PART III LAMINATES - Altair University · Version 1.0 5.3.1998 Theoretical Background of ESAComp...

PART III

LAMINATES

Version 1.0

HUT/LLS Theoretical Background of ESAComp Analyses 5.3.1998

III.0.1

III Laminates

Table of Contents

PART III � LAMINATES

Table of Contents

1 NOTATION AND CONVENTIONS....................................................................... 1.1

Symbols..................................................................................................................... 1.1

1.1 Laminate Coordinate System and Lay-up.......................................................... 1.3

1.2 Laminate Stresses and Strains ........................................................................... 1.5

1.2.1 Resultant forces and moments ............................................................. 1.5

1.2.2 Normalized stresses............................................................................. 1.7

1.2.3 Midplane and flexural strains .............................................................. 1.7

2 CLASSICAL LAMINATION THEORY................................................................ 2.1

Symbols..................................................................................................................... 2.1

2.1 General Assumptions ........................................................................................ 2.4

2.2 Strain-Displacement Relations .......................................................................... 2.4

2.3 Constitutive Equations ...................................................................................... 2.6

2.3.1 Stiffness matrices ................................................................................ 2.6

2.3.2 Compliance matrices ........................................................................... 2.7

2.3.3 Normalized stiffness and compliance matrices .................................... 2.8

2.3.4 Laminate engineering constants........................................................... 2.9

2.4 Internal Loads ................................................................................................. 2.11

2.4.1 Thermal loads.................................................................................... 2.11

2.4.2 Moisture loads................................................................................... 2.13

2.4.3 Initial strains ..................................................................................... 2.14

2.4.4 Constitutive equations in combined loading....................................... 2.16

2.5 Hygrothermal Expansion Coefficients............................................................. 2.17

2.5.1 Thermal expansion............................................................................ 2.17

2.5.2 Moisture expansion ........................................................................... 2.18

2.6 Laminate Load Response ................................................................................ 2.19

2.6.1 Laminate strain state applied ............................................................. 2.19

2.6.2 Laminate stress state applied ............................................................. 2.20

2.6.3 Laminate stresses applied in zero-curvature state............................... 2.20

2.7 Layer Strains and Stresses............................................................................... 2.21

2.7.1 Actual strains .................................................................................... 2.21

2.7.2 Strains and stresses due to applied forces and moments ..................... 2.22

2.7.3 Strains due to internal loads............................................................... 2.22

2.7.4 Free strains........................................................................................ 2.23

2.7.5 Residual strains and stresses.............................................................. 2.23

2.7.6 Equivalent strains and actual stresses................................................. 2.24

2.7.7 Sandwich face sheet stresses ............................................................. 2.25

2.8 Transverse Shear Stiffness .............................................................................. 2.25

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References ............................................................................................................... 2.27

3 FAILURE CRITERIA............................................................................................. 3.1

Symbols ..................................................................................................................... 3.1

3.1 General ............................................................................................................. 3.2

3.2 Isotropic Materials in Plane Stress State............................................................ 3.2

3.3 Fiber-Reinforced Plies in Plane Stress State ...................................................... 3.3

3.3.1 Maximum strain and stress criteria ...................................................... 3.4

3.3.2 Quadratic criteria................................................................................. 3.6

3.3.3 Partly interactive criteria ................................................................... 3.10

References ............................................................................................................... 3.12

4 CRITICALITY OF APPLIED LOADS.................................................................. 4.1

Symbols ..................................................................................................................... 4.1

4.1 Factors of Safety............................................................................................... 4.2

4.2 Margin to Failure .............................................................................................. 4.2

4.2.1 Constant and variable load approach ................................................... 4.2

4.2.2 Reserve factors.................................................................................... 4.3

4.2.3 Margins of safety................................................................................. 4.6

4.2.4 Infinite and indefinite failure margins.................................................. 4.6

4.2.5 Inverse reserve factors......................................................................... 4.8

References ................................................................................................................. 4.9

5 LINEAR LAMINATE FAILURE ANALYSES...................................................... 5.1

Symbols ..................................................................................................................... 5.1

5.1 First Ply Failure Analysis .................................................................................. 5.3

5.2 Degraded Laminate Failure Analysis................................................................. 5.4

5.2.1 General................................................................................................ 5.4

5.2.2 Degraded properties of layers .............................................................. 5.4

5.2.3 Selection between tensile and compressive behavior ........................... 5.7

5.2.4 Margin to failure ................................................................................. 5.9

References ............................................................................................................... 5.10

6 LOCAL INSTABILITY OF SANDWICH LAMINATES ..................................... 6.1

Symbols ..................................................................................................................... 6.1

6.1 Wrinkling.......................................................................................................... 6.3

6.1.1 Wrinkling stresses ............................................................................... 6.3

6.1.2 Reserve factors for wrinkling............................................................... 6.4

6.1.3 Wrinkling with FPF and DLF analyses ................................................ 6.6

References ................................................................................................................. 6.7

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7 TRANSVERSE SHEAR IN LAMINATE ANALYSES ......................................... 7.1

8 LAMINATE THERMAL CONDUCTIVITY ......................................................... 8.1

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1 Notation and Conventions

1 NOTATION AND CONVENTIONS

Markku Palanterä (HUT/LLS)

The laminate coordinate system, specification of layer orientations, and layer numbering convention are

introduced. The definitions of laminate resultant forces and moments and normalized stresses are given.

Laminate midplane and flexural strains are also defined.

SYMBOLS

h Laminate thickness

hC Thickness of sandwich laminate core

hFb Thickness of sandwich laminate bottom face sheet

hFt Thickness of sandwich laminate top face sheet

k Layer index (k = 1,�, n); layer interface index (k = 0,�, n)

{M}xy Laminate resultant moments

{N}xy Laminate resultant in-plane forces

n Number of layers in a laminate

Qx , Qy Transverse resultant shear forces

x, y, z Laminate coordinate system

γ Shear strain

ε Normal strain

{ε} Strain vector, contracted notation (i.e. engineering strains)

θ Rotation angle between the 12-coordinate system and the xy-

coordinate system around the z-axis

σ Normal stress

{σ} Stress vector

τ Shear stress

Subscripts


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xy In the xy-plane of the laminate xyz-coordinate system

Superscripts

b Bottom surface (of a laminate)

f Flexural (stress or strain)

t Top surface (of a laminate)

° Laminate midplane (strain); normalized in-plane (stress)

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1.1 LAMINATE COORDINATE SYSTEM AND LAY-UP

The laminate coordinate system is formed by the axes x, y, and z (Figure 1.1). The x- and y-

axes define the plane of the laminate, and the z-axis is thus normal to this plane. The origin of

the z-axis is fixed to the midplane of the laminate (Figure 1.2). The thickness of the laminate

is denoted by h. The laminate surface on the negative side of the z-coordinate is referred to as

the top surface (z = -½h) and the surface on the positive side as the bottom surface (z = ½h).

Figure 1.2 shows the layer numbering convention for laminates. The number of layers in a

laminate is n. The layer number of the top layer is 1 and the number of the bottom layer is n.

The z-coordinates referring to the top and bottom surface of the k'th layer are zk-1 and zk,

respectively.

The orientation of the k�th layer in a laminate is indicated by an angle θk. It is the angle

between the axes x and y of the laminate coordinate system and the axes 1 and 2 of the layer

coordinate system 123. Figure 1.3 defines the sign of the angle θ. The layer coordinate system

is identical to the ply coordinate system 123 introduced in Part I.

In sandwich laminates, the thickness of the core layer is denoted by hC and the thickness of

the top and bottom face sheet by hFt and hFb, respectively (Figure 1.4).

Figure 1.1 Laminate coordinate system xyz.

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Figure 1.2 Layer and layer interface numbering convention for laminates.

Figure 1.3 Positive rotation of the layer axes 1 and 2 with respect to the laminate axes x and y is defined by theright-hand-rule: when the thumb points in the z-direction, fingers indicate the positive direction of rotation.

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hFt

hFb

hCh

top face sheet

bottom face sheet

core

Figure 1.4 Notation for sandwich laminate core and face sheet thicknesses.

1.2 LAMINATE STRESSES AND STRAINS

For stresses and strains in individual layers of a laminate, the notation and conventions

introduced in Part I for plies apply. In the following, the laminate level stresses and strains are

defined.

1.2.1 Resultant forces and moments

Laminate resultant forces are forces per unit width corresponding to the stress state in the

laminate. Thus, the resultant in-plane forces are integrals of the in-plane stress components

over the laminate thickness:

{ } { } dzdz=

N

N

N

=N xy

h

hxy

y

xh

hxy

y

x

xy σ

τσσ

∫∫−−

=

2/

2/

2/

2/

(1.2.1)

The laminate resultant transverse shear forces are defined as

{ } dz=Q

Q=Q

zx

yzh

hx

y

xy

∫− τ

τ2/

2/

(1.2.2)

The order of the force components in the vector is due to the order of the shear stress

components in the stress vector for three-dimensional stress-state (Part I, Section 1.2). Since

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the transverse shear forces Qx and Qy are not considered in basic laminate analyses, the

resultant in-plane forces are simply referred to as the resultant forces later in this text.

The laminate resultant moments are defined in a way similar to the resultant forces to

characterize the moment effect of the laminate stresses:

{ } { } dzzdzz=

M

M

M

=M xy

h

hxy

y

xh

hxy

y

x

xy σ

τσσ

∫∫−−

=

2/

2/

2/

2/

(1.2.3)

The directions of the resultant forces and moments are shown in Figure 1.5.

Figure 1.5 Laminate resultant (a) in-plane forces, (b) transverse shear forces, and (c) moments.

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1.2.2 Normalized stresses

Laminate stress state can also be expressed in terms of so-called normalized stresses, which

are the resultant forces and moments normalized with respect to the laminate thickness.

Normalized in-plane stresses are the average stresses corresponding to the laminate in-plane

stress state. In terms of the laminate resultant forces, the normalized in-plane stresses are

{ } { }xy

xy

y

x

xy Nh

==1

ο

ο

ο

ο

τσσ

σ (1.2.4)

Similarly, the normalized transverse shear stresses are defined as

{ }xy

zx

yzQ

h=

1

ο

ο

ττ

(1.2.5)

Normalized flexural stresses are the stresses at the laminate surfaces that would, in case of a

linearly changing stress distribution with change of sign at the midplane, cause the same

laminate curvatures as the actual stress distribution. Thus, the normalized flexural stresses are

{ } { }xy

f

xy

f

y

f

x

xy

fM

h==

2

6

τσσ

σ (1.2.6)

The normalized in-plane and flexural stresses are illustrated in Figure 1.6a. Although the

actual stress distribution may be nonlinear and discontinuous, the linear distribution shown in

the figure is useful in describing the resultant effect of the stresses on the laminate level.

1.2.3 Midplane and flexural strains

In the classical lamination theory, laminate in-plane strains are assumed to change linearly as

functions of the z-coordinate (see Chapter 2). Thus, the strain state can be expressed in terms

of so-called midplane and flexural strains as shown in Figure 1.6b.

The laminate midplane strains are the mean values of the strains at the laminate top and

bottom surfaces:

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{ } { } { }( )t

xy

b

xy

xy

y

x

xy== εε

γεε

εο

ο

ο

ο +

2

1(1.2.7)

The laminate flexural strains describe the variation from a uniform strain distribution:

{ } { } { }( )t

xy

b

xy

f

xy

f

y

f

x

xy

f== εε

γεε

ε −

2

1(1.2.8)

Figure 1.6 (a) Normalized laminate in-plane and flexural stresses, and (b) laminate midplane and flexural

strains.

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Thus, in terms of the midplane and flexural strains, the strains at the laminate top and bottom

surfaces are

{ } { } { }{ } { } { }

xy

f

xy

b

xy

xy

f

xy

t

xy

=

=

εεε

εεεο

ο

+

−(1.2.9a,b)

Concerning the shear strains γxy, it should be reminded that engineering shear strains are used

in ESAComp (see Part I, Figure 1.3).

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III Laminates

2 Classical Lamination Theory

2 CLASSICAL LAMINATION THEORY


The basic assumptions of the classical lamination theory (CLT) are presented. Laminate stress-strain relations

are given in terms of stiffness and compliance matrices. Laminate moduli and coupling coefficients are further

derived. The handling of internal loads, i.e. thermal and moisture loads and initial strains, is introduced assuming

that the loads may vary linearly between the layer top and bottom surfaces. Hygrothermal expansion and

curvature coefficients of a laminate are also derived. Procedures for the computation of laminate load response

under combined external and internal loads are described. The types of external loads covered are laminate stress

state, strain state, and in-plane stress state for suppressed laminate curvature. Finally, the computation of layer

strains and stresses is described.

SYMBOLS

[A] In-plane stiffness matrix of a laminate

[a] In-plane compliance matrix of a laminate

[ã*] In-plane compliance matrix of a laminate for zero-curvature

(= [A*]-1)

[B] Coupling stiffness matrix of a laminate

[b] Coupling compliance matrix of a laminate

[D] Flexural stiffness matrix of a laminate

[d] Flexural compliance matrix of a laminate

E Young's modulus

G Shear modulus


hk Thickness of the k�th layer


{M}xy Laminate resultant moments

m Moisture content by weight

{N}xy Laminate resultant in-plane forces


[Q] Layer stiffness matrix, plane stress state, in the layer coordinate

system

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III Laminates


[ Q ] Layer stiffness matrix, plane stress state, in the laminate coordinate

system

T Temperature

[T] Transformation matrix

u, v, w Displacements in the x-, y-, and z-directions of the laminate

coordinate system


{α} Thermal expansion coefficients

{β} Moisture expansion coefficients

{Γ} Hygrothermal stress coefficients

γ Shear strain

ε Normal strain

{ε} Strain vector, contracted notation (i.e. engineering strains)

∆m Moisture content difference between the reference environment and

the operating environment

z

k

c

k mm ∆∆ , Constants describing the ∆m distribution within the k�th layer as a

function of z, Eqs. (2.4.11�12)

∆T Temperature difference between the reference environment and the

operating environment

z

k

c

k TT ∆∆ , Constants describing the ∆T distribution within the k�th layer as a

function of z, Eqs. (2.4.2�3)

{δ} Thermal curvature coefficients

{ε} Strains, contracted notation (i.e. engineering strains)

ηij,i Lekhnitskii's coefficient of the first kind (εi /γij , τij applied)

ηi,ij Lekhnitskii's coefficient of the second kind (γij /εi , σi applied)

{κ} Curvatures

νij Poisson�s ratio (-εj /εi , σi applied)

σ Normal stress

{σ} Stress vector

τ Shear stress

{φ} Moisture curvature coefficients

1, 2, 3 Layer coordinate system

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III Laminates


Subscripts

Fb Sandwich bottom face sheet

Ft Sandwich top face sheet


r Residual (due to an internal load)

ref Laminate reference environment


0 Free strain (layer initial strain or due to hygrothermal expansion)

12 In the 12-plane of the layer 123-coordinate system

Superscripts

A Apparent (force, moment, or normalized stress)

a Midplane of a layer

b Bottom surface (of a layer)

e Equivalent (strain)

f Flexural (engineering constant; stress or strain)

I Internal (due to combination of hygrothermal loads and initial

strains)

NM Due to applied forces and moments

s Stress/strain recovery plane (either top surface, bottom surface, or

midplane of a layer)

T Transpose matrix

t Top surface (of a layer)

∆m Due to moisture

∆T Thermal

ε0 Due to initial strain

κ=0 Zero-curvature

-T Inverse of a transpose matrix

-1 Inverse matrix

* Normalized (stiffness or compliance matrix)

° Laminate midplane (strain); normalized in-plane (stress)

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III Laminates


2.1 GENERAL ASSUMPTIONS

The linear laminated plate theory, also commonly known as the classical lamination theory

(CLT), is covered in numerous text books, e.g. [1�8]. The basic assumptions of the CLT are:

• The laminate is formed from layers that are perfectly bonded together.

• The material properties of each layer are constant through the thickness of the layer.

• The stress-strain relations of the layers are linear-elastic.

• The laminate is in plane stress state.

• Lines originally straight and normal to the midplane of the laminate remain straight and

normal in extension and bending.

• Laminate in-plane strains and curvatures are small compared to unity.

A laminate has to be relatively thin and flat to meet the requirements listed above. However,

the CLT has been successfully used for stress-strain analyses of sandwich laminates, in which

a core material with low in-plane modulus connects two face sheets. The transverse shear

effects have to be accounted for when determining the out-of-plane deformations of sandwich

constructions.

With caution, the CLT can be extended to analyses of plates with small curvatures (h/R « 1).

For accurate analyses of curved shells, the effects of curvatures have to be considered in the

strain-displacement equations.

2.2 STRAIN-DISPLACEMENT RELATIONS

Assuming small displacements, the laminate in-plane strains can be approximated with the

first order derivatives of the displacements:

{ }

∂∂

∂∂

∂∂∂∂

x

v+

y

u

y

v

x

u

==

xy

y

x

xy

γεε

ε (2.2.1)

Based on the CLT assumptions, the relations between laminate strains and curvatures can be

derived, e.g. [3]. Thus, in terms of laminate midplane strains and curvatures, laminate strains

can be expressed as

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III Laminates


{ } { } { }xyxyxy z+= κεε ο

(2.2.2)

According to the first equation, the midplane strains are

{ }

∂∂

∂∂

∂∂∂

∂

x

v+

y

u

y

v

x

u

==

xy

y

x

xy

οο

ο

ο

ο

ο

ο

ο

γεε

ε (2.2.3)

Laminate curvatures are second order derivatives of the midplane displacement in the z-

direction:

{ }

∂∂∂

∂∂

∂∂

−

yx

w

y

w

x

w

==

xy

y

x

xy

ο

ο

ο

κκκ

κ

2

2

2

2

2

2

(2.2.4)

The definition of laminate flexural strains was given in Section 1.2. The only assumption

made was a linear strain distribution between the top and bottom surfaces of a laminate.

Applying the CLT strain-displacement relations, the flexural strains can also be stated in

terms of laminate curvatures. This yields

{ } { }xyxy

f h= κε

2(2.2.5)

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2.3 CONSTITUTIVE EQUATIONS

2.3.1 Stiffness matrices

Applying the layer constitutive relations in the laminate coordinate system, and by

implementing the laminate linear strain distribution of Eq. (2.2.2), the laminate resultant in-

plane forces are

{ } { } [ ] { }

[ ] { } [ ] { }xyk

h

h

xyk

h

h

xyk

h

h

xy

h

h

xy

dzzQ+dzQ=

dzQ=dz=N

κε

εσ

ο2/

2/

2/

2/

2/

2/

2/

2/

∫∫

∫∫

−−

−−(2.3.1)

Similarly, the laminate resultant moments are

{ } { } [ ] { }

[ ] { } [ ] { }xyk

h

h

xyk

h

h

xyk

h

h

xy

h

h

xy

dzzQ+dzzQ=

dzzQ=dzz=M

κε

εσ

ο 2

2/

2/

2/

2/

2/

2/

2/

2/

∫∫

∫∫

−−

−−(2.3.2)

The layer stiffness matrices [ ]kQ were defined in Part I, Section 2.3.

For convenience, the given relations are written in the form

{ } [ ]{ } [ ]{ }{ } [ ]{ } [ ]{ }

xyxyxy

xyxyxy

D+B=M

B+A=N

κε

κεο

ο

(2.3.3a,b)

where

[ ] [ ]

[ ] [ ]

[ ] [ ] dzzQD

dzzQB

dzQA

k

h

h

k

h

h

k

h

h

2

2/

2/

2/

2/

2/

2/

∫

∫

∫

−

−

−

=

=

=

(2.3.4a�c)

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[A] is the in-plane, [B] the coupling, and [D] the flexural stiffness matrix of the laminate.

Since the layer stiffness matrices are assumed to be constant through the thickness of each

layer, the integrals can be replaced with summations. The contribution of each layer is

[ ] ( )[ ][ ] ( )[ ][ ] ( )[ ]

kkkk

kkkk

kkkk

Qzz=D

Qzz=B

Qzz=A

31

3

3

1

21

2

2

1

1

−

−

−

−

−

−

(2.3.5a�c)

Then, the matrices are summed to obtain the laminate stiffness matrices:

[ ] [ ]

[ ] [ ]

[ ] [ ]k

n

=k

k

n

=k

k

n

=k

D=D

B=B

A=A

∑

∑

∑

1

1

1

(2.3.6a�c)

The three matrices can be combined into a 6 by 6 stiffness matrix of the laminate. Hence, Eqs.

(2.3.3a,b) yield

xyxyDB

BA

=

M

N

−−

−−−−

−−

κ

ε ο

|

|

|

(2.3.7)

2.3.2 Compliance matrices

The laminate stress-strain relations can also be stated in terms of compliances, which leads to

xy

T

xyM

N

db

ba

=

−−

−−−−

−−

|

|

|

κ

ε ο

(2.3.8)

where [a], [b] and [d] are the in-plane, coupling, and flexural compliance matrices of the

laminate. The combined compliance matrix is the inverse of the 6 by 6 stiffness matrix:

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III Laminates


−−−−

−−−−

−

DB

BA

=

db

ba

T |

|

|

|

|

|1

(2.3.9)

2.3.3 Normalized stiffness and compliance matrices

The laminate constitutive relations can also be written in terms of normalized laminate

stresses. The normalized in-plane and flexural stresses relate to the laminate resultant forces

and moments as defined in Section 1.2. When curvatures are also replaced by laminate

flexural strains defined by Eq. (2.2.5), Eq. (2.3.7) leads to

xy

f

xy

f DB

BA

=

−−

−−−−

−−

ε

ε

σ

σ οο

**

**

|3

|

|

(2.3.10)

where [A*], [B*], and [D*] are the normalized in-plane, coupling, and flexural stiffness

matrices of the laminate. From the definitions of normalized stresses and laminate curvatures

it follows that

[ ] [ ]

[ ] [ ]

[ ] [ ]Dh

=D

Bh

=B

Ah

=A

*

*

*

3

2

12

2

1

(2.3.11a�c)

The inverse relation is

( )xy

fT

xy

f db

ba

=

−−

−−−−

−−

σ

σ

ε

ε οο

**

*

31*

|

|

|

(2.3.12)

where [a*], [b*], and [d*] are the normalized in-plane, coupling, and flexural compliance

matrices. The normalization procedure is inverse to that in Eqs. (2.3.11a�c), thus

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[ ] [ ]

[ ] [ ]

[ ] [ ]dh

=d

bh

=b

ah=a

*

*

*

12

23

2

(2.3.13a�c)

2.3.4 Laminate engineering constants

The in-plane engineering constants of a laminate can be determined using the analogy

between the ply compliance matrix (see Part I, Section 2.2) and the normalized in-plane

compliance matrix of a laminate. The Young's moduli and the in-plane shear modulus are

a=G

a=E

a=E *xy*y*x

662211

1;

1;

1(2.3.14a�c)

Similarly, the Poisson's ratios and the Lekhnitskii's coefficients are

a

a=

a

a=

*

*

yx*

*

xy

22

12

11

12; −− νν (2.3.15a,b)

and

a

a=

a

a=

a

a=

a

a=

*

*

yxy,*

*

xyy,

*

*

xxy,*

*

xyx,

66

26

22

26

66

16

11

16

;

;

ηη

ηη

(2.3.16a�d)

If there is coupling between the laminate in-plane and flexural behavior, i.e. there are non-

zero elements in the [B]-matrix, these engineering constants refer to the case where the

laminate is free to curve when loaded with in-plane forces.

The in-plane engineering constants may also be defined for the case of suppressed laminate

curvature, in-other words, for zero-curvature. The in-plane forces are now related to the in-

plane strains through the [A]-matrix alone. For convenience, the inverse of the normalized

[A]-matrix is denoted by

[ ] [ ] 1~−

A=a** (2.3.17)

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III.2.10

III Laminates


Thus, the in-plane engineering constants for zero-curvature are

a=G

a=E

a=E *xy*y*x ~

1;

~

1;

~

1

66

0

22

0

11

0 === κκκ (2.3.18a�c)

a

a=

a

a=

*

*

yx*

*

xy ~

~;

~

~

22

120

11

120 −− == νν κκ (2.3.19a,b)

a

a=

a

a=

a

a=

a

a=

*

*

yxy,*

*

xyy,

*

*

xxy,*

*

xyx,

~

~;

~

~

~

~;

~

~

66

260

22

260

66

160

11

160

ηη

ηη

κκ

κκ

==

==

(2.3.20a�d)

If all the elements in the [B]-matrix are zero, the in-plane engineering constants for free

curvature and zero-curvature are naturally the same.

The flexural engineering constants relate the laminate normalized flexural stresses and

flexural strains in a similar way as in-plane engineering constants relate the laminate

normalized in-plane stresses and midplane strains. Thus, the flexural engineering constants

are obtained by replacing the elements of the normalized in-plane compliance matrix with the

corresponding elements of the normalized flexural compliance matrix in the previous

equations:

d=G

d=E

d=E *

fxy*

fy*

fx

662211

1;

1;

1(2.3.21a�c)

d

d=

d

d=

*

*fyx*

*fxy

22

12

11

12; −− νν (2.3.22a,b)

d

d=

d

d=

d

d=

d

d=

*

*f

yxy,*

*f

xyy,

*

*f

xxy,*

*f

xyx,

66

26

22

26

66

16

11

16

;

;

ηη

ηη

(2.3.23a�d)

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III.2.11

III Laminates


2.4 INTERNAL LOADS

Besides external loads such as applied forces and moments, or given laminate strains and

curvatures, there may be loads that are internal by their nature. Hygrothermal expansion

causes this kind of loads. Initial strains may also be induced to layers of a laminate in some

other ways. In this section, the effects of internal loads are studied on the laminate level. The

resulting layer strains and stresses are covered later in Section 2.7.

Note: Analyses involving layer initial strains are to be introduced in ESAComp Version 2.0. For consistency, the

theoretical background for handling this type of loads is already presented here.

2.4.1 Thermal loads

In ESAComp, thermal strains and stresses in a laminate are assumed to be caused by

temperature differences between the operating temperature (T) and the reference temperature

(Tref). The stress-free temperature resulting from the curing process is usually used as the

reference temperature. However, a stress-free temperature may not be found, for instance, for

a laminate that is formed by bonding together cured sublaminates. Such cases can be handled

with layer initial strains introduced later in this section.

The reference temperature is assumed to be constant through the thickness of the laminate, but

the operating temperature may vary through the thickness of the laminate. Thus, the

temperature difference is also a function of the z-coordinate:

refTzT=zT −∆ )()( (2.4.1)

Further in this section, temperature differences are simply referred to as temperatures.

ESAComp assumes that the temperature distribution is a linear function of the z-coordinate

within each layer. Thus, the temperature distribution of a laminate can be given in terms of

the temperatures at the top (∆Tt) and bottom surfaces (∆T

b) of each layer. The temperature

distribution of the k'th layer is

( )z

k

c

k

kk

t

k

b

kk

t

kk

Tz+T

zz

TTzzTzT

∆∆=

−∆−∆

−+∆=∆−

−1

1)((2.4.2)

where

z

kk

t

k

c

k

kk

t

k

b

kz

k

TzT=T

zz

TT=T

∆−∆∆

−∆−∆

∆

−

−

1

1

(2.4.3a,b)

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III.2.12

III Laminates


For later use, the layer midplane temperatures are

( )b

k

t

k

a

k T+T=T ∆∆∆2

1 (2.4.4)

The equivalent thermal forces and moments are the forces and moments that, when acting

alone, cause the same laminate strain state as the thermal loading. Using the thermal stress

coefficients (Γ∆T) defined in Part I, Section 2.4, the equivalent thermal forces and moments

are

{ } { } { } ( )

{ } { }( )z

kk

Tc

kk

T

n

k

z

k

c

kkxy

Th

h

kkxy

Th

h

xy

T

TB+TA=

dzTz+T=dzT=N

∆∆

∆∆Γ∆Γ

∆∆

=

∆

−

∆

−

∆

∑

∫∫

1

,

2/

2/

,

2/

2/ (2.4.5)

and

{ } { } { } ( )

{ } { }( )z

kk

Tc

kk

T

n

k

z

k

c

kkxy

Th

h

kkxy

Th

h

xy

T

TD+TB=

dzzTz+T=dzzT=M

∆∆

∆∆Γ∆Γ

∆∆

=

∆

−

∆

−

∆

∑

∫∫

1

,

2/

2/

,

2/

2/ (2.4.6)

where

{ } ( ){ }{ } ( ){ }{ } ( ){ }

kxy

T

kkk

T

kxy

Tkkk

T

kxy

Tkkk

T

zz=D

zz=B

zz=A

,

31

3

31

,

21

2

21

,1

Γ−

Γ−

Γ−

∆−

∆

∆−

∆

∆−

∆

(2.4.7a�c)

Combining the expressions for equivalent thermal forces and moments yields

∆−−

∆

−−−−

−−

∆∆

∆∆

=∆

∆

∑k

z

c

TT

TT

k

n

k

xy

T

T

T

T

DB

BA

=

M

N

|

|

|

1

(2.4.8)

The laminate strains and curvatures in free expansion of a laminate can be calculated from the

equivalent loads with the aid of the compliance matrix of the laminate. Thus, the laminate

thermal strains and curvatures are

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III.2.13

III Laminates


xy

T

T

T

xy

T

T

M

N

db

ba

=

−−

−−−−

−−

∆

∆

∆

∆

|

|

|

κ

εο

(2.4.9)

2.4.2 Moisture loads

Moisture loads are handled in a way analogous with thermal loads. The moisture content

difference between the actual and the reference moisture content is

refmzm=zm −∆ )()( (2.4.10)

The reference moisture content is often assumed to be zero.

The expression for a linear moisture content distribution of the k'th layer is

z

k

c

kk mz+m=zm ∆∆∆ )( (2.4.11)

where

z

kk

t

k

c

k

kk

t

k

b

kz

k

mzm=m

zz

mm=m

∆−∆∆

−∆−∆

∆

−

−

1

1

(2.4.12a,b)

and the layer midplane values are

( )b

k

t

k

a

k m+m=m ∆∆∆2

1 (2.4.13)

The equivalent moisture forces and moments can be expressed as

∆−−

∆

−−−−

−−

∆∆

∆∆

=∆

∆

∑k

z

c

mm

mm

k

n

k

xy

m

m

m

m

DB

BA

=

M

N

|

|

|

1

(2.4.14)

where

Version 1.0


III.2.14

III Laminates


{ } ( ){ }{ } ( ){ }{ } ( ){ }

kxy

m

kkk

m

kxy

mkkk

m

kxy

mkkk

m

zz=D

zz=B

zz=A

,

31

3

3

1

,

21

2

2

1

,1

Γ−

Γ−

Γ−

∆−

∆

∆−

∆

∆−

∆

(2.4.15a�c)

Finally, the moisture strains and curvatures in free expansion of a laminate are

xy

m

m

T

xy

m

m

M

N

db

ba

=

−−

−−−−

−−

∆

∆

∆

∆

|

|

|

κ

ε ο

(2.4.16)

2.4.3 Initial strains

Internal loads may also be induced in a laminate by layer initial strains. For example,

laminates formed by bonding together sublaminates may be analyzed using this concept.

Strains due to hygrothermal expansion can also be thought of as initial strains; the product of

the temperature or moisture content difference and the expansion coefficient corresponds to

an initial strain. Unlike in hygrothermal expansion, shear strains may also be given as initial

strains.

Initial strains are assumed to change linearly within individual layers. The strain distributions

can thus be given as strains at the top and bottom surfaces of layers. Initial strains given in the

layer coordinate system are first transformed to the laminate coordinate system (see Part I,

Section 1.3):

{ } [ ] { }s

k

T

k

s

kxy T=,120,0 εε (2.4.17)

The superscript s refers either to the top or bottom surface of the layer.

The initial strain distribution of the k'th layer is

{ } { } { }kxy

z

kxy

c

kxyz=

,0,0,0εεε + (2.4.18)

where

{ } { } { }

{ } { } { } kxy

zk

t

kxykxy

c

kk

t

kxy

b

kxy

kxy

z

z=

zz=

,01,0,0

1

,0,0

,0

εεε

εεε

−

−

−

−

−

(2.4.19a,b)

Version 1.0


III.2.15

III Laminates


The layer initial midplane strains are

{ } { } { }( )b

kxy

t

kxy

a

kxy+=

,0,02

1

,0εεε (2.4.20)

The equivalent forces and moments due to initial strains can be written as

{ } [ ] { }

[ ] ( ){ } ( ){ }[ ]

[ ] { } [ ] { }( )kxy

z

kkxy

c

k

n

k

kxy

zkkkxy

ckkk

n

k

xyk

h

h

xy

B+A=

zz+zzQ=

dzQ=N

,0,0

1

,02

12

2

1

,01

1

0

2/

2/

0

εε

εε

εε

∑

∑

∫

=

−−=

−

−− (2.4.21)

and

{ } [ ] { }

[ ] ( ){ } ( ){ }[ ]

[ ] { } [ ] { }( )kxy

z

kkxy

c

k

n

k

kxy

zkkkxy

ckkk

n

k

xyk

h

h

xy

D+B=

zz+zzQ=

dzzQ=M

,0,0

1

,03

13

3

1

,02

12

2

1

1

0

2/

2/

0

εε

εε

εε

∑

∑

∫

=

−−=

−

−− (2.4.22)

where the stiffness submatrices [A]k, [B]k, and [D]k are defined by Eqs. (2.3.5a�c).

Combining the expressions gives

−−

−−−−

−− ∑

=

kxy

z

c

k

n

k

xyDB

BA

=

M

N

,0

0

1|

|

|

0

0

ε

ε

ε

ε

(2.4.23)

The laminate strains and curvatures due to initial strains are

xy

T

xyM

N

db

ba

=

−−

−−−−

−−

0

0

0

0

|

|

|

ε

ε

ε

εο

κ

ε(2.4.24)

Version 1.0


III.2.16

III Laminates


2.4.4 Constitutive equations in combined loading

Laminate constitutive equations were introduced in Section 2.3 for the case that only external

loads are applied. These equations are now modified to take into account the effects of

internal loads. The equivalent load vectors due to hygrothermal loads and initial strains are

first combined:

xyxy

m

m

xy

T

T

xy

I

I

M

N

M

N

M

N

=

M

N

−−+

−−+

−−

−−∆

∆

∆

∆

0

0

ε

ε

(2.4.25)

Actual laminate strains and curvatures are partly due to internal loads. Thus, the combined

equivalent load vector has to be subtracted from the right-hand-side of Eq. (2.3.7) to obtain

the externally applied forces and moments:

xy

I

I

xyxyM

N

DB

BA

=

M

N

−−−

−−

−−−−

−−

κ

ε ο

|

|

|

(2.4.26)

The equivalent loads can be moved to the left-hand-side of the equation, which yields the

apparent forces and moments:

xy

I

I

xyxy

A

A

M

N

M

N

=

M

N

−−+

−−

−− (2.4.27)

Now, the constitutive equations can be written as

xyxy

A

A

DB

BA

=

M

N

−−

−−−−

−−κ

ε ο

|

|

|

(2.4.28)

The laminate strains and curvatures caused by hygrothermal loads and initial strains can be

determined from the equivalent loads as described earlier in this section. These strains and

curvatures can further be combined:

Version 1.0


III.2.17

III Laminates


xyxy

m

m

xy

T

T

xy

I

I

=

−−+

−−+

−−

−−

∆

∆

∆

∆

0

0

ε

εοοοο

κ

ε

κ

ε

κ

ε

κ

ε(2.4.29)

The laminate strains and curvatures due to applied forces and moments are then

xy

I

I

xyxy

NM

NM

=

−−−

−−

−−

κ

ε

κ

ε

κ

ε οοο

(2.4.30)

Thus, the laminate constitutive equations can also be written in the form

xy

NM

NM

xyDB

BA

=

M

N

−−

−−−−

−−

κ

εο

|

|

|

(2.4.31)

As in Subsection 2.3.3, all the relations presented in this subsection can also be written in

terms of normalized in-plane and flexural stresses and midplane and flexural strains.

Consequently, the six by six stiffness matrix of the above equations must be replaced by the

normalized stiffness matrix of Eq. (2.3.10).

2.5 HYGROTHERMAL EXPANSION COEFFICIENTS

2.5.1 Thermal expansion

The thermal expansion coefficients of a laminate can be solved from the constitutive

equations for combined external and internal loading introduced in Subsection 2.4.4. An

equivalent thermal load vector corresponding to uniform unit temperature is applied to the

laminate and external forces and moments are set to zero. Midplane strains are replaced by the

laminate thermal expansion coefficients (α) and curvatures by the laminate thermal curvature

coefficients (δ). Hence

{ }0

|

|

|

=

−−−

−−

−−−−

∆

∆

xy

T

T

xyM

N

DB

BA

δ

α(2.5.1)

Version 1.0


III.2.18

III Laminates


where

k

T

T

n

kTT

TT

k

n

k

xy

T

T

B

A

DB

BA

=

M

N

−−=

−−

−−−−

−−

∆

∆

=∆∆

∆∆

=∆

∆

∑∑11

0

1

|

|

|

(2.5.2)

A∆T

and B∆T

are defined in Eqs. (2.4.7a�c). Solving for the expansion coefficients gives

−−

−−−−

−−

∆

∆

=∑

k

T

T

n

kT

xy B

A

db

ba

=1

|

|

|

δ

α(2.5.3)

In many structural applications, laminate curvatures are suppressed. Therefore, it is also

useful to determine the in-plane expansion coefficients for zero-curvature. Setting the in-plane

forces and curvatures to zero yields

xyk

T

T

n

k

xyMB

A

DB

BA

−−=

−−−

−−

−−−−

∆

∆

=

=

∑0

0|

|

|

1

0κα(2.5.4)

Solving for the thermal expansion coefficients gives

{ } [ ] { }

= ∆

=

−= ∑ k

Tn

k

xy AA1

10κα (2.5.5)

2.5.2 Moisture expansion

Moisture expansion is handled similarly as thermal expansion. The laminate moisture

expansion coefficients (β) and moisture curvature coefficients (φ) are

−−

−−−−

−−

∆

∆

=∑

k

m

m

n

kT

xy B

A

db

ba

=1

|

|

|

φ

β(2.5.6)

Version 1.0


III.2.19

III Laminates


A∆m

and B∆m

are determined according to Eqs. (2.4.15a�c).

The moisture expansion coefficients for zero-curvature are

{ } [ ] { }

= ∆

=

−= ∑ k

mn

k

xy AA1

10κβ (2.5.7)

2.6 LAMINATE LOAD RESPONSE

This section summarizes the procedures for solving the laminate stress-strain state under three

different types of external loads: applied strain state, applied stress state, and applied in-plane

stress state when laminate curvature is suppressed. In addition, it is assumed that internal

loads are applied simultaneously. The constitutive relations applied here were introduced in

Subsection 2.4.4.

2.6.1 Laminate strain state applied

In the following, it is assumed that the laminate strain state is given as midplane strains and

curvatures. The apparent forces and moments can directly be solved from

xyxy

A

A

DB

BA

=

M

N

−−

−−−−

−−κ

ε ο

|

|

|

(2.6.1)

The equivalent forces and moments corresponding to internal loads can be solved as described

in Section 2.4. Thus, the forces and moments applied externally to the laminate are

xy

I

I

xy

A

A

xyM

N

M

N

=

M

N

−−−

−−

−− (2.6.2)

The laminate strains and curvatures due to internal loads can be solved from the equivalent

loads as described in Section 2.4. Then, the strains and curvatures due to the applied forces

and moments are

Version 1.0


III.2.20

III Laminates


xy

I

I

xyxy

NM

NM

=

−−−

−−

−−

κ

ε

κ

ε

κ

ε οοο

(2.6.3)

These strains and curvatures could also be computed from the applied forces and moments

from the inverse relation of Eq. (2.4.31).

2.6.2 Laminate stress state applied

When the external laminate load is a given stress state, the equivalent forces and moments due

to internal loads are determined first as outlined in Section 2.4. Combining these to the

applied stress state, expressed as forces and moments, gives the apparent forces and moments:

xy

I

I

xyxy

A

A

M

N

M

N

=

M

N

−−+

−−

−− (2.6.4)

The actual laminate strains and curvatures are then

xy

A

A

T

xyM

N

db

ba

=

−−

−−−−

−−

|

|

|

κ

ε ο

(2.6.5)

As in the previous case, the strains and curvatures due to the applied forces and moments are

solved from Eq. (2.6.3).

2.6.3 Laminate stresses applied in zero-curvature state

Laminate response may also be solved for the case of suppressed laminate curvature while the

in-plane stress state of the laminate is specified. In the following, the external load is given as

resultant forces. The equivalent forces and moments due to internal loads are determined as in

the previous cases. Thus, the apparent forces are

{ } { } { }xy

I

xyxy

ANNN += (2.6.6)

Substituting the zero-values for curvatures gives

Version 1.0


III.2.21

III Laminates


xyxy

A

A

DB

BA

=

M

N

−−

−−−−

−−0|

|

| ε ο

(2.6.7)

Solving for the midplane strains in terms of apparent forces gives

{ } [ ] { }xy

A

xy NA1−=οε (2.6.8)

By substituting this back to the earlier equation, the expression for the apparent moments

becomes

{ } [ ]{ } [ ][ ] { }xy

A

xyxy

ANABBM

1−== οε (2.6.9)

Finally, the externally applied moments are solved from

{ } { } { }xy

I

xy

A

xy MMM −= (2.6.10)

The strains and curvatures due to the applied forces and moments are determined as in the

previous cases.

2.7 LAYER STRAINS AND STRESSES

2.7.1 Actual strains

Strains of individual layers can be determined after the laminate response in terms of

midplane strains and curvatures is first determined as described in Section 2.6. Laminate

strains vary linearly as functions of the z-coordinate according to Eq. (2.2.2). Thus, the actual

strains of the k'th layer in the laminate coordinate system are

{ } { } { }{ } { } { }{ } { } ( ){ }xykkxy

a

kxy

xyxy k

b

kxy

xyxy k

t

kxy

zz=

z=

z=

κεε

κεε

κεε

ο

ο

ο

12

1,

,

1,

−

−

++

+

+

(2.7.1a�c)

Version 1.0


III.2.22

III Laminates


where t denotes the top surface, b the bottom surface, and a the midplane of the layer. In the

following equations the superscript s refers to any of these planes.

Transformation of strains and stresses from one coordinate system to another was introduced

in Part I, Section 1.3. Actual layer strains transformed to the layer coordinate system are

{ } [ ] { }s

kxy

T

k

s

k T,,12

εε −= (2.7.2)

In the rest of this section, layer strains and stresses are presented only in the layer coordinate

systems. If needed, any strains or stresses can be transformed back to the laminate coordinate

system.

2.7.2 Strains and stresses due to applied forces and moments

Layer strains due to applied forces and moments correspond to the laminate strain-state

caused by the external forces and moments acting alone. Section 2.6 summarized the

computation of the laminate strains and curvatures due to applied forces and moments. The

transformation of these strains and curvatures to layer strains is equivalent to that described

previously for actual strains:

{ } [ ] { } { }( )xy

NMs

kxy

NMT

k

s

k

NMzT= κεε ο +−

,12(2.7.3)

From the ply constitutive relations (Part I, Section 2.3), the corresponding layer stresses due

to applied forces and moments are

{ } [ ] { }s

k

NM

k

s

k

NMQ

,12,12εσ = (2.7.4)

2.7.3 Strains due to internal loads

The layer stains due to internal loads, i.e. thermal strains, moisture strains, and strains due to

initial strains, are obtained from the corresponding laminate strains and curvatures derived in

Section 2.4. The transformation of the laminate strains and curvatures to layer strains goes as

in the previously described cases:

{ } [ ] { } { }( ){ } [ ] { } { }( ){ } [ ] { } { }( )

xy

s

kxy

T

k

s

k

xy

ms

kxy

mT

k

s

k

m

xy

Ts

kxy

TT

k

s

k

T

zT=

zT=

zT=

000

,12

,12

,12

εεοε

ο

ο

κεε

κεε

κεε

+

+

+

−

∆∆−∆

∆∆−∆

(2.7.5a�c)

Version 1.0


III.2.23

III Laminates


The layer strains due to combined internal loads are

{ } { } { } { }s

k

s

k

ms

k

Ts

k

I

,12,12,12,120εεεεε ++= ∆∆

(2.7.6)

2.7.4 Free strains

Free thermal strains and free moisture strains result directly from the expansion coefficients

and temperature or moisture content differences induced to the layers:

{ } { }{ } { } k

s

k

s

k

m

k

s

k

s

k

T

m

T

,12,120

,12,120

βε

αε

∆=

∆=

∆

∆

(2.7.7a,b)

By definition, layer initial strains (ε0) as such are the free strains applied to the layers. Hence,

the combined free strains are

{ } { } { } { }s

k

s

k

ms

k

Ts

k

I

,120,120,120,120 εεεε ++= ∆∆(2.7.8)

2.7.5 Residual strains and stresses

Residual strains are the layer strains due to the internal loads when a laminate is free to

expand, i.e. no external forces or moments are applied to the laminate. Residual strains are

obtained by subtracting the strains in free expansion of the layers from the thermal and

moisture strains:

{ } { } { }{ } { } { }{ } { } { }s

k

s

k

s

kr

s

k

ms

k

ms

k

m

r

s

k

Ts

k

Ts

k

T

r

,120,12,12

,120,12,12

,120,12,12

00 εεε

εεε

εεε

εε −=

−=

−=∆∆∆

∆∆∆

(2.7.9a�c)

The residual strains due to the different types of internal loads can be combined into

{ } { } { } { }s

kr

s

k

m

r

s

k

T

r

s

k

I

r ,12,12,12,120εεεεε ++= ∆∆

(2.7.10)

Version 1.0


III.2.24

III Laminates


Residual stresses are the layer stresses due to internal loads in free expansion of a laminate. In

terms of the corresponding residual strains, the residual stresses are

{ } [ ] { }{ } [ ] { }{ } [ ] { }s

krk

s

kr

s

k

m

rk

s

k

m

r

s

k

T

rk

s

k

T

r

Q

Q

Q

,12,12

,12,12

,12,12

00 εε εσ

εσ

εσ

=

=

=∆∆

∆∆

(2.7.11a�c)

The residual stresses can further be combined:

{ } { } { } { }s

kr

s

k

m

r

s

k

T

r

s

k

I

r ,12,12,12,120εσσσσ ++= ∆∆

(2.7.12)

2.7.6 Equivalent strains and actual stresses

Equivalent layer strains are the strains that induce actual layer stresses in a laminate. They are

obtained by subtracting the strains in free expansion of the layers from the actual layer strains,

that is

{ } { } { }s

k

Is

k

s

k

e

,120,12,12εεε −= (2.7.13)

Equivalent strains can also be expressed as the sum of the strains due to applied forces and

moments and the combined residual strains:

{ } { } { }s

k

I

r

s

k

NMs

k

e

,12,12,12εεε += (2.7.14)

Actual layer stresses are computed from the equivalent strains:

{ } [ ] { }s

k

e

k

s

k Q,12,12

εσ = (2.7.15)

Alike equivalent strains, actual layer stresses can also be presented as the sum of the stresses

due to applied forces and moments and the combined residual stresses, hence

{ } { } { }s

k

I

r

s

k

NMs

k ,12,12,12σσσ += (2.7.16)

Version 1.0


III.2.25

III Laminates


2.7.7 Sandwich face sheet stresses

The average stresses in the face sheets of sandwich laminates are needed in some analyses.

These stresses are obtained by integrating the actual layer stresses in the laminate coordinate

system over the thickness of the face sheet. Since the stress distribution is linear within each

layer, the integration can be written in the form of summation

{ }{ } { }( )

{ } FbFtFhh

=h

+h

=a

kxyk

FFk

F

b

kxy

t

kxyk

F

xyF ,,1

,

,,21

=∑∑∑

σσσ

σ (2.7.17)

where Ft and Fb refer to the top and bottom face sheet, respectively.

2.8 TRANSVERSE SHEAR STIFFNESS

Beyond the scope of the CLT, elements A44, A55, and A45 can be included in the laminate

stiffness matrix to describe the transverse shear stiffness of the laminate. These elements are

defined by

5,4,,

2/

2/

, == ∫−

jidzCA

h

h

kijij (2.8.1)

The elements ijC of the layer stiffness matrix are obtained as described in Part I, Section 2.2.

Further, the integral can be replaced with the summation

( ) 5,4,,1

,

1

1, ==−= ∑∑==

− jihCzzCAn

k

kkij

n

k

kkkijij (2.8.2)

The constitutive equations for transverse shear can now be written as

=

zx

yz

x

y

AA

AA

Q

Q

γγ

5545

4544(2.8.3)

However, the distributions of the transverse shear stresses in the thickness direction of the

laminate are not accounted for in this formulation. A coefficient k can be used to modify the

stiffness matrix for the stress distributions. Two separate coefficients may also be used for the

Version 1.0


III.2.26

III Laminates


principal directions of the laminate. This topic is covered in more detail as plate analyses are

introduced in ESAComp Version 2.0.

Version 1.0


III.2.27

III Laminates


REFERENCES

1. Agarwal, B.D. and Broutman, L.J., Analysis and Performance of Fiber Composites,

2nd edition. Wiley, New York, 1990.

2. Humphreys, E.A. and Rosen, B.W., �Properties Analysis of Laminates�, Engineered

Materials Handbook, Vol. 1, Composites, pp. 218�235. ASM International, Metals

Park, OH, 1987.

3. Jones, R.M., Mechanics of Composite Materials. Hemisphere, New York, 1975.

4. Structural Materials Handbook, Volume 1 � Polymer Composites, ESA PSS-03-203,

Issue 1. ESA Publications Division, ESTEC, Noordwijk, 1994.

5. Tsai, S.W., Composites Design, 4th edition. Think Composites, Dayton, OH, 1988.

(Replaced by: Tsai, S.W., Theory of Composite Design, Think Composites, Dayton,

OH, 1992.)

6. Tsai, S.W. and Hahn, H.T., Introduction to Composite Materials. Technomic,

Westport, CT, 1980.

7. Vinson, J.R. and Sierakowski, R.L., The Behavior of Structures Composed of

Composite Materials. Kluwer, Dordrecht, 1990.

8. Whitney, J.M., Structural Analysis of Laminated Anisotropic Plates. Technomic,

Lancaster, PA, 1987.

Version 4.4 III.3.1


III Laminates

3 Failure Criteria

3 FAILURE CRITERIA

The built-in failure criteria of the ESAComp system are introduced. The criteria for polymer matrix fiber-

reinforced composites can be divided into three groups: independent conditions (maximum strain and stress

criteria), quadratic criteria (Tsai-Wu, Hoffman, and Tsai-Hill), and physically based criteria (Puck and Hashin).

For isotropic materials, the maximum shear stress and Von Mises criteria are also included. All the criteria are

specified in the form of failure criterion functions. For major part of the criteria two options are provided;

traditional plane stress based criteria are supplemented with an option to include transverse shear in out-of-plane

direction. Also, special criteria are provided for core materials.

SYMBOLS

f Failure criterion function

Fi, Fij Coefficients in quadratic failure criteria, stress space

*

12F Interaction coefficient in the Tsai-Wu criterion

Q Shear failure stress in the 23-plane

R Shear failure stress in the 13-plane

S, Se Shear failure stress/strain in the 12-plane

X, Xe Failure stress/strain in the 1-direction

Y, Ye Failure stress/strain in the 2-direction

g Shear strain

e Normal strain

x, h, z Principal stress/strain coordinate system

s Normal stress

t Shear stress

1, 2, 3 Ply principal coordinate system

Subscripts

c Compressive

f Fiber failure modem Matrix failure mode

III.3.2 Version 4.4

Theoretical Background of ESAComp Analyses 3.12.2012

III Laminates

3 Failure Criteria

t Tensile

Version 4.4 III.3.3


III Laminates

3 Failure Criteria

3.1 GENERAL

The strength of materials and material systems under multiaxial loads can be predicted based

on different failure criteria. Failure criteria relate the material strengths, defined for simple

uniaxial load cases and shear, to the general stress-strain state due to multiaxial loads.

Typically failure criteria are presented as mathematical expressions called failure criterion

functions (f). Failure criterion functions are defined so that when no load is applied, the

function has the value zero. The value one corresponds to failure of the material (system).

This chapter concentrates on failure criteria commonly used for fiber-reinforced polymer

matrix composites. Although these criteria are mainly used for failure analyses under static

loads, some of them may also be applicable for preliminary fatigue analyses, provided that

appropriate fatigue strength values are used. Since metal sheets are frequently used in hybrid

composites, two widely used yield criteria for metals, the maximum shear stress and the Von

Mises criterion, are also introduced.

Absolute values are used for compressive strengths in this text. In other words, the values of

the compressive failure stresses Xc and Yc and failure strains Xec and Yec are positive.

It should be noted that when internal loads are involved in failure analyses of composite

laminates, failure predictions for layers must be based on actual stresses (s) and equivalent

strains (ee) introduced in Section 2.7.

3.2 ISOTROPIC MATERIALS

The maximum shear stress criterion (Tresca) is a commonly used yield criterion for

metallic materials. It is based on the equivalent shear stress

÷ø

öçè

æ -= hxhx sssst2

1,

2

1,

2

1maxeq (3.2.1)

where sx and sh, the principal stresses of the plane stress state, are

( ) ( )2

1

2

12

2

2121,4

1

2

1úû

ùêë

é +-±+= tsssss hx (3.2.2)

Even though the maximum shear stress is involved in the formulation of the criterion, axial

failure stresses of the material are typically used in failure prediction. From Eq. (3.2.1) it can

be seen that in uniaxial loading conditions teq corresponds to one half of the axial stress. Thus,

the failure criterion function of the maximum shear stress criterion can be written as

III.3.4 Version 4.4


III Laminates

3 Failure Criteria

X

feqt2

= (3.2.3)

In the maximum shear stress criterion, it has to be determined whether the tensile or

compressive failure stress is to be used as the value of X. This can be done on the basis of the

principal stresses defined by Eq. (3.2.2). If both sx and sh are positive, the tensile strength Xt

is used. Similarly, if both stresses are negative, Xc is used. The other stress component may

also be zero. If the stresses have different signs, the ratios of the principal stresses and the

corresponding failure stresses are the basis for selecting between tensile and compressive

stresses:

0,0; <>

=Þ-

<

=Þ-

³

hx

hx

hx

ssss

ss

c

ct

t

ct

XXXX

XXXX

(3.2.4)

The Von Mises criterion is based on an equivalent stress, which in plane stress state is

defined as:

( ) 2

122

hxhx sssss -+=eq (3.2.5)

The sign of the principal stresses in Eq. (1.2.5) and different failure strengths in tension and

compression are considered by scaling the principal stresses with respect to the tensile

strength:

xx ssi

t

X

X= (3.2.6)

where

ci

ti

XX

XX

=Þ<

=Þ>

0

0

x

x

ss

(3.2.7)

The other principal stress is managed similarly. The failure criterion function is the ratio of

the equivalent stress to the axial tensile strength of the material:

Version 4.4 III.3.5


III Laminates

3 Failure Criteria

X

feqs

= (3.2.8)

where X=Xt

With this approach the failure surface is smooth unlike for the maximum shear stress

criterion, which encounters a bump in the area where the reference strength X changes.

For the 3D Von Mises criterion the equivalent stress is defined as:

2222 )()()(2 xzzhhx sssssss -+-+-=eq (3.2.9)

Now the principal stresses are solved from the equation (s3=0)

0

032313

230212

131201

=

-

-

-

sstttsstttss

(3.2.10)

where s0 includes the three roots (sx,sh,sz).

Again, the failure criterion function is the ratio of the equivalent stress to the axial strength of

the material as per Eq. (3.2.8). Whether the tensile or compressive failure stress is to be used

as the value of X is determined by comparing quotients of the principal stresses and associated

axial strengths:

Vhxsssss Vhx

,,,0

0;,,max =

<=

³=÷÷ø

öççè

æ= j

whenci

whenti

XXXk

j

j

iii

(3.2.11)

If k is determined by the tensile failure stress, then X=Xt, else X=Xc. It should be noted that

with condition X=Xt=Xc maximum shear stress criterion and Von Mises criterion are

simplified to the traditional form of the criteria.

III.3.6 Version 4.4


III Laminates

3 Failure Criteria

3.3 FIBER-REINFORCED PLIES

Failure criteria primarily applicable to polymer matrix fiber-reinforced plies are introduced in

this section. Due to their general nature, at least the maximum strain and stress criteria can

also be applied to other materials. Unless otherwise specified, all the criteria are presented in

references [1] and [2]. Reference [1] also includes a comprehensive list of additional

references to the criteria introduced here.

3.3.1 Maximum strain and stress criteria

In the so-called independent conditions it is assumed that the stress or strain components of

the principal coordinate system do not interact in the failure mechanisms.

In the maximum strain criterion, the ratios of the actual strains to the failure strains are

compared in the ply principal coordinate system. The failure criterion function is written as

÷÷ø

öççè

æ=

eee

geeSYX

f 1221 ,,max (3.3.1)

where

ct

ct

YYYY

XXXX

eeee

eeee

eeee

=Þ<=Þ³

=Þ<=Þ³

0;0

0;0

22

11 (3.3.2)

For isotropic plies and for transversely isotropic plies with the plane of isotropy 12, such as

mat plies, the principal strains and the maximum shear strain have to be determined first. The

principal strains are

( ) ( ) ( ) ( )ïþ

ïýü

ïî

ïíì

úû

ùêë

é +-+±+-=2

1

2

12

2

2121,4

111

2

11tssnssne hx

E (3.3.3)

and the maximum shear strain is

( ) ( )

2

1

2

12

2

21max4

112úû

ùêë

é +-+

= tssn

gE

(3.3.4)

Version 4.4 III.3.7


III Laminates

3 Failure Criteria

The value of the failure criterion function is then computed according to Eqs. (3.3.1�2) by

replacing e1, e2, and g12 with ex, eh, and gmax, respectively. Due to the transverse isotropy, the

failure strains Xe and Ye are equal.

In the maximum stress criterion, the ratios of the actual stresses to the failure stresses are

compared in the ply principal coordinate system. Thus, the failure criterion function is

÷÷ø

öççè

æ=

QRSYXf 23131221 ,,,,max

tttss (3.3.5)

sig_1 MPa

sig_2

MPa

Max strain

Max stress

-1500 -1000 -500 0 500 1000 1500 2000

-300

-200

-100

0

100

Figure 3.1 s1�s2 failure envelopes of a carbon-epoxy unidirectional ply based on the maximum strain and stress criteria.

III.3.8 Version 4.4


III Laminates

3 Failure Criteria

eps_1 %

eps_2

%

Max strain

Max stress

-1 -0.5 0 0.5 1 1.5

-4

-3

-2

-1

0

1

Figure 3.2 e1�e2 failure envelopes of a carbon-epoxy unidirectional ply based on the maximum strain and stress

criteria.

where

ct

ct

YYYY

XXXX

=Þ<=Þ³

=Þ<=Þ³

0;0

0;0

22

11

ssss

(3.3.6)

As in the maximum strain criterion, isotropic plies and transversely isotropic plies with the

plane of isotropy 12 have to be considered separately. In plane stress state the principal

stresses are determined according to Eq. (3.2.2), and the maximum shear stress from the

expression

( )2

1

2

12

2

21max4

1úû

ùêë

é +-= tsst (3.3.7)

The value of the failure criterion function is then computed according to Eqs. (3.3.5�6) by

replacing s1, s2, and t12 with sx, sh, and tmax, respectively. Due to the transverse isotropy, the

failure stresses X and Y are equal.

Failure envelopes based on the maximum strain and stress criteria are illustrated in Figures

3.1 and 3.2 in stress and strain space, respectively.

Version 4.4 III.3.9


III Laminates

3 Failure Criteria

When 3D Maximum stress criterion (s3=0) is applied for isotropic plies, the principal stresses

are solved from the equation (3.2.10), and the failure function is

Vhxsstsss Vhx

,,,0

0;,,,max max =

<=

³=÷÷ø

öççè

æ= j

whenci

whenti

SXXXf

j

j

iii

(3.3.8)

where

)(2

1max Vx sst -= (3.3.9)

in which sx is the most positive and sz is the most negative principal stress.

3.3.2 Quadratic criteria

In quadratic criteria all the stress or strain components are combined into one expression.

Many commonly used criteria for fiber-reinforced composites belong to a subset of fully

interactive criteria called quadratic criteria. The general form of quadratic criteria can be

expressed as a second-degree polynomial. In stress space (s3=0), the polynomial is of the

form

22112112

2

1266

2

1355

2

2344

2

222

2

111 2 sssstttss FFFFFFFFf +++++++= (3.3.10)

The coefficients Fii and Fi are determined so that the value of the failure criterion function is

one when a unidirectional stress state corresponding to the material strength is applied.

In the Tsai-Wu criterion the coefficients F have the values

266

255244

222

111

1

11

111

111

SF

RF

QF

YYF

YYF

XXF

XXF

ctct

ctct

=

==

-==

-==

(3.3.11a�f)

Thus, the criterion can be written as

III.3.10 Version 4.4


III Laminates

3 Failure Criteria

sssstttss

2112212

212

2

213

2

223

22

21 2

1111F+

YY+

XX+

S+

R+

Q+

YY+

XX=f

ctctctct

÷ø

öçè

æ-÷

ø

öçè

æ- (3.3.12)

The coefficient F12 cannot be obtained directly from the failure stresses of uniaxial load cases.

For accurate results it should be determined through biaxial load tests. In practice, it is often

given in the form of a non-dimensional interaction coefficient

( )2

1

2211

12*

12FF

FF = (3.3.13)

from which

( )2

1

*

1212

ctct YYXX

FF = (3.3.14)

To insure that the criterion represents a closed conical failure surface, the value of *

12F has to

be within the range -1 < *

12F < 1. However, the value range for physically meaningful material

behavior is more limited. The often used value -½ corresponds to a �generalized Von Mises

criterion�. [3,4]

The Hoffman criterion is equivalent to the Tsai-Wu criterion, except that the value of F12 is

specified to be -½F11. Thus, the coefficients F are (plane stress state)

266

222

12111

1

111

2

1111

SF

YYF

YYF

XXF

XXF

XXF

ctct

ctctct

=

-==

-=-==

(3.3.15a�g)

It can be noted that for plies having identical behavior in the principal directions (Xt = Yt, Xc =

Yc), the Hoffman criterion is identical to the Tsai-Wu criterion with the value of interaction

coefficient *

12F = -½. On the other hand, for unidirectional plies with large differences in the

strengths in the principal directions, the Hoffman criterion leads to results close to those of the

Version 4.4 III.3.11


III Laminates

3 Failure Criteria

Tsai-Wu criterion with the value of coefficient *

12F = 0 [4]. This can be seen in the example

envelopes of Figures 3.3 and 3.4.

In the Tsai-Hill criterion, either tensile or compressive strengths are used for determining the

coefficients F depending on the loading condition. The coefficients are

266255244

2222

2121211

111

01

2

10

1

SF

RF

QF

FY

F

XFF

XF

===

==

-===

(3.3.16a�g)

sig_1 MPa

sig_2

MPa

Tsai-Wu, F_12*=-0.5

Tsai-Wu, F_12*=0

Hoffman

Tsai-Hill

-3000 -2000 -1000 0 1000 2000 3000

-300

-200

-100

0

100

Figure 3.3 Failure envelopes based on quadratic criteria in stress space. The material properties correspond to

the unidirectional carbon-epoxy ply used in the examples of Figures 3.1 and 3.2.



III Laminates

3 Failure Criteria

where the values of X and Y are

ct

ct

YYYY

XXXX

=Þ<=Þ³

=Þ<=Þ³

0;0

0;0

22

11

ssss

(3.3.17)

Hence, the Tsai-Hill failure criterion function can be written in the form

2

21

2

12

2

13

2

23

2

2

2

1

XS+

R+

Q+

Y+

X=f

sstttss -÷ø

öçè

æ÷ø

öçè

æ÷÷ø

öççè

æ÷ø

öçè

æ÷ø

öçè

æ (3.3.18)

In Figures 3.3 and 3.4, the Tsai-Hill criterion is compared to the other quadratic criteria.

eps_1 %

eps_2

%

Tsai-Wu, F_12*=-0.5

Tsai-Wu, F_12*=0

Hoffman

Tsai-Hill

-2 -1 0 1 2

-3

-2

-1

0

1

Figure 3.4 Failure envelopes based on quadratic criteria in strain space corresponding to the case of Figure 3.3.



III Laminates

3 Failure Criteria

3.3.3 Physically based criteria

Independent conditions and interactive criteria can be combined to allow different

formulations for different failure modes. These types of criteria are sometimes referred to as

physically based criteria as they distinguish the failure modes between fiber failure and

matrix (or inter fiber) failure. The criteria presented here have been developed for

unidirectional fiber-reinforced plies (R=S). Fibers are assumed to be aligned in the direction

of the axis 1.

In the Hashin criterion [5], criticality of tensile loads in the fiber direction is predicted with

the expression

2D: 0, 1

2

12

2

1 ³÷ø

öçè

æ÷÷ø

öççè

æs

tsS

+X

=ft

f (3.3.19a)

3D: 0, 12

2

13

2

12

2

1 ³+

÷÷ø

öççè

æs

ttsS

+X

=ft

f (3.3.20b)

Under compressive loads in the fiber direction, failure is predicted with an independent stress

condition

0, 11 <- s

s

c

fX

=f (3.3.21)

In the case of tensile transverse stress, the expression for predicting matrix failure is

2D: 0, 2

2

12

2

2 ³÷ø

öçè

æ÷÷ø

öççè

æs

tsS

+Y

=ft

m (3.3.22a)

3D: 0, 22

2

13

2

12

2

23

2

2 ³+

+÷÷ø

öççè

æ÷÷ø

öççè

æs

tttsSQ

+Y

=ft

m (3.3.23b)

A more complex expression is used when the transverse stress is compressive:

2D: 0,122

22

22

12

2

2 <úúû

ù

êêë

é-÷÷

ø

öççè

æ+÷

ø

öçè

æ+÷÷ø

öççè

æs

sts

c

cm

YQ

Y

SQ=f (3.3.24a)



III Laminates

3 Failure Criteria

3D: 0,122

22

2

2

2

13

2

12

2

23

2

2 <úúû

ù

êêë

é-÷÷

ø

öççè

æ+

++÷÷

ø

öççè

æ+÷÷

ø

öççè

æs

sttts

c

cm

YQ

Y

SQQ=f (3.3.25b)

The more critical of the two failure modes is selected:

( )mf fff ,max= (3.3.26)

When no shear load is applied, Hashin criteria act as the maximum stress criterion. Hence,

failure envelopes corresponding to the earlier examples are identical to those of the maximum

stress criterion in Figures 3.1 and 3.2.

The Puck criteria are not described here. The reference document [6] is available at

ESAComp support site: www.esacomp.com/support.

3.3.4 Sandwich Core Criteria

For core materials maximum out-of-plane shear criterion is provided, which considers out-

of-plane shear stresses only. The failure criterion function value is determined from

corecorthotropianforRQ

=f ,,max1323

÷÷ø

öççè

æ tt (3.3.27)

coreisotropicanforS

=f ,2

2

13

2

23 tt + (3.3.28)

The ROHACELL criterion [7] is specifically developed for PMI foam cores. The criterion is

based on an equivalent stress, which is defined as

( ) ( )

222

)44(4121212

12

11

2

1

2

121

2

2212

++

+++++++=

aa

IaIaaaaIaaeqs (3.3.29)

where stress invariants I1 and I2 are (s3=0)

211 ss +=I (3.3.30a)



III Laminates

3 Failure Criteria

( ))(33

1 2

23

2

13

2

1221

2

2

2

12 tttssss +++-+=I (3.3.31b)

and parameters

( )

tt

c

X

Sk

X

Xd

d

ka

d

dka 31

1 2

2

2

1 ==-=-

= (3.3.32a-d)

The failure criterion function of the ROHACELL criterion can be written as

t

eq

Xf

s= (3.3.33)

The applicability of the different failure criteria with respect to the physical nature of the ply

are summed-up in Table 3.1. Also, default failure criteria for the different ply types are

presented in the table.

Table 3.1



III Laminates

3 Failure Criteria

3.3.5 Laminate Interlaminar Shear Criterion

The interlaminar shear failure model is a special type of criterion, which is applied in the

laminate level. It is used to model delamination failure, i.e. to determine when the layers

become detached. Delamination is governed by the out-of-plane shear stresses and the out-of-

plane normal stress. If the out-of-plane normal stress is ignored, then failure function is:

2

2

13

2

23

ttt +

=f (3.3.34)

where t is the layer strength for shear delamination or the interlaminar shear strength.



III Laminates

3 Failure Criteria

REFERENCES

1. Failure criteria for an individual layer of a fibre reinforced composite laminate

under in-plane loading. ESDU 83014, Amendment A. Engineering Sciences Data

Unit, London, 1983/1986.

2. Structural Materials Handbook, Volume 1 � Polymer Composites. ESA PSS-03-203,


3. Tsai, S.W. and Hahn, H.T., Introduction to Composite Materials. Technomic,

Westport, CT, 1980.



OH, 1992.)

5. Hashin, Z., "Failure Criteria for Unidirectional Fiber Composites". Journal of

Applied Mechanics, 47 (1980), pp. 329�334.

6. Failure criteria for non-metallic materials, Implementation of Puck´s failure

criterion in ESAComp, FAIL-HPS-TN-003, European Agency Contract Report No.

16162/02/NL/CP, Braunschweig, 2004.

7. Advanced Material Models for the Creep Behaviour of Polymer Hard Foams; Latest

Advancements of Applied Composite Technology, Roth, M. A., Kraatz, A., Moneke,

M., Kolupaev, V., Proceedings 2006 of the SAMPE Europe, 27th International

Conference, Paris EXPO, Porte de Versailles, Paris, France, 27th � 29th March 2006.

ISBN 3-99522677-2-4. pp. 253 - 2258.

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III Laminates

4 Criticality of Applied Loads

4 CRITICALITY OF APPLIED LOADS


The concept of factors of safety is introduced shortly with emphasis to the practices commonly used in aerospace

applications. In addition, a stability factor is introduced for stability analyses. The principles of determining

margins to failure in the so-called constant and variable load approach are discussed without restricting to any

particular type of failure analysis. Definitions for different reserve factors and the corresponding margins of

safety are given. The inverse values of reserve factors are also discussed.

SYMBOLS

{F} Applied (nominal) load

{F}effective Effective load

{F}failure Failure load

FoS Factor of Safety

MoS Margin of Safety

RF Reserve Factor

SF Stability Factor

Superscripts

c Constant load

r Resultant load

v Variable load

v+c Reversed constant/variable load assumption

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4.1 FACTORS OF SAFETY

The factor of safety (FoS) is a parameter which takes into account the actual statistical

distribution of material strengths and applied loads, and possibly some design aspects. The

use of an appropriate factor of safety for the design provides the required confidence that

under the loads applied in service the structure will not fail with a given probability. The

determination of factors of safety for a given target probability requires the knowledge of the

distributions mentioned above. A thorough discussion on factors of safety is given in

references [1] and [2]. Factors of safety are used in practical design work without explicitly

considering these statistical aspects.

In failure analyses, the factor of safety can be taken into account by replacing the applied load

with the effective load, i.e. by multiplying the applied load with the factor of safety:

{ } { }FFoSF =effective

(4.1.1)

Practical experience may sometimes lead the designer to use factors of safety larger than

those requested by the statistics. In stability analyses, for example, the theoretical approaches

are based on ideal structures whereas real structures are always geometrically imperfect. It has

been shown, though, that with appropriate factors of safety the theoretical approaches may be

used in the design process.

Due to the above described nature of stability analyses, a separate stability factor (SF) is

introduced in ESAComp. The purpose of factors of safety is reserved for taking into account

uncertainties in loads and material strengths while the stability factor takes into account

imperfections in the structure. Hence, the effective load for stability analyses is

{ } { }FFoSSFF =effective

(4.1.2)

4.2 MARGIN TO FAILURE

4.2.1 Constant and variable load approach

In failure analyses of structures, the aim is not only to determine whether the structure can

withstand a given load, but also to give a quantitative measure on the margin to failure.

Failure is here interpreted in a broad sense. The concepts presented here are applicable to

various types of analyses including material failure under static loads and stability analyses.

In simple cases, the margin to failure can be expressed as strength/stress ratios. This requires

that the given load vector is assumed to have a fixed direction. In other words, the ratios of

the different load components with respect to each other must stay constant.

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Since loads applied to a structure are often due to independent physical phenomena, the

assumption on the constant ratios of the load components is not generally valid. Therefore, the

so-called constant and variable load approach [3,4] has been developed for measuring margins

to failure in the probable directions of load increase. The approach is based on the partitioning

of the applied load vector into a constant and variable load part:

{ } { } { }vcFFF += (4.2.1)

The basic assumption is to measure the margin to failure with respect to the increase of the

variable load vector. This is covered in more detail in the next subsection where reserve

factors are introduced.

Since the constant and variable load vectors may represent different types of loads (e.g. pre-

loads with low scatter and dynamic loads with large scatter), the factors of safety associated

with the load vectors may also be different. Hence, the effective load to be used in failure

analyses is defined as

{ } { } { }vvccFFoSFFoSF +=

effective(4.2.2)

As described in the previous section, a separate stability factor is used in stability analyses.

The effective load in the constant and variable load approach then becomes

{ } { } { }( )vvccFFoSFFoSSFF +=

effective(4.2.3)

In the following, margins to failure are studied assuming that no stability factor is involved,

i.e. SF = 1.

4.2.2 Reserve factors

The reserve factor (RF) is a measure of margin to failure. The effective load multiplied with

the reserve factor gives the failure load. Thus, reserve factor values greater than one indicate

positive margin to failure and values less than one indicate negative margin. The values of

reserve factors are always greater than zero.

In the constant and variable load approach, the margin to failure is primarily studied with

respect to the increase of the variable load. Thus, the load leading to failure can be expressed

as

{ } { } { } 0,=failure

>+ RFFFoSRFFFoSFvvcc

(4.2.4)

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III.4.4

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where RF is referred to as the primary reserve factor of the constant and variable load

approach (Figure 4.1).

For the constant and variable load applied alone, the reserve factors RFc and RF

v are defined

as

{ } { }{ } { } 0,=

0,=

failure

failure

>

>vvvvv

ccccc

RFFFoSRFF

RFFFoSRFF(4.2.5a,b)

If no constant load is applied, the value of RFv is equal to the value of the primary reserve

factor RF. In some instances, either of these reserve factors, or even both, may be less than

one while RF still indicates positive margin to failure. In that case, removing the other load

vector leads to failure. Failure could also occur if the loads are applied to the structure in

certain order instead of being applied simultaneously. Therefore, RFc and RF

v values below

one should normally not be accepted in design work.

RF c+v

= 1.62

MoS c+v

= 62 %

v

RF c = 1.76

MoS c = 76 %

RF v = 1.96

MoS v = 96 %

RF = 1.87

MoS = 87 %

c

FoS c {F

c}

{F c}

FoS v {F

v}{F

v}

RF r = 1.43

MoS r = 43 %

Figure 4.1 Graphical interpretation of the different reserve factors in the form of a so-called constant and

variable load envelope.

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Reversing the assumption on increasing load, i.e. presuming increasing constant load while

variable load stays constant, defines the reserve factor RFv+c

. The failure load is

{ } { } { } 0,=failure

>+ +++ cvvvcccvcvRFFFoSFFoSRFF (4.2.6)

A reserve factor can also be calculated by assuming that the applied load increases in the

direction of the resultant load vector. Thus, for RFr the previous expression becomes

{ } { } { }( ) 0,=failure

>+ rvvccrrRFFFoSFFoSRFF (4.2.7)

A graphical interpretation of the different reserve factors is shown in Figure 4.1.

Figure 4.2 illustrates that the above given equations for failure loads do not fully define the

reserve factors RF and RFv+c

. When the effective constant load is critical, RF could obtain

two real values that satisfy Eq. (4.2.4). Correspondingly, RFv+c

could obtain two real values if

the effective variable load is critical. The actual value of the reserve factor is the maximum of

the two values:

( )( )cvcvcv

RFRFRF

RFRFRF

+++21

21

,max=

,max=(4.2.8a,b)

This means that if the effective load is non-critical, the reserve factor indicates how much the

load can be increased. On the other hand, if the effective load is critical, the reserve factor

shows how much the load must be decreased to reach the failure load.

FoSc {Fc}

v

c

RFc < 1

RF1 RF2 = RF

FoSv {Fv}

Figure 4.2 From the two values RF1 and RF2 that satisfy the expression for the failure load in Eq. (4.2.4), the

larger one is the reserve factor RF.

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It should further be noted that, in ESAComp, the values of reserve factor equal to one are

considered non-critical. This interpretation is in line with the presented approach that

uncertainties in material properties and loads are taken into account with material strength

values and factors of safety. The convention has an effect on definitions given later in this

section for non-real values of reserve factors.

4.2.3 Margins of safety

The margin of safety (MoS) is an alternative for the reserve factor in indicating margin to

failure. The margin of safety is obtained from the corresponding reserve factor with the

relation

1−= RFMoS (4.2.9)

Identical expressions can be written for MoSc, MoS

v, MoS

v+c, and MoS

r.

A positive margin of safety indicates the relative amount that the applied load can be

increased before reaching failure load. Correspondingly, a negative margin of safety indicates

how much the applied load should be decreased. Margins of safety are typically expressed as

percentages.

4.2.4 Infinite and indefinite failure margins

In some cases, non-zero load vectors applied to a structure may have no actual load effect on

the part of the structure being studied. This happens, for instance, at the midplane of a

symmetric laminate in pure bending. To handle such situations, infinity (∞) is defined as a

value of the reserve factor indicating that the load vector can be increased without a limit.

For the primary reserve factor RF of the constant and variable load approach, the definition of

infinite values requires some further considerations. A natural requirement for RF to be

infinite is that RFv is also infinite. In addition, RF

c has to be non-critical. Otherwise, the

combined load case being studied would be critical. More precisely, for RF and RFv+c

this can

be stated as follows:

( )( )∞=≥∞=⇔∞=

∞=≥∞=⇔∞=+ vvccv

ccv

RFRFRFRF

RFRFRFRF

or1and

or1and(4.2.10a,b)

The value of a reserve factor is indefinite if the effective load causes failure and decreasing

the value of the reserve factor does not make the load case non-critical. This applies only to

RF and RFv+c

. The other reserve factors consider only a single load vector. Therefore,

decreasing the magnitude of the load vector, i.e. reducing the value of RFc, RF

v or RF

r, will

eventually make the load case non-critical.

Indefinite values of RF are demonstrated in Figure 4.3. Clearly, a necessary requirement for

the indefinite values of RF or RFv+c

is that the value of RFc or RF

v, respectively, is less than

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III.4.7

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one. Cases leading to indefinite values of reserve factor can also be depicted from Eq.

(4.2.10). If the first condition is true (RFv = ∞ or RF

c = ∞), but the latter one is not, the value

of the reserve factor is indefinite.

RF = indef.

v

RFc < 1

FoSv {Fv} (2)

FoSc {Fc} (1)

c

FoSv {Fv} (1)

FoSc {Fc} (2)

Figure 4.3 Loading conditions for which RF is indefinite. Decreasing the magnitude of the variable load does

not lead to a non-critical loading condition.

As stated earlier, when the value of a reserve factor is one, the applied load is considered non-

critical. To be more precise, for RF and RFv+c

this is true only when the failure envelope is

reached from the inside. When the failure envelope is reached from the outside, the load case

is considered critical and therefore the value of the reserve factors RF or RFv+c

is indefinite

instead of being one (Figure 4.4).

Margins of safety are infinite or indefinite when the corresponding reserve factors are.

RF = 1 (2)

RF = 1 (1)

FoSv {Fv} (3)

FoSv {Fv} (2)

FoSv {Fv} (1)

RF = indef. (3)

v

FoSc {Fc} (1)

c

FoSc {Fc} (2)

FoSc {Fc} (3)

Figure 4.4 When the effective load equals failure load, the value of RF is either one or indefinite depending on

whether the failure contour is approached from inside or outside of the envelope.

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III.4.8

III Laminates


4.2.5 Inverse reserve factors

The critical values of reserve factors lie between zero and one, whereas the non-critical values

range from one to infinity. Similarly, the critical values of margins of safety lie between

-100% and zero, and the non-critical values range from zero to infinity. When these values are

viewed in the form of charts, problems arise due to infinite values. Secondly, the scaling often

makes the non-critical values better represented in the charts than the critical values. For these

reasons, the inverse values of reserve factors (1/RF, 1/RFc, �) are often more practical in use.

The inverse value of an infinite reserve factor is zero. Hence, the non-critical values range

from zero to one, and the critical values from that on. Since reserve factors cannot be equal to

zero, inverse values never reach infinity. When the value of a reserve factor is indefinite, the

corresponding inverse reserve factor is also considered indefinite.

Inverse reserve factors can be given a special interpretation when they are used for assessing

criticality of applied loads based on failure criterion functions. As stated in Section 3.1, failure

criterion functions are also defined so that the value zero corresponds to zero-load and one

corresponds to failure. Thus, inverse reserve factors can be interpreted as �linearized failure

criterion functions�. Linearization refers here to the fact that, unlike values of failure criterion

functions in general, inverse reserve factors represent directly the margin to failure.

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III.4.9

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REFERENCES

1. De Mollerat, T. and Vidal, Ch., Evaluation of Design and Tests Safety Factors. Final

Report of ESTEC Contract No. 6370/85/NL/PB. Cannes, 1986.

2. Van Wagenen, R., A Guide to Structural Factors for Advanced Composites Used on

Spacecraft, NASA CR-186010. Washington, D.C., 1989.

3. Palanterä, M. and Klein, M., �Constant and Variable Loads in Failure Analyses of

Composite Laminates�, Computer Aided Design in Composite Material Technology

IV, pp. 221�228. Computational Mechanics Publications, Southampton, 1994.

4. Palanterä, M. and Karjalainen J.-P., �Failure Margins of Composite Laminates with

Constant and Variable Load Approach�. To be presented at the 8th European

Conference on Composite Materials (ECCM-8), 3�6 June, 1998, Naples, Italy.

Version 1.0


III.5.1

III Laminates

5 Linear Laminate Failure Analyses

5 LINEAR LAMINATE FAILURE ANALYSES


The first ply failure (FPF) and the degraded laminate failure (DLF) analyses for predicting laminate failure are

introduced. Both analyses are based on the laminate load response computed with the classical laminate theory.

In FPF analyses, failure is assumed to occur when failure first occurs in one layer or simultaneously in several

layers of the laminate. DLF analyses are used for predicting ultimate load carrying capability of laminates. The

approach is based on a simplified assumption that all layer stiffness properties can be replaced by degraded

values after which the analysis is performed in a way equivalent to the FPF analysis. The secant modulus method

for determining the degraded properties from the ultimate strengths of layers is presented.

SYMBOLS


RF Reserve Factor

S, Sε Shear failure stress/strain in the 12-plane (first failure)

X, Xε Failure stress/strain in the 1-direction (first failure)

Y, Yε Failure stress/strain in the 2-direction (first failure)

1, 2, 3 Layer coordinate system

Subscripts

c Compressive

DLF Degraded Laminate Failure

F First failure (failure stress or strain)

FPF First Ply Failure

k Layer index

t Tensile

U Ultimate (failure stress or strain)

Superscripts

a Midplane of a layer

b Bottom surface of a layer

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III.5.2

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c Constant load

D Degraded properties

r Resultant load

s Stress/strain recovery plane (either top surface, bottom surface, or

midplane of a layer)

t Top surface of a layer

v Variable load

v+c Reversed constant/variable load assumption

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III.5.3

III Laminates


5.1 FIRST PLY FAILURE ANALYSIS

In the first ply failure (FPF) analysis, the failure of a laminate is assumed to occur when

failure first occurs in one layer or simultaneously in several layers of the laminate. In the

ESAComp terminology, the analysis should actually be referred to as �first layer failure

analysis�. However, due to its wide use in the literature, the term FPF is also used here.

The layer failure stresses and strains used in FPF analyses usually correspond to the initiation

of matrix failure or fiber-matrix interface failure. Material behavior can be assumed to be

close to linear up to this point, and the classical lamination theory can be directly used for

solving layer equivalent strains and actual stresses in the laminate (see Sections 2.6�7).

As described in Section 2.7, the stress-strain state can be determined for the top and bottom

surfaces of a layer. Alternatively layer midplanes may be used as the stress/strain recovery

planes where the criticality of the applied load is evaluated. Especially when laminates with

large numbers of layers are analyzed, this may reduce computation time. However, when

laminates having only few layers are analyzed, using midplane values may lead to failure

margins that are considerably less conservative than the ones obtained based on the top and

bottom surfaces.

Criticality of the applied load is first determined separately for each layer and stress/strain

recovery plane in terms of reserve factors or margins of safety (see Chapter 4). Since the

layers of a laminate are generally in a multiaxial stress-strain state, failure criteria are needed

for determining the criticality of the layers (see Chapter 3). In ESAComp, the values of

reserve factors for many failure criteria are computed based on closed form solutions. For

instance, the procedure for quadratic criteria is outlined in [1]. The iterative line search

method of reference [2] is applied to some of the built-in criteria and to all the user specified

criteria. In either approach, the layer stress-strain states are determined separately for the

nominal constant and variable load vector after which the stress-strain states corresponding to

any combination of the load vectors are obtained from linear combinations.

The FPF reserve factor of the laminate is the minimum of the reserve factors computed for the

stress/strain recovery planes of the layers:

( )

==

=nk

abtsRFRF

s

kFPF,,1

or,,min

(5.1.1)

The layer or layers having the minimum value of the reserve factor are referred to as the

critical layer(s).

When both constant and variable loads are applied, all the reserve factors introduced in

Section 4.2 can be computed, i.e. RF, RFc, RF

v, RF

v+c, and RF

r. Depending on the type of

reserve factor, the resulting critical layers may be different.

The laminate FPF load can be computed once the reserve factor is known and its value is not

infinite or indefinite. For instance, from Eq. (4.2.4) the failure load corresponding to the

primary reserve factor RFFPF is

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III.5.4

III Laminates


{ } { } { }vv

FPF

cc

FPF FFoSRFFFoSF += (5.1.2)

Similarly, the FPF loads corresponding to the remaining four reserve factors of the constant

and variable load approach are computed in accordance with Eqs. (4.2.5�7).

5.2 DEGRADED LAMINATE FAILURE ANALYSIS

5.2.1 General

Fiber-reinforced layers have the capability of carrying significant loads even after the

initiation of matrix degradation. In other words, the FPF load may be exceeded. CLT based

analyses for determining the ultimate load carrying capability of laminates are generally

referred to as last ply failure (LPF) analyses. However, unlike in the case of FPF analyses,

there is not a single widely accepted approach for performing this type of analyses.

Since the stress-strain relations of reinforced plies are often highly nonlinear near the ultimate

strength, classical lamination theory cannot be directly used for LPF analyses. A simplified

procedure for performing LPF analyses has been introduced by Tsai [3]. The problem is

approached with linear analyses by replacing the Young's moduli and shear moduli of

degraded layers with reduced values. In Tsai's method it is further assumed that all the layers

in a laminate reach the totally degraded state before the laminate ultimate strength is reached.

This assumption is based on the observation that matrix cracks in one layer propagate to the

adjacent layers even when the stress states of these layers are below critical. Furthermore, it is

presumed that when the first layer in the degraded laminate reaches the failure load, the

ultimate load carrying capability of the laminate is reached.

The basic approach of ESAComp is identical to the Tsai�s method with improvements

introduced to the determination of degraded properties based on either tensile or compressive

behavior. The analysis is referred to as the degraded laminate failure (DLF) analysis. This is

to emphasize the nature of the analysis compared to other LPF approaches for determining

laminate ultimate strengths. Due to the many assumptions and simplifications of the method,

the DLF analysis should only be used as a tool for preliminary design studies.

5.2.2 Degraded properties of layers

The type of reinforcement has a significant influence on the behavior of layers beyond first

failure loads. For instance, matrix failure in a unidirectional layer loaded in the fiber direction

has a minor effect on the longitudinal modulus since fibers are the major load carrying

elements. For fabric and mat layers the reduction in moduli is more significant. Semi-

empirical or micromechanics approaches can be applied to estimate the properties of degraded

layers. ESAComp uses a secant modulus approach based on the ultimate layer strengths

obtained from material tests.

The current ESAComp approach considers the ply principal directions 1 and 2 independently.

Hence, the approach is mainly applicable to unidirectional and fabric plies. In the DLF

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analyses, no degradation is applied to plies having isotropic or 12-transversely isotropic

behavior.

A stress-strain curve for the direction 1 of a layer is shown in Figure 5.1. The degraded

modulus can be approximated with a secant modulus by assuming linear behavior from zero

stress-strain state up to the measured ultimate failure stress and strain. For the 1-direction, the

degraded moduli based on the tensile and compressive behavior, respectively, are

Uc

UcD

c

Ut

UtD

tX

XE

X

XE

,

,

1

,

,

1 ;εε

== (5.2.1a,b)

Xε t,F Xε t,U

Xc,U

Xc,F

ε1

σ1

Xt,U

Xt,F

Xε c,FXε c,U

ED

1t

E1

ED

1c

Figure 5.1 Determination of degraded moduli in the direction 1 based on the ultimate failure stresses and strains.

The shaded areas describe the feasible regions for ultimate values in the ESAComp system: The values of

ultimate stresses may not be less than the corresponding first failure stresses, and the value of a degradedmodulus defined by the ultimate values may not be greater than the intact modulus.

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III.5.6

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Correspondingly, for the 2-direction:

Uc

UcD

c

Ut

UtD

tY

YE

Y

YE

,

,

2

,

,

2 ;εε

== (5.2.2a,b)

The in-plane shear performance of orthotropic materials is not affected by the direction of the

stresses and strains. Hence the degraded shear modulus is

U

UD

S

SG

,

12

ε

= (5.2.3)

For the in-plane Poisson�s ratios, there are no proven degradation models applicable to

different types of layers. Therefore, the value of the intact layer is used for the in-plane major

Poisson�s ratio:

1212 νν =D(5.2.4)

For the directions 1 and 2, degraded properties based on tensile or compressive properties are

selected with a procedure described in the next subsection. Even though the aim is to make

�the correct choice�, there is a possibility that the layer stresses or strains of the degraded

laminate are negative when they were expected to be positive, or vice versa. Therefore, both

tensile and compressive failure strengths need to be specified for the degraded layers. The

failure stresses of degraded layers are always assumed to be equal to the given ultimate

values. Thus, the failure stresses in the 1-direction are

Uc

D

cUt

D

t XXXX ,, ; == (5.2.5a,b)

Figure 5.2 illustrates the determination of failure strains of a degraded layer when the

modulus is selected to be tensile. The tensile ultimate strain can be directly used for the

degraded layer, but the compressive value must be computed from the corresponding failure

stress:

D

t

UcD

cUt

D

tE

XXXX

1

,

, ; == εεε (5.2.6a,b)

Based on the same logic, if the compressive model is selected, the failure strains are

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Xε t,F Xε t,U

Xc,U

Xc,F

ε1

σ1

Xt,U

Xt,F

Xε c,FXε c,U

ED

1t

E1

XD

ε c

Figure 5.2 When the degraded modulus is determined based on tensile behavior, the compressive failure strain isresolved from the compressive failure stress and the degraded modulus.

Uc

D

cD

c

UtD

t XXE

XX ,

1

,; εεε == (5.2.7a,b)

The strengths in the 2-direction are handled in a way analogous to the 1-direction. Hence, X

and 1 in Eqs. (5.2.5-7) are replaced by Y and 2, respectively.

The shear strengths of the degraded layers are simply the given ultimate values:

U

D

U

DSSSS ,; εε == (5.2.8a,b)

5.2.3 Selection between tensile and compressive behavior

As the first step of a DLF analysis, an FPF analysis is performed with the same choice of

failure criteria, stress/strain recovery planes, and factors of safety as specified for the DLF

analysis. The stress-strain state of the laminate corresponding to the FPF load is used as the

basis for determining whether tensile or compressive values are used in the directions 1 and 2.

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Due to the multiaxial nature of layer stress-strain states, a layer stress component (σ1 or σ2)

and the corresponding strain (e

1ε or e

2ε ) do not always have the same signs. Therefore, the

choice of degraded properties for the DLF analyses can be selected to be either stress based or

strain based. In the following, the selection of tensile or compressive properties is described

for the stress based model. Strain based model is handled in a similar manner. To account for

the effect of internal loads, equivalent strains are used (see Subsection 2.7.6).

If layer midplanes are used as the stress/strain recovery planes, the selection of tensile or

compressive properties is simple. When the value of σ1 is positive or zero, D

tE1 is used.

Otherwise, D

cE1 is selected. An equivalent rule applies to the direction 2.

When layer top and bottom surfaces are used as stress/strain recovery planes, the selection of

appropriate degraded properties is somewhat more complicated. When the stress components

on the top and bottom surfaces have the same sign, tensile or compressive properties can be

selected accordingly. If the other component is zero, the non-zero component determines the

behavior. Tensile properties are used if both components are zero. When the stress

components on the two surfaces have different signs, the relative distance from the first

failure stress is used as the basis for the selection. For σ1 this can be expressed as follows:

ecompressiv

tensile

11

11

⇒−<

⇒−≥

c

c

t

t

c

c

t

t

XX

XX

σσ

σσ

(5.2.9a,b)

Redistribution of stresses and strains takes place in the degraded laminate. This may cause

layer stresses and strains to change their signs from the values corresponding to the initiation

of failure, i.e. the FPF load. Thus, the choice of tensile or compressive behavior in the 1- and

2-directions based on the results of the FPF analysis may prove to be incorrect. However, the

more loaded a layer in the intact laminate is, the less likely the stresses and strains are to

change signs. Consequently, the effect of layers having �incorrect� degraded properties is

likely to be small on the laminate behavior.

The above described approach for selecting between tensile and compressive behavior

requires that the stress-strain state corresponding to the FPF load can be determined. This,

however, is not possible when the value of the laminate FPF reserve factor is infinite or

indefinite (see Subsection 4.2.4). Infinite values can be handled easily since the laminate does

not fail at all and the DLF reserve factor will be infinite as well. This is discussed in more

detail in the next subsection. Indefinite values may only be encountered as values of RF and

RFv+c

when a constant load is involved. The stress-strain state corresponding to RFc and RF

v,

respectively, is used in these cases as the basis for deciding whether tensile and compressive

behavior dominates.

Version 1.0


III.5.9

III Laminates


5.2.4 Margin to failure

Following the determination of the degraded properties for each layer of the laminate, the FPF

analysis procedure described in Section 5.1 is applied to the degraded laminate. The minimum

of the layer reserve factors represents the reserve factor of the degraded laminate:

( )

==

=nk

abtsRFRF

sD

k

D

,,1

or,,min

,

(5.2.10)

The DLF load representing the ultimate load carrying capability of the laminate should

naturally be greater than or at least equal to the FPF load. However, the reserve factor of the

degraded laminate may sometimes be lower than the FPF reserve factor. This indicates that

the degradation of layer properties does not allow the load level to be increased beyond the

FPF load. Instead, a total loss of load carrying capability follows as the FPF load is exceeded.

In some cases, this physical interpretation may be incorrect, and the result merely indicates

that the DLF model does not properly describe the behavior of the laminate beyond the FPF

load.

Based on the above reasoning, either the reserve factors of the degraded laminate or the FPF

reserve factors are used to represent the DLF reserve factors for the laminate and the layers:

===

=⇒<

===

=⇒≥

nkabtsRFRF

RFRF

RFRF

nkabtsRFRF

RFRF

RFRF

s

kFPF

s

kDLF

FPFDLF

FPF

D

sD

k

s

kDLF

D

DLF

FPF

D

,,1;or,,

,,1;or,,

,,

,

,

(5.2.11a�d)

The equal sign is included in the first inequality for a specific reason. In ESAComp, a ply

ultimate stress can be specified equal to the first failure stress, whereas the corresponding

ultimate strain may be greater than the first failure strain (see Figure 5.1). As a natural

consequence, laminates may exhibit this type of ideal elastic-plastic behavior as well. This

can be demonstrated with a unidirectional laminate in the principal load cases. Although the

FPF and DLF failure strengths in terms of stresses are equal, the corresponding DLF failure

strain may be larger than the FPF value.

When both constant and variable loads are applied, the reserve factors c

DLFRF ,v

DLFRF ,cv

DLFRF+

,

and r

DLFRF can also be computed. In Eqs. 5.2.10�11, RFD and RFFPF are changed accordingly.

Furthermore, the FPF failure load and the corresponding stress-strain state used in

determining degraded properties changes for each reserve factor. As a result, the properties of

the degraded laminate may be different in the computation of each of the five reserve factors.

Version 1.0


III.5.10

III Laminates


REFERENCES

1. Palanterä, M. and Klein, M., �Constant and Variable Loads in Failure Analyses of

Composite Laminates�, Computer Aided Design in Composite Material Technology

IV, pp. 221�228. Computational Mechanics Publications, Southampton, 1994.

2. Kere, P. and Palanterä, M., �A Method for Solving Margins of Safety in Composite

Failure Analyses�, Proceedings of the 6th Finnish Mechanics Days, pp. 187�197.

University of Oulu, Finland, 1997.



OH, 1992.)

Version 1.0


III.6.1

III Laminates

6 Local Instability of Sandwich Laminates

6 LOCAL INSTABILITY OF SANDWICH LAMINATES


The approach used by ESAComp for predicting face sheet wrinkling of sandwich laminates is described.

Sandwich face sheets are considered as plates on an elastic foundation formed by a homogeneous or a

honeycomb core. The models developed for uniaxial compression are extended to take into account biaxial

compression with an interaction formula. To measure the criticality of applied loads, reserve factors with respect

to wrinkling are introduced. The use of wrinkling analyses in conjunction with laminate failure analyses is

discussed and combined reserve factors for wrinkling and FPF or DLF are defined.

SYMBOLS

E Young�s modulus

{F} Applied (nominal) load

{F}effective Effective load

f Failure criterion function


G Shear modulus

h Thickness

RF Reserve Factor

SF Stability Factor


ϕ Angle between the axes xy and ξη

ν Poisson�s ratio

ξ, η, ζ Principal stress coordinate system

σ Normal stress

τ Shear stress

Subscripts

C Core

DLF Degraded Laminate Failure analysis

Version 1.0


III.6.2

III Laminates


DLF/w Combined DLF and wrinkling analysis

F Sandwich laminate face sheet (top or bottom)

Fb Bottom face sheet

FPF First Ply Failure analysis

FPF/w Combined FPF and wrinkling analysis

Ft Top face sheet

w Wrinkling

Superscripts

c Constant load

D Degraded laminate

v Variable load

Version 1.0


III.6.3

III Laminates


6.1 WRINKLING

6.1.1 Wrinkling stresses

Wrinkling of sandwich face sheets is a local instability phenomenon, in which the face sheets

can be modeled as plates on an elastic foundation formed by the core. Simple formulas for

estimating wrinkling stresses of sandwich face sheets under uniaxial load have been

presented, for instance, in references [1] and [2]. Linear elastic material behavior is assumed.

Possible interaction of the top and bottom face sheets is not considered.

In the following, ξ, η, and ζ refer to a coordinate system in which the ξ-axis is in the direction

of compression and the ζ-axis is perpendicular to the face sheets. The subscript F and C

indicate the face sheet and the core, respectively.

For sandwich laminates with homogeneous cores, the wrinkling stress of a face sheet is

−−

ννσ

ηξξη

ζξζξξ

F,F,

C,C,F,

w,

GEEQ=

1

3

1

(6.1.1)

where the theoretical value of the so-called wrinkling coefficient Q is 0.825. The effects of

initial waviness and imperfections of the face sheet are normally accounted for by replacing

the theoretical value of the wrinkling coefficient with a lower value. References [1] and [2]

recommend to use a value Q = 0.5 as a safe design value for homogeneous cores. This is the

built-in value used by ESAComp. When reserve factors and margins of safety with respect to

wrinkling are determined, additional safety in the design may be obtained with the use of the

stability factor introduced in Section 4.1.

The wrinkling stresses for sandwich laminates with honeycomb cores are estimated with the

expression

( )

−−

h

hEEQ=

CF,F ,

FC,F,

w,

ννσ

ηξξη

ζξξ

1

2

1

(6.1.2)

The theoretical value of Q is 0.816. As in the case of homogeneous cores, a safe design value

Q = 0.33 [1,2] is used in ESAComp.

The prediction of wrinkling under multiaxial stress state is discussed in reference [2]. When

in-plane shear stresses exist, it is recommended that the principal stresses are determined first.

If the other of the two principal stresses is tensile, it is ignored and the analysis is based on the

equations given above. When biaxial compression is applied, wrinkling can be predicted with

an interaction formula. The condition for wrinkling is

Version 1.0


III.6.4

III Laminates


1

3

=

+

σσ

σσ

η

η

ξ

ξ

w,w,

(6.1.3)

where ξ is �the direction of maximum compression� [2]. For orthotropic sandwich face

sheets, ξ is more logically interpreted as the most critical of the two directions. The wrinkling

stresses σξ,w and ση,w are computed from the formulas for uniaxial compression by

considering the compressive stresses in the ξ- and η-directions independently.

6.1.2 Reserve factors for wrinkling

ESAComp uses reserve factors and margins of safety for indicating the margin to wrinkling

for a given load. The constant and variable load approach is applied, thus giving the margins

in the directions of the different load vectors (Section 4.2). As explained in Section 4.1, the

effective load used for predicting wrinkling is the nominal load multiplied by the factors of

safety and the additional stability factor:

{ } { } { }( )vvccFFoSFFoSSFF +=

effective(6.1.4)

The load response of sandwich laminates is computed with the CLT approaches described in

Chapter 2. From the layer stresses, the average face sheet stresses are obtained as shown in

Subsection 2.7.7. Here, the average stresses are simply denoted by σx, σy, and τxy. The

following procedure for the computation of reserve factors is then used independently for the

top and bottom face sheets.

If the shear stress τxy of the face sheet is zero, the normal stresses σx and σy are used directly

in the prediction of wrinkling. Otherwise, the principal stresses are determined first:

( ) ( ) ( )ηξηξ σστσσσσσ ≥

+−±+=

2

1

22

,4

1

2

1xyyxyx (6.1.5)

The orientation of the normalized principal stresses with respect to the xy-coordinate system is

−=

yx

xy

σστ

ϕ2

arctan2

1(6.1.6)

The in-plane moduli and Poisson's ratios of the face sheet are computed for the ξ- and η-

directions by considering the face sheet as a laminate and by applying the CLT analyses of

Chapter 2. For an unsymmetric face sheet, the moduli and Poisson's ratios are computed

Version 1.0


III.6.5

III Laminates


assuming suppressed curvature. The shear moduli Gζξ and Gηζ of a homogeneous core are

obtained through coordinate transformations described in Part I, Subsection 2.2.6.

To allow determination of reserve factors in a way analogous to FPF and DLF analyses, the

condition for wrinkling is presented in the form of a failure criterion function. The approach

for handling biaxial loads was outlined in the previous subsection. Thus, for the different

combinations of tensile and compressive principal stresses, the failure criterion is defined as

follows:

00

0=⇒

≥≥

wfη

ξ

σσ

(6.1.7)

w

wf,

0

0

η

η

η

ξ

σσ

σσ

=⇒

<≥

(6.1.8)

R<RR+R=f

RRR+R=f

w

w

≥

⇒

<<

ηξξη

ηξηξ

η

ξ

σσ

,

,

0

0

3

3

(6.1.9a,b)

where

σσ

σσ

η

ηη

ξ

ξξ

w,w,

=R=R ; (6.1.10a,b)

The wrinkling stress σξ,w is computed from Eq. (6.1.1) or (6.1.2). ση,w is obtained from the

same equations by switching the coordinates ξ and η. The values of reserve factors are solved

iteratively with the line search method introduced in reference [3]. In other words, the

magnitude of the load vector is varied to find the location where fw equals one. If both

constant and variable loads are applied, a change in the magnitude of the other load vector

may alter the orientations of the principal stresses. Therefore, new principal stresses must be

determined for each iteration round.

When the wrinkling load is reached in either the top or bottom face sheet, the critical

wrinkling load of the laminate is reached. Hence, the laminate reserve factor for wrinkling is

the minimum of the reserve factors for the two face sheets:

( )FbwFtww RFRFRF ,, ,min= (6.1.11)

Version 1.0


III.6.6

III Laminates


6.1.3 Wrinkling with FPF and DLF analyses

In the analysis and design of sandwich laminates, wrinkling is a potential failure mode that

has to be studied in conjunction with assessment of layer load carrying capabilities. The latter

can be based on the FPF and DLF analyses introduced in Chapter 5. If the FPF analysis is

used for predicting layer failures, the failure margin of a sandwich laminate can be expressed

as the minimum of the FPF and wrinkling reserve factors corresponding to the same applied

load:

( )wFPFwFPF RFRFRF ,min/ = (6.1.12)

Combining wrinkling with the DLF analysis is not as straightforward. In Section 5.2 the DLF

load was defined as the maximum of the FPF load and the failure load computed with the

degraded properties, i.e. the DLF load represents the ultimate load carrying capability. On the

other hand, Eqs. (6.1.1�2) reveal that if the laminate properties degrade, the wrinkling stresses

decrease in magnitude. This implies that degradation cannot increase the critical load with

respect to wrinkling, but wrinkling of the degraded laminate may instead become a limiting

factor in certain cases. First, for the intact laminate FPF has to be more critical than wrinkling,

i.e. degradation may occur in the layers when the load is further increased. Then, if RFD

computed for the degraded laminate indicates that the load carrying capability increases,

wrinkling of the degraded laminate needs to be considered to see whether it takes place at a

lower load level. With analogy to RFDLF, the value of the combined reserve factor RFDLF/w

cannot be lower than the value of RFFPF/w. This rationale leads to the following:

( )[ ]D

w

D

FPFwDLFFPFw

wwDLFFPFw

RFRFRFRFRFRF

RFRFRFRF

,min,max/

/

=⇒>

=⇒≤(6.1.13a,b)

RFwD is the wrinkling reserve factor computed with the same degraded properties as RF

D.

For the different reserve factors of the constant and variable load approach, similar equations

can be written by adding the corresponding superscripts to all RF�s.

Version 1.0


III.6.7

III Laminates


REFERENCES

1. Structural Materials Handbook, Volume 1 � Polymer Composites, ESA PSS-03-203,


2. Sullins, R.T. et al, Manual for Structural Stability Analysis of Sandwich Plates and

Shells, NASA CR-1457. 1969.

3. Kere, P. and Palanterä, M., �A Method for Solving Margins of Safety in Composite

Failure Analyses�, Proceedings of the 6th Finnish Mechanics Days, pp. 187�197.

University of Oulu, Finland, 1997.

Version 4.4 III.7.1


III Laminates

7 Transverse Shear in Laminate Analyses

7 TRANSVERSE SHEAR IN LAMINATE ANALYSES

Veera Skyttä (HUT/LLS, 2002)

The method for transverse shear stiffness and stress calculations is introduced and the failure criteria including

transverse shear terms is presented.

SYMBOLS

ijA In-plane stiffness matrix of a laminate, i, j = 1, 2, 6

ijA Elements of uncorrected transverse stiffness matrix of a laminate

(constant shear strain), i, j = 4, 5

a Width of a plate in the x-direction

Boolean matrices

Coupling compliance matrix of a laminate, i, j = 1, 2, 6

Bending compliance matrix

3x3 shape function matrix

Failure function

Thickness of a laminate

Laminate shear stiffness matrix, i, j = 4, 5

ijk Shear correction factors corresponding to laminate shear stiffness

components ijA , i, j = 4, 5

{ }M , iM Resultant bending moments, i=x, y, xy

{ }N , iN Resultant in-plane forces, i=x, y, xy

Shear resultant forces

[ ]Q , ijQ Layer stiffness matrix in the global coordinate system, i, j = 1, 2, 6

ijQ Layer transverse shear stiffness components in the global

coordinate system, i,j = 4, 5

Shear strengths in the 23, 31 and 12 planes

III.7.2 Version 4.4


III Laminates


[T] 2 x 2 transformation matrix

Shear strain energy

Basic strengths in 1, 2 and 3 direction of the material

{ }e In-plane strain vector

zxg , yzg Average transverse shear strains

{ }k Curvature vector

q Rotation angle between the xyz- and 123-coordinate system

xs , ys In-plane normal stresses

Dt Interlaminar shear strength

xyt In-plane shear stress

zxt , yzt Transverse shear stresses

t Non-dimensional shear stress

Subscripts

Core failure mode or compressive

Fiber failure mode

Value at the interface of layers k and k+1

m Matrix failure mode

Including transverse shear terms

Tensile

Differentiation with respect to x

Differentiation with respect to y

Superscripts

Layer index

Transpose

Inverse matrix

0 At the midplane surface z = 0

Version 4.4 III.7.3


III Laminates


7.1 INTRODUCTION

In this chapter the method for transverse shear stiffness and stress calculations is described.

The method is based on first order displacement field across the laminate thickness and can

therefore be called First Order Shear Deformation Theory (FSDT). This method was chosen

to ensure the compatibility with the commercial FE software since the simple shell element

formulations use the FSDT. Additionally, a point analysis does not provide enough

information for higher order theory applications. [1]

According to the strain-displacement relation, linear displacement field of the FSDT is

analogous to constant transverse shear strains across the laminate thickness. This assumption

results in a discontinuous, piecewise constant shear stress field if the shear stresses are

calculated from the shear strains. This is unrealistic because the transverse shear stress field

should be continuous through the thickness and vanish at the laminate surfaces. If the

transverse shear stresses are calculated from the equilibrium equations and not from the shear

strains, the results are quite accurate.

7.2 TRANSVERSE SHEAR CALCULATION METHOD

The calculation method is based on References [2-3]. The method is used to calculate the

transverse shear stiffness matrix and the transverse shear stresses resulting from the shear

forces. The calculation of transverse shear stresses straight from the shear forces is made

possible with assumptions that decrease the accuracy of the method. The accuracy of the

method and the effect of these assumptions are evaluated in Subsection 7.2.4.

7.2.1 Tranverse Shear Stresses

The shear stresses are calculated from the three dimensional equilibrium equations of

elasticity:

0

0

=¶

¶+

¶

¶+

¶¶

=¶

¶+

¶

¶+

¶

¶

zyx

zxy

zxxyx

yzxyy

tts

tts

(7.1.1)

In order to calculate the shear stresses from the equilibrium equations, the in-plane stresses

need to be derivated first and then integrated with respect to the thickness coordinate:

òþýü

îíì

+

+-=

þýü

îíì

dzyxyxx

xxyyy

zx

yz

,,

,,

tsts

tt

(7.1.2)

III.7.4 Version 4.4


III Laminates


In-plane stresses are piecewise continuous functions, so the integration needs to be done in

parts. Applying the stress-strain relation for in-plane stresses, Equation 7.1.2 takes the form:

( ) ( ){ }ò +++-=þýü

îíì

dzzQBzQB y

ky

k

x

kx

k

zx

yz

,

)(,

0)(

2,

)(,

0)(

1 kekett

(7.1.3)

where

úû

ùêë

é=

0

1

0

0

1

01B ú

û

ùêë

é=

1

0

0

1

0

02B (7.1.4)

Strain derivatives in Equation 7.1.3 can also be transformed to give an expression in terms of

force derivatives by applying the constitutive equations. In order to calculate the stresses

straight from the shear forces, some additional assumptions have to be made.

The influence of the in-plane force derivatives is neglected, that is:

{ } { }0=¶¶

Nx

{ } { }0=¶¶

Ny

(7.1.5)

Strain derivatives then reduce to a form:

{ } [ ] { }Mx

bx ¶

¶×=

¶¶ 0e { } [ ] { }M

xd

x ¶¶

×=¶¶

k (7.1.6)

{ } [ ] { }My

by ¶

¶×=

¶¶ 0e { } [ ] { }M

yd

y ¶¶

×=¶¶

k (7.1.7)

where and are the compliance matrices defined in Chapter 2.3.

The actual displacement fields are further simplified by assuming two separate cylindrical

bending modes. The moment derivatives then reduce to the simple resultant shear forces:

ïþ

ïý

ü

ïî

ïí

ì=

ïþ

ïý

ü

ïî

ïí

ì

¶¶

0

0

x

xy

y

x Q

M

M

M

x

ïþ

ïý

ü

ïî

ïí

ì

=ïþ

ïý

ü

ïî

ïí

ì

¶¶

0

0

y

xy

y

x

Q

M

M

M

y (7.1.8)

Version 4.4 III.7.5


III Laminates


Applying Equations 7.1.5 - 7.1.8 to Equation 7.1.3 yields:

[ ][ ] [ ][ ]ïþ

ïý

ü

ïî

ïí

ì+

ïþ

ïý

ü

ïî

ïí

ì=

þýü

îíì

0

0

)(

0

0)()(

2

)(

1

)(

y

k

x

k

k

zx

yzQzFB

Q

zFBtt

(7.1.9)

Equation 7.1.9 simplifies to:

þýü

îíì

úû

ùêë

é=

þýü

îíì

x

y

kk

zx

yz

Q

Q

FF

FF)(

1132

3122

)(

tt

(7.1.10)

in Equation 7.1.10 are the components of the 3x3 matrix . The

components of the -matrix are piecewise continuous, second order functions of z

determined by:

[ ] [ ]dznbzmzF k )()()( )( --= (7.1.11)

where

[ ] [ ][ ] [ ] 2)()(

)()(

2

1)()(

)()(

zQkNzdzQzn

zQkMdzQzm

kk

kk

+==

+==

ò

ò (7.1.12)

The functions and are defined by the continuity of stresses at the layer interfaces:

[ ] ( ) [ ]

[ ] ( ) [ ] 2)(

1

2

1

2)(

)(

1

1

)(

2

1

2

1)(

)(

k

kk

i

ii

i

k

kk

i

ii

i

zQzzQkN

zQzzQkM

--=

--=

å

å

=-

=-

(7.1.13)

It can be seen that the functions and (Equation 7.1.12) give the stiffness matrices A

and B at the lower surface of the laminate and zero at the top surface of the laminate:

III.7.6 Version 4.4


III Laminates


[ ][ ]Bhn

Ahm

hnhm

=

=

=-=-

)2/(

)2/(

0)2/()2/(

(7.1.14)

Therefore, the transverse shear stresses described by Equation 7.1.10 become zero for the top

and bottom surfaces and fulfill the boundary conditions that transverse shear stresses need to

vanish at the laminate surfaces.

The transverse shear stresses described by Equation 7.1.10 in the local coordinates are

determined by the 2 x 2 transformation matrix T:

þýü

îíì

úû

ùêë

é=þýü

îíì

zx

yz

xT t

ttt

2231

23 (7.1.15)

where

úû

ùêë

é -=úû

ùêë

éqqqq

cossin

sincos

22T

x

(7.1.16)

7.2.2 Transverse Shear Stiffness Matrix

The transverse shear stiffness matrix can be calculated using the strain energy principle and

the shear stresses described in the previous chapter.

The shear strain energy can be stated in a matrix form as follows:

{ } òþýü

îíì

úû

ùêë

é

þýü

îíì

=

-

dzQQ

QQW

zx

yz

T

zx

yz

tt

tt

1

5545

4544

2

1 (7.1.17)

where ijQ are elements of the layer shear stiffness matrix in the global coordinate system.

Global shear stiffness matrix is calculated with the 2 x 2 transformation matrix T:

T

xxTT QQ

QQ

QQ

QQ--

úû

ùêë

éúû

ùêë

éúû

ùêë

é=úû

ùêë

é

225545

4544

1

225545

4544 (7.1.18)

Version 4.4 III.7.7


III Laminates


Substituting Equation 7.1.10 to Equation 7.1.17 gives:

{ }þýü

îíì

úú

û

ù

êê

ë

éúû

ùêë

éúû

ùêë

éúû

ùêë

é

þýü

îíì

= ò-

x

y

TT

x

y

Q

Qdz

FF

FF

QQ

QQ

FF

FF

Q

QW

1132

3122

1

5545

4544

1132

3122

2

1 (7.1.19)

Shear energy can also be expressed in shear forces and laminate shear stiffness:

{ }þýü

îíì

úû

ùêë

é

þýü

îíì

=-

x

y

T

x

y

Q

Q

KK

KK

Q

QW

1

5545

4544

2

1 (7.1.20)

where [K] is the laminate shear stiffness matrix.

Combining Equations 7.1.19 and 7.1.20 gives an expression for the laminate shear stiffness

matrix:

[ ]1

1 1132

3122

1

5545

4544

1131

3222

1

-

=

-

úú

û

ù

êê

ë

éúû

ùêë

éúû

ùêë

éúû

ùêë

é= å ò

-

N

i

z

z

i

i

dzFF

FF

QQ

QQ

FF

FFK (7.1.21)

The shear stiffness matrix calculated this way is based on the actual shear stress distribution

shape and does not need shear correction factors.

7.2.3 Shear Correction Factors

Shear correction factors can be calculated inversely from the laminate shear stiffness matrix

K. The shear stiffness matrix corresponding to a constant shear strain and piecewise constant

shear stresses are defined by:

þýü

îíì

úû

ùêë

é=

þýü

îíì

=þýü

îíì

òzx

yz

zx

yz

x

y

AA

AAdz

Q

Q

gg

tt

5545

4544 (7.1.22)

where

III.7.8 Version 4.4


III Laminates


( )å --= 1

)(

kk

k

ijij zzQA 5,4, =ji (7.1.23)

These values overestimate the laminate shear stiffness and need to be corrected with a shear

correction factor. The corrected laminate shear stiffness is therefore:

úû

ùêë

é=

55554545

45454444

AkAk

AkAkK (7.1.24)

The shear correction factors resulting to the shear stiffness components given by Equation

7.1.21 can thus be calculated after the actual shear stiffness is known:

44

4444

A

Kk =

45

4545

A

Kk =

55

5555

A

Kk = (7.1.25)

7.2.4 Accuracy of the Results

The accuracy of the method was tested with three simple laminates: crossply [0/90/0],

unsymmetric angle-ply [15/-15] and symmetric angle-ply [30/-30/-30/30]. The results from

the method described in Subsection 7.2.1 are compared to the exact elasticity solutions

derived for cylindrical bending under a sinusoidal load [4, 5].

The transverse shear stress fields are plotted at the laminate edge (x=0). The results are shown

in Figures 7.1 to 7.3. The exact solution for the shear stress is presented by the dashed line

and the calculated result is the solid line.

a) b)

Figure 7.1 Shear stress tzx/(a/h) for [0/90/0] -laminate with a/h-ratio a) 4 and b) 10

Version 4.4 III.7.9


III Laminates


Figure 7.2 Shear stress tzx/(a/h) for [15/-15] �laminate

a) b)

Figure 7.2 Shear stress a) tzx/(q0 a/h) b) tyz/(q0 a/h) for symmetric [30/-30/-30/30] �laminate

As can be seen from the Figures, the accuracy of the method is good for cross-ply laminates

with moderate a/h-ratios and for unsymmetric angle-ply laminates with all a/h-ratios. For

symmetric angle-ply laminates, the shape of the shear stress field does not match with the

exact elasticity solution even with high a/h-ratios. This is due to the non-zero moment

derivatives that are assumed to be zero in the theory. The derivatives are dependent on the

coupling terms of the stiffness matrix, which decrease when the number of layers is increased.

It can thus be assumed that the accuracy of the method increases as the number of layers is

increased. [1]



III Laminates


REFERENCES

1. Skyttä, Transverse Shear in Laminate Analysis, Helsinki University of Technology,

Laboratory of Lightweight Structures, Report B-28, ISBN 951-22-6028-X, ISSN

0785-9511

2. Rolfes, Rohwer, Improved Transverse Shear Stresses in Composite Finite Elements

Based on First Order Shear Deformation Theory. Int. J. for Num. Meth. in Eng. Vol

40 1997 p. 51

3. Rohwer, Improved Transverse Shear Stiffnesses for Layered Finite Elements,

DFVLR-FB Braunschweig, 1988:88-132

4. Pagano, Exact Solutions for Composite Laminates in Cylindrical Bending, J. of

Composite Materials, Vol 3 1969 p.398

5. Pagano, Influence of Shear Coupling in Cylindrical Bending of Anisotropic

Laminates, J. of Composite Materials, Vol 4 1970 p.330

Version 4.4 III.8.1


III Laminates

8 Laminate Thermal Conductivity

8 LAMINATE THERMAL CONDUCTIVITY

Harri Katajisto (Componeering Inc.)

The theory used by ESAComp for determining laminate thermal conductivity is given. In the approach laminate

lay-up is homogenized to be able to specify the elements of the laminated plate thermal conductivity matrix. In

the plane of the laminate the material type is assumed to be anisotropic. In through the thickness direction of the

laminate thermal conductivity is determined using one-dimensional heat transfer analysis.

SYMBOLS

[A] In-plane stiffness matrix of a laminate



[T] Transformation matrix


l Thermal conductivity

[l] Thermal conductivity matrix

q Rotation angle between the 12-coordinate system and the xy-

coordinate system around the z-axis

1, 2, 3 Ply principal coordinate system

Subscripts

k Layer index (k = 1, �, n)


Superscripts

-1 Inverse matrix

* Normalized (stiffness matrix)

III.8.2 Version 4.4


III Laminates


8.1 COORDINATE TRANSFORMATIONS

Ply in-plane thermal conductivity transformation from the 12-coordinate system to the

laminate coordinate system is carried out using the equation:

[ ]þýü

îíì

=ïþ

ïý

ü

ïî

ïí

ì-

2

11

l

l

l

ll

T

xy

y

x

where the transformation matrix [T] is obtained from Part I, Section 1.3.2 by eliminating the

3rd

column of the matrix (1.3.7):

[ ]úúú

û

ù

êêê

ë

é

-

=-

qqqqqqqq

cossincossin

cossin

sincos22

22

1T

Similar transformation is presented in several sources dealing with thermal conductivity in

composite structures e.g. in [1].

8.2 LAMINATE IN-PLANE THERMAL CONDUCTIVITY

For a laminate, in-plane thermal conductivities are obtained integrating over the individual

layers of the laminate and normalizing the result:

{ }( ){ }

h

zzn

k

kxykk

xy

å=

--= 1

,1 ll

The approach is similar to the definition of the normalized in-plane stiffness matrix [A*] (see

part III, Section 2.3). Since the layer conductivity vectors are assumed to be constant through

the thickness of each layer, the integrals can be replaced with summations.

8.3 LAMINATE OUT-OF-PLANE NORMAL DIRECTION

Laminate thermal conductivity in the z-direction is defined using the equation:

Version 4.4 III.8.3


III Laminates


{ }å=

-

÷÷ø

öççè

æ -=

n

k k

kk

z

zz

h

1 ,3

1

l

l

This formula is widely referenced e.g. in [2]. An analogy for this formula is found in electrical

circuits, i.e. when resistors are connected in series.

8.4 THERMAL CONDUCTIVITY MATRIX FOR ANISOTROPIC

LAMINATE

The above-defined equations are used to define thermal conductivity matrix for a laminated

anisotropic plate:

[ ]úúú

û

ù

êêê

ë

é

=

zyzxz

yzyxy

xzxyx

lllllllll

l

Current approach does not cover the coupling terms lxz and lyz. However, in typical

applications of solid laminates thickness is small compared to in-plane dimensions. Therefore,

laminate through the thickness temperature distribution is constant and in-plane thermal

conduction capability is sufficient to describe thermal behavior of the structure.

III.8.4 Version 4.4


III Laminates


REFERENCES

1. Argyris J., Tenek L., and Oberg A., Multilayer Composite Triangular Element for

Steady-State Conduction/Convection/Radiation Heat Transfer in Complex Shells,

Computer Methods in Applied Mechanics and Engineering 120 (1995) 271-301.

2. Kulkarni M. R. and Brady R. P., A Model of Global Thermal Conductivity in

Laminated Carbon/Carbon composites, Composites Science and Technology 57

(1997) 277-285.

PART III LAMINATES - Altair University · Version 1.0 5.3.1998 Theoretical Background of ESAComp...

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