Kassapoglou’s methods (`composites.kassapoglou`)#

Methods based on Kassapoglou’s book: buckling loads of rectangular plates under compression, shear and combined compression and shear, and the effective width of a buckled plate, see the functions of this module.

Notes#

The functions implement the equations of Kassapoglou [1], chapters 6 and 7, whose numbers are given in each function.

References#

[1]

Kassapoglou, C., Design and Analysis of Composite Structures, Second Edition, John Wiley & Sons, 2013.

composites.kassapoglou.calc_Nxx_crit(a, b, m, n, D11, D12, D22, D66)#

Calculate uniaxial compression buckling for a composite plate

If \(m\) or \(n\) is set to None, the function searchers for the critical number of half-waves in the corresponding direction, up to 10 half-waves.

The load is evaluated for every value of \(m\) and \(n\), the None one replaced by 1, 2, ..., 10, and the minimum is returned. \(m\) and \(n\) may also be given as arrays, which are broadcast against each other.

Parameters:

a, bfloat: Plate length and width, dimension length.
m, nint or None: Number of half-waves along the plate length and width, respectively.
D11, D12, D22, D66float: Terms of the D matrix, dimension force times length.

Returns:

Nxx_critfloat: Critical uniaxial compression buckling load, force per unit length.

Raises:

NotImplementedError: If both \(m\) and \(n\) are None.

Notes

The output of this function is the result of Eq. 6.6, section 6.2 page 129 of [1], with \(AR = a/b\):

\[N_{xx,crit} = \frac{\pi^2}{a^2 m^2} \left( D_{11} m^4 + 2 (D_{12} + 2 D_{66}) m^2 n^2 AR^2 + D_{22} n^4 AR^4 \right)\]

i.e. the buckling load of a rectangular plate with the buckling mode \(\sin(m \pi x/a) \sin(n \pi y/b)\), exact for a specially orthotropic plate, \(D_{16} = D_{26} = 0\), simply supported on its four edges.

References

[1]

Kassapoglou, C., Design and Analysis of Composite Structures, Second Edition, John Wiley & Sons, 2013.

composites.kassapoglou.calc_Nxx_crit_combined_shear(k, a, b, D11, D12, D22, D66)#

Calculate combined uniaxial-shear buckling load for a composite plate

This function calculates the critical \(N_{xx}\) buckling load under the current level of shear load given by \(N_{xy} = k N_{xx}\). \(D16\) and \(D26\) are neglected.

The equation gives two roots, and the smaller of their absolute values is returned.

Parameters:

kfloat: Load ratio defined as \(k=N_{xy}/N_{xx}\), where \(N_{xy}\) is the current shear load, and \(N_{xx}\) the current compression load, both with the sign of Eq. 6.32 of Kassapoglou, positive in tension. Only \(k^2\) enters, such that the sign of \(k\) does not change the result. Dimensionless.
a, bfloat: Plate length and width, dimension length.
D11, D12, D22, D66float: Terms of the D matrix, dimension force times length.

Returns:

N0float: Critical \(N_{xx}\) buckling load under the current level of shear load given by \(N_{xy} = k N_{xx}\), force per unit length, non-negative.

See also

calc_Nxx_crit_combined_shear

Notes

Based on section 6.5 of [1], pp. 138-142, for a plate simply supported on its four edges (Fig. 6.12). The Rayleigh-Ritz method used therein is applied with the two terms of Eq. 6.31, \(w = w_1 \sin(\pi x/a) \sin(\pi y/b) + w_2 \sin(2 \pi x/a) \sin(2 \pi y/b)\), to the energy of Eq. 6.32 (p. 140), here including \(D_{16}\) and \(D_{26}\), with the resultants \(N_x\) and \(N_{xy}\) positive in tension:

\[\Pi = \int_0^a \int_0^b \left[ \frac{1}{2} \left( D_{11} w_{,xx}^2 + 2 D_{12} w_{,xx} w_{,yy} + D_{22} w_{,yy}^2 + 4 D_{66} w_{,xy}^2 + 4 D_{16} w_{,xx} w_{,xy} + 4 D_{26} w_{,yy} w_{,xy} \right) + \frac{1}{2} N_x w_{,x}^2 + N_{xy} w_{,x} w_{,y} \right] dx dy\]

As in Eq. 6.32a (p. 141), the compression load is \(N_o = -N_x\), positive in compression, and the shear load enters as \(k N_o\), with \(k = N_{xy}/N_x\), such that \(k N_o = -N_{xy}\). The argument Nxy of this function is the \(N_{xy}\) of Eq. 6.32, and the returned load is \(N_o\). The stationarity of \(\Pi\) with respect to \(w_1\) and \(w_2\) gives \(([a] - [b]) \{w_1, w_2\} = 0\), with

\[\begin{split}a_{11} = \frac{\pi^4 D_{11} b}{4 a^3} + \frac{\pi^4 D_{12}}{2 a b} + \frac{\pi^4 D_{22} a}{4 b^3} + \frac{\pi^4 D_{66}}{a b}, \quad a_{12} = a_{21} = -\frac{160 \pi^2 D_{16}}{9 a^2} - \frac{160 \pi^2 D_{26}}{9 b^2} \\ a_{22} = \frac{4 \pi^4 D_{11} b}{a^3} + \frac{8 \pi^4 D_{12}}{a b} + \frac{4 \pi^4 D_{22} a}{b^3} + \frac{16 \pi^4 D_{66}}{a b} \\ b_{11} = \frac{\pi^2 N_o b}{4 a}, \quad b_{22} = \frac{\pi^2 N_o b}{a}, \quad b_{12} = b_{21} = -\frac{32 k N_o}{9} = \frac{32 N_{xy}}{9}\end{split}\]

which, with \(D_{16} = D_{26} = 0\), is the generalized eigenvalue problem of Eqs. 6.33 (p. 141). \(\det([a] - [b]) = 0\) is a quadratic equation in \(N_o\), whose roots are

\[N_o = \frac{a \left( 36 a_{11} + 9 a_{22} \mp \sqrt{t} \right)}{18 \pi^2 b}, \quad t = 16384 N_{xy}^2 - 4608 N_{xy} (a_{12} + a_{21}) + 1296 a_{11}^2 - 648 a_{11} a_{22} + 1296 a_{12} a_{21} + 81 a_{22}^2\]

With \(D_{16} = D_{26} = 0\), \(a_{12} = a_{21} = 0\) and only \(N_{xy}^2 = (k N_o)^2\) enters, as in Eq. 6.34 (p. 142); otherwise the sign of the shear load changes the result.

References

[1]

Kassapoglou, C., Design and Analysis of Composite Structures, Second Edition, John Wiley & Sons, 2013.

composites.kassapoglou.calc_Nxy_crit(a, D11, D12, D16, D22, D66, rtol=1e-05, atol=1e-06, max_iter=50)#

Calculate shear buckling for a composite plate

Variables \(AR\) and \(\alpha\) of the buckling pattern are solved using a Newton-Raphson scheme. The iterations start from \(\alpha = \pi/6\) and stop when the residual is close to zero according to numpy.isclose(). \(D16\) is not used.

Parameters:

afloat: Width of the long plate, i.e. its short dimension, along \(x\), dimension length.
D11, D12, D16, D22, D66float: Terms of the D matrix, dimension force times length.
rtol, atolfloat, optional: Relative and absolute tolerances intended for the residual of the Newton-Raphson scheme. Default 1e-5 and 1e-6. They are not used: the convergence check uses the fixed values atol=1e-6 and rtol=1e-5.
max_iterint, optional: Maximum number of iterations used in the Newton-Raphson scheme. Default 50.

Returns:

Nxy_critfloat: Critical shear buckling load, force per unit length.

Raises:

RuntimeError: If the Newton-Raphson scheme does not converge within \(max_iter\) iterations.

Notes

The output of this function is the result of Eq. 6.28, section 6.4 page 137 of [1], for a long rectangular plate of width \(a\), simply supported on its long edges \(x = 0\) and \(x = a\), made of a symmetric laminate with \(D_{16} = D_{26} = 0\), under the shear load \(N_{xy}\) (pp. 136-137). The buckling mode, Eq. 6.20,

\[w = w_o \sin\frac{\pi x}{a} \sin\frac{\pi (y - x \tan\alpha)}{L}\]

is zero along the long edges and along lines inclined by the angle \(\alpha\) to the \(x\) axis, separated by the distance \(L\) along the long direction \(y\) (Fig. 6.10), such that the length of the plate does not enter, and \(AR = a/L\) (p. 137). Eq. 6.28 follows from the minimum of the total potential energy with respect to \(w_o\), Eqs. 6.21-6.27, and the critical \(AR\) and \(\alpha\) minimize it, Eqs. 6.29 and 6.30 (p. 138). Variables \(AR\) and \(\alpha\) are solved using a Newton-Raphson scheme that finds the solution of Eqs. 6.29 and 6.30 simultaneously:

\[\begin{split}AR = \left( \frac{D_{11}}{D_{11} \tan^4\alpha + 2 (D_{12} + 2 D_{66}) \tan^2\alpha + D_{22}} \right)^{1/4} \\ r(\alpha) = 3 D_{11} AR^4 \tan^4\alpha + (6 D_{11} AR^2 + 2 (D_{12} + 2 D_{66}) AR^4) \tan^2\alpha - (D_{11} + 2 (D_{12} + 2 D_{66}) AR^2 + D_{22} AR^4) = 0 \\ N_{xy,crit} = \frac{\pi^2}{2 AR^2 a^2 \tan\alpha} \left( D_{11} (1 + 6 \tan^2\alpha AR^2 + \tan^4\alpha AR^4) + 2 (D_{12} + 2 D_{66}) (AR^2 + AR^4 \tan^2\alpha) + D_{22} AR^4 \right)\end{split}\]

The tolerances rtol and atol are intended for the residual \(r\) of Eq. 6.30. At each iteration \(AR\) is updated with Eq. 6.29 and \(\alpha\) is corrected by \(-r/(\partial r/\partial \alpha)\), the derivative being taken at constant \(AR\).

References

[1]

Kassapoglou, C., Design and Analysis of Composite Structures, Second Edition, John Wiley & Sons, 2013.

composites.kassapoglou.calc_beff(b, Px, Pcr, A11, A12, A22)#

Calculate the effective width of a plate

The effective width \(b_{eff}\) is defined as:

\(\int_y N_{xx} dy = 2{N_x}_{max} b_{eff}\)

\(Px\) and \(Pcr\) enter only through the ratio Pcr/Px, and the terms of the A matrix only through ratios between them, such that the result has the dimension of \(b\).

Parameters:

bfloat: Plate width, dimension length.
Pxfloat: Applied compressive force, of the same dimension as \(Pcr\).
Pcrfloat: Critical buckling force, of the same dimension as \(Px\).
A11, A12, A22float: Terms of the A matrix, dimension force per unit length.

Returns:

befffloat: Effective width of the plate, dimension length.

Notes

The output of this function is the result of Eq. 7.15, section 7.1 page 164 of [1]:

\[b_{eff} = \frac{b}{2 \left( 1 + 2 \left( 1 + \frac{A_{12}}{A_{11}} \right) \left( 1 - \frac{P_{cr}}{P_x} \right) \frac{A_{11}}{A_{11} + 3 A_{22}} \right)}\]

At \(P_x = P_{cr}\), \(b_{eff} = b/2\), i.e. the whole width is effective; the expression is meant for the post-buckling regime, \(P_x \ge P_{cr}\).

References

[1]

Kassapoglou, C., Design and Analysis of Composite Structures, Second Edition, John Wiley & Sons, 2013.

Kassapoglou’s methods (composites.kassapoglou)#

Notes#

References#

Kassapoglou’s methods (`composites.kassapoglou`)#