The analyzed rectangular plate is assumed clamped along two opposite edges. The flexibility method (or force method) of the linear beam theory was applied in this analysis whereby the unknowns were the bending moments along the opposite edges. In this regard the primary system was the plate simply supported along the above mentioned opposite edges and the redundant system was the plate subjected to bending moments along those edges. The compatibility equations (vanishing of the slopes along the opposite edges) were used to determine the unknowns.
2.2.1. Primary problem: plate simply supported along two opposite edges
The plate dimensions in x- and y-direction are denoted by a and b, respectively. The rectangular plate is assumed simply supported along the edges x = 0 and x = a. The solution by Lévy [
4] that satisfies the boundary conditions at these edges is considered for the deflection curve w(
x,y) as follows:
The Fourier sine series of the transverse applied load is given by
Substituting Equations (4) and (5) into (1) yields the differential equation
The solution to Equation (6) is as follows:
where the coefficients A
m, B
m, C
m, and D
m are determined by satisfying the boundary and continuity conditions in y-direction, and F
mp(y) is a particular solution to the differential equation.
Substituting Equation (4) into (2a-c) yields the equations of bending moments and twisting moments as follows:
With regard to the boundary conditions at y = 0 and y = b the equations for the slope ∂w/∂y and the bending moment myy are set using Equations (4), (7), and (8b) as follows
Furthermore, with regard to the boundary conditions at y = 0 and y = b the Kirchhoff shear force V
y is calculated by substituting Equations (4) and (7) into (3b) as follows:
In summary, the boundary conditions at y = 0 and y = b needed to determine the coefficients Am, Bm, Cm, and Dm.are expressed using the equations for the deflection (Equations (4) and (7)), the slope ∂w/∂y (Equation (9a)), the bending moment myy (Equation (9b)), and the Kirchhoff shear force Vyy (Equation (9c)).
The flexibilities δ
j0 (slopes at positions j of the opposite edges where the compatibility equations will be set) for the primary problem are calculated using Equations (4) and (7) as follows:
For the primary problem the bending moments myy are calculated using Equations (9b), and the bending moments mxx and twisting moments mxy are calculated using Equations (7) and (8a,c) as follows
2.2.2. Redundant problem: plate subjected to a distributed moment along the edge x = 0
The distribution of the bending moments X(y) along the clamped edge x = 0 depends on the support conditions along the edges y = 0 and y = b, namely supported (simply supported or clamped) or free.
Edges y = 0 and y = b simply supported or clamped Redundant problem Xi = 1
The edges y = 0 and y = b are assumed simply supported or clamped. Observing that the bending moments vanish at angles supported in two directions (in this case at y= 0 and y = b), the distribution of the bending moments X(y) along the clamped edge x = 0 can be described with the following trigonometric series
Here, the redundant effort is a distributed bending moment sin (iπy/b) along the edge x = 0 according to Equation (12).
To account for the edge moments the solution by Lévy [
4] for the transverse displacement can be modified as follows
The second term on the right-hand side of Equation (13) is the displacement function of a plate strip simply supported at its ends and subjected at the edge x = 0 to a distributed moment sin (iπy/b). It is noted that Equation (13) satisfies the boundary conditions at edges x = 0 and x = a. Substituting Equation (13) into (1) yields
The following functions contained in Equation (14) are expanded in Fourier sine series
Substituting Equations (15) into (14) yields
Given that Equation (16) holds for any value of x, it results the following differential equation
The solution to Equation (17) is identical to Equation (7) whereby the particular solution is given by
Combining Equations (7), (13) and (18) yields the transverse displacement function as follows
To satisfy the boundary conditions at y = 0 and y = b, the Fourier series of Equation (15) is used; Equation (19) becomes
With respect to the boundary conditions at y = 0 and y = b the equations for the slope ∂w/∂y and the bending moment myy are set using Equations (9a-b) and (20) as follows
In summary, the boundary conditions at y = 0 and y = b needed to determine the coefficients Ami, Bmi, Cmi, and Dmi.are expressed using Equations (20) and (21a-b)
The flexibilities δ
ij (slopes at relevant positions j (xj, yj) of the opposite edges where the compatibility equations will be set) for the redundant problem X
i = 1 are calculated using Equation (19); it yields
The positions j must be chosen such that they are regularly distributed along the clamped edge. Recalling that there should be as many redundants as compatibility equations, for an edge with n redundants considered the positions y
i = k×b/ (n + 1) with k = 1, 2, 3 …n as represented in
Figure 2a can be taken.
In case of system and loading symmetrical with respect to axis y = b/2 only odd values of i need be considered, the redundants being zero for even values; and in case of system symmetrical and loading anti symmetrical with respect to axis y = b/2 only even values of i need be considered, the redundants being zero for odd values. In both cases, the flexibilities and compatibility equations can be set in half of the structure in y–direction. Given the half edge having n redundants, the positions y
i = k×b/2n with k = 1, 2, 3 …n as represented in
Figure 2b can be taken.
For the redundant problem X
i = 1 the bending moments m
xx and m
yy , and the twisting moments m
xy are calculated using Equations (2a-c) and (19):
According to the flexibility method of the theory of elasticity, the compatibility equations are established so as to restore the geometric boundary conditions of the clamped edges (vanishing of slopes ∂w/∂x at selected positions of the edge). Assuming a plate with n redundant, the compatibility equations for the position y
i can be expressed as follows
δ
i0 and δ
ij being the flexibility coefficients in the primary problem and in the redundant problems, respectively. Equation (24) is established at any selected position and so the redundant efforts are determined. The efforts in the plate are then calculated as follows
S0 and Sj being the efforts in the primary problem and in the redundant problem, respectively.
Edge y = 0 simply supported or clamped, and edge y = b free Redundant problem Xi = 1
The edge y = 0 is simply supported or clamped and y = b is free. The analysis in this section is valid if the edge y = 0 is free and y = b is simply supported or clamped; the results must be appropriately inverted.
Observing that the bending moments X(y) vanish at y = 0 (angle supported in two directions), their distribution along the clamped edge x = 0 can be described with the following trigonometric series
i being an odd number . The redundant effort is a distributed bending moment sin (iπy/2b) along the edge x = 0 according to Equation (26).
To account for the edge moments the solution by Lévy [
4] for the transverse displacement can be modified as follows
The analysis continues similarly to
Section 2.2.2.1. The differential equation (Equation (17)) becomes
The solution to Equation (28) is identical to Equation (7) whereby the particular solution is given by
Therefore the transverse displacement function is as follows
To satisfy the boundary conditions at y = 0 and y = b, the Fourier series of Equation (15) is used; Equation (30) becomes
With respect to the boundary conditions at y = 0 and y = b the equations for the slope ∂w/∂y and the bending moment myy are set using Equations (8b) and (31) as follows
Furthermore, with regard to the boundary conditions at y = b the Kirchhoff shear force is calculated using Equations (31) and (9c). It is noted that the first and third derivatives of the term with sin (iπy/2b) contain cos (iπy/2b) ¨that vanishes at y = b. It results
In summary the boundary conditions at y = 0 and y = b are expressed using Equations (31), (32a-b), and (33); they permit to determine the coefficients Ami, Bmi, Cmi, and Dmi.
The flexibilities δ
ij (slopes at relevant positions j (xj, yj) of the opposite edges where the compatibility equations will be set) for the redundant problem X
i = 1 are calculated using Equation (30) as follows
Given n redundants considered, the positions y
i = k×b/n with k = 1, 2, 3 …n as represented in
Figure 3 can be taken.
For the redundant problem the bending moments m
xx and m
yy and the twisting moments m
xy are calculated using Equations (2a-c) and (30):
The compatibility equations and the efforts in the plate are calculated using Equations (24) and (25).
Edges y = 0 and y = b free Redundant problem Xi = 1
The edges y = 0 and y = b are free. Observing that the bending moments X(y) have non zero values at the angles, their distribution along the clamped edge x = 0 can be described with the following trigonometric series
The redundant effort is a distributed bending moment cos (iπy/b) along the edge x = 0 according to Equation (36).
To account for the edge moments the solution by Lévy [
4] for the transverse displacement can be modified as follows
The analysis continues similarly to
Section 2.2.2.1. The differential equation (Equation (17)) becomes
The solution to Equation (38) is identical to Equation (7) whereby the particular solution is given by
Therefore, the transverse displacement function is as follows
To satisfy the boundary conditions at y = 0 and y = b, the Fourier series of Equation (15) is used; Equation (40) becomes
With regard to the boundary conditions at y = 0 and y = b the equation for the bending moment myy is set using Equations (8b) and (41) as follows
Furthermore, with regard to the boundary conditions at y = 0 and y = b the Kirchhoff shear force is calculated using Equations (41) and (9c). It is noted that the first and third derivatives of the term with cos (iπy/b) contain sin (iπy/b) that vanishes at y = 0 and y = b. It results at y = 0 and y = b
In summary the boundary conditions at y = 0 and y = b are expressed using Equations (42) and (43); they permit to determine the coefficients Am, Bm, Cm, and Dm.
The flexibilities δ
ij (slopes at relevant positions j (xj, yj) of the opposite edges where the compatibility equations will be set) for the redundant problem X
i = 1 are calculated using Equation (40) as follows
Given n redundants considered, the positions y
i = k×b/(n – 1) with k = 0, 1, 2, 3 …n - 1 as represented in
Figure 4a can be taken.
In case of loading symmetrical with respect to axis y = b/2 only even values of i need be considered, the redundants being zero for odd values; and in case of loading anti symmetrical with respect to axis y = b/2 only odd values of i need be considered, the redundants being zero for even values. In both cases, the flexibilities and compatibility equations can be set in half of the structure in y–direction. Given the half edge having n redundants, the positions y
i = k×b/2(n – 1) with k = 0, 1, 2, 3 …n - 1 as represented in
Figure 4b can be taken.
For the redundant problem X
i = 1 the bending moments m
xx and m
yy and the twisting moments m
xy are calculated using Equations (2a-c) and (40):
The compatibility equations and the efforts in the plate are calculated using Equations (24) and (25).