The out-of-plane shear stress components are obtained by integration of expression (II.1.42) along the thickness. This leads to:
One assumes zero shear stress along the bottom surface of the laminate. This corresponds to a free surface, or at least to a surface that receives no contact forces in direction and . This assumption leads to the following expression:
in which one introduces a new matrix notation:
and a new vector notation for the gradient of bending moments:
The new matrix is of course a matrix.
An explicit expression of the integrated matrix is calculated ply-by-ply, from bottom layer to top layer. If :
In expression (II.1.44), one introduced new matrices that are calculated as follows:
(II.1.45) |
Note that the expression above involve the a priori unknown quantity . To calculate this expression, one uses the continuity of across ply interfaces:
This relation corresponds to the continuity of out-of-plane shear stress at each interface between two consecutive plies. One develops the relation as follows.
The last line of this development allows to calculate recursively the components of from bottom ply to top ply. For bottom ply, the condition leads to the following expressions:
Then, it becomes possible to calculate recursively the matrices.
One checks easily that the condition ensures also that . Indeed, one has:
The last line of previous equation contains twice the integral of along the laminate thickness. One develops this integral as follows:
On the other hand, equation (II.1.33) allows to write:
(The “lam” subscript has been omitted for concision sake.) The identification of the right upper corner of the last expression with the integration of along the laminate thickness shows that this integral must be zero. Consequently, one also has:
It is interesting to remark that the ply out-of-plane shear moduli have not been used in the calculations to obtain (II.1.43). The out-of-plane shear stresses depend only on laminate in-plane bending moments and ply in-plane material properties. One shows in section II.1.6.6 that on the other hand, the calculation of out-of-plane shear strains caused by out-of-plane shear forces requires the knowledge of ply out-of-plane material constants.