Composite Materals Design - Stress Distribution in Laminates
Determination of strains and stresses in laminates subjected to normal and bending loadsIn this exercise we will show how to define an imposed set of normal loads and bending loads and use these loads to calculate the strains within the laminate and hence how to obtain th estress distribution within the individual plies that make up the laminate. We will also show how to apply the Halpin-Tsai failure criterion so that it is possible to calculate the first ply failure of the laminate. Since there are 6 possible imposed loads, it is necessary to define 5 of the loads and solve for the 6th. In simple axial tension and simple bending, there is only one applied load (the remainder are zero) and hence the axial strength and bending strength can be determined.To define the force resultants N, we recall that N is the force per unit width of the laminate, in other words it is the applied stress multiplied by the thickness of the laminate (H) while the moments M, are simply a force per unit width x distance.  Again, we should note that N and M have different units and that to combine the two vectors into a single vector we need to multiply the M vector by distance before stacking the two vectors to form a 1x6 column. The uniform strains in the laminate  and the curvatures of the laminate  , as a whole, are calculated from  from which we can obtain the actual strains at any point in the cross-section . In MathCAD, the {eo} and {k} matrices are stacked into a single vector so we have to do the addition of the two components manually using a program block . Normally {eo} and {k} have different units, strain being dimensionless and curvature having the units of distance-1. However, all the terms in the  vector, as generated by MathCAD, have the same dimension and thus we need to divide the curvatures by distance (metres) to correct the dimensions before mutliplying by the vertical displacement z and adding the resulting strain onto the uniform strain. To obtain the stress state at any point in the laminate all we need do is multiply the strain at that point by the stiffness tensor of the individual ply - thus we need a simply way to determine which ply a given vertical displacement, z, is located in. The function Pw(z) does just that by sequentially checking whether the value of z lies between the lower and upper interface of each individual ply and exiting when the test is true and returning the index of the ply. Once we know which ply a given point is located within, we can multiply the appropriate stiffness tensor by the strain tensor to obtain the stresses. You should note that the functions are written in terms of the volume fraction of fibres, f, an arbtrary imposed rotation of the laminate {f}, the vertical displacement, z, and the imposed force and moment resultants. The logical IF statement in the program block is there simply to catch values of z which are not within the laminate and return zero for the stress.  The variation of the axial, transverse and shear stresses through the thickness of the laminate can now be plotted. Any changes to the imposed loads, fibre fraction or orientation of the laminate, defined at the top of this exercise, will be reflected in the graph below. It should be apparent that the stress state can change abruptly between plies.  Strength in Simple TensionStrength in Simple Bending |
