Tsai-Hill

The strength of laminates is usually predicted from a combination of laminated plate theory and a failure criterion for the individual larnina. A general treatment of composite failure criteria is beyond the scope of the present discussion. Broadly, however, composite failure criteria are of two types noninteractive, such as maximum stress or maximum strain, in which the lamina is taken to fail when a critical value of stress or strain is reached parallel or transverse to the fibers in tension, compression, or shear or interactive, such as the Tsai-Hill or Tsai-Wu (1,7) type, in which failure is taken to be when some combination of stresses occurs. Generally, the ply materials do not have the same strengths in tension and compression, so that five-ply strengths must be deterrnined ... 

In practice the second term in the above equation is found to be small relative to the others and so it is often ignored and the reduced form of the Tsai-Hill Criterion becomes... 

If the fibres are aligned at 15° to the jc-direction, calculate what tensile value of Ox will cause failure according to (i) the Maximum Stress Criterion (ii) the Maximum Strain Criterion and (iii) the Tsai-Hill Criterion. The thickness of the composite is 1 mm. 

For an applied stress of 1 MN/m and letting X be the multiplier on this stress, we can determine the value of X to make the Tsai-Hill equation become equal to 1. 

Solving this gives X = 169. Hence a stress of Ox = 169 MN/m would cause failure. It is more difficult with the Tsai-Hill criterion to identify the nature of the failure ie tensile, compression or shear. Also, it is generally found that for fibre angles in the regions 5°-15 and 40 -90 , the Tsai-Hill criterion predictions are very close to the other predictions. For angles between 15° and 40° the Tsai-Hill tends to predict more conservative (lower) stresses to cause failure. 

Determine whether failure would be expected to occur according to (a) the Maximum Stress (b) the Maximum Strain and (c) the Tsai-Hill criteria. 

The Tsai-Hill criteria gives the following values... 

The failure in the 30° plies is thus confirmed. The Tsai-Hill criteria also predicts failure in the 0° plies and it may be seen in Fig. 3.31 that this is... 

A single ply glass/epoxy composite has the properties Usted below. If the fibres are aligned at 30° to the x-direction, determine the value of in-plane stresses, a, which would cause failure according to (a) the Maximum Stress criterion (b) the Maximum Strain criterion and (c) the Tsai-Hill criterion. 

A carbon/epoxy composite with the stacking arrangement [0/ - 30/30]j has the properties listed below. Determine the value of in-plane stress which would cause failure according to the (a) Maximum Strain (b) Maximum Stress and (c) Tsai-Hill criteria. 

Hence once again a stress of Ojt = 92 MN/m is the limiting condition, (iii) Tsai-Hill Criterion Letting X be the multiplier for... 

For the Tsai-Hill Criterion, letting X = multiplier on stress a, then solving... 

The limiting condition is thus a, = 583 MN/m as predicted by Tsai-Hill. 

Results for this criterion are plotted in Figure 2-40 along with the experimental data for E-glass-epoxy. The agreement between the Tsai-Hill failure criterion and experiment is quite good. Thus, a suitable failure criterion has apparently been found for E-glass-epoxy laminae at various orientations in biaxial stress fields. 

The Tsai-Hill failure criterion appears to be much more applicable to failure prediction for this E-glass-epoxy composite material than either the maximum stress criterion or the maximum strain failure criterion. Other less obvious advantages of the Tsai-Hill failure criterion are ... 

Rgure 2-40 Tsai-Hill Failure Criterion (After Tsai [2-21])... 

For E-glass-epoxy, the Tsai-Hill failure criterion seems the most accurate of the criteria discussed. However, the applicability of a particular failure criterion depends on whether the material being studied is ductile or brittle. Other composite materials might be better treated with the maximum stress or the maximum strain criteria or even some other criterion. 

To account for different strengths in tension and compression, Hoffman added linear terms to Hill s equation (the basis for the Tsai-Hill criterion) [2-23] ... 

A single failure criterion is used in all quadrants of o,-02 space instead of the segments in separate quadrants for the Tsai-Hill failure criterion because of different strengths in tension and compression. 

The terms that are linear in the stresses are useful in representing different strengths in tension and compression. The terms that are quadratic in the stresses are the more or less usual terms to represent an ellipsoid in stress space. However, the independent parameter F,2 is new and quite unlike the dependent coefficient 2H = 1/X in the Tsai-Hill failure criterion on the term involving interaction between normal stresses in the 1- and 2-directions. 

At this point, recali that all interaction between normal stresses o, and 02 iri the Tsai-Hill failure criterion is related to the strength in the 1-direction ... 

Thus, the Tsai-Wu tensor failure criterion is obviously of more general character than the Tsai-Hill or Hoffman failure criteria. Specific advantages of the Tsai-Wu failure criterion include (1) invariance under rotation or redefinition of coordinates (2) transformation via known tensor-transformation laws (so data interpretation is eased) and (3) symmetry properties similar to those of the stiffnesses and compliances. Accordingly, the mathematical operations with this tensor failure criterion are well-known and relatively straightforward. 

Find the Tsai-Hill failure criterion for pure shear loading at various angles B to the principal material directions, i.e., the shear analog of Equation (2.134). 

Note that the lamina failure criterion was not mentioned explicitly in the discussion of Figure 4-36. The entire procedure for strength analysis is independent of the actual lamina failure criterion, but the results of the procedure, the maximum loads and deformations, do depend on the specific lamina failure criterion. Also, the load-deformation behavior is piecewise linear because of the restriction to linear elastic behavior of each lamina. The laminate behavior would be piecewise nonlinear if the laminae behaved in a nonlinear elastic manner. At any rate, the overall behavior of the laminate is nonlinear if one or more laminae fail prior to gross failure of the laminate. In Section 2.9, the Tsai-Hill lamina failure criterion was determined to be the best practical representation of failure... 

The procedure of laminate strength analysis outlined in Section 4.5.2, with the Tsai-Hill lamina failure criterion will be illustrated for cross-ply laminates that have been cured at a temperature above their service or operating temperature in the manner of Tsai [4-10]. Thus, the thermal effects discussed in Section 4.5.3 must be considered as well. For cross-ply laminates, the transformations of lamina properties are trivial, so the laminate strength-analysis procedure is readily interpreted. 

The failure criterion must be applied to determine the maximum values of Nx and AT that can be sustained without failure of any layer. Actually, the failure criterion is applied to each layer separately. For the special orientation of cross-ply laminates, the Tsai-Hill failure criterion for each layer can be expressed as... 

Note that the stresses Cy are very small in comparison to the shearing stresses. Thus, the Tsai-Hill lamina failure criterion can be simplified for this lamina to... 

The Tsai-Hill criterion governs failure of a lamina (the strength-analysis procedure could, of course, involve another criterion). 

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