2) ]

[ Kg(2) ]

[ T ]0.00 -2231.25 0.00 0.00 2231.25 0.002.17 0.00 433.13 -2.17 0.00 433.13

=433.13 0.00 115500.00 -433.13 0.00 57750.00

0.00 2231.25 0.00 0.00 -2231.25 0.00-2.17 0.00 -433.13 2.17 0.00 -433.13

433.13 0.00 57750.00 -433.13 0.00 115500.00

f2 4.40g2 -0.85m2

= [ T ] [ U ] =-226.24

f3 -4.40g3 0.85m3 -112.88

Reaksi Tumpuan Lokal

f2 4.40 0.00 4.40g2 -0.85 3.91 3.06m2

=-226.24

+390.63

=164.39

f3 -4.40 0.00 -4.40g3 0.85 3.91 4.75m3 -112.88 -390.63 -503.50

GAMBAR BMD, SFD, NFD

164.39 503.50

164.39

( BMD )

-81.95

3.06

1.52

4.75

[ Kg(1) ]

[ Kg(1) ]

( SFD )

-0.49

4.40 4.40

-1.87

( NFD )

-1.87