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Statika & Mek Bhn

Analisis Struktur

Peranc.Str. Beton Peranc.Str. Baja

Din.Str.&Rek.Gempa

S1

S2

S3

S4 S5

S7

S6

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Peranc.Str. Bangunan Sipil

Analisis Struktur (4SKS)

Menghitung defleksi/perpind

ahan titik

Analisis Struktur Statis Tak Tentu

Beam & Frame

Castigliano Method

Virtual Load Method

Double Integration

Conjugate Beam

Moment Area Method

Truss/Rangka Batang

Castigliano Method

Virtual Load Method

Beam & Frame

Force Method

Slope-Deflection Eq. Method

Moment Distribution Method

Truss/Rangka Batang

Force Method

W1

W2

W3

W4

W5

W6

W7-9

W10

W11,12

W13-15

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Pertemuan – 1, 2

Prinsip Perpindahan Maya

Mata Kuliah : Analisis Struktur

Kode : CIV - 209

SKS : 4 SKS

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• Kemampuan Akhir yang Diharapkan

• Mahasiswa dapat menjelaskan prinsip kerja dan Energi dalam perhitungan deformasi struktur

• Sub Pokok Bahasan :

• Prinsip Dasar Metode Energi

• Kerja dan Energi

• Prinsip Konservasi Energi

• Virtual work

• Aplikasi kerja maya

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• Text Book :

• Hibbeler, R.C. (2010). Structural Analysis. 8th edition. Prentice Hall. ISBN : 978-0-13-257053-4

• West, H.H., (2002). Fundamentals of Structural Analysis. John Wiley & Sons, Inc. ISBN : 978-0471355564

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Deflections

• Deflections of structures can occur from various sources, such as loads, temperature, fabrication errors, or settlement.

• In design, deflections must be limited in order to provide integrity and stability of roofs, and prevent cracking of attached brittle materials such as concrete, plaster or glass.

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• Before the slope or displacement of a point on a beam or frame is determined, it is often helpful to sketch the deflected shape of the structure when it is loaded in order to partially check the results.

• This deflection diagram represents the elastic curve of points which defines the displaced position of the centroid of the cross section along the members.

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• Kerja Prinsip Konservasi Energi (conservation of energy principle) :

Kerja akibat seluruh gaya luar yang bekerja pada sebuah struktur (external forces) Ue, menyebabkan terjadinya gaya-gaya dalam pada struktur (internal work or strain energy) Ui seiring dengan deformasi yang terjadi pada struktur.

Apabila tegangan yang terjadi tidak melebihi batas elastis material struktur tersebut, elastic strain energy akan mengembalikan bentuk struktur ke tahap awal sebelum terjadinya pembebanan, jika gaya-gaya luar yang bekerja dihilangkan.

(1)

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External work-Axial force

L, E, A

F

dx

P

F

x D

xP

F D

DD

DD

PdxxP

dxFU e2

1

00

Kerja yang dilakukan oleh gaya luar P

(2)

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External work-Bending moment

M

m

f q

fq

M

m

qffq

fqq

MdM

dmU e2

1

00

Kerja yang dilakukan oleh momen lentur M

m

df

(3)

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Strain Energy – Axial force

Gaya P yang bekerja pada sebuah Bar seperti yang terlihat pada Gambar, dikonversikan menjadi strain energy yang menyebabkan pertambahan panjang pada batang sebesar ∆ dan timbulnya tegangan 𝜎. Mengingat Hukum Hooke : 𝜎 = 𝐸𝜖.

Maka persamaan defleksi dapat dituliskan menjadi:

Subsitusikan persamaan 4 ke dalam persamaan 2, maka didapat energi regangan yang tersimpan dalam batang :

AE

PLD (4)

AE

LPU i

2

2

(5)

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Strain Energy – Bending Moment

Energi regangan yang tersimpan pada balok :

(6)

dxEI

Md q Lihat Mekanika Bahan pt.11.

dxEI

MdMdU i

22

1 2

q

EI

LMdx

EI

MU

L

i22

2

0

2

Dari pers.(3)

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External Work, Ue Internal Work, Ui

Axial Force 1

2𝑃∆

𝑃2𝐿

2𝐴𝐸

Bending Moment 1

2𝑀𝜃

𝑀2𝐿

2𝐸𝐼

• Resume

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Castigliano Theorem • Italian engineer Alberto Castigliano (1847 – 1884) developed

a method of determining deflection of structures by strain energy method.

• His Theorem of the Derivatives of Internal Work of Deformation extended its application to the calculation of relative rotations and displacements between points in the structure and to the study of beams in flexure.

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Castigliano’s Theorem for Beam Deflection • For linearly elastic structures, the partial derivative of the

strain energy with respect to an applied force (or couple) is equal to the displacement (or rotation) of the force (or couple) along its line of action.

i

ii

i

ii

M

U

P

U

D qor (7)

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• Subtitusi persamaan 6 ke persamaan 7 :

• Jika sudut rotasi q, yang hendak dicari :

D

LL

EI

dx

P

MM

EI

dxM

P00

2

2(8)

L

EI

dx

M

MM

0

q (9)

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Example 1

• Determine the displacement of point B of the beam shown in the Figure.

• Take E = 200 GPa, I = 500(106) mm4.

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Example 1 • A vertical force P is placed on the beam at B

• Internal moments : (taken from the right side of the beam)

• From Castigliano’s theorem :

PxxM

xPx

xM

M

26

02

12

0

26 0 since xMP

xP

M

m,

EI

mkN

EI

dxxx

EI

dx

P

MM

L

B 150010156 3310

0

2

0

D

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Example 2 • Determine the slope at point B of the beam shown in Figure.

• Take E = 200 GPa, I = 60(106) mm4.

• Since the the slope at B is to be determined, an external couple M’ is placed on the beam at B.

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1

53 053 0

For

0

3 03 0

For

2

2222

2

1

1111

1

M

M

xMMxMMM

x

M

M

xMxMM

x

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rad10375960200

5112

mkN 511215303

3

25

0

22

5

0

11

0

,,

EI

,

EI

dxx

EI

dxx

EI

dx

M

MM

B

L

B

q

q

Setting M’ = 0, its actual value, and using Castigliano Theorem, we have : The negative sign indicates that qB is opposite to the direction of the couple moment M’.

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Example 3 • Determine the vertical displacement of point C of the beam.

• Take E = 200 GPa, I = 150(106) mm4.

External Force P. A vertical force P is applied at point C. Later this force will be set equal to a fixed value of 20 kN. Internal Moments M. In this case two x coordinates are needed for the integration, since the load is discontinuous at C.

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22

22

22

2

11

2111

11

11

1

50

508

0508 0

For

50

45024

02

85024 0

For

x,P

M

xP,M

xP,MM

x

x,P

M

xxP,M

Mx

xxP,M

x

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• Applying Castigliano’s Theorem. Setting P = 20 kN :

mm 14,2 m 01420150200

7426

mkN7426mkN192mkN7234

501850434

333

4

0

222

4

0

112

11

0

D

,,

EI

,

EIEI

,

EI

dxx,x

EI

dxx,xx

EI

dx

P

MM

L

Cv

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Soal Latihan (Chapter IX, Hibbeler)

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• Prinsip perpindahan maya (virtual work) Prinsip ini dikembangkan oleh John Bernoulli pada tahun

1717 dan lebih dikenal dengan nama Unit Load Method. General Statement : • If we take a deformable structure of any shape

or size and apply a series of external loads P to it, it will cause internal loads u at points throughout the structure.

• It is necessary that the external and internal loads be related by the equations of equilibrium.

• As a consequence of these loadings, external displacements D will occur at the P loads and internal displacements d will occur at each point of internal load u.

Gambar 2.1

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Gambar 2.2

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1. ∆ = 𝑢 . 𝑑𝐿

Dimana :

𝑃′ = 1 = beban maya luar yang bekerja searah dengan ∆

∆ = perpindahan yang disebabkan oleh beban nyata

u = beban dalam maya yang bekerja dalam arah dL

dL = deformasi dalam benda yang disebabkan oleh beban nyata.

(1)

Virtual loading Real displacement

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Dengan cara yang sama, apabila kita ingin menentukan besar sudut rotasi pada lokasi tertentu dari sebuah benda, kita dapat mengaplikasikan beban momen maya M’ sebesar 1 satuan, lalu mengintegrasikannya dengan persamaan rotasi akibat beban momen nyata, sehingga :

1. 𝜃 = 𝑢𝜃 . 𝑑𝐿

Dimana :

𝑀′ = 1 = beban maya luar yang bekerja searah dengan ∆

𝜃 = perpindahan rotasi yang disebabkan oleh beban nyata

𝑢𝜃 = kerja dalam maya yang bekerja dalam arah dL

dL = deformasi dalam benda yang disebabkan oleh beban nyata.

(2)

Virtual loading Real displacement

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• Kerja Maya Pada Balok/Frame

D

L

dxEI

mM

0

1 (3)

dxEI

MddL q

D udL1

Virtual Loads

Real Displ.

mu

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• Kerja Maya Pada Balok/Frame

L

dxEI

Mm

0

1 qq (4)

dxEI

MddL q

dLuqq1

Virtual Loads

Real Displ.

qmu

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Example 4

• Determine the displacement of point B of the beam shown in the Figure.

• Take E = 200 GPa, I = 500(106) mm4.

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• The vertical displacement of point B is obtained by placing a virtual unit load of 1 kN at B.

Real Moment, M Virtual Moment, mq

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mm 150 m 1500500200

00015

mkN 0001561kN 1

3210

0

2

0

D

D

,.

EI

.

EI

dxxxdx

EI

mM

B

L

B

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Example 5 • Determine the slope at point B of the beam shown in Figure.

• Take E = 200 GPa, I = 60(106) mm4.

• The slope at B is determined by placing a virtual unit couple of 1 kN.m at B.

• Calculate virtual momen mq and real moment M

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Real Moment, M Virtual Moment, mq

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• The slope at B, is thus given by :

rad103759rad 009375060200

5112

5112531301

3

2

5

0

22

5

0

11

0

,,,

mkNEI

,dx

EI

xdx

EI

xdx

EI

Mm

B

L

B

q

q q

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Example 7 • Determine the tangential

rotation at point C of the frame shown in figure.

• Take E = 200 GPa,

I = 15(106) mm4

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rad 00875015200

2526mkN 2526152511

5715211

2

2

0

2

3

0

11

0

,,

EI

,

EIEI

,

EI

dx,

EI

dxx,dx

EI

Mm

C

L

C

q

q q

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Soal Latihan (Chapter IX, Hibbeler)