This would yield a flexural to tensile strength ratio of 1.52 for an E-glass unidirectional (equal volumes of stressed material in flexure and tension, i.e. Vf = Vt and = 15).
As mentioned, the flexural:tensile strength ratio for CSM/WR averaged over fibre content by mass of 30- 50% as given by properties from Annex C of [1] would be 1.47. This suggests that equation (4) may provide at least a starting point for correcting the conventionally used in-plane analysis.
strength for use in a laminate stack
Strictly, equation (4) should not be applied to a single ply in a laminate stack since it assumes that the ply is subjected to the normal bending stress distribution (i.e. zero at mid thickness, reversal top and bottom). Instead of integrating the stress function along the span and through the thickness as was done by the authors of [7, 8] to obtain equation (4), the integration should be along the span only (as each ply is a fixed distance from the neutral axis and may be treated at having constant stress through its thickness).
This would effectively remove the ‘square’ of the term from equation (4).
Including the thickness effect from
equation (3) gives an approximate correction factor which may relate the strength of a thin ply under bending as part of a laminate stack to the tensile strength as obtained from a thicker test coupon, i.e.
ff tt
t
2 1
tV
1/ 2( 1) V 1/ (5)
Equation (4) as taken directly from references [7, 8] is based on the assumption that failure in flexure is directly related to the normal stress on the tensile face.
laminar effects are not considered. In the case of most carbon and aramid plies and quite a few E glass based continuous fibres, the compressive strength will be much less than the tensile strength.
The use of strain-based design raises the question of differences
According to
between in-plane and published
equations between plies. the flexural
modulus of CSM/WR is about 20% lower than the mean of the tensile and compressive moduli. One explanation for this may be the presence of less stiff zones at the resin-rich interface
flexural modulus. [2],
Possible shear
distortion may also account for some loss of stiffness, although conventional shear deflection is easy to correct for and not a problem if the coupon span:depth ratio is suitably dimensioned as per the appropriate standard.
Inter- In addition for some
carbon and most aramid plies, the bending strength will be intermediate between the tensile and compressive.
Equation (5) is only presented here as a speculative device to attempt to account for thickness and non- uniform stress distribution effects on ply strength.
It requires only a briefest acquaintance with composites to appreciate that the most reliable design friendly results come from combining theory and experiment.
Free-
standing reliable computational methods tend to be too sophisticated for routine marine design use and wholly empirical methods often mask parametric trends unless the database is very extensive. ‘Modified rule of mixtures’ is probably a good illustration of a semi- empirical approach at its best.
One thing is fairly clear. While the strength of a structure is governed by localized effects, the deflection is generally a result of some kind of ‘averaged’ stiffness. If stiffness is a governing factor, the stress levels are usually low enough that the presence of flaws has minimal effect on stiffness. As far as simple design of single skin composites is
concerned, not only is
difference between in-plane and flexural moduli much smaller than the corresponding
difference between
strengths, the effect on thickness is smaller due to the cubed-root associated with the stiffness driven version of equation (2a). Hence a 20% variation in modulus gives only 7% difference in thickness. Stiffness limits tend to be more arbitrary than strength limits.
Indeed the ISO
standard does not specify deflection limits for single skin, nor does reference (2).
Given that the variations in flexural v in-plane
strengths/strains are far more significant than differences between in-plane and flexural moduli, it is assumed here that Eflexure = Ein-plane. This is the same assumption made
Hence equation (5) will require the use of published data in order to allow it to be evaluated in a usable form.
It
will then be necessary to verify the veracity of the equation against independent measurements on laminate stacks. This is discussed in section 3.
2.3
A SIMPLIFIED APPROACH TO OBTAINING ‘APPARENT’ PLY STRAINS FOR USE IN A STACK ANALYSIS
It is more logical to work with failure strains rather than stresses, given that the intended application is for laminate stacks which consist of plies of different modulus. These strains are ‘apparent ply failure strains’ obtained by dividing the ultimate strength by the initial elastic modulus and are generally quoted as a percentage.
Actual failure strains will be higher than apparent ply failure strains but use of the actual strains would necessitate a knowledge of the relationship between tangent modulus and strain.
This would elevate the
simple laminate stack analysis to a non-linear procedure and this is not considered to be appropriate for the intended application in small marine craft design within limited design budgets.
© 2008: Royal Institution of Naval Architects
B-33
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