Seminar05 MPE

8/13/2019 Seminar05 MPE

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1.1 Example

kinematic boundary conditions: v(0) = 0

v(L) = 0

natural boundary conditions: v (0) = 0

v (L) = 0

1. a11(x) =Z1(x) =A0+A1x+A2x2

kinematic boundary conditions: Z1(0) != 0 A0= 0

Z1(L) != 0 A2= A1

1

L

Z1(x) =A1 x

1

x

L= a11(x) (4)

2. a22(x) =Z2(x) =A3x3 +A4x

4

kinematic boundary conditions: Z2(0) != 0 fulfilled

Z2(L) != 0 A4= A3

1

L

Z2(x) =A3 x3

1

x

L

= a22(x) (5)

3. to n. all other parts analogously

yN(x) =i=1

aii(x) =x

1 x

L

a1+a2x2 +a3x

4 +a4x6 +

(6)

in case of complete series: strict fulfilment of equilibrium conditions(in every point of body = local)

in case of abortion: weak fulfilment of equilibrium conditions(only in weighted integral mean value)

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1.2 Discretization

p(x)

S0

s(x)

x, u

y, v

L

p0

L1

L2

L3

x1

x2

x3I3

I2

I1

discretization e.g. by

mean value

Ii=I(xi= 0) +I(xi= Li)

2

integral mean value

Ii=

Li0

I(xi) dxi

Li

ansatz function for each section

vN(x1) = a0+a1x1+a2x2

1

vN(x2) = b0+b1x2+b2x2

2

vN(x3) = c0+c1x3+c2x2

3

boundary/matching condtions

v(x1= 0) = 0; v(x1= L1) =v(x2= 0) ; v(x2= L2) =v(x3= 0) ;

v (x1= 0) = 0; v (x1= L1) =v

(x2= 0) ; v (x2= L2) =v

(x3= 0) ;

e.g. for L1= L2= L3

vN(x1) = a2x2

vN(x2) = a2

3L

1

3L+ 2x

+b2x

2

vN(x3) = a2

3L (L+ 2x) +

b2

3L+c2x

2

internal potential (without axial strain energy)

i=3

k=1

12

EIk

Lk0

vN(xk)2

dxk

external potential (1st order theory)

ex= 3

k=1

Lk0

p (xk) vN(xk) dx

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2 Minimum potential energy of a plate

x

y

LX

p(x,y) LY

w(x,y)

applied load p (x, y)

displacement field w (x, y)

plate stiffness B= Et3

12(1 2)

potential of internal energy for a plate (derived in lecture)

i= B

2

A

(wxx+wyy)

2 2 (1 )

wxxwyy w

2

xy

dx dy (7)

whereaswxx=

2w

x2, wyy =

2w

y2 and wxy=

2w

xy

potential of external energy

e=

A

p (x, y) w (x, y) dx dy (8)

2.1 Example 1

p (x, y) =p0 sin

xLx

sin

yLy

E, , t, p0= const.

ansatz-function

boundary conditions

w (x= 0, y) = 0 and w (x, y= 0) = 0

w (x= Lx, y) = 0 w (x, y= Ly) = 0

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strong solution (Fourier series)

w (x, y) =a sin

x

Lx

sin

y

Ly

(9)

approximate solution (power series)

wN(x, y) =a

xLx 2 x

3

L3x+ x

4

L4x

yLy 2 y

3

L3y+ y

4

L4y

(10)

strong solution

derivations of displacement function

wx(x, y) = a

Lx cos

x

Lx

sin

y

Ly

wy(

x, y) =

a

Ly

sinx

Lx

cosy

Ly

wxx(x, y) = a

Lx

2 sin

x

Lx

sin

y

Ly

wyy(x, y) = a

Ly

2 sin

x

Lx

sin

y

Ly

wxy(x, y) = a 2

LxLy cos

x

Lx

cos

y

Ly

(11)

internal potential

i = B

2

A

w2xx+ 2wxxwyy+ w

2

yy 2 (1 )

wxxwyy w2

xy

dx dy

= B

2 [I1+ 2I3+I2 2 (1 ) (I3 I4)] (12)

integral I1

I1 =

A

w2xxdx dy = a2

Lx

4

A

sin2

x

Lx

sin2

y

Ly

dx dy

= a2

Lx

4

12

xLx4

sin

2 Lx

x

Lx

0

12

y Ly4

sin

2 Ly

y

Ly

0

= a2

Lx

41

4LxLy (13)

Integral I2

I2=

A

w2yy dx dy = a2

Ly

41

4LxLy (14)

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Integral I3

I3 =

A

wxx wyy dx dy = a2

4

L2xL2y

A

sin2

x

Lx

sin2

y

Ly

dx dy

= a2 4

L2

xL2

y

1

4

LxLy (15)

Integral I4

I4 =

A

w2xydx dy = a2

4

L2xL2y

A

cos2

x

Lx

cos2

y

Ly

dx dy

= a2 4

L2xL2y

1

2x+

Lx

4sin

2

Lxx

Lx

0

1

2y+

Ly

4sin

2

Lyy

Ly

0

= a2 4

L2xL2y

1

4LxLy (16)

finally

i = a2

4

4LxLy

B

2

1

L4x+

2

L2xL2y

+ 1

L4y 2 (1 )

1

L2xL2y

1

L2xL2y

= a24

8LxLyB

1

L2x+

1

L2y

2(17)

external potential

a = A

p (x, y) w (x, y) dx dy = p0aA

sin2

xLx

sin2

yLy

dx dy

= p0a 1

4LxLy (18)

complete potential

=1

4LxLy

a24

B

2

1

L2x+

1

L2y

2p0a

(19)

minimisation principal

a = 0 =

1

4LxLy

2a4

B

2

1

L2x+

1

L2y

2p0

(20)

determination of unknown parameter a

a= p0

B4

1

L2x+ 1

L2y

2 (21)

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2.2 Example 2

ansatz-function

boundary conditions

w (x= 0, y) = 0 and w (x, y= 0) = 0

w (x= Lx, y) = 0 w (x, y= Ly) = 0

w (x= 0, y) = 0

approximate solution (power series)

wN(x, y) =a

x2

L2x

x3

L3x

y

Ly 2

y3

L3y+

y4

L4y

(22)

3 Comparison of strong and approximate solution (Ritz)

structure/system stiffness of approximate solution higher than reality

strong approx

(w) (wN) (23)

displacements of approximate solution lower than reality

w wN (24)

w

n

strong solution

approximatesolution

with displacement w and number of coordinate functions n

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Seminar05 MPE