OptiStruct supports analysis and optimization of plates and shells made of layered composites, in which several layers of different materials (plies) are bonded together to form a cohesive structure. Typically, the plies are made of unidirectional fibers or of woven fabrics and are joined together by a bonding medium (matrix). In OptiStruct composite shells, the plies are assumed to be laid in layers parallel to the middle plane of the shell. Each layer may have a different thickness and different orientation of fiber directions.
Four-layer composite with ply angle shown.
OptiStruct uses classical lamination theory to calculate effective stiffness and mass density of the composite shell. This is done automatically within the code using the properties of individual plies. The homogenized shell properties are then used in the analysis.
After the analysis, the stresses and strains in the individual layers and between the layers can be calculated from the overall shell stresses and strains. These results may then be used to assess the failure indices of individual plies and of the bonding matrix.
In OptiStruct, analysis of composite shells is very similar to the solution of standard shell elements. The primary difference is the use of the PCOMP or PCOMPG property card, instead of PSHELL, to specify shell element properties. From the ply information specified on the PCOMP entry, OptiStruct automatically calculates the effective properties of the shell element.
After the analysis, the available results include shell-type stresses as well as stresses, strains, and failure indices for individual plies and their bonding. These results are controlled by the results flags on the PCOMP or PCOMPG entry and the usual I/O control cards.
PCOMP and PCOMPG define the composite lay-up in two different ways.
PCOMP defines the structure and properties of a composite lay-up which is then assigned to an element. The plys are only defined for that particular property and there is no relationship of plys that reach across several properties.
PCOMPG defines the structure and properties of a composite lay-up allowing for global ply identification which is then assigned to an element. The plys of different PCOMPG definitions can have a relationship because of the use of global ply IDs.
Some remarks are in place regarding the specifics of composite analysis:
The most typical material type used for composite plies is MAT8, which is planar orthotropic material. OptiStruct also supports the use of isotropic MAT1 or general anisotropic MAT2 for ply properties.
While it is possible to specify ply angles relative to the element coordinate system, the results become strongly dependent upon the node numbering in individual elements. Thus, it is advisable to prescribe a material coordinate system for composite elements and specify ply angles relative to this system.
Depending on the specific lay-up structure, the composite may be offset from the reference plane of the shell element, i.e. have more material below than above the reference plane (or vice versa).
Stress results for composites include both shell-type stresses and individual ply stresses. Importantly, shell-type stresses are calculated using homogenized properties and thus only represent the overall stress-state in the shell. To assess the actual stress-state in the composite, individual ply results need to be examined.
OptiStruct calculates a number of composite-specific results for composite shell elements. Due to the specialized nature of these results, some explanation is in place regarding their meaning.
· Ply Stresses and Strains
Classical lamination theory assumes two-dimensional stress-state in individual plies (so-called membrane state). The values of stresses and strains are calculated at the mid-plane of each ply, i.e. halfway between its upper and lower surface. For sufficiently thin plies, these values can be interpreted as representing uniform stress in the ply.
Ply stresses and strains are calculated in coordinate
systems aligned with ply material angles as specified on the PCOMP card.
In particular, 
correspond to the primary ply direction,
is orthogonal to it, and 
represents in-plane
shear stress.
· Inter-laminar Stress
Inter-laminar bonding matrix usually has different
material properties and stress-state than the individual plies. The primary
stress that is of importance here is inter-laminar shear with two components:
· Failure Indices
To facilitate prediction of potential failure of the laminate, OptiStruct calculates failure indices for plies and bonding material. While there are several theories available for such calculations, their common feature is that failure indices are scaled relative to allowable stresses or strains, so that:
the value of a failure index lower than 1.0 indicates that the stress/strain is within the allowable limits (as specified on the material data card), and
a failure index above 1.0 indicates that the allowable stress/strain has been exceeded.
Here, we provide a brief summary of failure theories available in OptiStruct.
Hill's Theory of Ply Failure
According to Hill's theory, the ply failure index is calculated as:
where X is
the allowable stress in the ply material direction (1), Y
is the allowable stress in the ply material direction (2), and S
is the allowable in-plane shear stress. If the you provide different values
of X and Y
for tension and compression (prescribed as 
on the material
data card), then the appropriate values are used depending on the signs
of and respectively.
Hoffman's Theory of Ply Failure
In Hoffman's theory, the ply failure index is calculated as:
Tsai-Wu Theory of Ply Failure
In Tsai-Wu theory, the ply failure index is calculated as:
where 
is a factor to be determined
experimentally.
Maximum Strain Theory of Ply Failure
In maximum strain theory, the ply failure index is calculated as the maximum ratio of ply strains to allowable strains:
where 
is the allowable strain in the
ply material direction (1), 
is the allowable strain in
the ply material direction (2), and 
is the allowable
in-plane engineering shear strain. If you provide different values of
and for tension and compression, the
appropriate values are used depending on the signs of 
,
respectively. Note that if you prescribe allowable stresses rather than
strains on the material data card, then the allowable strains are calculated
via simple division by the relevant material modulae.
Bonding Material Failure
The primary failure mode of the bonding material is due to inter-laminar shear. The corresponding failure index is calculated as:
where SB is the allowable shear in the bonding material.
Final Failure Index for Composite Element
After calculation of failure indices for individual plies, OptiStruct calculates potential failure index for the composite shell element. This is based on the premise that failure of a single layer qualifies as failure of the composite. Thus, failure index for composite element is calculated as a maximum of all computed ply and bonding failure indices (note that only plies with requested stress output are taken into account here).
Comparison of laminate modeled with PCOMP and PCOMPG