Previous studies have shown that composite fibre orientations can be optimised for specific load cases such as longitudinal or inplane loading. However, the methodologies utilised in these studies cannot be used for general analysis of such problems. In this research, an extra term is added to the optimisation penalty function in order to consider the transverse shear effect. This modified penalty function leads to a new methodology whereby the thickness of laminated composite plate is minimized by optimising the fibre orientations for different load cases. Therefore, the effect of transverse shear forces is considered in this study. Simulated annealing (SA) is used to search for the optimal design. This optimisation algorithm has been shown to be reliable as it is not based on the starting point, and it can escape from the local optimum points. In this research, the TsaiWu failure and maximum stress criteria for composite laminate are chosen. By applying two failure criteria at the same time the results are more reliable. Experimentally generated results show a very good agreement with the numerical results, validating the simulated model used. Finally, to validate the methodology the numerical results are compared to the results of previous research with specific loading.
The demand for high strength, high modulus and low density industrial materials has generated an increased number of applications for fibre laminated composite structures in many different fields such as in submarines, sport equipment, medical instruments, civil engineering, enabling technologies, primary and secondary marine and aerospace structures, astronavigation and many more industries [
In the last half century, the use of composite materials has grown rapidly. These materials are ideal for structural applications that require high strength and low weight. They have good fatigue characteristics and are resistant to corrosion. They provide some flexibility in design through the variation of the fibre orientation or stacking sequence of fibre and matrix materials [
Optimum strength designs of continuous fibrereinforced composite laminates have been used since the early days of these materials. The first research to investigate the fibre orientation of a unidirectional lamina yielding maximum strength under inplane stress conditions has been carried out by Sandhu [
Previous studies have shown that composite fibre orientation angles can be optimised by different optmisation methods for specific load cases such as longitudinal or inplane loading [
In this paper, the thickness of laminated composite plates is minimised by optimising the fibre orientation angles for different load cases. The novelty of the research presented in this paper is that the effect of transverse shear forces, and therefore, the induced twist angle are considered.
The state of stress at a point in a general continuum can be represented by nine stress components
State of stress at a point of a continuum [
In the most general case the stress and strain components are related by the generalised Hook’s law as follows:
or
By considering the symmetry of the stress and strain tensors and the energy relations, it is proven that the stiffness matrices are symmetric. Thus, the state of stress (strain) at a point can be described by six components of stress (strain), and the stressstrain equations are expressed in terms of 21 independent stiffness constants [
The simplest equivalent singlelayer (ESL) laminated plate theory, based on the displacement field, is the classical laminated plate theory (CLPT) [
For solving inplane stress normally the classical laminate theory is used. It is assumed that plane stress components are taken as zero. With respect to the coordinate system shown in (Figure
A scheme of composite plate under inplane stress [
The principal stiffness terms,
This method works relatively well for structures that are made out of a symmetric and balanced laminate, experiencing pure bending or pure tension. The error induced/introduced by neglecting the effect of transverse shear stresses becomes trivial on or close to the edges and corners of thicksectioned configurations. The induced error increases for thick plates made of composite layers, for which the ratio of longitudinal to transverse shear elastic modulii is relatively large compared to isotropic materials [
As discussed in Section
In the following, referred to as firstorder shear deformation laminate plate theory, the assumption of normality of straight lines is removed compared to CLPT. On the other hand straight lines normal to the middle surface remain straight but not normal to that surface after deformation [
For outplane stress, (
where the components of this section stiffness matrix are given by
As it is shown in previous research [
Maximum stress criterion is one of the simplest failure methods to apply. According to this criterion, failure is predicted whenever one of the principal stress components exceeds its corresponding strength. It is expressed in the form of the following subcriteria:
The TsaiWu failure criterion is one of the most reliable static failure criteria as it provides a simple analytical expression taking components. Reddy [
By applying assumptions, some of
An optimised composite laminate requires finding the minimum number of layers, and the best fibre orientation and thickness for each layer. Several optimisation methods have been introduced to solve this challenging problem, which is often nonlinear, nonconvex, multimodal, and multidimensional. Nowadays usually stochastic nonlinear optmisation methods are utilised for this problem as they can avoid the local minimums. One of the best algorithms in this category is simulated annealing (SA) method which is used in similar problems [
Kirkpatrick et al. [
At each iteration of the simulated annealing algorithm, a new point is randomly generated. The distance of the new point from the current point, or the extent of the search, is based on a probability distribution with a scale proportional to the temperature. The algorithm accepts all new points that lower the objective, but also, with a certain probability, points that raise the objective. The algorithm avoids being trapped in local minima, by accepting points that raise the objective, and is able to explore globally for more possible solutions. An annealing schedule is selected to systematically decrease the temperature as the algorithm proceeds. As the temperature decreases, the algorithm reduces the extent of its search to converge to a minimum.
If a set of configurations is considered, in each iteration the speed convergency would be increased. In this paper the SA proposed by Erdal and Sonmez [
In this step a penalty function is expressed, and then this function has to be optimized:
This penalty function is the same as the one defined by Akbulut and Sonmez [
Implying several tests by finite element method (FEM) software shows that maximum twist for each material happens at the specific angles
For a proper
The reason that the objective is reduced for safe designs is that there may be many feasible designs with the same minimum thickness. Of these designs, the optimum is defined as the one with the largest failure load. Accordingly, the objective function is linearly reduced in proportion to the failure margin [
The penalty value due to the violation of the maximum stress and TsaiWu criteria are calculated in (
In order to validate the FEM model some experimental tests have been performed. For each case six similar laminated plates are manufactured. One of the samples is shown in Figure
Process of making laminated plate.
Square laminated composite panels (
Boundary condition and loading (the plate is clamped on side a. Displacements are measured on side b).
In Figure
An example for one case;
The experimental tests have been carried out for 15 different cases with different loads and layups. In Figure
Percentage difference of deformation between tests and FEM results.
Percentage difference of (
In this Section, two case studies are considered. The first case study compares the obtained results with those which were found by Akbulut and Sonmez [
In this case the maximum acceptable twist angle is
Optimum lamina orientations for material T300/5308 under different loads.
Loading: 
Optimum lamina orientations  Safety factor  


Akbulut and Sonmez [ 
Present work 
Akbulut and Sonmez [ 
Present work  
(Figure 
Max. stress  TsaiWu  Max. stress  TsaiWu  
10/5/0 


1.0277  1.0068  1.1309  1.1001 
20/5/0 


1.1985  1.0208  1.3305  1.1560 
40/5/0 


1.5381  1.0190  1.6504  1.1903 
80/5/0 


1.2213  1.0113  1.2302  1.0120 
120/5/0 


1.0950  1.0030  1.0951  1.0030 
Optimum lamina orientations for material T300/5308 under different loads for constant thickness.
Loading: 
Optimum lamina orientations  Safety factor  

(KPa m) (Figure 
Akbulut and Sonmez [ 
Present work  Akbulut and Sonmez [ 
Present work 
200/200/0 


2.14  2.14 
200/0/200 


4.84  4.59 
400/200/0 


1.64  1.42 
200/200/200 




In the second case study a highly anisotropic material is considered (material II). The elements of stiffness matrix are
In Figure
Twist angle for a first layer of material (II).
This test is for the first layer in order to find the
Optimum lamina orientations for second case study material under different loads for constant thickness.
Loading: 
Optimum lamina orientations  Safety factor  

Max. stress  Present work  
100/100/0/20 

1.2720  1.0931 
100/0/100/20 

1.5402  1.2112 
200/100/0/20 

1.3200  1.111 
100/100/100/20 

1.3401  1.0376 
In this study, an optimisation methodology of composite plates was presented. A method was proposed in order to overcome the difficulties and shortcomings faced by the previous research. In previous work the effect of transverse shear was neglected, and therefore the induced twist angle is ignored. In some applications the twist angle, which is the direct effect of transverse shear, is undesirable. Therefore, in this research, after optimising the fibre orientations, by considering the induced twist angle as well as safety factor, the induced twist angle always stays less than the acceptable twist angle. One of the other weakness in previous work was that the plate was optimised under specific loads, such as longitudinal or inplane loading. By the proposed method in this research the outplane stress optimisation can be solved as well as the inplane stresses. In order to have a reliable optimisation, simulated annealing, which is one of the stochastic optimisation methods and can escape the local minima is applied and the penalty function for this optimisation method is modified. This modified penalty function forces the induced twist to stay under a predefined induced twist. In addition, two TsaiWu and maximum stress failure criteria are used in the algorithm individually to avoid false optimal design.