3.1. Comparison with Literature Results
A first part of the section is devoted to the validation of the present Ritz-based procedure with results available in the literature and finite element calculations.
The first comparison is presented against the 3D exact solutions derived by Noor and Burton [
12] for a set of angle-ply, straight fiber configurations. Despite the focus of this preliminary example not being expressly directed towards the case of variable-stiffness plates, it is of particular interest because the reference solutions are derived by means of an exact three-dimensional approach.
The plates are square and different ratios are considered for illustrating the role played by transverse shear deformability. Pre-buckling constraints Case-Tx2 are considered, i.e., the normal motion of the four edges is prevented, whilst simply supported boundary conditions are considered along the four edges for solving the buckling eigenvalue problem.
An orthotropic material is considered, having the following non-dimensional thermoelastic properties:
The lay-up is characterized by the stacking of 10 plies oriented at
. It should be noted that, for the problem at hand, the exact elasticity solution of the pre-buckling problem (see also [
12]) is characterized by a membrane state of internal stresses, where the only not null components of the stress tensor are those reported in Equation (
6). It follows that no approximations are introduced in the pre-buckling analysis if Equation (
30) is used.
The results are summarized in
Table 1 for different values of
, ply angles and equivalent single layer kinematic theories of order 2,3 and 4. Based on a preliminary convergence analysis, Ritz results are computed using 12 functions along both directions.
As seen from
Table 1, close agreement can be achieved between the 3D results reported in [
12] and those calculated with the present Ritz formulation. As expected, thicker plates demand increased orders in the kinematic model to guarantee adequate accuracy of the predictions. When
is equal to 100, no substantial improvement is observed by refining the theory from the second to fourth order. On the contrary, the critical temperature for thick plates with
of 20/3 can be accurately estimated using the third- or fourth-order theories ED3 and ED4. In general, it can be noted that the discrepancy with the reference solution is higher for increasing values of
. This behaviour is due to the skewness of the in-plane displacement pattern when the plies are rotated in the off-axis direction. In these cases, the convergence of the solution is slower, and a proper description of the buckling pattern requires a larger number of trial functions [
45].
A second test case is taken from [
37], where the thermal buckling analysis is presented for variable-stiffness plates subjected to restrained pre-buckling thermal expansion and simply supported boundary conditions. The material properties are those of an orthotropic material, whose non-dimensional thermoelastic parameters are given as:
= 40.0,
=
= = 0.6,
= 0.5,
= = = 0.25,
= 1 10C,
= 2.0.
Three stacking sequences are considered, with plies oriented at , and , where the angles are measured with respect to the x-axis, and the variation is linear with x.
The analysis of [
37] is conducted based on finite element models, using FSDT as underlying kinematic theory with a shear factor equal to 5/6.
To further validate the results, finite element computations are performed with Abaqus finite element simulations. To this aim, a mesh size of 50 50 S4R shell elements was found adequate to obtain converged results. Note that the mesh is constructed such that each element is characterized by a constant orientation, thus the fiber steering is represented in a stepwise manner. Given the linear variation of the fiber angle, each row of elements at share the same orientation, with a discrete increase of the angle when moving along the x-coordinate.
The results are summarized in
Table 2, where the present Ritz formulation is exploited by considering three different kinematic theories. In addition, the convergence of the method is assessed by considering an increasing number of trial functions
.
The convergence of the Ritz solutions is clearly seen from
Table 2, where the critical temperatures get smaller and and smaller as the number of trial function increases. It is observed that the number of integration points for evaluating the Ritz in-plane integrals is taken as
. Due to the numerical integration process, convergence of the temperatures from above cannot be guaranteed a priori, although it is generally experienced [
28]. For the case at hand, a few degrees of freedom are necessary, in most cases, to reach convergence up to the first two digits. Overall, the convergence is faster for the straight fiber configuration, whose orientation angle at 15 is responsible for a slight amount of buckling mode skewness. The convergence is slower when variable-stiffness panels are of concern, with increasing need to consider more trial functions as the steering of the fibers is stronger.
As seen by the results in
Table 2, a substantial agreement is obtained with the predictions of [
37], although in some cases the discrepancies can be non-negligible. For instance, the Ritz results relative to the
lay-up are higher with respect to those of [
37], irrespective of the kinematic theory. For this reason, Abaqus analyses were performed, demonstrating, on the contrary, very close agreement. Note that a local buckling mode, characterized by several short half waves in proximity of the boundaries, is predicted as the first instability by Ritz and Abaqus analyses for the plate with lay-up
and
. Local instabilities are not of concern in this study, so the values in the table refer to the first global mode. For the configurations with
equal to 40, Abaqus and Ritz results are almost identical. Clearly, no substantial advantages are associated with the adoption of higher-order theories, and an ED2 model is generally sufficient. The matching with finite element results is clear even for thicker configurations. In these cases, the adoption of ED4 or LD2 theories can be beneficial to obtain accurate predictions, whilst a second-order theory ED2 is generally responsible for large errors. It is noted that LD2 results generally lead to smaller critical temperatures with respect to Abaqus analyses based on shell elements, as one may expect. In a few cases, this behaviour is not respected due an over-correction associated with the evaluation of shear corrections factor performed by Abaqus for non-symmetric lay-ups. Similar issues were observed in [
46]. It is useful to note that the higher-order theories available in the variable-kinematic formulation do not require the definition of a shear factor, and the effect of transverse shear flexibility is inherently accounted for by the kinematic models themselves.
An extensive set of results for the thermal buckling of thin variable-stiffness panels is provided by Duran and co-workers [
36]. The authors analyzed several composite materials, focusing on the sensitivity of the optimal configuration with respect to the material properties. The analyses of [
36] are based on finite element analyses using four-node, Kirchhoff finite elements, so they are restricted to thin-plate configurations.
A set of results is presented in
Table 3, where square plates are analyzed, with non-dimensional ratio
=148. The pre-buckling boundary conditions are those denoted here as Case-Tx1, while the flexural ones refer to the simply supported assumptions. Different materials and lay-ups are considered. For the sake of conciseness, the material properties are not reported here but can be found in [
36]. The lay-ups are characterized by a linear variation of the fiber angle along the direction
x, with stacking sequence given by
. It is noted that the configurations of
Table 3 are those maximizing the critical temperatures according to the optimizations performed in [
36] and, for this reason, the angles
and
are non-integer values. The results are reported by considering the equivalent single layer theories ED3 and ED4; furthermore, the second-order layer-wise theory LD2 is adopted for furnishing lower bound reference results.
As observed, close agreement is noted between the temperatures obtained by Duran et al. [
36] and the present formulation. In most cases, the differences are of a few percent points, thus confirming the correct implementation of the Ritz code. The same cannot be claimed for the Kevlar/Epoxy and Carbon/Epoxy configurations, where the discrepancies are very high. To clarify this aspect, Abaqus results were performed for all the configurations: as seen from
Table 3, the differences between Ritz and Abaqus results are very small for all the cases. It is then concluded that even the evaluation of the Kevlar/Epoxy and Carbon/Epoxy plates is correct. As a further verification, the comparison against Abaqus analyses is reported in
Figure 4 for the Kevlar/Epoxy plate in terms of pre-buckling stress resultants and buckled shape. The pre-buckling forces are evaluated for a unitary temperature increase, and are characterized by a uniform
distribution, and a nonlinear one for
, according to Equation (
23). The quality of the semi-analytical Ritz predictions can be noticed, also with reference to the skew buckled pattern, which is correctly predicted using a few degrees of freedom.
Following the study of [
36], the Graphite/Epoxy configuration is analyzed by evaluating the critical temperatures for all the combinations of
and
. between –90 and 90. Note that the current analyses do not consider any technological requirement, nor the effects of strong fiber steering on the constitutive law.
The results are calculated using ED3 theory with 12
12 functions, and are reported in the plot of
Figure 5.
The contour plot is very similar to the one reported in [
36], to which the interested reader is referred. The symmetry of the plot can be noted, due to the plate configuration, which is such that
. By considering an angle step of 2.5, the configuration guaranteeing the maximum is characterized by the lay-up
. The orientation of the angle at the plate center is close to the optimal one reported in
Table 3, while a slight difference is noted with respect to the fiber orientation at the plate edge.
Overall, this kind of analysis illustrates the potentialities of a Ritz-based procedure. Indeed, the possibility of exploring the design spaces offered by variable-stiffness configurations is subjected to the availability of computationally efficient analysis tools. For instance, the total number of analyses for generating the plot of
Figure 5 is, in this case, beyond 5000. The Ritz approach, due to the few degrees of freedom required to achieve convergence, is particularly suited for this, and the time for the analysis is on the order of few seconds.
3.2. Pre-Buckling and Buckling Response of Carbon/Epoxy VSP
Having demonstrated the capabilities of the Ritz formulation, further results are discussed in this section. The present study focuses on carbon/epoxy composites, which are commonly employed in advanced constructions such as in the aerospace field. The material properties are those relative to the Graphite/Epoxy material of
Table 3, and taken from [
36]. For convenience, they are reported here below:
= 155,000 MPa,
= = 8070 MPa,
= = = 4550 MPa,
= = = 0.22,
= 1/C,
= 2.0.
It can be noted that the orthotropy ratio is approximately 20; the thermal coefficient along the fiber direction is slightly negative, meaning that a shortening is experienced by the material when subjected to heating, while the value is positive along the transverse direction, i.e., the matrix expands when heated. The distribution of thermal and elastic properties influences the pre-buckling stress distribution, and, in general, can be tailored by allowing the fibers to change their orientation in the panel domain. The pre-buckling distribution of internal membrane forces, in turn, affects the buckling response of the plate. Furthermore, the overall response depends upon the boundary conditions applied to the plate, with in-plane ones having the twofold effect of influencing the pre-buckling response and buckling one. Note that, for symmetric thin laminates, in-plane boundary conditions are relevant only in terms of pre-buckling response, as the out-of-plane buckling equation is uncoupled from the in-plane ones. For thick plates, whenever the assumption of in-plane stress constitutive equation is not adequate to model the plate elastic response, an inherent coupling between in-plane and out-of-plane behaviour is recovered, and the buckling behaviour is affected by the in-plane boundary conditions as well.
A set of panels is studied below here by considering the orientation of the fibers to vary along the y-axis, and the in-plane boundary conditions Case-Ty1 and Case-Ty2. The panels are square, with dimensions equal to 150 mm, and are obtained by the stacking of four plies oriented at , where the angles are measured with respect to the x axis. To illustrate how the panel response is affected by the fiber path, the angles are varied between 0 and 90, while two distinct angles are considered, and taken equal to 15 and 75. The first case corresponds to a plate with a relatively high stiffness along the x direction, at least in the middle strips of the plate close to y=0; in the second case the central region is much weaker along the x-direction due to the almost transverse orientation of the fibers.
Firstly, the results are assessed in terms of pre-buckling behaviour; then, the buckling response is illustrated, reporting also the comparison with finite element calculations.
3.2.1. Pre-Buckling Analysis
Results relative to the boundary conditions Case-Ty1 are presented in
Figure 6. It is recalled that the approach developed in the previous section is exact. The comparison against finite element calculations is not presented for the sake of conciseness. However, the correctness of the derivation and its implementation has been verified through an extensive set of comparisons against finite element results.
The membrane forces per unit length are calculated for a unitary temperature increase according to Equation (
32), and are reported by illustrating the resultant
in the left portion of the plot, and the transverse resultant
in the right part. Positive values denote traction, while negative ones are associated with compression. Note that the curves are reported for half of the plate’s width. The remaining part can be easily recovered by exploiting the symmetry of the internal distribution with respect to the vertical axis.
Referring to
Figure 6a, one can note that low values of
are associated with fiber paths running closer to the longitudinal direction. Given the high value of the ratio
, the thermal expansion mostly relates to the transverse direction of the laminate. Due to the prescribed null displacement along the longitudinal edges, a larger force
is introduced with respect to those cases with higher angles
. At the same time, the compressive force
tends to increase its value for increasing orientations of the fiber angle
at the panel’s edge. Looking, for instance, at the case with
equal to 90, it is possible to notice the higher compressive values at the outer edge, where the thermoelastic response is dictated by the response along the matrix direction. The compressive force becomes smaller when moving towards the center of the panel, i.e., when the fibers are progressively rotated along the axial direction
x.
The second case of
Figure 6b is relative to a panel with fibers oriented at
at the panel’s center. In this case, the steering to small values of
(see, e.g., the case for
) determines a matrix-dominated response along the
y-direction. The thermal dilatation is thus reacted with higher values of
. As it concerns the longitudinal direction, one can note that compressive forces are milder on the outer regions, as far as the fibers are oriented with an angle
, which is smaller than
: in these cases, the steering is such that the matrix contribution to thermal dilatation increases when moving towards the outer regions. An opposite response is observed for
equal to 90, where the thermal dilation coefficient at the outer part is equal to that of the matrix, and larger with respect to the center, and similarly the force
.
A second set of results is available in
Figure 7, where the in-plane boundary conditions Case-Ty2 are considered. The displacements along the normal direction are prevented at the transverse edges, whilst the longitudinal ones are free to expand or contract. This set of constraints is representative of a design situation where two sides of the plate are restrained with stringers characterized by high in-plane bending stiffness, such that the expansion is prevented. On the contrary, the in-plane bending stiffness of the two other parallel sides is much smaller, and the motion along the normal direction is not restrained.
From the results of Equation (
36), it is seen that no membrane forces develop along the transverse direction as a response to a uniform heating or cooling. The distribution of internal forces is then purely uniaxial. The behaviour of plates characterized by
equal to 15 is summarized in
Figure 7a. One can note that steering the angle
from 0 to 90 has the effect of increasing the laminate thermal coefficient, thus increasing the amount of compression introduced at the outer edges. For the case at hand, the smallest values of
(0, 22.5) are characterized by a state of internal traction: thermal buckling is thus possible only if the plate is cooled. The same conclusion holds for
equal to 45 and 67.5, as the compression region is relatively small. When
is equal to 90, an internal state of compression is promoted, thus making the onset of instability phenomena possible due to heating.
The results of
Figure 7b are relative to configurations where the fiber angle
is equal to 75. This set of configurations is particularly interesting, as a mechanical buckling-driven design would suggest configurations where the mid angle is approximately perpendicular to the load direction in order to allow load re-distribution towards the edges. In this case, fiber angles
equal to 90, 67.5 and 45 are associated with compressive pre-buckling force per unit length
. In other words, the matrix contribution to thermal expansion is dominant along the direction
y, resulting in a compressive force in response to the prevented displacement along
y at the two transverse edges. For relatively high values of fiber steering, i.e., when the external fibers are oriented at 0 or 22.5, the small thermal expansion associated with the fiber direction is such that the external parts of the plate experience a traction force resultant, whilst the inner part still is subjected to the matrix-dominated compression forces. Due to the contemporary presence of compressive and tensile internal forces, this latter configuration can undergo instability phenomena due to thermal heating.
3.2.2. Buckling Analysis
Non-dimensional buckling temperatures are reported in
Table 4 for the plates discussed in the previous section. In-plane boundary conditions Case-Ty1 and Case-Ty2 are considered, while simply supported assumptions are introduced for analyzing the buckling problem.
The results are presented by considering a third-order equivalent single layer theory, ED3, while 14 trial functions are considered along both the orthogonal directions.
The results illustrate the dependence of the critical temperature with respect to the fiber path, boundary conditions and geometry of the plate. As expected, smaller values of , i.e., thicker plates, lead to higher critical temperatures. In particular, it is noted that the pre-buckling internal forces do not depend on the panel relative thickness , but they are a function of the thermoelastic properties only, i.e., the free thermal strains are not affected by the geometry of the plate. As a consequence, plates with different values of , and heated with a unitary temperature increase, are subjected to the same internal resultants. On the contrary, the buckling condition depends upon the square of the ratio , meaning that thicker plates require higher levels of internal membrane forces to buckle, and thus higher temperature increase. Based on these remarks, one can observe that the ratio between the critical temperatures associated with values of equal to 100 and 50 are slightly smaller than four due to effects of transverse shear flexibility. Similarly, the ratio of the temperatures of plates characterized by equal to 50 and 20 is slightly smaller than 6.25.
Looking at the role played by in-plane boundary conditions, it can be noted that, for a given non-dimensional parameter
, all of the critical temperatures fall in a relatively small range of value when the plate is constrained with Case-Ty1 conditions. In particular, the maximum ratio between the highest and the lowest temperatures is equal to 2, approximately. This is due to the inherent biaxiality of the internal pre-buckling forces. As noted, from the plots of
Figure 6, the fiber steering has, in general, the effect of mitigating the internal compression along one direction, while increasing that on the mutually orthogonal direction, and vice versa.
A different response is observed for the in-plane conditions identified by Case-Ty2. In this case, the uniaxiality of the internal forces leads to a more pronounced dependence on the fiber path. Some configurations do not buckle under thermal heating, while they are prone to buckling as far as the fiber is steered along the transverse direction at panel edges.
For clarity, a design chart is presented in
Figure 8, where the critical temperatures are reported for Case-Ty2 conditions and different stacking sequences.
The results for straight fiber configurations display, with close agreement, the behaviour illustrated by Nemeth [
4]: heating-induced instability is possible in the range between
equal to 44 and 90 degrees, while cooling is the only mechanism promoting buckling in the range between 0 and 43 degrees.
The curves of
Figure 8 highlight the effect of steering the fibers. One can note that the region of instability due to heating tends to shift on the upper-right part of the plot as
is reduced. Indeed, resistance against thermal heating is provided by longitudinally directed fibers. On the contrary, the effect of progressively rotating the center plate fibers along the longitudinal direction is that of shifting the cooling stability region to the bottom left of the plot: the matrix-dominated behaviour, at least in the central part of the plate, has the effect of improving the resistance to cooling. It is interesting to highlight that, for some range of values, buckling is possible both for heating and cooling, for temperatures that may reasonably be retained in the field of material linear response. For instance, when
is equal to 75 and
is 0, positive and negative critical non-dimensional temperatures are possible with absolute values of 24.74 and 50.68, respectively. The co-existence of positive and negative buckling multipliers is not possible for straight-fiber configurations, where the uniform pre-buckling condition is strictly compressive or tensile. The outlined response is thus peculiar of variable-stiffness plates, where the variability of the internal pre-buckling forces allows for the presence of non-uniform membrane force resultants with compressive and tensile regions. With similar motivations, one can observe that thermal buckling due to heating is always possible for the configurations with
equal to 75, irrespective of the values of the fiber orientation at the edge
.
Additional results are presented in
Figure 9, where the comparison is presented in terms of pre-buckling stress resultants and buckled shape for three variable-stiffness configurations with lay-up
, and one constant stiffness panel with
. The internal stress distribution is evaluated referring to Equation (
36) by considering a unitary temperature increase. As seen from
Figure 9a–d, the internal stress non-uniformity is progressively reduced as the angle
is increased up to 90. Note that, for clarity, the same color scale is adopted in all of the plots. With the exception of the case
, which is the buckling configuration associated with the plate cooling, all the other buckled surfaces are characterized by two half-waves, with increased skewness for smaller values of
due to increased bending-twisting coupling. Despite the similarity of the buckling modes, it is noted that the critical temperatures, available from
Figure 8, are drastically affected by the steering of the fibers.
The case of clamped conditions is summarized in
Figure 10 to quantify the effect of flexural boundary conditions.
As expected, the effect of adding constraints is that of raising the values of critical temperatures. It can be noted that the same trends observed for the simply supported case are essentially preserved, the main difference regarding a translation of the stability regions towards higher absolute values of the temperature.
The effects of normal and transverse shear deformability are investigated in
Table 5 by presenting the critical temperatures for different values of
. All the results are obtained referring to a layer-wise, second-order LD2 theory, which guarantees the possibility of achieving quasi-3D predictions. Aiming at highlighting the role played by higher-order deformability effects, the results are now presented according to the following non-dimensional form [
4]:
In other words, the non-dimensional temperature is expressed in terms of the buckling coefficient , which, for thin plate theory, eliminates the dependence on the geometric parameter . In the general case of higher-order theories and for fixed plate configuration, any variation of is only ascribable to normal and transverse shear flexibility effects.
The results provide a clear insight into the importance of properly accounting for shear deformation effects, when thick and moderately thick plates of concern. For the configurations at hand, the buckling coefficients associated with equal to 10 are between 0.50 and 0.78 times the values registered for the thin configurations with equal to 1000.
It is interesting to note that the influence of the fiber orientation on transverse shear effects is relatively weak when the plate is constrained with Case-Ty1 conditions. This behaviour is ascribable to the inherent biaxiality of the internal membrane forces, which tends to reduce transverse shear deformation effects.
When the plate is constrained with Case-Ty2 conditions, the internal forces are uniaxial, and the effects of transverse shear deformability are more pronounced. Referring to
Table 5, it can be noted that the relevance of transverse shear effects is higher for increasing intensities of fiber steering. The effect is at a maximum for the lay-up
, where the buckling coefficient associated with
= 10 is almost half of the one predicted for the thin plate counterpart. In those cases, the need for adopting proper kinematic theories is evident. At the same time, even the analysis of moderately thick plates in the range of
between 50 and 20 is susceptible to noticeable approximation if thin plate theory is adopted, especially for variable-stiffness configurations with aggressive steering.