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| import numpy as np from sympy import *
def integrate2(f, x, y): g = integrate(f, x) g = integrate(g, y) return g
def diffn(f, x, n): while n != 0: f = diff(f, x) n = n - 1 return f
ST = 0
BC = 3
m = n = 1 C11 = 263E9 C12 = 154E9 C44 = 127E9
h = 0.8 a = 10 b = 10
E = 25.5E9
v = 0.2
k = Symbol('k') x, y, z = symbols('x y z')
w_xx, w_yy, w_xy, w_x, w_y = symbols('w_xx w_yy w_xy w_x w_y') theta_xx, theta_yy, theta_xy, theta_yx, theta_x, theta_y = symbols('theta_xx theta_yy theta_xy theta_yx theta_x theta_y') lambda_xx, lambda_yy, lambda_xy, lambda_yx, lambda_x, lambda_y = symbols('lambda_xx lambda_yy lambda_xy lambda_yx lambda_x lambda_y') phi_xx, phi_yy, phi_xy, phi_yx, phi_x, phi_y = symbols('phi_xx phi_yy phi_xy phi_yx phi_x phi_y')
Gxy = G = E / 2 / (1 + v)
Diff = E * h ** 3 / 12 / (1 - v ** 2) DF = E * h ** 3 NCr = k * pi ** 2 * Diff / (a ** 2) kv = 0
print('Plate Thickness: h=', h, 'm') print('Length to thickness ratio: a/h=', a / h) print('length to width ratio: a/b=', a / b)
if BC == 1: print('Boundary Condition: SSSS') alphaM = m * pi / a beltaN = n * pi / b Xm = sin(alphaM * x) Yn = sin(beltaN * y)
D1Xm = cos(x * alphaM) * alphaM ** 1 D2Xm = -sin(x * alphaM) * alphaM ** 2 D3Xm = -cos(x * alphaM) * alphaM ** 3 D4Xm = sin(x * alphaM) * alphaM ** 4
D1Yn = cos(y * beltaN) * beltaN ** 1 D2Yn = -sin(y * beltaN) * beltaN ** 2 D3Yn = -cos(y * beltaN) * beltaN ** 3 D4Yn = sin(y * beltaN) * beltaN ** 4
if BC == 2: print('Boundary Condition: CCCC') alphaM = (m + 0.5) * pi / a beltaN = (n + 0.5) * pi / b Xm = sin(alphaM * x) - sinh(alphaM * x) - (sin(alphaM * a) - sinh(alphaM * a)) /\ (cos(alphaM * a) - cosh(alphaM * a)) * (cos(alphaM * x) - cosh(alphaM * x)) Yn = sin(beltaN * y) - sinh(beltaN * y) - (sin(beltaN * b) - sinh(beltaN * b)) /\ (cos(beltaN * b) - cosh(beltaN * b)) * (cos(beltaN * y) - cosh(beltaN * y))
D1Xm = cos(x * alphaM) * alphaM - cosh(x * alphaM) * alphaM - (sin(a * alphaM) - sinh(a * alphaM))\ * (-sin(x * alphaM) - sinh(x * alphaM) * alphaM) / (cos(a * alphaM) - cosh(a * alphaM)) D2Xm = -sin(x * alphaM) * alphaM ** 2 - sinh(x * alphaM) * alphaM ** 2 - (sin(a * alphaM) - sinh(a * alphaM)) \ * (-cos(x * alphaM) * alphaM ** 2 - cosh(x * alphaM) * alphaM ** 2) / (cos(a * alphaM) - cosh(a * alphaM)) D3Xm = -cos(x * alphaM) * alphaM **3 - cosh(x * alphaM) * alphaM ** 3 - (sin(a * alphaM) - sinh(a * alphaM)) \ * (sin(x * alphaM) * alphaM ** 3 - sinh(x * alphaM) * alphaM ** 3) / (cos(a * alphaM) - cosh(a * alphaM)) D4Xm = sin(x * alphaM) * alphaM ** 4 - sinh(x * alphaM) * alphaM ** 4 - (sin(a * alphaM) - sinh(a * alphaM)) \ * (cos(x * alphaM) * alphaM ** 4 - cosh(x * alphaM) * alphaM ** 4) / (cos(a * alphaM) - cosh(a * alphaM))
D1Yn = cos(y * beltaN) * beltaN - cosh(y * beltaN) * beltaN - (sin(b * beltaN) - sinh(b * beltaN))\ * (-sin(y * beltaN) * beltaN - sinh(y * beltaN) * beltaN) / (cos(b * beltaN) - cosh(b * beltaN)) D2Yn = -sin(y * beltaN) * beltaN ** 2 - sinh(y * beltaN) * beltaN ** 2 - (sin(b * beltaN) - sinh(b * beltaN)) \ * (-cos(y * beltaN) * beltaN ** 2 - cosh(y * beltaN) * beltaN ** 2) / (cos(b * beltaN) - cosh(b * beltaN)) D3Yn = -cos(y * beltaN) * beltaN ** 3 - cosh(y * beltaN) * beltaN ** 3 - (sin(b * beltaN) - sinh(b * beltaN)) \ * (sin(y * beltaN) * beltaN ** 3 - sinh(y * beltaN) * beltaN ** 3) / (cos(b * beltaN) - cosh(b * beltaN)) D4Yn = sin(y * beltaN) * beltaN ** 4 - sinh(y * beltaN) * beltaN ** 4 - (sin(b * beltaN) - sinh(b * beltaN)) \ * (cos(y * beltaN) * beltaN ** 4 - cosh(y * beltaN) * beltaN ** 4) / (cos(b * beltaN) - cosh(b * beltaN))
if BC == 3: print('Boundary Conditions: CCSS') alphaM = (m + 0.5) * pi / a beltaN = n * pi / b Xm = sin(alphaM * x) - sinh(alphaM * a) - (sin(alphaM * a) - sinh(alphaM * a)) / (cos(alphaM * a) - cosh(alphaM * a))\ * (cos(alphaM * x) - cosh(alphaM * x)) Ym = sin(beltaN * y)
D1Xm = cos(x * alphaM) * alphaM - cosh(x * alphaM) * alphaM - (sin(a * alphaM) - sinh(a * alphaM))\ * ((-sin(x * alphaM) * alphaM - sinh(x * alphaM) * alphaM)) / (cos(a * alphaM) - cosh(a * alphaM)) D2Xm = -sin(x * alphaM) * alphaM ** 2 - sinh(x * alphaM) * alphaM ** 2 - (sin(a * alphaM) - sinh(a * alphaM))\ * (-cos(x * alphaM) * alphaM ** 2 - cosh(x * alphaM) * alphaM ** 2) / (cos(a * alphaM) - cosh(a * alphaM)) D3Xm = -cos(x * alphaM) * alphaM ** 3 - cosh(x * alphaM) * alphaM ** 3 - (sin(a * alphaM) - sinh(a * alphaM))\ * (sin(x * alphaM) * alphaM ** 3 - sinh(a * alphaM) * alphaM ** 3) / (cos(a * alphaM) - cosh(a * alphaM)) D4Xm = sin(x * alphaM) * alphaM ** 4 - sinh(x * alphaM) * alphaM ** 4 - (sin(a * alphaM) - sinh(a * alphaM))\ * (cos(x * alphaM) * alphaM ** 4 - cosh(x * alphaM) * alphaM ** 4) / (cos(a * alphaM) - cosh(a * alphaM))
D1Yn = cos(y * beltaN) * beltaN D2Yn = -sin(y * beltaN) * beltaN ** 2 D3Yn = -cos(y * beltaN) * beltaN ** 3 D4Yn = sin(y * beltaN) * beltaN ** 4
PT = 1
for i in range(0, 1, 2): kar = 1 if PT == 2: kar = 5 / 6
R1 = R2 = R3 = 0 Rz1 = Rz2 = Rz3 = 0 Rp1 = Rp2 = Rp3 = 0 Rn1 = Rn2 = Rn3 = 0 Rz1p = Rz2p = Rz3p = 0 Rz1n = Rz2n = Rz3n = 0
print('Plate Theory: Kirchoff Plate')
varepsilon_xx = (R1 - z) * w_xx + R1 * phi_xx + R2 * theta_xx + R3 * lambda_xx varepsilon_yy = (R1 - z) * w_yy + R1 * phi_yy + R2 * theta_yy + R3 * lambda_yy gamma_xy = 2 * (R1 - z) * w_xy + R1 * (phi_xy + phi_yx) + R2 * (theta_xy + theta_yx) + R3 * (lambda_xy + lambda_yx) gamma_xz = Rz1 * (w_x + phi_x) + Rz2 * theta_x + Rz3 * lambda_x gamma_yz = Rz1 * (w_y + phi_y) + Rz2 * theta_y + Rz3 * lambda_y
sigma_xx = E / (1 - v ** 2) * (varepsilon_xx + v * varepsilon_yy) sigma_yy = E / (1 - v ** 2) * (varepsilon_yy + v * varepsilon_xx) sigma_xy = G * gamma_xy sigma_xz = G * gamma_xz sigma_yz = G * gamma_yz
varepsilon_xxsp = (Rp1 - h / 2) * w_xx + Rp1 * phi_xx + Rp2 * theta_xx + Rp3 * lambda_xx varepsilon_yysp = (Rp1 - h / 2) * w_yy + Rp1 * phi_yy + Rp2 * theta_yy + Rp3 * lambda_yy gamma_xysp = 2 * (Rp1 - h / 2) * w_xy + Rp1 * (phi_xy + phi_yx) + Rp2 * (theta_xy + theta_yx) + Rp3 * (lambda_xy + lambda_yx) gamma_xzsp = Rz1p * (w_x + phi_x) + Rz2p * theta_x + Rz3p * lambda_x gamma_yzsp = Rz1p * (w_y + phi_y) + Rz2p * theta_y + Rz3p * lambda_y varepsilon_xxsp = 1 / 2 * gamma_xysp varepsilon_xzsp = 1 / 2 * gamma_xzsp varepsilon_yzsp = 1 / 2 * gamma_xzsp
varepsilon_xxsn = (Rn1 + h / 2) * w_xx + Rn1 * phi_xx + Rn2 * theta_xx + Rn3 * lambda_xx varepsilon_yysn = (Rn1 + h / 2) * w_yy + Rn1 * phi_yy + Rn2 * theta_yy + Rn3 * lambda_yy gamma_xysn = 2 * (Rn1 + h / 2) * w_xy + Rn1 * (phi_xy + phi_yx) + Rn2 * (theta_xy + theta_yx) + Rn3 * (lambda_xy + lambda_yx) gamma_xzsn = Rz1n * (w_x + phi_x) + Rz2n * theta_x + Rz3n * lambda_x gamma_yzsn = Rz1n * (w_y + phi_y) + Rz2n * theta_y + Rz3n * lambda_y varepsilon_xysn = 1 / 2 * gamma_xysn varepsilon_xzsn = 1 / 2 * gamma_xzsn varepsilon_yzsn = 1 / 2 * gamma_yzsn
Mxx = integrate(sigma_xx * (R1 - z), (z, -h / 2, h / 2)) Myy = integrate(sigma_yy * (R1 - z), (z, -h / 2, h / 2)) Mxy = integrate(sigma_xy * (R1 - z), (z, -h / 2, h / 2))
Pxx1 = integrate(sigma_xx * R1, (z, -h / 2, h / 2)) Pxx2 = integrate(sigma_xx * R2, (z, -h / 2, h / 2)) Pxx3 = integrate(sigma_xx * R3, (z, -h / 2, h / 2))
Pyy1 = integrate(sigma_yy * R1, (z, -h / 2, h / 2)) Pyy2 = integrate(sigma_yy * R2, (z, -h / 2, h / 2)) Pyy3 = integrate(sigma_yy * R3, (z, -h / 2, h / 2))
Pxy1 = integrate(sigma_xy * R1, (z, -h / 2, h / 2)) Pxy2 = integrate(sigma_xy * R2, (z, -h / 2, h / 2)) Pxy3 = integrate(sigma_xy * R3, (z, -h / 2, h / 2))
Qx1 = integrate(kar * sigma_xz * Rz1, (z, -h / 2, h / 2)) Qx2 = integrate(kar * sigma_xz * Rz2, (z, -h / 2, h / 2)) Qx3 = integrate(kar * sigma_xz * Rz3, (z, -h / 2, h / 2))
Qy1 = integrate(kar * sigma_yz * Rz1, (z, -h / 2, h / 2)) Qy2 = integrate(kar * sigma_yz * Rz2, (z, -h / 2, h / 2)) Qy3 = integrate(kar * sigma_yz * Rz3, (z, -h / 2, h / 2))
Axx = Mxx.coeff(w_xx) Bxx = Mxx.coeff(w_yy) C1xx = Mxx.coeff(phi_xx) C2xx = Mxx.coeff(theta_xx) C3xx = Mxx.coeff(lambda_xx) D1xx = Mxx.coeff(phi_yy) D2xx = Mxx.coeff(theta_yy) D3xx = Mxx.coeff(lambda_yy) E1xx = Mxy.coeff(w_xy) F1xx = Mxy.coeff(phi_xy) F2xx = Mxy.coeff(theta_xy) F3xx = Mxy.coeff(lambda_xy)
G1xx = Pxx1.coeff(w_xx) H1xx = Pxx1.coeff(w_yy) I11xx = Pxx1.coeff(phi_xx) I12xx = Pxx1.coeff(theta_xx) I13xx = Pxx1.coeff(lambda_xx)
J11xx = Pxx1.coeff(phi_yy) J12xx = Pxx1.coeff(theta_yy) J13xx = Pxx1.coeff(lambda_yy)
G2xx = Pxx2.coeff(w_xx) H2xx = Pxx2.coeff(w_yy) I21xx = Pxx2.coeff(phi_xx) I22xx = Pxx2.coeff(theta_xx) I23xx = Pxx2.coeff(lambda_xx)
J21xx = Pxx2.coeff(phi_yy) J22xx = Pxx2.coeff(theta_yy) J23xx = Pxx2.coeff(lambda_yy)
G3xx = Pxx3.coeff(w_xx) H3xx = Pxx3.coeff(w_yy) I31xx = Pxx3.coeff(phi_xx) I32xx = Pxx3.coeff(theta_xx) I33xx = Pxx3.coeff(lambda_xx)
J31xx = Pxx3.coeff(phi_yy) J32xx = Pxx3.coeff(theta_yy) J33xx = Pxx3.coeff(lambda_yy)
K1xx = Pxy1.coeff(w_xy) L11xx = Pxy1.coeff(phi_xy) L12xx = Pxy1.coeff(theta_xy) L13xx = Pxy1.coeff(lambda_xy)
K2xx = Pxy2.coeff(w_xy) L21xx = Pxy2.coeff(phi_xy) L22xx = Pxy2.coeff(theta_xy) L23xx = Pxy2.coeff(lambda_xy)
K3xx = Pxy3.coeff(w_xy) L31xx = Pxy3.coeff(phi_xy) L32xx = Pxy3.coeff(theta_xy) L33xx = Pxy3.coeff(lambda_xy)
S1xx = Qx1.coeff(w_x) S2xx = Qx2.coeff(w_x) S3xx = Qx3.coeff(w_x)
T11xx = Qx1.coeff(phi_x) T12xx = Qx1.coeff(theta_x) T13xx = Qx1.coeff(lambda_x)
T21xx = Qx2.coeff(phi_x) T22xx = Qx2.coeff(theta_x) T23xx = Qx2.coeff(lambda_x)
T31xx = Qx3.coeff(phi_x) T32xx = Qx3.coeff(theta_x) T33xx = Qx3.coeff(lambda_x)
A11 = integrate2((I11xx * D3Xm * Yn + L11xx * D1Xm * D2Yn - T11xx * D1Xm * Yn) * D1Xm * Yn, (x, 0, a), (y, 0, b)) A12 = integrate2((I12xx * D3Xm * Yn + L12xx * D1Xm * D2Yn - T12xx * D1Xm * Yn) * D1Xm * Yn, (x, 0, a), (y, 0, b)) A13 = integrate2((I13xx * D3Xm * Yn + L13xx * D1Xm * D2Yn - T13xx * D1Xm * Yn) * D1Xm * Yn, (x, 0, a), (y, 0, b)) A14 = integrate2(((J11xx + L11xx) * D1Xm * D2Yn) * D1Xm * Yn, (x, 0, a), (y, 0, b)) A15 = integrate2(((J12xx + L12xx) * D1Xm * D2Yn) * D1Xm * Yn, (x, 0, a), (y, 0, b)) A16 = integrate2(((J13xx + L13xx) * D1Xm * D2Yn) * D1Xm * Yn, (x, 0, a), (y, 0, b)) A17 = integrate2((G1xx * D3Xm * Yn + (H1xx + K1xx) * D1Xm * D2Yn - S1xx * D1Xm * Yn) * D1Xm * Yn, (x, 0, a), (y, 0, b))
A21 = integrate2((I21xx * D3Xm * Yn + L21xx * D1Xm * D2Yn - T21xx * D1Xm * Yn) * D1Xm * Yn, (x, 0, a), (y, 0, b)) A22 = integrate2((I22xx * D3Xm * Yn + L22xx * D1Xm * D2Yn - T22xx * D1Xm * Yn) * D1Xm * Yn, (x, 0, a), (y, 0, b)) A23 = integrate2((I23xx * D3Xm * Yn + L23xx * D1Xm * D2Yn - T23xx * D1Xm * Yn) * D1Xm * Yn, (x, 0, a), (y, 0, b)) A24 = integrate2(((J21xx + L21xx) * D1Xm * D2Yn) * D1Xm * Yn, (x, 0, a), (y, 0, b)) A25 = integrate2(((J22xx + L22xx) * D1Xm * D2Yn) * D1Xm * Yn, (x, 0, a), (y, 0, b)) A26 = integrate2(((J23xx + L23xx) * D1Xm * D2Yn) * D1Xm * Yn, (x, 0, a), (y, 0, b)) A27 = integrate2((G2xx * D3Xm * Yn + (H2xx + K2xx) * D1Xm * D2Yn - S2xx * D1Xm * Yn) * D1Xm * Yn, (x, 0, a), (y, 0, b))
A31 = integrate2((I31xx * D3Xm * Yn + L31xx * D1Xm * D2Yn - T31xx * D1Xm * Yn) * D1Xm * Yn, (x, 0, a), (y, 0, b)) A32 = integrate2((I32xx * D3Xm * Yn + L32xx * D1Xm * D2Yn - T32xx * D1Xm * Yn) * D1Xm * Yn, (x, 0, a), (y, 0, b)) A33 = integrate2((I33xx * D3Xm * Yn + L33xx * D1Xm * D2Yn - T33xx * D1Xm * Yn) * D1Xm * Yn, (x, 0, a), (y, 0, b)) A34 = integrate2(((J31xx + L31xx) * D1Xm * D2Yn) * D1Xm * Yn, (x, 0, a), (y, 0, b)) A35 = integrate2(((J32xx + L32xx) * D1Xm * D2Yn) * D1Xm * Yn, (x, 0, a), (y, 0, b)) A36 = integrate2(((J33xx + L33xx) * D1Xm * D2Yn) * D1Xm * Yn, (x, 0, a), (y, 0, b)) A37 = integrate2((G3xx * D3Xm * Yn + (H3xx + K3xx) * D1Xm * D2Yn - S3xx * D1Xm * Yn) * D1Xm * Yn, (x, 0, a), (y, 0, b))
A41 = integrate2(((J11xx + L11xx) * D2Xm * D1Yn) * Xm * D1Yn, (x, 0, a), (y, 0, b)) A42 = integrate2(((J12xx + L12xx) * D2Xm * D1Yn) * Xm * D1Yn, (x, 0, a), (y, 0, b)) A43 = integrate2(((J13xx + L13xx) * D2Xm * D1Yn) * Xm * D1Yn, (x, 0, a), (y, 0, b)) A44 = integrate2((I11xx * Xm * D3Yn + L11xx * D2Xm * D1Yn - T11xx * Xm * D1Yn) * Xm * D1Yn, (x, 0, a), (y, 0, b)) A45 = integrate2((I12xx * Xm * D3Yn + L12xx * D2Xm * D1Yn - T12xx * Xm * D1Yn) * Xm * D1Yn, (x, 0, a), (y, 0, b)) A46 = integrate2((I13xx * Xm * D3Yn + L13xx * D2Xm * D1Yn - T13xx * Xm * D1Yn) * Xm * D1Yn, (x, 0, a), (y, 0, b)) A47 = integrate2((G1xx * Xm * D3Yn + (H1xx + K1xx) * D2Xm * D1Yn - S1xx * Xm * D1Yn) * Xm * D1Yn, (x, 0, a), (y, 0, b))
A51 = integrate2(((J21xx + L21xx) * D2Xm * D1Yn) * Xm * D1Yn, (x, 0, a), (y, 0, b)) A52 = integrate2(((J22xx + L22xx) * D2Xm * D1Yn) * Xm * D1Yn, (x, 0, a), (y, 0, b)) A53 = integrate2(((J23xx + L23xx) * D2Xm * D1Yn) * Xm * D1Yn, (x, 0, a), (y, 0, b)) A54 = integrate2((I21xx * Xm * D3Yn + L21xx * D2Xm * D1Yn - T21xx * Xm * D1Yn) * Xm * D1Yn, (x, 0, a), (y, 0, b)) A55 = integrate2((I22xx * Xm * D3Yn + L22xx * D2Xm * D1Yn - T22xx * Xm * D1Yn) * Xm * D1Yn, (x, 0, a), (y, 0, b)) A56 = integrate2((I23xx * Xm * D3Yn + L23xx * D2Xm * D1Yn - T23xx * Xm * D1Yn) * Xm * D1Yn, (x, 0, a), (y, 0, b)) A57 = integrate2((G2xx * Xm * D3Yn + (H2xx + K2xx) * D2Xm * D1Yn - S2xx * Xm * D1Yn) * Xm * D1Yn, (x, 0, a), (y, 0, b))
A61 = integrate2(((J31xx + L31xx) * D2Xm * D1Yn) * Xm * D1Yn, (x, 0, a), (y, 0, b)) A62 = integrate2(((J32xx + L32xx) * D2Xm * D1Yn) * Xm * D1Yn, (x, 0, a), (y, 0, b)) A63 = integrate2(((J33xx + L33xx) * D2Xm * D1Yn) * Xm * D1Yn, (x, 0, a), (y, 0, b)) A64 = integrate2((I31xx * Xm * D3Yn + L31xx * D2Xm * D1Yn - T31xx * Xm * D1Yn) * Xm * D1Yn, (x, 0, a), (y, 0, b)) A65 = integrate2((I32xx * Xm * D3Yn + L32xx * D2Xm * D1Yn - T32xx * Xm * D1Yn) * Xm * D1Yn, (x, 0, a), (y, 0, b)) A66 = integrate2((I33xx * Xm * D3Yn + L33xx * D2Xm * D1Yn - T33xx * Xm * D1Yn) * Xm * D1Yn, (x, 0, a), (y, 0, b)) A67 = integrate2((G3xx * Xm * D3Yn + (H3xx + K3xx) * D2Xm * D1Yn - S3xx * Xm * D1Yn) * Xm * D1Yn, (x, 0, a), (y, 0, b))
A71 = integrate2((C1xx * D4Xm * Yn + (D1xx + 2 * F1xx) * D2Xm * D2Yn - T11xx * D2Xm * Yn) * Xm * Yn, (x, 0, a), (y, 0, b)) A72 = integrate2((C2xx * D4Xm * Yn + (D2xx + 2 * F2xx) * D2Xm * D2Yn - T12xx * D2Xm * Yn) * Xm * Yn, (x, 0, a), (y, 0, b)) A73 = integrate2((C3xx * D4Xm * Yn + (D3xx + 2 * F3xx) * D2Xm * D2Yn - T13xx * D2Xm * Yn) * Xm * Yn, (x, 0, a), (y, 0, b)) A74 = integrate2((C1xx * Xm * D4Yn + (D1xx + 2 * F1xx) * D2Xm * D2Yn - T11xx * Xm * D2Yn) * Xm * Yn, (x, 0, a), (y, 0, b)) A75 = integrate2((C2xx * Xm * D4Yn + (D2xx + 2 * F2xx) * D2Xm * D2Yn - T12xx * Xm * D2Yn) * Xm * Yn, (x, 0, a), (y, 0, b)) A76 = integrate2((C3xx * Xm * D4Yn + (D3xx + 2 * F3xx) * D2Xm * D2Yn - T13xx * Xm * D2Yn) * Xm * Yn, (x, 0, a), (y, 0, b)) A77 = integrate2((Axx * (D4Xm * Yn + Xm * D4Yn) + 2 * (Bxx + E1xx) * D2Xm * D2Yn - S1xx * (Xm * D2Yn + D2Xm * Yn)) * Xm * Yn, (x, 0, a), (y, 0, b))
e1 = integrate2(D2Xm * Yn * Xm * Yn, (x, 0, a), (y, 0, b)) e2 = integrate2(Xm * D2Yn * Xm * Yn, (x, 0, a), (y, 0, b)) e3 = integrate2(D4Xm * Yn * Xm * Yn, (x, 0, a), (y, 0, b)) e4 = integrate2(D2Xm * D2Yn * Xm * Yn, (x, 0, a), (y, 0, b)) e5 = integrate2(Xm * D4Yn * Xm * Yn, (x, 0, a), (y, 0, b))
AP77 = (e1 + e2) * NCr
AK = np.array([[A77 + AP77]]) AK = Matrix(AK) kk = solve(AK.det(), k) print('Bulkling intensity factor for Kirchoff plate is k=', kk) kv = kk[0]
print('The critical pressure is: Ncr=', (pi ** 2 * Diff / (a ** 2) * kv).evalf(6))
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