The New Approach to Analysis of Thin Isotropic Symmetrical Plates
Abstract
:1. Introduction
- Simplicity of structure modeling. Equations of equilibrium are solved once independently of the boundary conditions and solution remains unchanged throughout the calculation process. This allows the reduction of structure modeling to set kinematic and static boundary conditions in the separate points at the plate contour.
- Possibility to provide better accuracy in comparison with finite element method. It is achieved only by increasing of number of the approximations (parameter K) or replace nodes at the contour. This procedure is performed automatically with author’s program to solve of the plate structure.
- Possibility to define the load as a function of two variables. Because not all FEA programs provide such functionality, ABAQUS software have been chosen to compare the results.
- Possibility of implementing of suggested method in higher-order theory, say Mindlin-Reissner theory.
- Discretize of the structure into set of finite elements,
- Aggregate of this set into discrete structure,
- Introduce local and global coordinate system,
- Numerate of nodes. Solution of equilibrium equation can be obtained with given accuracy and it is independent of the number of approximations.
2. Literature Review of the Methods for Solving Plate Bending Problems
3. Materials and Methods
4. Results
- 1
- Rectangular plate with two opposite edges simply supported and the other free,
- 2
- Rectangular plate simply supported at the contour,
- 3
- Plate clamped at the contour.
0.0 | 0.8 | 1.6 | 2.4 | 3.2 | 4.0 | |
0.0 | 0.4 | 0.8 | 1.2 | 1.6 | 2.0 |
4.1. Rectangular Plate with Two Opposite Edges Simply Supported and the Other Free
4.2. Rectangular Plate Simply Supported at the Contour
4.3. Rectangular Plate Clamped at the Contour
5. Discussion
- Simplicity of structure modeling,
- Possibility to define static, kinematic and mixed boundary conditions,
- High accuracy and efficiency of calculation,
- Meshless approach for solving the problem.
- It cannot be used directly to solve fracture mechanics problems,
- It is worked for simply connected structures; for each additional contour, it is necessary to introduce the new block of basic functions,
- It gives good results for regular plates; plates with irregular geometry are solved with less accuracy,
- Unlike FEM, system matrix is consisted from zero and non-zero blocks.
- Presented method is based on Kirchhoff theory of the plates which is deformation theory of zeroth-order where deformations of a cross-section are omitted. Instead generalized shearing force as a sum of shearing forces and derivative of twisting moments is introduced.
- ABAQUS package is based on Mindlin plate theory which is deformation theory of first-order where deformations of a cross-section are taken into account. Twisting moments and shearing forces are independent ones. It is reason that obtained values of bending and twisting moments are different.
- Relatively small number of finite elements. The Student Edition of the ABAQUS package is restricted to 1000 nodes.
6. Conclusions
- to obtain with high accuracy particular solution of equilibrium equations at the separate surface nodes using boundary collocation method,
- to generate set of the initial points at the coordinate axis and then their distribution at the plate contour,
- to write boundary conditions at each edge nodes; number of these nodes always corresponds to number of unknown parameter of the model,
- to solve system of boundary equations,
- to calculate the desired magnitudes,
- to prepare the results.
- The mathematical model of thin isotropic plates is constructed. Basic shape and force functions are introduced.
- Within the model equilibrium equations are performed exactly.
- Expansion of the deflection within the area of the plate is obtained as sum of shape and force functions multiplied by unknown parameters.
- Displacements, slopes, moments and shearing forces are obtained using method of automatic differentiation. Union of presented relations create mathematical model of the plate.
- Within constructed model, a method to solving plate of arbitrary geometry have been suggested.
- All operations are performed automatically with author’s program.
- Effectiveness of method were illustrated at the examples of rectangular plates with various boundary conditions. Obtained results have been compared with numerical ones computed in ABAQUS package.
- With help of suggested method three kinds of the rectangular plates, namely: plate with two opposite edges simply supported and the other free; plate simply supported and clamped at the contour were solved. It has been shown that suggested method performs all boundary conditions. Finite element method performs exactly only kinematic ones. Static boundary conditions are performed with less accuracy.
Author Contributions
Funding
Conflicts of Interest
Abbreviations
, , | Rectangular coordinates |
h | Thickness of a plate |
Intensity of a continuously distributed load | |
E | Modulus of elasticity in tension and compression |
Poisson’s ratio | |
D | Flexural rigidity of a plate |
w | Deflection of the plate |
, | Slopes of the deflection |
, | Bending moments per unit length of sections of a plate perpendicular to and axes, respectively |
Twisting moment per unit length of sections of a plate perpendicular to axis | |
, | Shearing forces parallel to axis per unit length of sections of a plate perpendicular to and axes, respectively |
, | Generalized shearing forces parallel to axis per unit length of sections of a plate perpendicular to and axes |
Appendix A. Results from ABAQUS
- 1
- Rectangular plate with two opposite edges simply supported and the other free,
- 2
- Rectangular plate simply supported at the contour,
- 3
- Rectangular plate clamped at the contour.
Appendix A.1. Example 1
Appendix A.2. Example 2
Appendix A.3. Example 3
Appendix B. Listing from Author’S Program
import autograd.numpy as np from autograd import elementwise_grad as grad
import matplotlib.pyplot as plt from matplotlib import cm from matplotlib.ticker import LinearLocator from mpl_toolkits.mplot3d import Axes3D
# ==========
# Parameters
# ==========
# Number of approximations of the solution
_K = 5
# Dimensions a_1 = 4. # m a_2 = 2. # m h = 0.2 # m
# Young’s modulus E = 3e10 # Pa # Poisson ratio nu = 0.2
# Intensity of the load q_0 = 10_000.00 # N
# Flexural rigidity of a plate
D = (E * h**3) / (12*(1-nu**2))
# ================
# Helper functions
# ================
def a(s): if s == 1: return a_1 elif s == 2: return a_2
def x(s): def wrapper(x_1, x_2): if s == 1: return x_1 elif s == 2: return x_2 return wrapper
def gamma(k, s): return (k*np.pi) / a(s)
def delta(k, s): return ((2*k-1)*np.pi) / (2*a(s))
def kappa(k, p, s): if p == 1: return gamma(k, s) elif p == 2: return delta(k, s)
def T(k, p, s): def wrapper(x_1, x_2): return np.cos(kappa(k, p, s) * x(s)(x_1, x_2)) return wrapper
def calculate(fs, solution, x_1, x_2): coeffs = np.array([f(x_1, x_2) for f in fs]) return np.einsum(’ijk,i->jk’, coeffs, solution)
def particular_solution(C, f): def wrapper(x_1, x_2): return C * f(x_1, x_2) return wrapper
def final(general_solution, particular_solution): result = [] for f in general_solution: result.append(f) result.append(particular_solution) return result
# ====================================================
# Relations known from theory of thin isotropic plates
# ====================================================
def phi_1(w): def wrapper(x_1, x_2): return grad(w, 0)(x_1, x_2) return wrapper
def phi_2(w): def wrapper(x_1, x_2): return grad(w, 1)(x_1, x_2) return wrapper
def M_11(w): def wrapper(x_1, x_2): return -D * (grad(grad(w, 0), 0)(x_1, x_2) + nu*grad(grad(w, 1), 1)(x_1, x_2)) return wrapper
def M_22(w): def wrapper(x_1, x_2): return -D * (grad(grad(w, 1), 1)(x_1, x_2) + nu*grad(grad(w, 0), 0)(x_1, x_2)) return wrapper
def M_12(w): def wrapper(x_1, x_2): return -D * (1-nu) * grad(grad(w, 0), 1)(x_1, x_2) return wrapper
def Q_1(w): def wrapper(x_1, x_2): return -D * (grad(grad(grad(w, 0), 0), 0)(x_1, x_2) + grad(grad(grad(w, 0), 1), 1)(x_1, x_2)) return wrapper
def Q_2(w): def wrapper(x_1, x_2): return -D * (grad(grad(grad(w, 0), 0), 1)(x_1, x_2) + grad(grad(grad(w, 1), 1), 1)(x_1, x_2)) return wrapper
def V_1(w): def wrapper(x_1, x_2): return -D * (grad(grad(grad(w, 0), 0), 0)(x_1, x_2) + (2-nu)*grad(grad(grad(w, 0), 1), 1)(x_1, x_2)) return wrapper
def V_2(w): def wrapper(x_1, x_2): return -D * (grad(grad(grad(w, 1), 1), 1)(x_1, x_2) + (2-nu)*grad(grad(grad(w, 0), 0), 1)(x_1, x_2)) return wrapper
# ===============
# Force functions
# ===============
def force_functions(m, n, p, q): def wrapper(x_1, x_2): return T(m, p, 1)(x_1, x_2) * T(n, q, 2)(x_1, x_2) return wrapper
# ===============
# Shape functions
# ===============
# Base functions of the solution
def B(k, p, s, ni): def wrapper(x_1, x_2): if ni == 1: return (np.cosh(kappa(k, p, 3-s) * x(s)(x_1, x_2))) elif ni == 2: return ((x(s)(x_1, x_2) / a(s)) * (np.sinh(kappa(k, p, 3-s) * x(s)(x_1, x_2)))) return wrapper
# Shape functions
def W(k, p, s, ni): def wrapper(x_1, x_2): return B(k, p, s, ni)(x_1, x_2) * T(k, p, 3-s)(x_1, x_2) return wrapper
def shape_functions(K): result = [] for k in range(1, K+1): for p in range(1, 3): for s in range(1, 3): for ni in range(1, 3): result.append(W(k, p, s, ni)) return result
# ===========
# Build model
# ===========
W_g = shape_functions(_K) W_p = force_functions(1, 1, 2, 2)
# ================
# General solution
# ================
# Calculate derivatives of shape function
U_g = [phi_1(f) for f in W_g]
V_g = [phi_2(f) for f in W_g]
X_g = [M_11(f) for f in W_g]
Y_g = [M_22(f) for f in W_g]
Z_g = [M_12(f) for f in W_g]
H_g = [Q_1(f) for f in W_g]
G_g = [Q_2(f) for f in W_g]
K_g = [V_1(f) for f in W_g]
L_g = [V_2(f) for f in W_g]
# ===================
# Particular solution
# ===================
# Calculate derivatives of force function
U_p = phi_1(W_p)
V_p = phi_2(W_p)
X_p = M_11(W_p)
Y_p = M_22(W_p)
Z_p = M_12(W_p)
H_p = Q_1(W_p)
G_p = Q_2(W_p)
K_p = V_1(W_p)
L_p = V_2(W_p)
C = q_0 / (D*(delta(1, 1)**2 + delta(1, 2)**2)**2)
W_s = particular_solution(C, W_p) U_s = particular_solution(C, U_p) V_s = particular_solution(C, V_p) X_s = particular_solution(C, X_p) Y_s = particular_solution(C, Y_p) Z_s = particular_solution(C, Z_p) H_s = particular_solution(C, H_p) G_s = particular_solution(C, G_p) K_s = particular_solution(C, K_p) L_s = particular_solution(C, L_p)
# ==============
# Final solution
# ==============
W = final(W_g, W_s) U = final(U_g, U_s) V = final(V_g, V_s) X = final(X_g, X_s) Y = final(Y_g, Y_s) Z = final(Z_g, Z_s) H = final(H_g, H_s) G = final(G_g, G_s) K = final(K_g, K_s) L = final(L_g, L_s)
# ===================
# Boundary conditions
# ===================
# Initial points
p_1 = np.linspace(0, a_1, 2*_K)
p_2 = np.linspace(0, a_2, 2*_K)
# Boundary points
b_11 = np.full_like(p_2, a_1)
b_12 = p_2
b_21 = p_1
b_22 = np.full_like(p_1, a_2)
# Shape of blocks
m = len(W)
n = 2*_K
# Initialize matrix blocks
B_1 = np.zeros((m, n))
B_2 = np.zeros((m, n))
B_3 = np.zeros((m, n))
B_4 = np.zeros((m, n))
# Fill matrix blocks with values for boundary conditions
for i in range(m):
B_1[i] = W[i](b_11, b_12)
B_2[i] = U[i](b_11, b_12)
B_3[i] = W[i](b_21, b_22)
B_4[i] = V[i](b_21, b_22)
# ============
# Solve system
# ============
# Augmented matrix M = np.hstack((B_1, B_2, B_3, B_4)) M = M.T # Coefficient matrix A = M[:, :-1] # Column vector of constant terms b = M[:, -1] # Solve a linear matrix equation R = np.linalg.solve(A, -b) R = np.append(R, 1)
# ==========================
# Calculate and plot results
# ==========================
# Prepare points for calculating the values in them
x_1 = np.linspace(-a_1, a_1, num=51)
x_2 = np.linspace(-a_2, a_2, num=51)
X_1, X_2 = np.meshgrid(x_1, x_2)
for f in [W, U, V, X, Y, Z, H, G, K, L]: X_3 = calculate(f, R, X_1, X_2) fig = plt.figure() ax = fig.gca(projection=’3d’) ax.set_xlabel(’$x_1$’) ax.set_ylabel(’$x_2$’) ax.set_zlabel(’$x_3$’) surf = ax.plot_surface(X_1, X_2, X_3, cmap=cm.coolwarm, linewidth=0, antialiased=True) ax.zaxis.set_major_locator(LinearLocator(10)) ax.set_zlim(ax.get_zlim()[::-1]) boundaries = np.linspace(np.min(X_3), np.max(X_3), 25) fig.colorbar(surf, shrink=0.75, aspect=5, boundaries=boundaries) plt.show() --------------------------------------------------------------------------------
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Present Method | Abaqus | RE | |||||
---|---|---|---|---|---|---|---|
Magnitude | Unit | Min | Max | Min | Max | Min | Max |
w | 0 | 0.013 | 0 | 0.013 | 0% | 0% | |
0 | 41,490.63 | 1944 | 41,310 | — | 0.44% | ||
0 | 5516.33 | −78.02 | 5311 | — | 3.87% | ||
−3263.79 | 3263.79 | −2402 | 2402 | 35.88% | 35.88% | ||
−15,343.36 | 15,343.36 | — | — | — | — | ||
−3229.42 | 3229.42 | — | — | — | — | ||
−14,440.95 | 14,440.95 | — | — | — | — | ||
−2677.45 | 2677.45 | — | — | — | — | ||
−0.005 | 0.005 | −0.005 | 0.005 | 0% | 0% | ||
−0.0005 | 0.0005 | −0.0005 | 0.0005 | 0% | 0% |
Magnitude | Edge | Unit | Present Method | Abaqus |
---|---|---|---|---|
w | Simply supported | 0 | 0 | |
Simply supported | 0 | 1966.02 | ||
Free | 0 | 288.97 | ||
Free | 0 | — |
Present Method | Timoshenko | Abaqus | RE | ||||||
---|---|---|---|---|---|---|---|---|---|
Magnitude | Unit | Min | Max | Min | Max | Min | Max | Min | Max |
w | 0.00 | 0.0008 | 0.00 | 0.0008 | 0.00 | 0.0008 | 0% | 0% | |
0.00 | 4668.88 | 0.00 | 4668.88 | −252.4 | 4655 | — | 0.30% | ||
0.00 | 10,894.05 | 0.00 | 10,894.05 | −221.6 | 10,930 | — | −0.33% | ||
−4150.12 | 4150.12 | −4150.12 | 4150.12 | −4041 | 4041 | — | 2.70% | ||
−5092.96 | 5092.96 | −5092.96 | 5092.96 | — | — | — | — | ||
−10,185.92 | 10,185.92 | −10,185.92 | 10,185.92 | — | — | — | — | ||
−8352.45 | 8352.45 | −8352.45 | 8352.45 | — | — | — | — | ||
−11,815.66 | 11,815.66 | −11,815.66 | 11,815.66 | — | — | — | — | ||
−0.0003 | 0.0003 | −0.0003 | 0.0003 | −0.0003 | 0.0003 | 0% | 0% | ||
−0.0006 | 0.0006 | −0.0006 | 0.0006 | −0.0006 | 0.0006 | 0% | 0% |
Magnitude | Edge | Unit | Present Method | Timoshenko | Abaqus |
---|---|---|---|---|---|
w | Simply supported 1 | 0 | 0 | 0 | |
Simply supported 1 | 0 | 0 | 416.46 | ||
w | Simply supported 2 | 0 | 0 | 0 | |
Simply supported 2 | 0 | 0 | 1170.5 |
Present Method | Abaqus | RE | |||||
---|---|---|---|---|---|---|---|
Magnitude | Unit | Min | Max | Min | Max | Min | Max |
w | 0.00 | 0.0002 | 0.00 | 0.0002 | 0% | 0% | |
−4174.73 | 1909.82 | −3326 | 1877 | 25.52% | 1.75% | ||
−9305.64 | 5180.73 | −7727 | 5080 | 20.43% | 1.98% | ||
−1190.87 | 1190.87 | −1120 | 1120 | 6.33% | 6.33% | ||
−6806.21 | 6806.21 | — | — | — | — | ||
— | — | — | — | ||||
−7268.18 | 7268.18 | — | — | — | — | ||
— | — | — | — | ||||
0.00 | 0.00 | 0.00 | 0.00 | 0% | 0% | ||
−0.0002 | 0.0002 | −0.0002 | 0.0002 | 0% | 0% |
Magnitude | Edge | Unit | Present Method | Abaqus |
---|---|---|---|---|
w | Clamped 1 | 0 | 0 | |
Clamped 1 | 0 | 0 | ||
w | Clamped 2 | 0 | 0 | |
Clamped 2 | 0 | 0 |
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Delyavskyy, M.; Rosiński, K. The New Approach to Analysis of Thin Isotropic Symmetrical Plates. Appl. Sci. 2020, 10, 5931. https://doi.org/10.3390/app10175931
Delyavskyy M, Rosiński K. The New Approach to Analysis of Thin Isotropic Symmetrical Plates. Applied Sciences. 2020; 10(17):5931. https://doi.org/10.3390/app10175931
Chicago/Turabian StyleDelyavskyy, Mykhaylo, and Krystian Rosiński. 2020. "The New Approach to Analysis of Thin Isotropic Symmetrical Plates" Applied Sciences 10, no. 17: 5931. https://doi.org/10.3390/app10175931
APA StyleDelyavskyy, M., & Rosiński, K. (2020). The New Approach to Analysis of Thin Isotropic Symmetrical Plates. Applied Sciences, 10(17), 5931. https://doi.org/10.3390/app10175931