Effects of Surface-Parallel Edge Restraints and Inter-laminar Shear on the Responses of Doubly Curved General Cross-Ply Panels

This study presents first a relatively lesser studied topic of the role played by surface-parallel restraints in determining the response of simply supported thick to thin doubly curved cross-ply panels of rectangular plan-form, modeled using a third order shear deformation theory, quantified by way of the difference between full and absent surface-parallel edge restraints. Mathematically speaking, this corresponds to the difference between complementary solutions to mixed boundary-value problems, resulting from two extreme sets of surface-parallel restraints. Of special interest is the three-way interaction of the membrane action due to curvature with the surface-parallel boundary constraint as well as the higher-order (resp. first-order) bending-stretching coupling producing beam-column/tie-bar type softening/hardening effects in thick (respectively thin) asymmetric cross-ply panels. Comparison with other popular shear deformation theories, such as the layer-wise constant shear-angle theory or zig-zag theory and first order shear deformation theory, also constitutes an important focus of this investigation. Results for cross-ply plates are regenerated in order to show the severity of the effect of curvature, especially in the thin shell regime.


Motivation and Objective
This paper presents boundary-discontinuous double Fourier series solution to a class of self-adjoint differential system of five highly coupled fourth order linear partial differential equations, arising out of the afore-mentioned TSDT-based formulation for general cross-ply doubly curved panels, subjected to SS4 boundary conditions (full surface-parallel edge restraints). Novel numerical results are presented to understand the complex deformation behavior of simply supported (SS4) antisymmetric cross-ply panels with the membrane action due to the curvature effect. Although similar interaction of the in-plane edge restraint with the bending-stretching coupling has already been investigated in detail [100], the intricacies of the three-way interaction of the SS4 type simply supported boundary condition, which entails full surface-parallel boundary constraints, with the effect of curvature as well as the bending-stretching coupling, arising out of the asymmetry of lamination, has so far remained an enigma in the literature. Interaction of the membrane action due to curvature with the higher-order (respectively, first-order) bending-stretching coupling in thick (respectively, thin) cross-ply panels constitutes an important focus of this investigation.
The significance of four different types of simply supported boundary constraints is well-known in structural mechanics literature [109]. While the SS1 boundary constraint is a two-dimensional (mathematical model) extension of the roller type simple support of a beam, the SS4 can be considered to be a similar extension of the hinge type simple support of beamlike (e.g., bridge) structures. The single most important factor to the commercial and military aircraft and rocket motor case designers alike is, of course, the design flexibility inherent in these composite laminates, known as tailoring, which is essentially exploiting the possibility of obtaining optimum design through a combination of structural/material concepts, such as stacking sequence, choice of the component phases, etc., panel-edge restraints (or lack thereof) to meet specific design requirements. In the context of simply supported panel edge constraints, the responses of panels with the SS1 and the SS4 boundary conditions represent the two bounds, and the rest (i.e., SS2 or SS3) lies somewhere in between.
Of special interest here is the role played by surface-parallel restraints in determining the response of simply supported thick to thin doubly curved cross-ply panels of rectangular plan-form, quantified by way of the difference between full (SS4) and absent (SS1) surface-parallel edge restraints. Mathematically speaking, this corresponds to the difference between complementary solutions, to mixed boundary-value problems, resulting from two extreme sets of surface-parallel restraints, namely SS1 and SS4. Results for cross-ply plates are regenerated in order to show the severity of the effect of curvature, especially in the thin shell regime [100]. Finally, comparison with other popular shear deformation theories, such as the layerwise constant shear-angle theory (LCST) or zig-zag theory, e.g., [30,[110][111][112][113][114] and FSDT , also constitutes an important focus of the present investigation. It may be noted that Oktem and Chaudhuri [101] have shown that the elastic responses of the present TSDT-based cross-ply spherical panels are in qualitative agreement with their zig-zag or LCST (layer-wise constant shear angle theory)-based counterparts, computed using finite element methods (FEM) [113].

Outline
Section 2 presents the statement of the problem under investigation, while some of the details thereof are described in Appendix A, Appendix B and Appendix C. Specifically, the Euler-Lagrange equations representing the equations of equilibrium, and the associated admissible boundary constraints for the present TSDT, both derived using the principle of virtual work, are presented in Appendix A. Appendix B shows some of the constants that appear in the equations, while the classification of four different types of simply supported or clamped boundary constraints for the present TSDT, obtained from Appendix A, is presented in Appendix C. Section 3 supplemented by Appendix D presents the implementation of the boundary-continuous double Fourier series to analytically (exact in the limit) the boundary-value problem stated above. In Section 4, novel numerical results are presented to understand the complex deformation behavior of simply supported (SS4) antisymmetric doubly curved cross-ply panels. The intricacies of the interaction of the SS4 type simply supported boundary condition, which entails full surface-parallel boundary constraints, with the effect of curvature (membrane action) have so far remained largely unaddressed in the literature. Interaction of the membrane action due to the curvature effect with the third order (respectively, firstorder) bending-stretching coupling producing beam-column/tie-bar type softening/hardening effects in thick (respectively, thin) cross-ply panels also constitutes an important focus of this investigation. Some interesting conclusions drawn from the present investigation are presented in Sub-Sec. 5, In addition, extensive suggestions for future research of laminated shells are provided in Section 6. Figure 1 depicts a laminated doubly curved panel of rectangular planform, of length a, width b and thickness h. The crossply shell under consideration is composed of a finite number of orthotropic layers of uniform thickness. Strain-displacement relations from the theory of elasticity in curvilinear coordinates are given by [9]:   ( , , ) ,

Statement of the Problem
In Eq. (1), ( 1,2,3,4,5,6) represents the components of the strain tensor, and i u (i = 1, 2, 3) denotes the components of the displacement vector along the (ξ 1 , ξ 2 , ξ 3 = ζ) coordinates at a point, 1 2 3 ( , , ) The principal radii of normal curvature of the reference (middle) surface are denoted by R 1 and R 2 , while R G1 and R G2 represent the geodesic curvatures of ξ 1 and ξ 2 curves, respectively [114]. In order to model the kinematic behavior of the shell, an additional set of simplifying assumptions are invoked: (i) Transverse inextensibility, (ii) Moderate shallowness (in regards to the normal curvatures), and (iii) Negligibility of geodesic curvature. The last two assumptions permit the use of curved panel coordinates x 1 , x 2 , x 3 = ζ, attached to the panel mid-surface. It may be noted that for a cylindrical shell, the lines of principal curvature coincide with the surfaceparallel coordinate lines, while for a spherical or hyperbolic-paraboloidal shell, the same can be assumed upon negligence of the geodesic curvatures of the coordinate lines.
The surface-parallel displacements can be expanded in power series of ξ 3 = ζ as suggested by Basset [85]. Only keeping the cubic terms and satisfying the conditions of transverse shear stresses (and hence strains) vanishing at a point (ξ 1 ξ 2 ±h/2) on the outer (top) and inner (bottom) surfaces of the shell, yields [19,86,96] where ( 1,2,3) i u i = denotes the displacements of a point on the middle surface, while 1 ϕ and 2 ϕ are the rotations at ζ = 0 with respect to the ξ 2 and ξ 1 axes, respectively. The corresponding kinematic relations are given by The stress resultants, moment resultants and higher-order moment and shear resultants, given by Eqs. (A5) in terms of components of displacement and rotation, can now be written as follows:    15 3,22 , 1  19 3  22 2,2  13 1,1  16 2,2  15 3,11 17 3,22 , The constants, , , Substitution of Eqs. (7) into equilibrium equations given by Eqs. (A3) supplies the following five highly coupled fourth-order governing partial differential equations [89,96]. 11  In what follows, the above system of five partial differential equations is solved in conjunction with the SS4 type simply supported boundary condition, prescribed at the edges, 1 0, x a = , and 2 0, x b = , which are given as follows: SS4 type simply supported boundary condition, prescribed at the edges, 1 0, x a = and 2 0, x b = which are given as follows (Refer to Figure 19d, and Eqs. (C1) and (C2d) in Appendix C):

Solution Methodology
The solution methodology is based on the boundary discontinuous double Fourier series approach [54,55] to solve boundary value problems, involving partial differential equations (with constant coefficients) of arbitrary orders subjected to corresponding admissible boundary conditions, of plate/shell type structural components of rectangular planform, with the starting point of selecting particular solution functions. These are assumed as follows: It may be noted that the assumed surface-parallel displacement functions for all edges SS4, while being identical to their SS1 counterparts [101], admit boundary discontinuities (complementary boundary constraints) different from the latter. The transverse displacement and the two rotations do not, as before, exhibit any boundary discontinuity, and are identical to their SS1 counterparts.
The total number of unknown Fourier coefficients introduced in Eqs. (10) enumerate to 5mn + 2m + 2n. The next operation is comprised of partial differentiation of the assumed particular solution functions. The procedure for differentiation of these functions is based on Lebesgue integration theory that introduces boundary Fourier coefficients arising from discontinuities, known as complementary boundary constraints [54,55], of the particular solution functions at the edges x 1 = 0, a, and x 1 = 0, b. As has been noted by Chaudhuri [55], the boundary Fourier coefficients serve as complementary solution to the problem under investigation. The procedure imposes certain boundary constraints in the form of equalities and complementary boundary constraints in the form of inequalities, the details of which are available in Chaudhuri [54,55], and will not be further discussed here in the interest of brevity of presentation. The partial derivatives, which cannot be obtained by term-wise differentiation, are obtained as follows. The above function u 1 and its first partial derivative, 1,1 u obtained by term-wise differentiation, are not satisfied at the edges, x 1 = 0, a, thus violating the boundary constraints and complementary boundary constraints, respectively, at these edges. Therefore, u 1 is forced to vanish at these edges (boundary constraints), while for further differentiation, 1,11 u is expanded in double Fourier series, in the form suggested by Chaudhuri [54,55], in order to satisfy the complementary boundary constraint (inequality). The partial derivatives are then obtained as follows [55].
In a manner similar to 1 2 , u u , is forced to vanish at the edges, 2 0 x = and b. The derivatives of 2 u are given as follows:  can be obtained by term-wise differentiation. The above step generates additional 2m + 2n (unknown) boundary Fourier coefficients.
Introduction of the displacement functions and their appropriate derivatives into the governing partial differential equations will supply the following 5mn + 2m + 2n equations: Satisfying the geometric boundary conditions given by Eqs. (9c, d), i.e., those pertaining to vanishing of 1 u and 2 u at the appropriate edges, and equating the coefficients of below [89,96,100], in a manner outlined in Figure 2.
These Fourier coefficients are then substituted in natural boundary conditions, given by Eqs. (17) Figure 2: Flow chart for numerical solution [95,96]. of the matrix to be inverted by orders of magnitude (from a system of 5mn + 4m + 4n to that of 2m + 2n equations in as many unknowns) depending on the cut-off values of m and n. Resulting

Numerical Results and Discussions
In what follows, the numerical results are presented using the following material properties:  [109,116]. An expression for the nonlinear shear modulus, G 12 (=G 13 ) is derived, by Chaudhuri [116] based on the assumption of uniform distribution of micro-kinks and fiber misalignment defects. Fiber micro-kinking is caused by crystallite disorientations, as detected by the Raman [117] and X-Ray measurements, inside a carbon fiber. This said, this type of material nonlinearity can be ignored for small deflection analysis of polymer matrix composite (PMC) structural elements without much loss of accuracy (in the interest of analytical convenience). However, material nonlinearity must be accounted for analysis of metal matrix composite (MMC) structural elements, e.g., boron/aluminum tubes [118]. The following normalized quantities are defined: , of a moderately thick (a/h = 10) antisymmetric cross-ply [0°/90°] square shell, computed using the present TSDT with the material type I for SS4 type boundary condition is displayed in Figure 3. Unlike its SS1 counterpart (see Figure 2 of Oktem and Chaudhuri [101], the convergence plot of the central moment,   discontinuities on the boundaries introduced by the SS4 boundary condition [100]. The convergence rate decreases when there are more discontinuities in the problem, which is a well-known fact in Fourier series analysis. Figure 4 shows that the spherical panels exhibit progressively stiffer response with the a/h ratio in the thinner shell regime. As can be seen from these curves, the membrane action (curvature effect) is more pronounced in the deeper [0°/90°] spherical panel regime (R/a < 40). These results are also presented in Table 1, wherein variations of normalized central deflections, * 3 u , of antisymmetric [0/90] moderately deep (R/a =10) spherical panels as well as their flat plate counterparts, with a/h ratio, ranging from very thick (a/h = 5) to very thin (a/h = 100), are compared for the SS4 boundary condition. Effect of membrane action, quantified in the form of % difference between spherical and flat panels vary from some membrane action for a/h = 5 to very high membrane action for a/h = 100.

Effects of Edge restraints and curvature
Oktem and Chaudhuri [93][94][99][100], and Chaudhuri and Seide [111] have solved boundary-value problems relating to thick to thin cross-ply plates by employing the boundary-discontinuous double Fourier series analytical technique and the finite element method, respectively, while Chaudhuri and Kabir [72] have studied their moderately thick counterparts, and Whitney and Leissa  Table 1 is considerably higher than its FSDT-based SS2 counterpart, 11.55, shown in Table 4 of Chaudhuri and Kabir [72], the difference being as large as 38.54%.
Chaudhuri and Kabir [80,115], and Seide and Chaudhuri [113] have investigated moderately thick and thick cross-ply shells using the boundary-discontinuous double Fourier series analytical technique and the finite element method, respectively. The present computed value of 14.881 for central * 3 u for a/h = 10 and R/a = 10 presented in Table 1 is also considerably higher than its FSDT counterpart, 10.11, shown in Table 2 of Chaudhuri and Kabir [115], the difference being as large as 32.06%. Furthermore, as illustrated in Table 1, the difference between spherical panel and plate normalized (central) deflections are    8.56%, 20.83%, 48.46%, 85.34% and 96.09%, for a/h = 5, 10, 20, 50 and 100, respectively. These results suggest that while the inter-laminar shear deformation or thickness-shear effect is dominant in the thick shell regime (a/h ≤ 10), the curvature effect (membrane action) dominates in its thinner counterpart.  Table 2 is substantially higher than its FSDT counterpart, 77.53, shown in Table 2 of Chaudhuri and Kabir [115], the difference being a whopping 43.15%. Furthermore, as illustrated in Table 2, the difference between spherical panel and plate (central) moments for SS4 boundary condition are 8.52%, 0.17 %, 23.41%, 67.04% and 85.68% for a/h = 5, 10, 20, 50 and 100, respectively. These results suggest that while the inter-laminar shear deformation effect is dominant in the thicker shell regime (a/h ≤ 20), the curvature effect (membrane action) progressively dominates in its thinner counterpart.  Table 3 (respectively, Table 4), is in reasonably close agreement with its FSDT counterpart, 9.17 (respectively, 106.79), shown in Table 3 of Chaudhuri and Kabir [115], the difference being 2.32% (resp. -0.64%). Furthermore, as illustrated in Table 3 (respectively, Table 4), the difference between spherical panel and plate (central)  deflections (respectively, moments) are 7.51% (respectively, 7.42%), 13.88% (respectively, 13.73%), 31.59% (respectively, 31.25%), 73.05% (respectively, 72.27%) and 92.58% (respectively, 91.59%) for a/h = 5, 10, 20, 50 and 100, respectively.  effect (membrane action) is also responsible for the decreased normalized response in both symmetric and antisymmetric type spherical panels for R/a ≤ 20. This notwithstanding, these results provide a clear indication of the tie-bar effects, arising out of bending-stretching coupling, which has a complex interaction with the membrane action, caused by the curvature effect, in the presence of surface-parallel edge restraint (SS4 boundary condition). It is interesting to point out here that there are two types of bending-stretching coupling in the TSDT: (i) Third-order bending-stretching, which arises out of non-vanishing E ij 's, that can only be captured by the TSDT, and (ii) First-order bending-stretching coupling, caused by non-vanishing B ij 's, which is generally covered by all laminated shell theories. The latter type has been investigated in-depth in the context of axi-symmetric deformation of thin [119][120][121][122][123] as well as moderately thick [124][125][126] variously laminated cylindrical shells, in addition to static and dynamic tests on flat laminates [62,106,127]. It is further noteworthy in this context that the bending-stretching coupling may either cause surface-parallel compression ("beam-column" effect) or surface-parallel stretching (called "tie-bar" effect) in a panel subjected to bending. The former has a softening effect while the latter has a stiffening or hardening effect on the laminated panel response [123]. It also is important to note that for the normalized central moment, M , the influence of curvature becomes progressively less dominant for R/a > 20. Figure 11 depicts the variation of   The relative difference in central deflections, * 3 u , between the two boundary conditions tends to increase with the increasing values of a/h (thinner structures). This difference is, however, somewhat less pronounced in case of symmetric laminations than their asymmetric counterpart.

Comparison between SS1 and SS4 Boundary Conditions
Variations of the relative difference in central deflections,  Figure 15 and Figure 16. For comparison, antisymmetric [0/90] plate results have also been regenerated [100]. As shown in both the plots, membrane (curvature) action exerts an important effect on the surface-parallel boundary constraints particularly for R/a < 20. This effect diminishes with the increase of R/a, as shown in Figure 16 (flat panel case). These results are also presented in Table 5, wherein variations of normalized central deflections, * 3 u , of antisymmetric [0/90] moderately deep (R/a = 10) spherical panels as well as their flat plate counterparts, with a/h ratio, ranging from very thick (a/h = 5) to very thin (a/h = 100), are compared for three different surface-parallel edge constraints: SS1, SS3 and SS4. Effect of membrane action, quantified in the form of % difference between spherical and flat panels vary from some membrane action for a/h = 5 to very high membrane action for a/h = 100. Effect of surface-parallel edge restraint is seen to be insignificant in the entire range of a/h ratios. It may be pointed out in this context, that "FSDT-based convergence plots of an antisymmetric cross-ply spherical panel, with the SS2 boundary condition…, are numerically too close to their SS4 counterparts…. This is because both the conditions prescribe u n = 0 (instead of N n = 0) at a boundary -a choice that appears to have a major influence on the response of antisymmetric cross-ply panels under uniform loads. Likewise, the convergence characteristics of the response quantities for antisymmetric cross-ply panels, with SS1 and SS3 boundary conditions…, where N n = 0, instead of u n = 0, is prescribed at a boundary, are numerically very close" [80,115]. Figure 17 and Figure 18 show the relative difference in central moment, * 1 M , of both symmetric and antisymmetric moderately deep (R/a = 10) spherical panels with respect to a/h and R/a ratios, respectively. The aforementioned curvature effect is also clearly visible in Figure 18. It is noteworthy that the membrane action due to the effect of curvature has a complex  interaction with the bending-stretching type coupling effect, caused by the asymmetry of lamination. This interaction is stronger in the case of surface-parallel displacement boundary constraint, i.e., 0 n u = prescribed at an edge n x = constant (e.g., SS4) as compared to its absence, i.e., 0 n N = (e.g., SS1), which is an illustration of the beam-column effect. For example, the bendingstretching type coupling has a highly pronounced interaction with the type of surface-parallel boundary constraint, imposed by e.g., SS4 as compared to SS1 type simply supported boundary conditions, prescribed at all four edges.
Comparisons of results, computed using three leading shear deformation theories, and the classical lamination theory  Table 5 shows that the LCST or zig-zag theory yields much greater shear-flexibility than the TSDT in terms of the computed normalized central deflection, * 3 u (344.295% vs. 231.514%) for a very thick [0/90/0] plate (a/h = 4). The difference in the degree of shear flexibility decreases with the increase of a/h ratio. Table 6 shows that the LCST or zig-zag theory yields somewhat greater shear-flexibility than the TSDT in terms of the computed normalized central moment, Again, the difference in the degree of shear flexibility diminishes with the increase of a/h ratio. Computed zig-zag theory and TSDT (0.149% vs. -0.839%) results are very close for a/h = 20. Table 7 and Table 8 Table 7 shows that the LCST or zig-zag theory yields much greater shear-flexibility than the TSDT in terms of the computed normalized central deflection, * 3 u (46.079% vs. 9.767%) for a very thick [0°/90°/0°] cylindrical panel (a/h = 4). The difference in the relative degree of shear flexibility progressively diminishes with the increase of a/h ratio. Table 8 shows that the LCST or zig-zag theory yields somewhat greater shear-flexibility than the TSDT in terms of the computed normalized central moment,   Table 9 shows that the LCST or zig-zag theory yields much greater shear-flexibility than the TSDT in terms of the computed normalized central deflection,

Comments on the reliability of the finite element methods (FEM)
Finally, reliability of the finite element method, based on the assumed displacement potential energy approach, in the neighborhood of stress discontinuities and stress singularities has been discussed by Whitcomb, et al. [128] (at the free edge in a laminated composite), and Chaudhuri and co-workers [129][130][131][132][133][134][135][136][137] (at the circumferential corner line of an internal circular cylindrical hole). Whitcomb, et al. [128] have concluded that "the finite element method yielded accurate solutions everywhere except in a region involving the elements closest to the stress discontinuity or singularity and that this region can be made arbitrarily small by refining the finite element model". Chaudhuri [129][130][131][132][133] has probed into the issue much deeper. His conclusion re-affirms, to a rather milder degree, the conclusion reached in the case of its laminate counterpart [133][134][135][136][137] in regards to the accuracy (or lack thereof) of the stresses (more accurately, the stress gradients) computed using the conventional (assumed displacement potential energy based) finite element analysis and computed FEM-based post-processing analysis results for transverse shear stresses in the vicinity of a stress singularity, such as the circumferential corner line of an internal circular/elliptical cylindrical hole.    In conclusion, numerical solutions, based on finite elements methods (FEM), finite difference, boundary elements methods (BEM) and so on, have been presented by numerous authors. However, their resolution is not fine enough to yield definitive results at a line of discontinuity. Only the boundary-discontinuous Fourier analysis of the present type can reproduce precisely the discontinuities in displacement functions and/or their partial derivatives at the edges.

Conclusions
The key conclusions that emerge from the present numerical results can be summarized as follows: i. The central deflection shows a rapid and monotonic convergence, while the central moment shows an initially oscillatory convergence. The latter is possibly due to the presence of a discontinuity (complementary boundary constraint) in the derivative of the displacement in expression of the moment. In addition, because of the same reason, the computed solutions for SS4 boundary conditions converge less rapidly than their SS1 counterpart.
ii. The length-to-thickness ratio, / a h , has a significant effect on the computed results in the moderately thick to thick panel regime (a/h < 20). This is due to the effect of transverse (interlaminar) shear deformation.
iii. The radius-to-length ratio / R a has a pronounced effect on the response of curved cross-ply panels. This effect is progressively more pronounced in the deeper shell regime ( / 40) R a < for a given a/h. iv. The effect of the transverse shear deformation is compensated to a certain extent by the bending-stretching coupling effect -a characteristic of antisymmetric laminates.
v. The difference (normalized) in central deflection between the SS1 and SS4 boundary conditions increases with the increasing values of a/h (thinner structures). This difference is, however, somewhat less pronounced in case of symmetric laminates than their asymmetric counterpart.
vi. The membrane action due to the effect of curvature has a complex interaction with the bending-stretching type coupling effect, caused by the asymmetry of lamination. This interaction is stronger in the case of surface-parallel displacement boundary constraint, i.e., 0 n u = prescribed at an edge n x = constant (e.g., SS4) as compared to its absence, i.e., 0 n N = (e.g., SS1). This is an illustration of the beam-column/tie-bar effect. For example, the bending-stretching type coupling has a highly pronounced interaction with the type of surface-parallel boundary constraint, imposed by e.g., SS4 as compared to SS1 type simply supported boundary conditions, prescribed at all four edges.
vii. The layer-wise constant shear-angle theory or zig-zag theory yields much greater shear-flexibility than the TSDT in terms of the computed normalized central deflection, viii. The layer-wise constant shear-angle theory or zig-zag theory yields somewhat greater shear-flexibility than the third order shear deformation theory in terms of the computed normalized central moment,

Suggestion for Future Research
First, although the present double Fourier series solutions are confined to general cross-ply thick hyperbolic-paraboloidal panels subjected to SS4 type simply supported boundary condition, the present methodology is general enough to be able to solve arbitrarily laminated doubly curved panels subjected to any combination of admissible boundary conditions. These problems need to be solved in the near future to provide bench-mark analytical solutions for a variety of numerical solutions including those computing using the FEM.
Fourth, computation of the normalized central deflections by employing both commercial FEA codes (e.g., ABAQUS and ANSYS) and the current MATLAB code based on the boundary continuous double Fourier series, for all parameters, and demonstrating the main differences thus obtained, is an important topic for further research.
Fifth, experimental results are needed to validate the present as well as other higher order shear deformation theory based models for laminated anisotropic panels of negative Gaussian curvature, in a manner that can be considered as extensions of Refs. [62,[106][107][108]127].

Funding
This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.