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Fig. 6. Numerical example 4.1(b) (Circular plate with simply supported edge): Convergence rate of displacement (L, error) with R= 10 (left), R = 100 (mid- dle) and R = 1000 (right). Fig. 5. Numerical example 4.1(a) (Circular plate with clamped edge): Normalized deflection and rotation angle along the radial direction for R = 10 (left) and R = 1000 (right).

Figure 6 Numerical example 4.1(b) (Circular plate with simply supported edge): Convergence rate of displacement (L, error) with R= 10 (left), R = 100 (mid- dle) and R = 1000 (right). Fig. 5. Numerical example 4.1(a) (Circular plate with clamped edge): Normalized deflection and rotation angle along the radial direction for R = 10 (left) and R = 1000 (right).

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among all deflections u(x, x») and rotation angles g,(x,,x,) satisfying the kinematic boundary conditions. Here da = dx,dx,, 0 = —4—- = A+2u I-v? with A, w and v being the Lame constants and Poisson’s ratio, respec- tively. Note that g, are the rotation angles due to pure shear deforma- tion, py are the measures of bending and f is the external transverse load. They are given by
among all deflections u(x, x») and rotation angles g,(x,,x,) satisfying the kinematic boundary conditions. Here da = dx,dx,, 0 = —4—- = A+2u I-v? with A, w and v being the Lame constants and Poisson’s ratio, respec- tively. Note that g, are the rotation angles due to pure shear deforma- tion, py are the measures of bending and f is the external transverse load. They are given by
Although not relevant for the FE-implementation, the strong formu- lation of the problem is also stated for the sake of completeness. Under the assumptions that u and y, are doubly differentiable, we apply the partial integration to Eq. (14) and put it in the form where n, is the unit normal outward to the boundary and ds is the length element. It is assumed here that the remaining part 0, of the plate edge is free. Because of the arbitrariness of the variations dy, and 6u in Q and on d,, from (17) follow the second order partial differential equations Denote the space of admissible functions as K = {(v, x,)|(v, %q)| a= 0}. The weak formulation of the problem is stated as follows: Given f, find (u,y,) € K such that Eq. (16) is satisfied for all (6u,dy,) € K. For the integrals in (15) to be meaningful, v and y, must belong at least to the Sobolev’s space of square integrable functions with the square integrable first derivatives, H 1(Q). However, since the desired asymptotic accuracy of FSDT may require higher smoothness of u and y,, the continuity assumption in K still remains unspecified.
Although not relevant for the FE-implementation, the strong formu- lation of the problem is also stated for the sake of completeness. Under the assumptions that u and y, are doubly differentiable, we apply the partial integration to Eq. (14) and put it in the form where n, is the unit normal outward to the boundary and ds is the length element. It is assumed here that the remaining part 0, of the plate edge is free. Because of the arbitrariness of the variations dy, and 6u in Q and on d,, from (17) follow the second order partial differential equations Denote the space of admissible functions as K = {(v, x,)|(v, %q)| a= 0}. The weak formulation of the problem is stated as follows: Given f, find (u,y,) € K such that Eq. (16) is satisfied for all (6u,dy,) € K. For the integrals in (15) to be meaningful, v and y, must belong at least to the Sobolev’s space of square integrable functions with the square integrable first derivatives, H 1(Q). However, since the desired asymptotic accuracy of FSDT may require higher smoothness of u and y,, the continuity assumption in K still remains unspecified.
with the elements of the stiffness matrix being the blocks given in Eqs. (25)-(27), and F“, F’!, FY’? the first terms in (24) representing the external forces. 3.3. Isogeometric analysis
with the elements of the stiffness matrix being the blocks given in Eqs. (25)-(27), and F“, F’!, FY’? the first terms in (24) representing the external forces. 3.3. Isogeometric analysis
Fig. 2. Structure of the computational code. 4. Numerical examples the patch information matches at the interface, including the parametric (knot) information and the location of the control points, the multi- patch is naturally connected and the C° continuity is guaranteed on the patch interfaces, i.e. strong coupling. However, patch-wise parametric matching is not the necessary condition. There are various methods to maintain the patch continuity in a NURBS multipatch, such as the penalty method [50], Nitsche method [51] or bending strip method [52]. These methods impose weak coupling condition and are more so- phisticated to implement. 4.1. Circular plate under uniform transverse load
Fig. 2. Structure of the computational code. 4. Numerical examples the patch information matches at the interface, including the parametric (knot) information and the location of the control points, the multi- patch is naturally connected and the C° continuity is guaranteed on the patch interfaces, i.e. strong coupling. However, patch-wise parametric matching is not the necessary condition. There are various methods to maintain the patch continuity in a NURBS multipatch, such as the penalty method [50], Nitsche method [51] or bending strip method [52]. These methods impose weak coupling condition and are more so- phisticated to implement. 4.1. Circular plate under uniform transverse load
Fig. 4. Numerical example 4.1(a) (Circular plate with clamped edge): Conver- gence rate of displacement (L, error) with R= 10 (left), R = 100 (middle) and R= 1000 (right). It can be seen that the cubic order discretization gives very good results even at coarse mesh. Fig. 4 visualizes the convergence rate of analysis with second order (p = 2, q = 2) and cubic order (p = 3, gq = 3) discretiza- tion. We observe that the error increases with increasing R, yet the convergence rate is close to optimal, showing that the proposed for- mulation does not exhibit shear-locking behavior. Although rescaling inflates the domain, the number of elements and degrees of freedom required to obtain a small error are not significantly affected by this rescaling. A slight reduction of convergence rate is observed for R = 10 of the cubic order analysis. This can be attributed to the loss of accu- racy when the L, error is very small (~ 10-°) and the machine precision starts to affect the computation.
Fig. 4. Numerical example 4.1(a) (Circular plate with clamped edge): Conver- gence rate of displacement (L, error) with R= 10 (left), R = 100 (middle) and R= 1000 (right). It can be seen that the cubic order discretization gives very good results even at coarse mesh. Fig. 4 visualizes the convergence rate of analysis with second order (p = 2, q = 2) and cubic order (p = 3, gq = 3) discretiza- tion. We observe that the error increases with increasing R, yet the convergence rate is close to optimal, showing that the proposed for- mulation does not exhibit shear-locking behavior. Although rescaling inflates the domain, the number of elements and degrees of freedom required to obtain a small error are not significantly affected by this rescaling. A slight reduction of convergence rate is observed for R = 10 of the cubic order analysis. This can be attributed to the loss of accu- racy when the L, error is very small (~ 10-°) and the machine precision starts to affect the computation.
Numerical example 4.1(a) (Circular plate with clamped edge): Deflection computed at the center of the plate and error. u, is the error of deflection compared to the analytical solution. is worth noting that to evaluate q, the derivatives of u at the integration point are calculated and then transferred to the control points using an L,-global projection algorithm [58]. Table 1
Numerical example 4.1(a) (Circular plate with clamped edge): Deflection computed at the center of the plate and error. u, is the error of deflection compared to the analytical solution. is worth noting that to evaluate q, the derivatives of u at the integration point are calculated and then transferred to the control points using an L,-global projection algorithm [58]. Table 1
However, this is still not the true average transverse displacement of the plate. The latter should be computed in accordance with Eq. (12) giving
However, this is still not the true average transverse displacement of the plate. The latter should be computed in accordance with Eq. (12) giving
Fig. 7. Numerical example 4.2(a) (Rectangular plate with one clamped edge and three free edges): Geometry and boundary condition for (left) the analysis using FSDT plate element and (right) the 2D analysis using solid element.
Fig. 7. Numerical example 4.2(a) (Rectangular plate with one clamped edge and three free edges): Geometry and boundary condition for (left) the analysis using FSDT plate element and (right) the 2D analysis using solid element.
Fig. 8. Numerical example 4.2(a) (Rectangular plate with one clamped edge and three free edges): Average displacement and rotation angle for L = 3 (left) and L=10 (right).
Fig. 8. Numerical example 4.2(a) (Rectangular plate with one clamped edge and three free edges): Average displacement and rotation angle for L = 3 (left) and L=10 (right).
Fig. 9. Numerical example 4.2(b): A plate with two simply supported edges and two free edges.
Fig. 9. Numerical example 4.2(b): A plate with two simply supported edges and two free edges.
Fig. 10. Numerical example 4.2(b) (Rectangular plate with simply supported boundary condition): Average displacement and rotation angle for L = 3 (left) anc L=10 (right). of the FSDT. Since the difference between the analytical solution of the FSDT (red solid line) and the solution resulting from the classical Kirchhoff (beam-like) theory (blue dashed line) becomes significant for moderate L, the latter is not applicable for moderately thick plates. A detail worth mentioning is that the true average displacement calcu- lated according to (12) does not exactly satisfy the kinematic boundary condition: The last correction term in (12), although small, does not vanish at x, = 0 (cf. Eq. (51)). The reason for this is well known: The FSDT correction might not work in a thin boundary layer near the edge of the plate [18]. Far from the edge, this correction term of order h? //? compared to unity is essential for ensuring the asymptotic accuracy of the FSDT up to that order. Another detail worth mentioning is that the elastic 2-D analysis does not use an approximation, but evaluates the solution fields in a brute force method with a very fine mesh. The average transverse displacements and rotation angles obtained by cal- culating integrals (13) over the thickness in a post-processing represent only the integral characteristics of the FSDT. The comparison between the detailed distributions of displacements across the thickness is left for future investigations.
Fig. 10. Numerical example 4.2(b) (Rectangular plate with simply supported boundary condition): Average displacement and rotation angle for L = 3 (left) anc L=10 (right). of the FSDT. Since the difference between the analytical solution of the FSDT (red solid line) and the solution resulting from the classical Kirchhoff (beam-like) theory (blue dashed line) becomes significant for moderate L, the latter is not applicable for moderately thick plates. A detail worth mentioning is that the true average displacement calcu- lated according to (12) does not exactly satisfy the kinematic boundary condition: The last correction term in (12), although small, does not vanish at x, = 0 (cf. Eq. (51)). The reason for this is well known: The FSDT correction might not work in a thin boundary layer near the edge of the plate [18]. Far from the edge, this correction term of order h? //? compared to unity is essential for ensuring the asymptotic accuracy of the FSDT up to that order. Another detail worth mentioning is that the elastic 2-D analysis does not use an approximation, but evaluates the solution fields in a brute force method with a very fine mesh. The average transverse displacements and rotation angles obtained by cal- culating integrals (13) over the thickness in a post-processing represent only the integral characteristics of the FSDT. The comparison between the detailed distributions of displacements across the thickness is left for future investigations.
Note that the average transverse displacement according to the classical Kirchhoff’s plate theory in this beam-like model is given by
Note that the average transverse displacement according to the classical Kirchhoff’s plate theory in this beam-like model is given by
Fig. 11. Numerical example 4.2(c) (Square concrete plate with clamped boundary condition): Deflection of the plate.
Fig. 11. Numerical example 4.2(c) (Square concrete plate with clamped boundary condition): Deflection of the plate.
Related topics:
Finite Element MethodsPlates and shells
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