skma4113 laminate analysis 141119

SKMA 4113 Aircraft Structures II Analysis of Unidirectional Composites - p1 © Ainullotfi Abdul Latif, UTM

SKMA4113 Aircraft Structures II Analysis of Unidirectional Composites

Laminate Analysis

Laminates consist of two or more laminae bonded together to be one integrated structural element. Each lamina may be arranged in any direction desired, giving the laminate characteristics as determined by the designer.

Laminate Code

A laminate consisting of several unidirectional laminae arranged in various orientations is designated by a code which defines the order of the lamina stacking and its orientation, such as:

[03/90

2/45/-45

3]S

This code indicates that the laminate is symmetrical (from the subscript S at the end, outside the square bracket), and consists of three laminae in the 0º direction vis-a-vis x-axis, two laminae at 90º, one at 45º and three at -45º, in the lower half of its thickness. Friom symmetry, this is followed by three more laminae at -45º, one at 45º, two at 90º and three at 0º in the uer half of the thickness. This laminate has 18 laminae.

In its full form without the use of the symmetry sign, the laminate may also be written as:

[03/90

2/45/-45

6/45/90

2/0

3]

This code starts off with the bottom-most lamina (i.e. with the lowest z-coordinate). Each lamina is made from unidirectional composites.

In another example, a non-symmetrical laminate is:

[453/0

2/-45

2/90

2/45

3]

This laminate has twelve laminae, i.e. three at 45º, two 0º, two -45º, two 90º and three 45º. The cross-section view of this laminate, with thickness t, is as follows.

Lamina Stress-Strain Relationship

The on-axis stress-strain relationship for an orthotropic material (i.e. with three planes of symmetry for the material properties) is as follows:

z

-½t

½t 3 lamina 45º

2 lamina 90º 2 lamina -45º 2 lamina 0º

3 lamina 45º


)1(Q000QQ0QQ

6

2

1

66

2221

1211

6

2

1

In the off-axis direction this becomes:

)2(QQQQQQQQQ

s

y

x

sssysx

ysyyyx

xsxyxx

s

y

x

For the k-th lamina in a multi-lamina laminate, these matrix equations can be expressed as:

{}k = [Q]

k {}

k

The Variation of Stresses and Strains Within A Laminate

To obtain the variation of stresses and strains within a laminate, it is assumed that every lamina is binded perfectly, i.e. the displacements across the laminate is continuous and there is no slipping occuring between adjacent laminae.

Also, the laminate is assumed to be a thin plate, with planes normal to the middle surface remain normal under any deformation of the plate.

Consider a thin plate laminate in the laminate x-y plane. A point at (x,y,0) in the middle plane of the plate geometry displaces u

0, v

0 and w

0 respectively in the x-, y- and z-directions.

For any point P at z above the middle plane, the displacements in the x- and y-directions (refer to the previous diagram) are:

ywzvv(z)

xwzuu(z)

00

00

z

x

z

u0

P

z

For 0 tan = w0/x

u(z)

w0

P


From basic mechanics:

xv

yu

yv

xu

syx

Thus:

yxwz2

xv

yu

ywz

yv

xwz

xu

02

00s

20

20

y

20

20

x

In terms of vectors and matrices:

s

y

x

0s

0y

0x

s

y

x

kkk

z

or:

{} = {0} + z {k}

where {0} is the middle (neutral) plane strain vector, and {k} is the laminate middle plane curvature vector (as in plate analysis):

yxw2

ywxw

kkk

02

20

2

20

2

s

y

x

k

Susbstituting this strain equation into equation (2) which relates the strains with the stresses in the kth lamina, we have:

)3(kkk

zQQQQQQQQQ

ks

y

x

0s

0y

0x

ksssysx

ysyyyx

xsxyxx

ks

y

x

Since the modulus components Qij can differ between one lamina and the other (since the fibres do not have to be oriented in the same direction), the variation of stresses along the laminate thickness can be of the form of a step function, as shown in the diagram below for a laminate consisting of four laminae. This happens even though the strains across the laminate is continuous under the applied loadings (for example, pure bending).


Laminate Force and Moment Intensities

Force intensities (i.e. force per unit width of laminate) can be obtained by integrating the stresses along the thickness of the laminate. For examle, in the x-direction:

2t

2t

dzN xx

Similarly, the moment intensities per unit width of the laminate cab be obtained by integrating the product of stress and moment arm from the neutral plane, as follows (for the x-direction case):

2t

2t

dzzM xx

Integrating along the laminate thickness is actually the same as summing the integrals along the thicknesses of each lamina in the laminate, thus for the whole laminate we have for the force intensities:

n

1k

z

z

ks

y

x

s

y

x

s

y

xk

1k

2t

2t

dzdzNNN

And for the moment intensities:

n

1k

z

z

ks

y

x

s

y

x

s

y

xk

1k

2t

2t

dzzdzzMMM

In the above equations t is the thickness of the whole laminate, and the limits for the integration, zk and zk-1, are defined as below:

Laminate Applied A Load

Qij

Lamina Qij (Different)

Strains Variation

(Continuous)

Stress Variation


Substituting equation (3) for stresses:

n

1k

z

z

ks

y

xz

z

k0s

0y

0x

ksssysx

ysyyyx

xsxyxx

s

y

xk

1k

k

1kdzz

kkk

dzQQQQQQQQQ

NNN

n

1k

z

z2

ks

y

xz

z

k0s

0y

0x

ksssysx

ysyyyx

xsxyxx

s

y

xk

1k

k

1kdzz

kkk

dzzQQQQQQQQQ

MMM

Before this it has been shown that 1º,

2º and

6º, and k

1, k

2 and k

6 are the middle (neutral) plane

strains and curvatures, which do not at all depend on the position z. As such these terms can be taken out of the summation. Once the equations have been integrated they can be written as:

where:

Take note that the existence of B

ij indicates that there is coupling between extension (due to the force

N) and flexure (as a result of the moment M) for the laminate.

[A] is the extensional stiffness matrix, and [D] the flexural striffness matrix, while [B] is the coupling stiffness matrix (between extension and flexure).

© Assoc Prof Ainullotfi Abdul Latif, 19 November 2014.

t zk-1 zk kth Lamina

z

n

k 11kkkijij zzQA

n

k 1

31-k

3kkij3

1ij zzQD

n

k 1

21-k

2kkij2

1ij zzQB

s

y

x

sssysx

ysyyyx

xsxyxx

0s

0y

0x

sssysx

ysyyyx

xsxyxx

s

y

x

kkk

BBBBBBBBB

AAAAAAAAA

NNN

s

y

x

sssysx

ysyyyx

xsxyxx

0s

0y

0x

sssysx

ysyyyx

xsxyxx

s

y

x

kkk

DDDDDDDDD

BBBBBBBBB

MMM

skma4113 laminate analysis 141119

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