Laminated plates:

Governing differential equation for a general laminate:

 

Governing Differential Equations for Classical Laminate Theory
The equilibrium equations for a laminate are

clip_image001

(5.133)

In the laminate, in general, we consider that the transverse shear stresses are vanishing at the top and bottom of the laminate, that is clip_image002  at clip_image003  and clip_image004. Now, integrate Equation (5.133) with respect to z. The first two of the above equation give us

clip_image005

(5.134)

The third of the Equation (5.133) gives

clip_image006

(5.135)

where,

clip_image007 and clip_image008

Now, multiply the first of Equation (5.133) with z and integrate with respect to z to get

clip_image009

(5.136)

Now, let us write

clip_image010

Now recalling that clip_image011  at clip_image003[1]  and clip_image004[1] we can write for the third term in Equation (5.136) as

clip_image013 and clip_image014

Thus, Equation (5.136) becomes

clip_image015

(5.137)

Similarly, we can write

clip_image016

(5.138)

Now putting Equation (5.137) and Equation (5.138) in Equation (5.135) we get

clip_image017

(5.139)

Note that this equation is identical with the homogeneous plate theory. However, in these equations the definition of the resultants is different.
One can express the moment resultants in terms of A, B and D matrices and the derivatives of mid plane displacements as given below.
Equation (5.137) can be written as

image

(5.140)

Equation (5.138) becomes

image

(5.141)

And Equation (5.139) becomes

image

(5.142)

 

Cross Ply Laminate:

A laminate in which the ply orientations are oriented at right angles to each other, with ply orientations limited to 0° and 90° only. It is usually best to arrange stacking sequences with fibers oriented in different directions.

Angle Ply Laminate:

One possessing equal plies with positive and negative angles. This is a bi-directional orthotropic laminate, such as a [±45°].

Composite Failure Criteria:

Failure criteria for composite materials are significantly more complex than yield criteria for metals because composite materials can be strongly anisotropic and tend to fail in a number of different modes depending on their loading state and the mechanical properties of the material. While theories which reflect detailed mechanisms of failure are currently being developed, empirical criteria based on test data have been used for decades. These criteria have been incorporated in the MSC.Laminate Modeler to allow rapid evaluation of the strength of a structure according to the current industry standards. The user can also define custom criteria using PCL functions for use in specialized applications.

Nomenclature:

Failure criteria compare the loading state at a point (stress or strain) with a set of values reflecting the strength of the material at that point (often referred to as the material allowable). Both loading and strength values should be reflected in the same material coordinate system. For unidirectional materials, this is typically in the direction of the fibres. However, for woven and knitted fabrics, this direction is not obvious, and might change as the material is formed to shape.

In general, the load is represented by a full stress or strain tensor having six independent components. By convention, for lamina materials the material X axis lies in the direction of the warp fibres while the Z axis lies in the through-thickness direction of the sheet. Note than in Patran, shear strains are stored in tensor rather than engineering notation, and any experimental failure strengths should reflect this.

STRESS

σxyzxyyzxz

STRAIN

εxyzxyyzxz

The strength of a composite can be expressed by an arbitrarily large number of values, depending on the complexity of the failure criterion. However, lamina materials, used in composites, are often assumed to be orthotropic; the through‑thickness stresses or strains are ignored and it is assumed that there is negligible interaction between the different failure modes. The strength of the material can therefore be represented by seven independent variables:

TX

tensile strength along the X axis

0 < TX

CX

compressive strength along the X axis

0 < CX

TY

tensile strength along the Y axis

0 < TY

CY

compressive strength along the Y axis

0 < CY

SXY

shear strength in the XY plane

0 < SXY

SYZ

shear strength in the YZ plane

0 < SYZ

SXZ

shear strength in the XZ plane

0 < SXZ

In the Tsai-Wu criterion, these values have been supplemented by an interaction term which reflects the interdependence of failure modes due to loading along both the X and Y material directions.

IXY

interaction between X and Y directions

-1< IXY <1

Note that the above values can be applied to either stress or strain.

The form of the failure criterion is typically described as a mathematical function of the above variables which reaches the value of unity at failure as follows.

Failure Index = FI (load, strength) = 1

The strength of a structure can be given as a Strength Ratio (SR), which is the ratio by which the load must be factored to just fail. (Note that the Strength Ratio is not necessarily the reciprocal of the Failure Index.) Alternatively, the Margin of Safety (MoS), where MoS = SR – 1, is used.

Maximum Criterion

This criterion is calculated by comparing the allowable load with the actual strength for each component. Mathematically, it is defined by:

FI = max (σx/TX, -σx/CX, σy/TY, -σy/CY,

abs(τxy)/SXY, abs(γyz)/SYZ, abs(γxz)/SXZ)

In this case,

SR = 1/FI

Hill Criterion

The Hill criterion was one of the first attempts to develop a single formula to account for the widely different strengths in the various principal directions:

FI = FXX σx2 + FYY σy2 + 2 FXY σx σy + FSS τxy2

where

FXX =

1/(TX TX)

if σx >= 0

1/(CX CX)

if σx <0

FYY =

1/(TY TY)

if σy >= 0

1/(CY CY)

if σy <0

FXY =

-1/(2 TX TX)

if σxσy >= 0

-1/(2 CX CX)

if σxσy <0

FSS =

1 / (SXY SXY)

 

Because this failure theory is quadratic:

SR = 1 / sqrt (FI)

In the Laminate Modeler, the Tsai-Wu criterion for in-plane loads (representing fiber failure) has been supplemented by a maximum load theory for out-of-plane shear loads (representing matrix failure):

FI = max( abs(γyz)/SYZ, abs(γxz)/SXZ )

In this case,

SR = 1/FI

For every ply, the lower of the Margins of Safety for fibre and matrix failure is calculated and displayed.

Tsai-Wu Criterion

The Tsai-Wu failure criterion is an unashamed, empirical criterion based on the sum of the linear and quadratic invariants as follows:

Fi σi + Fij σi σj = 1

i,j = 1…6

where Fi and Fij are dependent on the material strengths. For the restrictions of lamina materials, this equation reduces to:

FI = FX σx + FY σy + FXX σx2 + FYY σy2 + 2 FXY σx σy + FSS τxy2

where:

FX = 1/TX – 1/CX

FY = 1/TY – 1/CY

FXX = 1/(TX CX)

FYY = 1/(TY CY)

FXY = IXY sqrt(FXX FYY) = IXY / sqrt(TX CX TY CY)

FSS = 1 / (SXY SXY)

Because this failure theory is quadratic, the Strength Ratio (SR) = 1/FI. However, multiplying the failure criterion by SR and rearranging gives

a SR2 + b SR – 1 = 0

where

a = FXX σx2 + FYY σy2 + 2 FXY σx σy + FSS τxy2

b = FX σx + FY σy

Therefore

SR = [-b + sqrt (b2 + 4a)] / 2a

In the Laminate Modeler, the Tsai-Wu criterion for in-plane loads (representing fiber failure) has been supplemented by a maximum load theory for out-of-plane shear loads (representing matrix failure):

FI = max( abs(γyz)/SYZ, abs(γxz)/SXZ )

In this case,

SR = 1/FI

For every ply, the lower of the Margins of Safety for fibre and matrix failure is calculated and displayed.

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