This new work aims to develop a full coupled thermomechanical method including both the temperature profile and displacements as primary unknowns of the model. This generic full coupled 3D exact shell model permits the thermal stress investigation of laminated isotropic, composite and sandwich structures. Cylindrical and spherical panels, cylinders and plates are analyzed in orthogonal mixed curved reference coordinates. The 3D equilibrium relations and the 3D Fourier heat conduction equation for spherical shells are coupled and they trivially can be simplified in those for plates and cylindrical panels. The exponential matrix methodology is used to find the solutions of a full coupled model based on coupled differential relations with respect to the thickness coordinate. The analytical solution is based on theories of simply supported edges and harmonic relations for displacement components and sovra-temperature. The sovra-temperature magnitudes are directly applied at the outer faces through static state hypotheses. As a consequence, the sovra-temperature description is assumed to be an unknown variable of the model and it is calculated in the same way as the three displacements. The final system is based on a set of coupled homogeneous differential relations of second order in the thickness coordinate. This system is reduced in a first order differential relation system by redoubling the number of unknowns. Therefore, the exponential matrix methodology is applied to calculate the solution. The temperature field effects are evaluated in the static investigation of shells and plates in terms of displacement and stress components. After an appropriate preliminary validation, new benchmarks are discussed for several thickness ratios, geometrical data, lamination sequences, materials and sovra-temperature values imposed at the outer faces. Results make evident the accordance between the uncoupled thermo-mechanical model and this new full coupled thermo-mechanical model without the need to separately solve the Fourier heat conduction relation. Both effects connected with the thickness layer and the related embedded materials are included in the conducted thermal stress analysis.
The monitoring of temperature gradients in the aerospace structures is one of the main topics in the stress analysis. The study of temperature contribution on strains and stresses is very important for modern aircraft, spaceships, launchers, high-tech propulsion devices, pressure vessels for nuclear energy applications and other installations for industries. In each proposed application, a dedicated structural thermal analysis must be conducted where several features must be included, e.g., heat transfer conditions, transitory and static state thermal stress components, production and remaining stress components, vibrations of heated plate and shell structures, large deflections, post-buckling behavior and analyses of shell and plate geometries [
The most recent works in literature about the thermo-elastic investigation of laminated and one-layered shells and plates are discussed in four different sections: the first one is related to 3D exact solutions, the second one is on 3D numerical models, the third one is related to 2D exact solutions and the last part is on 2D numerical models. Therefore, the innovative points of this new three-dimensional full coupled thermo-elastic shell theory for the thermal stress investigation of sandwich and multilayered structures are properly focused.
Bhaskar et al. [
Numerical 3D models are more general than exact 3D models but even more complicated. They permit the analysis of general boundary, loading and lamination conditions. Moleiro et al. [
Important simplified assumptions are performed through the thickness direction of the structures in 2D exact models. The 2D exact model for shells proposed in [
In the framework of two-dimensional numerical theories, the work [
The present new 3D coupled thermo-elastic shell model has the following innovative points. It is general for cylindrical and spherical shells, cylinders and plates including different composite, orthotropic and isotropic layers. The closed-form solution of equations is written using simply-supported hypotheses for all the sides as boundary conditions and harmonic structures for thermal and elastic primary unknowns. The system of differential relations is solved using the exponential matrix methodology, along the thickness direction. In this way, a layer-wise theory is possible where equilibrium and compatibility conditions can be easily imposed. Moreover, the included temperature evaluation permits taking into account the thermal thickness layer effect and the thermal material layer effect without separately solving the 3D Fourier heat conduction equation. In the open literature, there are not 3D exact full coupled thermo-elastic models based on the exponential matrix methodology able to analyze several benchmarks using the same tool. The proposed three-dimensional exact coupled thermo-elastic shell theory can be seen as a generalization of the pure elastic model already proposed by Brischetto in [
The proposed three-dimensional methodology is able to consider plate and shell geometries by means of a generic and unique theory where several structures can be analysed by means of the same tool. The innovative point with respect to the similar 3D model proposed by Brischetto et al. in [
The value
For shells with constant curvature radii, the parameters
The geometries proposed are loaded by means of a sovratemperature field
The constitutive equation allows the six strain components to be linked with the six stress components via the elastic coefficient matrix
The analytical form of
The closed-form solution of the coupled relations
The introduction of the harmonic expression of the displacements and temperature profile (
This system of equations can be decreased to a first order one via the methodology discussed in [
A general homogeneous system of first order differential relations can be proposed as:
In analogy, the resolution of
The requirements specified in
In
The diagonal part composed of 1 indicates the congruence conditions expressed as displacement components and the continuity of temperature written in
All structures taken into account have simply supported sides, this boundary condition is automatically imposed via the harmonic forms for all the primary unknowns:
In addition, load boundary requirements must be imposed at the top and bottom of the structures. These conditions can be written as:
In order to include
Using
Matrix
After the calculation of the displacements at the bottom,
The first part of this section shows three preliminary validation results to verify the developed general three-dimensional exact coupled thermo-elastic shell theory. Comparisons with [
Assessment 1 | Assessment 2 | Assessment 3 | |
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a [m] | 2, 4, 10, 20, 50, 100 |
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1 |
b [m] | 2, 4, 10, 20, 50, 100 | 1 | 1 |
h [m] | 1 | 2, 1, 0.2, 0.1, 0.01 | 0.1 |
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10 | 5, 10, 50 |
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5, 10, 50 |
h1 | h/3 | h/2 | h/2 |
h2 | h/3 | h/2 | h/2 |
h3 | h/3 | - | - |
Lamination | 0 |
Al2024/Ti22 | 0 |
|
+1 | +1 | +0.5 |
|
−1 | 0 | −0.5 |
m | 1 | 1 | 1 |
n | 1 | 1 | 1 |
B1 | B2 | B3 | B4 | B5 | B6 | B7 | B8 | |
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a [m] | s | s | 2 |
2 |
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|
b [m] | s | s | 30 | 30 | 20 | 20 |
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|
h [m] | 1 | 1 | t | t | t | t | t | t |
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|
10 | 10 | 10 | 10 | 10 | 10 |
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|
10 | 10 |
h1 [m] | h | h/3 | h | h/2 | h | 0.1 h | h | h/3 |
h2 [m] | - | h/3 | - | h/2 | - | 0.8 h | - | h/3 |
h3 [m] | - | h/3 | - | - | - | 0.1 h | - | h/3 |
Lamination | 0 |
0 |
Al2024 | Al2024/Ti22 | Ti22 | Al2024/PVC/Al2024 | Steel | Al2024/Ti22/Steel |
|
+0.5 | +0.5 | +0.5 | +0.5 | +1 | +1 | +1 | +1 |
|
−0.5 | −0.5 | −0.5 | −0.5 | 0 | 0 | 0 | 0 |
m | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 |
n | 1 | 1 | 1 | 1 | 0 | 0 | 1 | 1 |
M | 300 | 300 | 300 | 300 | 300 | 300 | 300 | 300 |
N | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |
Al2024 |
Ti22 |
|
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|
|
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|
73 | 110 | 3 | 210 | 25 | 25 | 172.72 |
|
73 | 110 | 3 | 210 | 1 | 1 | 6.909 |
|
73 | 110 | 3 | 210 | 1 | 1 | 6.909 |
|
0.3 | 0.32 | 0.4 | 0.3 | 0.25 | 0.25 | 0.25 |
|
0.3 | 0.32 | 0.4 | 0.3 | 0.25 | 0.25 | 0.25 |
|
0.3 | 0.32 | 0.4 | 0.3 | 0.25 | 0.25 | 0.25 |
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|
0.5 | 0.5 | 3.45 |
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|
0.5 | 0.5 | 3.45 |
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|
0.2 | 0.2 | 1.38 |
|
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|
1 | 1 |
|
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|
1125 | 3 |
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|
|
|
1125 | 1 |
|
|
130 | 21.9 | 0.18 | 60 | 36.42 | 36.42 | 36.42 |
|
130 | 21.9 | 0.18 | 60 | 0.96 | 0.96 | 0.96 |
|
130 | 21.9 | 0.18 | 60 | 0.96 | 0.96 | 0.96 |
a/h | 2 | 4 | 10 | 20 | 50 | 100 |
---|---|---|---|---|---|---|
|
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3D( |
48.87 | 32.12 | 16.39 | 11.93 | 10.47 | 10.25 |
|
||||||
3D-u- |
48.85 | 32.11 | 16.40 | 11.93 | 10.47 | 10.25 |
|
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3D-u- |
48.86 | 32.11 | 16.40 | 11.93 | 10.47 | 10.25 |
|
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3D-u- |
48.86 | 32.11 | 16.40 | 11.93 | 10.47 | 10.25 |
|
||||||
3D( |
487.6 | 796.8 | 948.0 | 961.8 | 964.3 | 964.5 |
|
||||||
3D-u- |
510.5 | 796.9 | 948.0 | 961.8 | 964.3 | 964.6 |
|
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3D-u- |
497.0 | 796.8 | 948.0 | 961.8 | 964.3 | 964.6 |
|
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3D-u- |
495.8 | 796.8 | 948.0 | 961.8 | 964.3 | 964.6 |
|
||||||
3D( |
142.9 | 119.4 | 71.96 | 56.46 | 51.27 | 50.50 |
|
||||||
3D-u- |
142.9 | 119.4 | 71.96 | 56.46 | 51.28 | 50.50 |
|
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3D-u- |
142.9 | 119.4 | 71.96 | 56.46 | 51.28 | 50.50 |
|
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3D-u- |
142.9 | 119.4 | 71.96 | 56.46 | 51.28 | 50.50 |
|
5 | 10 | 50 | 100 | 1000 |
---|---|---|---|---|---|
|
|||||
3D( |
0.0002 | 0.0010 | 0.0060 | 0.0129 | 0.0424 |
|
|||||
3D-u- |
0.0002 | 0.0010 | 0.0060 | 0.0129 | 0.0424 |
|
|||||
3D-u- |
0.0002 | 0.0010 | 0.0060 | 0.0129 | 0.0424 |
|
|||||
3D-u- |
0.0002 | 0.0010 | 0.0060 | 0.0129 | 0.0424 |
|
|||||
3D( |
−0.0031 | −0.0031 | −0.0027 | −0.0023 | 0.0009 |
|
|||||
3D-u- |
−0.0031 | −0.0031 | −0.0027 | −0.0023 | 0.0009 |
|
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3D-u- |
−0.0031 | −0.0031 | −0.0027 | −0.0023 | 0.0009 |
|
|||||
3D-u- |
−0.0031 | −0.0031 | −0.0027 | −0.0023 | 0.0009 |
|
50 | 100 | 500 |
---|---|---|---|
|
|||
3D( |
1.0142 | 1.0780 | 1.1000 |
|
|||
3D-u- |
1.0142 | 1.0780 | 1.1000 |
|
|||
3D-u- |
1.0142 | 1.0780 | 1.1000 |
|
|||
3D-u- |
1.0142 | 1.0780 | 1.1000 |
a/h | 2 | 5 | 10 | 20 | 50 | 100 |
---|---|---|---|---|---|---|
v[ |
||||||
3D( |
0.5261 | 0.5861 | 0.5918 | 0.7981 | 1.7020 | 3.3184 |
3D( |
0.5261 | 0.5861 | 0.5918 | 0.7981 | 1.7020 | 3.3184 |
3D( |
0.2718 | 0.4771 | 0.5572 | 0.7856 | 1.6977 | 3.3162 |
3D-u- |
0.2718 | 0.4771 | 0.5572 | 0.7856 | 1.6977 | 3.3162 |
w[ |
||||||
3D( |
0.0936 | 0.7345 | 2.3955 | 8.7090 | 52.699 | 209.77 |
3D( |
0.0936 | 0.7345 | 2.3955 | 8.7090 | 52.699 | 209.77 |
3D( |
0.0958 | 0.6221 | 2.2634 | 8.5750 | 52.565 | 209.64 |
3D-u- |
0.0958 | 0.6221 | 2.2634 | 8.5750 | 52.565 | 209.64 |
|
||||||
3D( |
13.240 | 10.687 | 10.238 | 10.105 | 10.065 | 10.059 |
3D( |
13.240 | 10.687 | 10.238 | 10.105 | 10.065 | 10.059 |
3D( |
2.7489 | 7.2060 | 9.1706 | 9.8214 | 10.019 | 10.047 |
3D-u- |
2.7480 | 7.2058 | 9.1705 | 9.8214 | 10.019 | 10.048 |
|
||||||
3D( |
1.0518 | 0.9566 | 0.5860 | 0.3108 | 0.1265 | 0.0634 |
3D( |
1.0518 | 0.9566 | 0.5860 | 0.3108 | 0.1265 | 0.0634 |
3D( |
0.5278 | 0.7701 | 0.5501 | 0.3057 | 0.1261 | 0.0633 |
3D-u- |
0.5278 | 0.7701 | 0.5501 | 0.3057 | 0.1261 | 0.0633 |
|
||||||
3D( |
1955.6 | 136.85 | 11.229 | 0.7601 | 0.0199 | 0.0013 |
3D( |
1955.6 | 136.85 | 11.229 | 0.7601 | 0.0199 | 0.0013 |
3D( |
−503.79 | −10.923 | 0.4205 | 0.0598 | 0.0018 | 0.0001 |
3D-u- |
−503.73 | −10.923 | 0.4205 | 0.0598 | 0.0018 | 0.0001 |
a/h | 2 | 5 | 10 | 20 | 50 | 100 |
---|---|---|---|---|---|---|
u[ |
||||||
3D( |
−0.0791 | −0.1779 | −0.3388 | −0.6635 | −1.6472 | −3.2909 |
3D( |
−0.0791 | −0.1779 | −0.3388 | −0.6635 | −1.6472 | −3.2909 |
3D( |
−0.0400 | −0.1451 | −0.3191 | −0.6532 | −1.6430 | −3.2888 |
3D-u- |
−0.0401 | −0.1451 | −0.3191 | −0.6532 | −1.6430 | −3.2888 |
w[ |
||||||
3D( |
0.1159 | 0.8948 | 2.6731 | 9.0380 | 53.045 | 210.12 |
3D( |
0.1159 | 0.8948 | 2.6731 | 9.0380 | 53.045 | 210.12 |
3D( |
0.1132 | 0.7622 | 2.5294 | 8.9002 | 52.910 | 209.99 |
3D-u- |
0.1132 | 0.7622 | 2.5294 | 8.9002 | 52.910 | 209.99 |
|
||||||
3D( |
6.9043 | 9.8311 | 10.943 | 11.336 | 11.459 | 11.477 |
3D( |
6.9043 | 9.8311 | 10.943 | 11.336 | 11.459 | 11.477 |
3D( |
9.4299 | 10.275 | 11.024 | 11.352 | 11.461 | 11.478 |
3D-u- |
9.4301 | 10.275 | 11.024 | 11.352 | 11.462 | 11.478 |
|
||||||
3D( |
1.1161 | 0.9225 | 0.5754 | 0.3091 | 0.1263 | 0.0634 |
3D( |
1.1161 | 0.9225 | 0.5754 | 0.3091 | 0.1263 | 0.0634 |
3D( |
0.5190 | 0.7375 | 0.5398 | 0.3040 | 0.1260 | 0.0633 |
3D-u- |
0.5189 | 0.7375 | 0.5398 | 0.3040 | 0.1260 | 0.0633 |
|
||||||
3D( |
1704.4 | 124.34 | 11.042 | 0.7725 | 0.0205 | 0.0013 |
3D( |
1704.4 | 124.34 | 11.042 | 0.7725 | 0.0205 | 0.0013 |
3D( |
−512.45 | −16.459 | 0.3721 | 0.0744 | 0.0024 | 0.0002 |
3D-u- |
−512.40 | −16.459 | 0.3721 | 0.0744 | 0.0024 | 0.0002 |
|
2 | 5 | 10 | 20 | 50 | 100 |
---|---|---|---|---|---|---|
u[ |
||||||
3D( |
3.0170 | 1.8209 | 0.9969 | 0.5167 | 0.2105 | 0.1059 |
3D( |
3.0170 | 1.8209 | 0.9969 | 0.5167 | 0.2105 | 0.1059 |
3D( |
3.0032 | 1.8196 | 0.9967 | 0.5167 | 0.2105 | 0.1059 |
3D-u- |
3.0032 | 1.8196 | 0.9967 | 0.5167 | 0.2105 | 0.1059 |
w[ |
||||||
3D( |
4.7030 | 2.3414 | 1.2126 | 0.6117 | 0.2453 | 0.1227 |
3D( |
4.7030 | 2.3414 | 1.2126 | 0.6117 | 0.2453 | 0.1227 |
3D( |
4.7108 | 2.3416 | 1.2126 | 0.6117 | 0.2453 | 0.1227 |
3D-u- |
4.7108 | 2.3416 | 1.2126 | 0.6118 | 0.2453 | 0.1227 |
|
||||||
3D( |
4.3550 | 3.4394 | 2.0110 | 1.0738 | 0.4451 | 0.2250 |
3D( |
4.3550 | 3.4394 | 2.0110 | 1.0738 | 0.4451 | 0.2250 |
3D( |
4.2954 | 3.4341 | 2.0102 | 1.0737 | 0.4451 | 0.2250 |
3D-u- |
4.2954 | 3.4341 | 2.0102 | 1.0737 | 0.4451 | 0.2251 |
|
||||||
3D( |
−8.4089 | −4.3683 | −2.3272 | −1.1947 | −0.4846 | −0.2434 |
3D( |
−8.4089 | −4.3683 | −2.3272 | −1.1947 | −0.4846 | −0.2434 |
3D( |
−8.3275 | −4.3624 | −2.3264 | −1.1946 | −0.4846 | −0.2434 |
3D-u- |
−8.3275 | −4.3624 | −2.3264 | −1.1946 | −0.4846 | −0.2434 |
|
||||||
3D( |
14.109 | 5.3127 | 2.5599 | 1.2521 | 0.4937 | 0.2456 |
3D( |
14.109 | 5.3127 | 2.5599 | 1.2521 | 0.4937 | 0.2456 |
3D( |
13.978 | 5.3055 | 2.5591 | 1.2520 | 0.4937 | 0.2456 |
3D-u- |
13.978 | 5.3055 | 2.5591 | 1.2520 | 0.4937 | 0.2457 |
|
2 | 5 | 10 | 20 | 50 | 100 |
---|---|---|---|---|---|---|
v[ |
||||||
3D( |
1.4618 | 0.8157 | 0.8649 | 0.9555 | 1.0315 | 1.0604 |
3D( |
2.9972 | 2.7826 | 3.1004 | 3.3448 | 3.5181 | 3.5802 |
3D( |
2.8572 | 2.7563 | 3.0931 | 3.3429 | 3.5178 | 3.5801 |
3D-u- |
2.8572 | 2.7563 | 3.0931 | 3.3429 | 3.5178 | 3.5801 |
w[ |
||||||
3D( |
0.9037 | −0.6784 | −1.4092 | −1.7945 | −2.0284 | −2.1065 |
3D( |
−3.1677 | −5.4294 | −6.3630 | −6.8341 | −7.1132 | −7.2051 |
3D( |
−2.8414 | −5.3676 | −6.3469 | −6.8300 | −7.1125 | −7.2050 |
3D-u- |
−2.8414 | −5.3676 | −6.3469 | −6.8300 | −7.1125 | −7.2050 |
|
||||||
3D( |
−60.434 | −77.267 | −82.907 | −85.559 | −87.072 | −87.561 |
3D( |
−103.96 | −121.98 | −127.17 | −129.39 | −130.57 | −130.94 |
3D( |
−100.49 | −121.39 | −127.02 | −129.35 | −130.57 | −130.94 |
3D-u- |
−100.49 | −121.39 | −127.02 | −129.35 | −130.57 | −130.94 |
|
||||||
3D( |
−9.4212 | −4.4541 | −2.2626 | −1.1307 | −0.4509 | −0.2251 |
3D( |
−10.034 | −4.5094 | −2.2221 | −1.0907 | −0.4298 | −0.2137 |
3D( |
−9.8714 | −4.5011 | −2.2213 | −1.0906 | −0.4298 | −0.2137 |
3D-u- |
−9.8714 | −4.5011 | −2.2213 | −1.0906 | −0.4298 | −0.2137 |
|
||||||
3D( |
12.392 | 4.6425 | 2.2412 | 1.0977 | 0.4333 | 0.2156 |
3D( |
13.659 | 4.7199 | 2.2012 | 1.0583 | 0.4129 | 0.2047 |
3D( |
13.403 | 4.7109 | 2.2004 | 1.0582 | 0.4129 | 0.2047 |
3D-u- |
13.403 | 4.7109 | 2.2004 | 1.0582 | 0.4129 | 0.2047 |
|
2 | 5 | 10 | 20 | 50 | 100 |
---|---|---|---|---|---|---|
u[ |
||||||
3D( |
−3.5730 | −1.7728 | 0.6652 | 5.4209 | 19.623 | 43.275 |
3D( |
−3.5730 | −1.7728 | 0.6652 | 5.4209 | 19.623 | 43.275 |
3D( |
−3.2200 | −1.7140 | 0.6772 | 5.4223 | 19.622 | 43.274 |
3D-u- |
−3.2200 | −1.7140 | 0.6772 | 5.4223 | 19.622 | 43.276 |
w[ |
||||||
3D( |
2.0687 | 6.3251 | 13.444 | 27.651 | 70.233 | 141.18 |
3D( |
2.0687 | 6.3251 | 13.444 | 27.651 | 70.233 | 141.18 |
3D( |
2.1059 | 6.3144 | 13.431 | 27.642 | 70.229 | 141.18 |
3D-u- |
2.1059 | 6.3144 | 13.431 | 27.642 | 70.228 | 141.18 |
|
||||||
3D( |
7.0586 | 0.0892 | −0.5280 | −0.4182 | −0.2053 | −0.1117 |
3D( |
7.0586 | 0.0892 | −0.5280 | −0.4182 | −0.2053 | −0.1117 |
3D( |
−5.0098 | −2.1901 | −1.1261 | −0.5712 | −0.2301 | −0.1179 |
3D-u- |
−5.0100 | −2.1902 | −1.1261 | −0.5712 | −0.2305 | −0.1156 |
|
||||||
3D( |
1.5486 | −0.2240 | −0.2448 | −0.1544 | −0.0694 | −0.0351 |
3D( |
1.5486 | −0.2240 | −0.2448 | −0.1544 | −0.0694 | −0.0351 |
3D( |
−1.7924 | −0.7635 | −0.3792 | −0.1879 | −0.0747 | −0.0364 |
3D-u- |
−1.7924 | −0.7635 | −0.3792 | −0.1879 | −0.0746 | −0.0372 |
|
||||||
3D( |
346.96 | −27.698 | −51.184 | −19.624 | −3.8665 | −0.9796 |
3D( |
346.96 | −27.698 | −51.184 | −19.624 | −3.8665 | −0.9796 |
3D( |
14.887 | −285.50 | −91.627 | −25.148 | −4.2371 | −1.0266 |
3D-u- |
14.887 | −285.50 | −91.627 | −25.147 | −4.2213 | −1.0709 |
|
2 | 5 | 10 | 20 | 50 | 100 |
---|---|---|---|---|---|---|
u[ |
||||||
3D( |
−10.134 | −6.2106 | 0.7010 | 14.818 | 57.457 | 128.61 |
3D( |
−10.547 | −6.0658 | 1.7093 | 17.534 | 65.279 | 144.94 |
3D( |
−10.428 | −6.0160 | 1.7227 | 17.536 | 65.279 | 144.94 |
3D-u- |
−10.428 | −6.0161 | 1.7227 | 17.536 | 65.278 | 144.95 |
w[ |
||||||
3D( |
2.8059 | 15.568 | 37.159 | 80.036 | 208.30 | 421.89 |
3D( |
3.0068 | 17.794 | 42.099 | 90.160 | 233.78 | 472.91 |
3D( |
3.4353 | 17.879 | 42.112 | 90.159 | 233.78 | 472.90 |
3D-u- |
3.4353 | 17.879 | 42.112 | 90.159 | 233.78 | 472.93 |
|
||||||
3D( |
11.276 | 5.8276 | 5.0898 | 4.8953 | 4.8300 | 4.8193 |
3D( |
6.0584 | 0.8660 | 0.1630 | −0.0221 | −0.0839 | −0.0936 |
3D( |
3.7887 | 0.6018 | 0.1018 | −0.0369 | −0.0863 | −0.0941 |
3D-u- |
3.7886 | 0.6018 | 0.1018 | −0.0369 | −0.0860 | −0.0969 |
|
||||||
3D( |
2.4049 | 0.8769 | 0.4277 | 0.2118 | 0.0843 | 0.0421 |
3D( |
2.3775 | 0.8540 | 0.4130 | 0.2036 | 0.0808 | 0.0403 |
3D( |
1.8442 | 0.8126 | 0.4077 | 0.2029 | 0.0808 | 0.0403 |
3D-u- |
1.8442 | 0.8126 | 0.4077 | 0.2029 | 0.0808 | 0.0403 |
|
||||||
3D( |
−22.828 | 0.1782 | 0.1182 | 0.0657 | 0.0278 | 0.0142 |
3D( |
−728.70 | 0.2087 | 0.1288 | 0.0698 | 0.0292 | 0.0148 |
3D( |
−273.08 | 0.2005 | 0.1273 | 0.0696 | 0.0292 | 0.0148 |
3D-u- |
−273.07 | 0.2005 | 0.1273 | 0.0696 | 0.0292 | 0.0148 |
|
2 | 5 | 10 | 20 | 50 | 100 |
---|---|---|---|---|---|---|
u[ |
||||||
3D( |
−2.9176 | −1.7996 | −0.8976 | −0.3635 | −0.1096 | −0.0478 |
3D( |
−2.9176 | −1.7996 | −0.8976 | −0.3635 | −0.1096 | −0.0478 |
3D( |
−2.4554 | −1.7161 | −0.8837 | −0.3617 | −0.1095 | −0.0478 |
3D-u- |
−2.4554 | −1.7161 | −0.8837 | −0.3617 | −0.1095 | −0.0478 |
w[ |
||||||
3D( |
2.5188 | 5.1544 | 6.5432 | 6.7022 | 6.3930 | 6.2141 |
3D( |
2.5188 | 5.1544 | 6.5432 | 6.7022 | 6.3930 | 6.2141 |
3D( |
2.1970 | 4.9750 | 6.4703 | 6.6807 | 6.3894 | 6.2132 |
3D-u- |
2.1970 | 4.9750 | 6.4703 | 6.6807 | 6.3894 | 6.2132 |
|
||||||
3D( |
−63.025 | −43.070 | −17.981 | −5.0090 | −0.6002 | −0.0429 |
3D( |
−63.025 | −43.070 | −17.981 | −5.0090 | −0.6002 | −0.0429 |
3D( |
−50.132 | −40.541 | −17.589 | −4.9637 | −0.5977 | −0.0426 |
3D-u- |
−50.132 | −40.541 | −17.589 | −4.9637 | −0.5977 | −0.0426 |
|
||||||
3D( |
16.767 | 3.5155 | −2.2579 | −3.0027 | −1.7068 | −0.9349 |
3D( |
16.767 | 3.5155 | −2.2579 | −3.0027 | −1.7068 | −0.9349 |
3D( |
6.5367 | 2.3933 | −2.3892 | −3.0150 | −1.7074 | −0.9350 |
3D-u- |
6.5366 | 2.3933 | −2.3892 | −3.0150 | −1.7074 | −0.9350 |
|
||||||
3D( |
1.4441 | 2.9960 | 3.6512 | 2.6452 | 1.2471 | 0.6508 |
3D( |
1.4441 | 2.9960 | 3.6512 | 2.6452 | 1.2471 | 0.6508 |
3D( |
4.8006 | 3.0618 | 3.6338 | 2.6400 | 1.2466 | 0.6507 |
3D-u- |
4.8006 | 3.0618 | 3.6338 | 2.6400 | 1.2466 | 0.6507 |
|
2 | 5 | 10 | 20 | 50 | 100 |
---|---|---|---|---|---|---|
v[ |
||||||
3D( |
0.6479 | 1.6264 | 1.3680 | 0.8175 | 0.3458 | 0.1743 |
3D( |
1.3282 | 2.0430 | 1.5665 | 0.8946 | 0.3680 | 0.1838 |
3D( |
1.3841 | 2.0187 | 1.5592 | 0.8934 | 0.3679 | 0.1838 |
3D-u- |
1.3841 | 2.0187 | 1.5592 | 0.8934 | 0.3679 | 0.1838 |
w[ |
||||||
3D( |
2.9557 | 5.8248 | 7.4366 | 7.7817 | 7.6037 | 7.4685 |
3D( |
3.1261 | 6.2203 | 7.6461 | 7.7384 | 7.3669 | 7.1651 |
3D( |
2.9405 | 6.1003 | 7.5992 | 7.7248 | 7.3647 | 7.1646 |
3D-u- |
2.9405 | 6.1003 | 7.5992 | 7.7248 | 7.3647 | 7.1646 |
|
||||||
3D( |
−30.353 | −39.796 | −63.025 | −94.912 | −121.52 | −131.22 |
3D( |
−16.005 | −31.260 | −63.948 | −101.82 | −131.33 | −141.73 |
3D( |
−49.975 | −38.883 | −65.959 | −102.30 | −131.40 | −141.75 |
3D-u- |
−49.976 | −38.883 | −65.959 | −102.30 | −131.40 | −141.75 |
|
||||||
3D( |
13.084 | 5.4242 | 0.6419 | −0.6211 | −0.4972 | −0.2884 |
3D( |
12.141 | 3.6918 | −0.6355 | −1.3225 | −0.7766 | −0.4262 |
3D( |
11.798 | 3.6010 | −0.6378 | −1.3213 | −0.7765 | −0.4262 |
3D-u- |
11.798 | 3.6010 | −0.6378 | −1.3213 | −0.7765 | −0.4262 |
|
||||||
3D( |
−2.1848 | −0.6175 | 0.5564 | 0.6729 | 0.3720 | 0.2022 |
3D( |
2.6622 | 1.1518 | 1.5228 | 1.1590 | 0.5600 | 0.2944 |
3D( |
3.3096 | 1.1724 | 1.5195 | 1.1579 | 0.5599 | 0.2944 |
3D-u- |
3.3096 | 1.1724 | 1.5195 | 1.1579 | 0.5599 | 0.2944 |
A number of three preliminary validation results are employed for the validation of this new three-dimensional general exact coupled thermo-elastic shell model, indicated as 3D-u-
The first preliminary validation result considers a square plate having simply-supported sides. The geometrical data can be seen in the second column of
The second preliminary validation result shows a two-layered isotropic cylindrical shell having simply-supported sides. The geometrical data can be seen in the third row of
The last and third preliminary validation result proposes a simply-supported spherical shell. The configuration and the geometrical data can be visualized in
The next section proposes new benchmarking analyses where different geometrical data, thickness ratios, lamination stacking sequences, materials, temperature profiles and displacement-stress studies are proposed. Even if the preliminary results suggest
Eight benchmarking analyses are here discussed for cylindrical and spherical panels, cylinders and plates. Temperature is enforced at the outer faces with different amplitudes and half-wave numbers (see
The first benchmark represents a square one-layered composite plate with simply-supported sides (see
The second benchmark shows a square composite plate, equally divided in three layers, having simply-supported sides (in
The third benchmark shows an isotropic homogeneous cylinder having simply supported sides (see
The benchmark number four discusses an isotropic two-layered cylinder with simply supported sides (see
The fifth benchmark analyses an isotropic one-layered cylindrical panel having simply supported sides (see
The benchmark number six analyses the thermal stress investigation of a sandwich cylindrical panel (in
The benchmark seven proposes an isotropic one-layered spherical panel having simply-supported sides (see
The last and eighth benchmark shows an isotropic three-layered spherical panel having simply-supported sides (consult
An overall full coupled thermo-elastic 3D shell model in closed-form solution for the thermal stress investigation of single-and multi-layered isotropic, sandwich and composite plate and shell geometries has been discussed. The sovra-temperature maximum values have been directly applied at the highest and lowest point of outer faces in static conditions. The temperature profile is then evaluated through the thickness direction. The temperature profile is a primary variable as the displacements thanks to the coupling between the 3D Fourier heat conduction equation and the 3D equilibrium equations for shells. This sovra-temperature profile considers both the effects connected with the thickness layer and the embedded material for each possible case, without the necessity to separately solve the 3D or 1D form of the Fourier heat conduction equation. The global coupled system given by the three-dimensional equilibrium equations for shells and the three-dimensional Fourier heat conduction relation for shells is exactly solved via the exponential matrix methodology. Different results, given as displacement components, in-plane and out-of-plane stress components and sovra-temperature evaluations have been discussed for several thickness ratio values, geometrical properties, lamination sequences, temperature impositions and materials. These analyses showed a complete match between the 3D uncoupled model that uses the 3D Fourier heat conduction relation and the 3D full coupled one. This last new method has the advantage that takes into account both the thickness and the material layer effects using a mathematical formulation that is simpler and more elegant because the three-dimensional Fourier heat conduction relation is not separately solved. Moreover, a reduced number of artificial layers M is requested in comparison with the uncoupled 3D model.
The authors received no specific funding for this study.
The authors declare that they have no conflicts of interest to report regarding the present study.