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Fig. 10 Nonlinear center deflection versus load for a clamped, cross- ply (0° / 90° / 90° / 0°) square plate under uniform load (a = b = 12in., h = 0.096in., Ey = 1.8282 x 10° psi, Er = 1.8315 x 10° psi, viz = 0.24, Giz = G13 = G23 = 0.3125 x 10° psi)

Figure 10 Nonlinear center deflection versus load for a clamped, cross- ply (0° / 90° / 90° / 0°) square plate under uniform load (a = b = 12in., h = 0.096in., Ey = 1.8282 x 10° psi, Er = 1.8315 x 10° psi, viz = 0.24, Giz = G13 = G23 = 0.3125 x 10° psi)

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where S_, is a group of nodes whose shape function supports cover node L. It has been demonstrated [24] that the smoo- thing gradient Eq. (17) satisfies the integration constraints in nodal integration. where I"; is the boundary of the representative domain of node L and nis the outward normal of boundary I’, as shown in Fig. 2. For a two-dimensional problem, introducing RPIM shape functions into Eq. (15) yields
where S_, is a group of nodes whose shape function supports cover node L. It has been demonstrated [24] that the smoo- thing gradient Eq. (17) satisfies the integration constraints in nodal integration. where I"; is the boundary of the representative domain of node L and nis the outward normal of boundary I’, as shown in Fig. 2. For a two-dimensional problem, introducing RPIM shape functions into Eq. (15) yields
Fig. 1 Problem domain represented by irregular nodes. a Voronoi diagram. b Nodal domain by Delaunay triangulation shell theory, the displacement field is expressed as
Fig. 1 Problem domain represented by irregular nodes. a Voronoi diagram. b Nodal domain by Delaunay triangulation shell theory, the displacement field is expressed as
A modified nonlinear strain-displacement equations [12 of the Sanders nonlinear shell theory [27] is expressed as Fig. 3. Geometry of a cylindrical shell panel
A modified nonlinear strain-displacement equations [12 of the Sanders nonlinear shell theory [27] is expressed as Fig. 3. Geometry of a cylindrical shell panel
A, D, Band As represent matrices of extensional stiff- nesses, bending stiffnesses, bending-extensional coupling stiffnesses and transverse shear stiffnesses, and are defined
A, D, Band As represent matrices of extensional stiff- nesses, bending stiffnesses, bending-extensional coupling stiffnesses and transverse shear stiffnesses, and are defined
The external work done due to surface traction and body force is given by
The external work done due to surface traction and body force is given by
Substituting Eq. (32) into Eq. (31) and taking variation to the energy functional yield the equilibrium equation ‘he discrete nodes are generated freely on the x — y space in he parametric coordinate system. The nodal representative lomains based on Delaunay triangulation are obtained on he x — y space. For a shell panel domain discretized by set of nodes x7, 7 = 1,..., NP, the approximations of isplacements and rotations of the mid-plane of the panel re expressed using RPIM shape functions as
Substituting Eq. (32) into Eq. (31) and taking variation to the energy functional yield the equilibrium equation ‘he discrete nodes are generated freely on the x — y space in he parametric coordinate system. The nodal representative lomains based on Delaunay triangulation are obtained on he x — y space. For a shell panel domain discretized by set of nodes x7, 7 = 1,..., NP, the approximations of isplacements and rotations of the mid-plane of the panel re expressed using RPIM shape functions as
where x; and A, are node coordinate and the nodal repre- sentative area, respectively, NPb is the number of nodes on the natural boundary, and s;, are the weights associated with the boundary point obtained from Delaunay triangulation. The stiffness matrix in Eq. (37) is evaluated using the stabilized nodal integration technique introduced in Sect. 3, while Eqs. (38) to (41) are computed by the direct nodal integration. The approximations of Eqs. (37) to (41) are given as follows
where x; and A, are node coordinate and the nodal repre- sentative area, respectively, NPb is the number of nodes on the natural boundary, and s;, are the weights associated with the boundary point obtained from Delaunay triangulation. The stiffness matrix in Eq. (37) is evaluated using the stabilized nodal integration technique introduced in Sect. 3, while Eqs. (38) to (41) are computed by the direct nodal integration. The approximations of Eqs. (37) to (41) are given as follows
B>( x L), B? 7(X L), Br (x L), BS 7 (XL), H an d G are giv en b y In order to solve the nonlinear equation system Eq. (33), an iterative algorithm, either Newton—Raphson method or modified Newton—Raphson method, is typically used. An incremental form of Eq. (33), required by the Newton-type methods, should be provided. Equation (33) is rearranged as
B>( x L), B? 7(X L), Br (x L), BS 7 (XL), H an d G are giv en b y In order to solve the nonlinear equation system Eq. (33), an iterative algorithm, either Newton—Raphson method or modified Newton—Raphson method, is typically used. An incremental form of Eq. (33), required by the Newton-type methods, should be provided. Equation (33) is rearranged as
Fig. 4 A square plate subjected to uniform transverse load Convergence studies are performed based on the nodal dis- tributions of a simply supported plate (a / h = 10) anda clamped plate (a / h = 100), which are subjected to trans- verse uniform load, as shown in Fig. 4. The Poisson ratio of the plates is v = 0.3. Table 1 shows the variation of nondi- mentional center deflection w = woE 3 / qoa* with nodal distributions. It is seen that a relatively stable convergent trend is observed for both the simply supported and clamped plates.
Fig. 4 A square plate subjected to uniform transverse load Convergence studies are performed based on the nodal dis- tributions of a simply supported plate (a / h = 10) anda clamped plate (a / h = 100), which are subjected to trans- verse uniform load, as shown in Fig. 4. The Poisson ratio of the plates is v = 0.3. Table 1 shows the variation of nondi- mentional center deflection w = woE 3 / qoa* with nodal distributions. It is seen that a relatively stable convergent trend is observed for both the simply supported and clamped plates.
Table 2 Transverse center displacement of a pinched cylindrical shell (L = 600in, R = 300in, h = 3in, E = 3 x 10®psi, v = 0.3)
Table 2 Transverse center displacement of a pinched cylindrical shell (L = 600in, R = 300in, h = 3in, E = 3 x 10®psi, v = 0.3)
Table 1 Transverse center displacement t) = w Eh? /qoa* of a square plate (a = b = 10in., v = 0.3) Fig. 5 Shear locking test for a simply supported square plate (v = 0.3)
Table 1 Transverse center displacement t) = w Eh? /qoa* of a square plate (a = b = 10in., v = 0.3) Fig. 5 Shear locking test for a simply supported square plate (v = 0.3)
Fig. 6 Shear locking test for a clamped square plate (v = 0.3)
Fig. 6 Shear locking test for a clamped square plate (v = 0.3)
Fig. 7 Geometry of the pinched circular cylinder (L = 600in. R = 300in., h = 3in., E =3 x 10°, v = 0.3, P=1)
Fig. 7 Geometry of the pinched circular cylinder (L = 600in. R = 300in., h = 3in., E =3 x 10°, v = 0.3, P=1)
Fig. 8 Nonlinear center deflection parameter versus load parame- ter for simply supported, isotropic square plate under uniform load (a=b=10in,h = lin, E =7.8 x 10°, v = 0.3)
Fig. 8 Nonlinear center deflection parameter versus load parame- ter for simply supported, isotropic square plate under uniform load (a=b=10in,h = lin, E =7.8 x 10°, v = 0.3)
Fig. 9 Nonlinear center deflection versus load for a clamped, ortho- tropic square plate under uniform load (a = b = 12 in., h = 0.138in., E, = 3.0 x 10°psi, E> = 1.28 x 10° psi, viz = 0.32, Gi2 = Gi3 = G23 = 0.37 x 10°psi)
Fig. 9 Nonlinear center deflection versus load for a clamped, ortho- tropic square plate under uniform load (a = b = 12 in., h = 0.138in., E, = 3.0 x 10°psi, E> = 1.28 x 10° psi, viz = 0.32, Gi2 = Gi3 = G23 = 0.37 x 10°psi)
Fig. 12 Nonlinear response of a clamped cylindrical shell panel (L = 20in., R = 100in., h = 0.125in., 09 = 0.2rad, E = 4.5 x 10°, v = 0.3) Fig. 11 Isotropic clamped cylindrical shell panel under uniform radial pressure go
Fig. 12 Nonlinear response of a clamped cylindrical shell panel (L = 20in., R = 100in., h = 0.125in., 09 = 0.2rad, E = 4.5 x 10°, v = 0.3) Fig. 11 Isotropic clamped cylindrical shell panel under uniform radial pressure go
Fig. 13. Hinged cylindrical shell under point load
Fig. 13. Hinged cylindrical shell under point load
Fig. 14 Nonlinear response of point loaded cylindrical shell panel (L = 20in., h = 1.0in., R = 100in., 6) = 0.2rad, E = 4.5 x 10° psi, v = 0.3)
Fig. 14 Nonlinear response of point loaded cylindrical shell panel (L = 20in., h = 1.0in., R = 100in., 6) = 0.2rad, E = 4.5 x 10° psi, v = 0.3)
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Mechanical EngineeringCivil EngineeringComputational MechanicsInterdisciplinary EngineeringMeshless methodNodal Integration
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