The 'indirect' transverse tensile strengths, σA, of several 52% Vf uniaxially aligned glass fibre/polyester pultrusions were measured using the diametral compression of disc-shaped samples between concave loading anvils. In the vicinity of the disc centre there is an axial compressive stress, -3σA; thus the test, when performed at atmospheric pressure, is biaxial. In an attempt to identify the failure criterion, triaxial test was carried out by superimposing hydrostatic pressure, -H, which yielded the principal stresses σ1 = σA + H σ2 = H σ3 = -3σA + H For the cases where there was no interaction between the pressurizing medium and the composite, the linear pressure dependence of σA was c. -0·1 H, i.e. for -H > 30 MPa, extensile failure from the fibre/resin interfaces took place when all the principal stresses, though unequal, were compressive. This rules out maximum tensile stress as a transverse tensile (interfacial) failure criterion; the results, however, were consistent with that of a critical tensile strain. Using this criterion the (simple) transverse tensile strengths were estimated to lie in the range 59-101 MPa. For the cases of environmental attack by the pressurizing medium (a synthetic diester), the relevant criterion appeared to be that of maximum deviatoric stress, and on this hypothesis the (atmospheric) strengths of the glass fibre/polyester interfaces were estimated to lie in the range 47-57 MPa. © 1990.