Regangan Bidang Analisa Struktur Lanjutan

Prof. Dr.Ing. Johannes Tarigan Teknik Sipil USU

Kuliah ANALISA STRUKTUR LANJUTAN Tegangan Bidang (Plane Stress)y zy z yz zx yx xy x Z Y X xy = yx xz = xz yz = zy This state of stress in called triaxial

xz

General three dimensional state of stress. In plane stress, two faces of the cubic element are free of stress. It is conventional to assume the stress-free faces to be the front and the back faces of the cubic element. There for z = zx = zy = 0 y yz = xy xy

.yx

x

An examples of plane stress we can see as follows F1 F2

P P MT

EQUATIONS FOR THE TRANSFORMATION

Analisa Struktur Lanjutan hal.

1


OF PLANE Stress

Y dA.Cos

xy x

X

x xy

dA

yx dA.Sin y

Equilibrium equation :

X = 0 Y = 0

x . dA - x (dA . Cos ) Cos - xy (dA . Cos ) . Sin - y (dA . Sin ) Sin - xy (dA . Sin ) Cos = 0

xy . dA + x (dA . Cos ) . Sin - xy (dA . Cos ) . Cos - y (dA . Sin ) Cos + xy (dA . Sin ) Sin = 0

x = x . Cos2 + y . Sin2 + 2 xy . Sin . Cos xy = - (x - y) (Sin . Cos ) + xy (Cos2 - Sin2 )Using the trigonometri relations : Cos2 = 1 + Cos 2 1 Cos 2 : Sin2 = 2 2 Sin 2 = 2 Sin . Cos Cos 2 = Cos2 - Sin2 we have : x =

x + y + x - y . Cos 2 + xy . Sin 22 2

+ xy = - x 2 y . Sin 2 + xy . Cos 2


2


Prinsipal Stresses

x =

x + y x - y . Cos 2 + xy . Sin 2 + 2 2 x + y x - y . Cos 2 + . Sin 2 ) = 0 xy + 2 2

dx d = d d

- (x - y) . Sin 2 + 2 xy . Cos 2 = 0

It can be solved for as 2 xy tg 2p = - x y the angle p call be called principal planes. we can also interpretive with this figure

xy 2 p x - y 2 2 xy x - y

tg 2 p =


3


MAXIMUM SHEAR STRESS

xy = -

x - y . Sin 2 + . Cos 2 xy 2

d xy d = d d

- x - y . Sin 2 + xy . Cos 2) = 0 2

- (x - y) Cos 2 - 2 xy . Sin 2 = 0 tg 2s = - x - y 2 xy s the angle which define the maximum shear stress.

R 2 s

-

x - y 2

xy

tg 2s = -

x - y 2 xy


4


Mohrs circle for Plane stress Analysis

x - x + y = x - y . Cos 2 + xy . Sin 2 2 2 xy = x - y . Sin 2 + . Cos 2 xy22 2

x - x + y2

+ (xy - 0)2 =

x - y2

+ (xy)2

(x - ave)2 + (xy - 0)2 = R2 (x - a)2 + (y - b)2 = r2

(y, xy)

ave

(x, - xy)

a = ave b=0 r=R=

x - y2

2

+ (xy)2


5


xy

xy

+How to draw the mohr circle ?

-

y

x

x xy y

Step : 1. Write down the two stress coordinates X,Y to be plotted. X = (x, xy) ; Y = (y, - xy) 2. Determine the center C of the circle from C = ave =

x + y2

Where C is always on the horizontal axis


6


3. Sketch the circle using points X, Y, and C, as shows in figure bellow.

X

S X 2p

2s P2 C D

P1

Y (y, - xy)

4. Determine the radius R of the circle by observing from circle that2 2 R = CD + XD

Tg 2 s =

CD XC

Tg 2 p = XD CD


7


Contoh soal :

9.000

N Cm2

24.000

N x Cm2

xy9.000

24.000 N Cm2

N Cm2

yx

max maxCara I :

?

max :tg 2p =

18.000 2 . 9000 x - y = 24.000 - 0 = 24.000 = 0,75 p = 18,4

2 yx

2p = 36,80

max =p

x + y + x - y . Cos 2 + xy . Sin 22 2 18,40 24.000 + 0 24.000 + 0 . Cos 36,8 + 9.000 . Sin 36.8 + 2 2N

max =

= 27.000 /cm2


8


max :tg 2 s = -

x - y - 24.000 - 0 = = - 1,33 2 . 9.000 2xy

2 s = 53.10

max = -

x - y . Sin 2 s + . Cos 2 s xy2

0 0 = - 24.000 - 0 . Sin 53,1 + 9.000 . Cos 53,1 2

= 15.000 /cm2 Cara II : Lingkaran Mohr

N

(N/cm2)

O

S 2 s 2 p C D

max.

X (24.000, 9.000)

(N/cm2) max.

Y (0,9.000)

xX= (24.000,

xy9.000)

yY= (0,

- xy-9.000)


9


R = CD2 + XD2 = 122 + 92 = 15.000

OC = ave = 24.000 + 0 2 = 12.000

max. = 15 N 2Cm

max. = OC + R= 12 + 15 = 27 tg 2s = CD = XD 12 9

2s = 53.10 tg 2p = XD = CD 2p = 36.80 9 12

Analisa Struktur Lanjutan hal. 10


PLANE STRAIN (Regangan Bidang) Y 2 x Y y 1 X x . dx y . dy dy X 3

Y

1 dx

2

xy

3 X

x = x dx x = x . dx d = 1 + 2 + 3

y = y dy y = y . d y

d = x . dx . Cos + y . dy . Sin + xy dy d d d x = dL = x . dx . Cos + y . y . Sin + xy . y . Cos d d d dx = Cos d dy = Sin d x = x . Cos2 + y . Sin2 + xy . Sin . Cos Using trigonometric relation

x =

x + y x - y . Cos + xy . Sin 2 + 2 2 2



1 =

x . dx . Sin d

;

2 =

y . dy . Cos d

;

1 =

xy dy . Sin d

1 = x . Cos . Sin ; 2 = y . Sin . Cos ; 3 = xy . Sin2

= - 1 + 2 - 3 + 900 + 900 = - (x y) . Sin ( + 900 ) . Cos ( + 900 ) - xy . Sin2 ( + 900 ) Using the Trigonometric : Sin ( + 900 ) = Cos Cos ( + 900 ) = - Sin + 900 = (x y) . Cos . Sin - xy . Cos2 xy = - + 900 = - 2 (x y) Sin . Cos + xy . (Cos2 - Sin2) Using the Trigonometric identities : Cos2 = Sin2 = 1 + Cos2 2 1 - Cos2 2 Sin2 2 d X = - (x y) Sin . Cos - xy . Sin2)

Sin . Cos =

xy = (x y) Sin 2 + xy . Cos 2



Prinsipal Angle p : xy tg 2p = x y

The angle assosiated with the maximum shen stain is tg 2s = x y xy

Morhs circle for plane strain 1. X = (x, xy ) ; Y = (y, 2 x + y 2 xy ) 2 - xy 2

2. C = ave =

3. Gambar titik X, Y dan C

4. R = CD2 + XD2xy/2

(y, xy ) 2

S (ave, max ) 2

D C 2 p

X (y, -

xy ) 2

xy xy 2 XD = x - y = x - y 2 p = CD 2



x = 400 ; y = 300 ; xy = 600

xX= (400,

yY= (300,

xy 2 600 2 xy 2 600 2

2

Y (300, 300)

2 p

X (400, - 300) 350 x - y 400 + 300 = 2 2

C = ave = = 350



Kesimpulan : Plane Stress :

1,2 = x + y 2

2

x - y2

2

+ (xy)

2

max =

x - y2

2 + (xy) = R

max = -

x - y2

2

2 + (xy)

Plane Strain :

+ 1,2 = x 2 y max = 2

2

x - y2 + xy 2

2

+2

xy 2

x - y2


Regangan Bidang Analisa Struktur Lanjutan

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