The paper proposes a threedimensional elastic analysis of the free vibration problem of onelayered spherical, cylindrical, and flat panels. The exact solution is developed for the differential equations of equilibrium written in orthogonal curvilinear coordinates for the free vibrations of simply supported structures. These equations consider an exact geometry for shells without simplifications. The main novelty is the possibility of a general formulation for different geometries. The equations written in general orthogonal curvilinear coordinates allow the analysis of spherical shell panels and they automatically degenerate into cylindrical shell panel, cylindrical closed shell, and plate cases. Results are proposed for isotropic and orthotropic structures. An exhaustive overview is given of the vibration modes for a number of thickness ratios, imposed wave numbers, geometries, embedded materials, and angles of orthotropy. These results can also be used as reference solutions to validate twodimensional models for plates and shells in both analytical and numerical form (e.g., closed solutions, finite element method, differential quadrature method, and global collocation method).
Exact solutions for the threedimensional analysis of plates and shells have been developed by several researchers. In particular, threedimensional static and dynamic analysis of cylinders and cylindrical/spherical shells has received considerable attention in the past few years. The development of plate and shell elements highlights the importance of research on the design of these geometries because they are fundamental for the analysis of general structures in the engineering field. For this reason, plate and shell elements need accurate validation. Threedimensional exact solutions allow such validations and checks to be made, and they also give further details about threedimensional effects and their importance. In the literature, works about exact threedimensional solutions analyze separately the various geometries and they do not give a general overview of plate and shell elements. The present paper aims to fill this gap by proposing a general formulation for the equations of motion in orthogonal curvilinear coordinates that is valid for square and rectangular plates, cylindrical shell panels, spherical shell panels, and cylinders. A general overview is given for those readers interested in both plate and shell analysis. This paper exactly solves the equations of motion in general curvilinear orthogonal coordinates including an exact geometry for shell structures without simplifications. To the best of the present author’s knowledge, this is the first time that this solution is proposed by means of the exponential matrix method for the threedimensional elastic free vibration analysis of plates and shells.
The next paragraph discusses the most relevant works about threedimensional shell analysis. Chen et al. [
Threedimensional analysis of plates is usually performed by using simpler equations that do not allow the analysis of shell geometries. Demasi [
The revision of the literature about threedimensional analysis of shells and plates demonstrates that there are a variety of interesting works concerning plate or shell geometry, and they include static and dynamic analysis, functionally graded, composite and piezoelectric materials, displacement, and mixed models. The present work gives a general formulation for all the geometries (square and rectangular plates, cylindrical and spherical shell panels, and cylindrical closed shells). The equations of motion for the dynamic case are written in general orthogonal curvilinear coordinates by using an exact geometry for shells. The system of second order differential equations is reduced to a system of first order differential equations, and afterwards it is exactly solved by using the exponential matrix method. Similar approaches have been used in [
A shell is a threedimensional body bounded by two closely spaced curved surfaces where one dimension (the distance between the two surfaces) is small in comparison with the other two dimensions in the plane directions. The middle surface of the shell is the locus of points which lie midway between these surfaces. The distance between the surfaces measured along the normal to the middle surface is the
Notation and reference system for shells.
In this work, we will focus on shells with constant radii of curvature (e.g., cylindrical and spherical geometries). The geometrical relations written for shells with constant radii of curvature are a particular case of the straindisplacement equations of threedimensional theory of elasticity in orthogonal curvilinear coordinates (see also [
Geometrical relations (see (
The three differential equations of equilibrium written for the case of free vibration analysis of spherical shells with constant radii of curvature
Equation (
The system of second order differential equations can be reduced to a system of first order differential equations by using the method described in [
In the case of plate geometry coefficients
Thickness coordinates for plates and shells.
In the case of shell geometry matrices
The structures are simply supported and free stresses at the top and at the bottom; this feature means:
The free vibration problem means finding the nontrivial solution of
A certain number of circular frequencies
It is possible to find
The threedimensional exact solution proposed has been validated by means of a comparison with three assessments given in the literature for a onelayered isotropic plate [
Geometries considered for the assessments and benchmarks: (a) square plate, (b) rectangular plate, (c) cylindrical shell panel, (d) cylinder, and (e) spherical shell panel.
The assessment proposed in Vel and Batra [
Assessment 1. Simply supported aluminum square plate with Poisson ratio
Mode  

I  II  III  IV  V  VI  


Present 3D ( 
5.3370  27.554  46.493  154.39  168.09  268.36 
Present 3D ( 
5.7769  27.554  46.502  192.91  197.98  329.35 
Present 3D ( 
5.7769  27.554  46.502  196.21  201.87  308.34 
Present 3D ( 
5.7769  27.554  46.502  196.77  201.34  358.23 
Present 3D ( 
5.7769  27.554  46.502  196.77  201.34  357.42 
Present 3D ( 
5.7769  27.554  46.502  196.71  201.27  355.40 
Present 3D ( 
5.7769  27.554  46.502  196.77  201.34  351.90 
Present 3D ( 
5.7769  27.554  46.502  196.77  201.34  357.42 
3D [ 
5.7769  27.554  46.503  196.77  201.34  357.42 
3D [ 
5.7769  27.554  46.502  196.77  201.34  357.42 


 
Present 3D ( 
/  8.7132  14.182  17.512  /  / 
Present 3D ( 
4.6598  8.7132  14.463  20.987  23.999  29.925 
Present 3D ( 
4.6582  8.7132  14.463  21.383  25.009  31.399 
Present 3D ( 
4.6581  8.7132  14.463  21.343  24.830  34.049 
Present 3D ( 
4.6581  8.7132  14.463  21.343  24.830  33.982 
Present 3D ( 
4.6580  8.7132  14.463  21.337  24.821  33.939 
Present 3D ( 
4.6581  8.7132  14.463  21.343  24.830  33.923 
Present 3D ( 
4.6581  8.7132  14.463  21.343  24.830  33.982 
3D [ 
4.6582  8.7132  14.463  21.343  24.830  33.982 
3D [ 
4.6582  8.7134  14.462  21.343  24.830  33.982 
The assessment proposed by Khalili et al. [
Assessment 2 (part I). Simply supported aluminum cylinder with Poisson ratio










Present 3D ( 
0.03724  0.02323  0.02175  0.02846 
Present 3D ( 
0.03730  0.02360  0.02464  0.03687 
Present 3D ( 
0.03730  0.02359  0.02463  0.03686 
Present 3D ( 
0.03730  0.02359  0.02462  0.03686 
3D [ 
0.03730  0.02359  0.02462  0.03686 




Present 3D ( 
0.05769  0.04772  0.04229  0.04272 
Present 3D ( 
0.05853  0.04979  0.04790  0.05546 
Present 3D ( 
0.05853  0.04978  0.04789  0.05545 
Present 3D ( 
0.05853  0.04978  0.04789  0.05545 
3D [ 
0.05853  0.04978  0.04789  0.05545 




Present 3D ( 
0.08017  0.07732  0.07308  0.06667 
Present 3D ( 
0.10057  0.10234  0.10688  0.11528 
Present 3D ( 
0.10057  0.10234  0.10688  0.11528 
Present 3D ( 
0.10057  0.10234  0.10688  0.11528 
3D [ 
0.10057  0.10234  0.10688  0.11528 




Present 3D ( 
0.48156  0.48619  0.49378  0.50419 
Present 3D ( 
0.27489  0.27848  0.28446  0.29286 
Present 3D ( 
0.27491  0.27849  0.28447  0.29287 
Present 3D ( 
0.27491  0.27849  0.28447  0.29287 
3D [ 
0.27491  0.27849  0.28447  0.29287 
Assessment 2 (part II). Simply supported aluminum cylinder with Poisson ratio










Present 3D ( 
0.05613  0.03607  0.03241  0.02570 
Present 3D ( 
0.05652  0.03930  0.04998  0.07822 
Present 3D ( 
0.05652  0.03929  0.04996  0.07821 
Present 3D ( 
0.05652  0.03929  0.04996  0.07821 
3D [ 
0.05652  0.03929  0.04996  0.07821 




Present 3D ( 
0.08833  0.07237  0.05717  0.01965 
Present 3D ( 
0.09403  0.08546  0.09094  0.11205 
Present 3D ( 
0.09402  0.08545  0.09093  0.11205 
Present 3D ( 
0.09402  0.08545  0.09093  0.11205 
3D [ 
0.09402  0.08545  0.09093  0.11205 




Present 3D ( 
0.36506  0.37946  0.40165  0.43001 
Present 3D ( 
0.18895  0.19467  0.20616  0.22449 
Present 3D ( 
0.18894  0.19467  0.20616  0.22450 
Present 3D ( 
0.18894  0.19467  0.20616  0.22450 
3D [ 
0.18894  0.19467  0.20616  0.22450 




Present 3D ( 
0.72230  0.72911  0.74032  0.75570 
Present 3D ( 
0.50335  0.50934  0.51931  0.53321 
Present 3D ( 
0.50338  0.50937  0.51934  0.53325 
Present 3D ( 
0.50338  0.50937  0.51934  0.53325 
3D [ 
0.50338  0.50937  0.51934  0.53325 
The assessment proposed by Soldatos and Ye [
Assessment 3. Simply supported orthotropic cylinder with thickness ratio
Mode  

I  II  III  I  II  III  
( 
( 

Present 3D ( 
1.02407  2.55671  4.66395  0.20406  0.49393  0.91564 
Present 3D ( 
0.32274  0.41827  3.29830  0.30493  0.81346  7.51166 
Present 3D ( 
0.99999  1.29840  1.85418  0.95308  1.00000  3.04558 
Present 3D ( 
1.00000  1.29714  2.48260  0.95308  1.00000  3.25249 
Present 3D ( 
1.00000  1.29714  2.48260  0.95308  1.00000  3.25249 
3D [ 
1.00000  1.29730  2.48260  0.95307  1.00000  3.25250 
All the new benchmarks proposed in Section
Ten different benchmarks are proposed to show a complete overview of the free vibration analysis of onelayered plates and shells. Five different geometries are taken into account in Figure
Benchmark 1. Simply supported isotropic square plate. First three vibration modes in term of nodimensional circular frequency

100  50  10  5 

( 

I mode  2.9861  2.9846  2.9360  2.8012 
II mode  194.83  97.417  19.483  9.7417 
III mode  329.33  164.66  32.908  16.415 


( 

I mode  2.9861  2.9846  2.9360  2.8012 
II mode  194.83  97.417  19.483  9.7417 
III mode  329.33  164.66  32.908  16.415 


( 

I mode  5.9713  5.9650  5.7769  5.3036 
II mode  275.54  137.77  27.554  13.777 
III mode  465.73  232.86  46.502  23.136 


( 

I mode  14.920  14.881  13.805  11.645 
II mode  435.66  217.83  43.566  21.783 
III mode  736.37  368.14  73.350  36.157 


( 

I mode  14.920  14.881  13.805  11.645 
II mode  435.66  217.83  43.566  21.783 
III mode  736.37  368.14  73.350  36.157 


( 

I mode  23.860  23.761  21.214  16.882 
II mode  551.07  275.54  55.107  27.554 
III mode  931.42  465.63  92.545  45.071 
Benchmark 2. Simply supported orthotropic square plate. First three vibration modes in term of nodimensional circular frequency

100  50  10  5  100  50  10  5 




I mode  2.8553  2.8537  2.8058  2.6731  10.007  9.9712  9.0075  7.2059 
II mode  227.77  113.88  22.777  11.388  227.77  113.88  22.777  11.388 
III mode  314.89  157.44  31.457  15.679  1104.7  552.35  110.41  46.861 





I mode  10.007  9.9712  9.0075  7.2059  2.8553  2.8537  2.8058  2.6731 
II mode  227.77  113.88  22.777  11.388  227.77  113.88  22.777  11.388 
III mode  1104.7  552.35  110.41  46.861  314.89  157.44  31.457  15.679 





I mode  11.367  11.325  10.226  8.2183  11.367  11.325  10.226  8.2183 
II mode  382.03  191.01  38.169  19.032  382.03  191.01  38.169  19.032 
III mode  1130.2  565.09  112.94  49.636  1130.2  565.09  112.94  49.636 





I mode  17.719  17.648  15.817  12.656  41.011  40.436  29.614  19.220 
II mode  652.40  326.17  65.004  32.100  549.29  274.64  54.882  27.380 
III mode  1204.5  602.22  120.31  56.667  2222.2  1111.0  190.33  53.344 





I mode  41.011  40.436  29.614  19.220  17.719  17.648  15.817  12.656 
II mode  549.29  274.64  54.882  27.380  652.40  326.17  65.004  32.100 
III mode  2222.2  1111.0  190.33  53.344  1204.5  602.22  120.31  56.667 





I mode  45.302  44.664  32.873  21.768  45.302  44.664  32.873  21.768 
II mode  764.05  381.98  76.128  37.625  764.05  381.98  76.128  37.625 
III mode  2260.4  1130.1  198.54  59.836  2260.4  1130.1  198.54  59.836 
Benchmark 3. Simply supported isotropic rectangular plate. First three vibration modes in terms of nodimensional circular frequency

100  50  10  5 



I mode  0.3318  0.3318  0.3312  0.3293 
II mode  64.944  32.472  6.4944  3.2472 
III mode  109.78  54.888  10.977  5.4870 




I mode  2.9861  2.9846  2.9360  2.8012 
II mode  194.83  97.417  19.483  9.7417 
III mode  329.33  164.66  32.908  16.415 




I mode  3.3179  3.3159  3.2562  3.0923 
II mode  205.37  102.69  20.537  10.269 
III mode  347.14  173.56  34.685  17.297 




I mode  4.3130  4.3097  4.2099  3.9450 
II mode  234.16  117.08  23.416  11.708 
III mode  395.80  197.89  39.537  19.699 




I mode  12.270  12.243  11.497  9.9094 
II mode  395.04  197.52  39.504  19.752 
III mode  667.72  333.83  66.559  32.908 




I mode  13.264  13.233  12.369  10.572 
II mode  410.74  205.37  41.074  20.537 
III mode  694.26  347.10  69.186  34.170 
Benchmark 4. Simply supported orthotropic rectangular plate. First three vibration modes in terms of nodimensional circular frequency

100  50  10  5  100  50  10  5 




I mode  0.3173  0.3173  0.3167  0.3148  1.1130  1.1126  1.0987  1.0586 
II mode  75.922  37.961  7.5922  3.7961  75.922  37.961  7.5922  3.7961 
III mode  104.98  52.483  10.495  5.2459  368.24  184.12  36.822  18.408 





I mode  10.007  9.9712  9.0075  7.2059  2.8553  2.8537  2.8058  2.6731 
II mode  227.77  113.88  22.777  11.388  227.77  113.68  22.777  11.388 
III mode  1104.7  552.35  110.41  46.861  314.89  157.44  31.457  15.679 





I mode  10.127  10.091  9.1126  7.2879  3.4236  3.4213  3.3511  3.1611 
II mode  249.70  124.85  24.967  12.480  312.77  156.38  31.251  15.586 
III mode  1107.6  553.78  110.69  47.184  441.11  220.55  44.099  22.031 





I mode  10.538  10.500  9.4774  7.5865  6.1047  6.0947  5.8032  5.1319 
II mode  306.16  153.08  30.604  15.284  341.88  170.94  34.157  17.031 
III mode  1116.1  558.04  111.54  48.130  774.35  387.17  77.407  38.652 





I mode  40.001  39.445  28.902  18.694  11.868  11.841  11.093  9.5179 
II mode  466.89  233.44  46.684  23.337  560.57  280.28  56.021  27.943 
III mode  2210.8  1105.3  187.77  51.133  656.59  328.31  65.435  32.318 





I mode  40.364  39.801  29.152  18.872  13.685  13.649  12.644  10.634 
II mode  499.39  249.69  49.921  24.938  625.52  312.73  62.343  30.811 
III mode  2215.1  1107.5  188.74  51.982  882.21  441.09  88.126  43.879 
Benchmark 5. Simply supported isotropic cylindrical shell panel. First three vibration modes in terms of nodimensional circular frequency

1000  100  10  5 



I mode  1067.1  106.71  10.683  5.3415 
II mode  1068.3  106.83  10.690  5.3727 
III mode  1855.7  185.57  18.557  9.2797 




I mode  2.5478  2.5184  2.4853  2.3948 
II mode  2040.3  204.03  20.411  10.216 
III mode  3635.3  363.51  36.229  17.917 




I mode  222.72  22.503  3.8758  3.2156 
II mode  2364.5  236.45  23.656  11.842 
III mode  4017.4  401.73  40.060  19.846 




I mode  552.89  55.574  7.7851  5.7590 
II mode  3031.5  303.16  30.328  15.181 
III mode  5048.7  504.86  50.367  24.983 




I mode  70.382  14.052  11.533  10.021 
II mode  4230.8  423.08  42.325  21.174 
III mode  7214.2  721.38  71.724  35.120 




I mode  232.90  27.451  13.793  11.643 
II mode  4639.2  463.92  46.414  23.226 
III mode  7847.2  784.68  78.020  38.182 
Benchmark 6. Simply supported orthotropic cylindrical shell panel. First three vibration modes in term of nodimensional circular frequency

1000  100  10  5  1000  100  10  5 




I mode  1248.9  124.89  12.489  6.2444  1096.5  109.67  11.281  6.0108 
II mode  1721.6  172.16  17.230  8.6360  1248.9  124.89  12.489  6.2444 
III mode  3858.5  385.85  38.590  19.293  6057.8  605.78  60.570  30.272 





I mode  8.4450  8.4392  7.6484  6.1872  2.3272  2.4079  2.3754  2.2863 
II mode  2385.2  238.52  23.861  11.941  2385.2  238.52  23.861  11.943 
III mode  12194  1219.4  121.48  51.358  3476.0  347.58  34.633  17.114 





I mode  230.37  24.568  8.0953  6.3709  352.48  35.490  5.3604  4.1227 
II mode  2939.9  294.00  29.454  14.782  3588.6  358.85  35.788  17.735 
III mode  12257  1225.7  122.10  52.106  6559.0  655.90  65.574  32.756 





I mode  641.70  64.887  10.877  6.8670  644.27  65.496  12.428  9.0851 
II mode  4155.7  415.58  41.604  20.824  4164.3  416.43  41.609  20.709 
III mode  12445  1244.4  123.94  54.240  12370  1237.0  123.62  54.652 





I mode  79.744  39.112  28.352  18.587  160.90  20.150  11.545  9.9019 
II mode  5062.8  506.29  50.681  25.371  6476.9  647.66  64.461  31.673 
III mode  23492  2349.1  205.91  56.149  7983.5  798.33  79.684  39.571 





I mode  243.62  46.084  29.034  19.077  364.43  40.611  16.527  13.080 
II mode  5849.5  584.96  58.566  29.293  6951.1  695.08  69.132  33.884 
III mode  23598  2359.6  208.24  58.169  13115  1311.5  131.01  62.003 
Benchmark 7. Simply supported isotropic cylinder. First three vibration modes in term of nodimensional circular frequency

1000  100  10  5 

( 

I mode  7691.7  769.17  76.917  38.459 
II mode  12342  1234.2  123.42  61.710 
III mode  41586  4158.6  416.30  208.81 




I mode  2445.9  244.59  24.492  12.295 
II mode  27270  2727.0  272.81  136.56 
III mode  59189  5918.8  591.08  294.28 




I mode  7712.1  771.23  77.252  38.820 
II mode  33375  3337.5  333.85  167.09 
III mode  61353  6135.3  612.74  305.15 




I mode  13536  1353.7  135.69  68.323 
II mode  40081  4008.1  400.89  200.56 
III mode  65490  6548.9  654.17  325.96 
Benchmark 8. Simply supported orthotropic cylinder. First three vibration modes in terms of nodimensional circular frequency

1000  100  10  5  1000  100  10  5 




I mode  8991.9  899.19  89.919  44.960  8991.9  899.19  89.919  44.959 
II mode  12402  1240.2  124.16  62.281  39097  3909.7  391.39  196.33 
III mode  138826  13883  1389.6  696.66  44039  4403.9  440.45  220.32 





I mode  2486.4  248.65  24.921  12.541  5295.1  529.53  53.084  26.738 
II mode  31697  3169.8  317.14  158.81  50390  5039.0  503.83  251.80 
III mode  196438  19643  1961.3  974.72  58023  5802.3  579.62  288.86 





I mode  8071.9  807.21  80.903  40.715  11839  1184.0  119.10  60.562 
II mode  38989  3898.9  390.16  195.44  56607  5660.7  565.47  281.81 
III mode  196773  19677  1964.6  976.14  92402  9240.2  924.00  461.97 





I mode  14640  1464.1  146.76  73.903  17656  1765.9  178.90  92.578 
II mode  48229  4823.0  482.66  241.82  58769  5876.8  587.41  293.24 
III mode  197336  19733  1970.0  978.55  134283  13428  1342.8  671.29 
Benchmark 9. Simply supported isotropic spherical shell panel. First three vibration modes in terms of nodimensional circular frequency

1000  100  10  5 



I mode  999.01  99.924  10.213  5.3986 
II mode  2040.3  204.02  20.360  10.115 
III mode  3785.7  378.55  37.677  18.556 




I mode  999.01  99.924  10.213  5.3986 
II mode  2040.3  204.02  20.360  10.116 
III mode  3785.7  378.55  37.677  18.556 




I mode  1046.3  104.76  11.564  6.9554 
II mode  2885.4  288.53  28.793  14.302 
III mode  5111.6  511.13  50.821  24.937 




I mode  1076.3  108.52  16.906  12.400 
II mode  4562.2  456.21  45.524  22.598 
III mode  7857.9  785.72  77.933  37.847 




I mode  1076.3  108.52  16.906  12.400 
II mode  4562.2  456.21  45.524  22.598 
III mode  7857.9  785.72  77.933  37.847 




I mode  1084.1  110.77  23.273  17.513 
II mode  5770.8  577.07  57.581  28.565 
III mode  9869.7  986.87  97.645  46.869 
Benchmark 10. Simply supported orthotropic spherical shell panel. First three vibration modes in terms of nodimensional circular frequency

1000  100  10  5  1000  100  10  5 




I mode  2385.2  238.51  23.802  11.825  1037.9  104.08  12.585  7.7554 
II mode  2902.0  290.20  29.026  14.513  2385.2  238.51  23.802  11.824 
III mode  4371.6  437.15  43.559  21.518  12223  1222.3  121.65  51.888 





I mode  1037.9  104.08  12.583  7.7554  2385.2  238.51  23.802  11.825 
II mode  2385.2  238.51  23.802  11.824  2902.0  290.20  29.026  14.513 
III mode  12223  1222.3  121.65  51.888  4371.6  437.15  43.559  21.518 





I mode  1209.7  121.32  14.754  9.1755  1209.7  121.32  14.754  9.1755 
II mode  4016.5  401.63  40.041  19.819  4016.5  401.63  40.041  19.819 
III mode  12456  1245.5  123.88  54.506  12456  1245.5  123.88  54.506 





I mode  1682.9  169.01  21.986  14.305  1114.0  117.90  30.685  19.645 
II mode  6838.7  683.84  68.035  33.373  5758.3  575.82  57.401  28.398 
III mode  13147  1314.6  130.57  60.847  23599  2359.8  208.50  58.572 





I mode  1114.0  117.90  30.685  19.645  1682.9  169.01  21.986  14.305 
II mode  5758.3  575.82  57.401  28.398  6838.7  683.84  68.035  33.373 
III mode  23599  2359.8  208.50  58.572  13147  1314.6  130.57  60.847 





I mode  1260.9  133.19  34.372  22.411  1260.9  133.19  34.372  22.411 
II mode  8009.4  800.89  79.625  39.004  8009.4  800.89  79.625  39.004 
III mode  23985  2398.3  216.62  65.349  23985  2398.3  216.62  65.349 
Benchmark 1, simply supported isotropic square plate. First three vibration modes in terms of displacement components through the thickness for thickness ratio
Benchmark 2, simply supported orthotropic square plate
Benchmark 2, simply supported orthotropic square plate (
Benchmark 7, simply supported isotropic cylinder. First three vibration modes in terms of displacement components through the thickness for thickness ratio
Benchmark 8, simply supported orthotropic cylinder
Benchmark 8, simply supported orthotropic cylinder
Benchmark 9, simply supported isotropic spherical shell panel. First three vibration modes in terms of displacement components through the thickness for thickness ratio
Benchmark 10, simply supported orthotropic spherical shell panel
Benchmark 10, simply supported orthotropic spherical shell panel
Table
Table
Tables
Table
Tables
Tables
The parametric coefficient effects, in particular the
Effects of

1000  100  10  5 



I mode  1067.1  106.71  10.683  5.3415 
II mode  1068.3  106.83  10.690  5.3727 
III mode  1855.7  185.57  18.557  9.2797 
I mode ( 
1067.1  106.71  10.686  5.3486 
II mode ( 
1068.3  106.83  10.688  5.3684 
III mode ( 
1855.7  185.56  18.550  9.2651 




I mode  2.5478  2.5184  2.4853  2.3948 
II mode  2040.3  204.03  20.411  10.216 
III mode  3635.3  363.51  36.229  17.917 
I mode ( 
2.7747  2.7473  2.7029  2.5828 
II mode ( 
2040.3  204.03  20.403  10.201 
III mode ( 
3635.3  363.52  36.303  18.076 




I mode  222.72  22.503  3.8758  3.2156 
II mode  2364.5  236.45  23.656  11.842 
III mode  4017.4  401.73  40.060  19.846 
I mode ( 
222.73  22.535  4.0440  3.3738 
II mode ( 
2364.5  236.45  23.668  11.868 
III mode ( 
4017.4  401.73  40.115  19.965 
Effects of

1000  100  10  5  1000  100  10  5 




I mode  2385.2  238.51  23.802  11.825  1037.9  104.08  12.585  7.7554 
II mode  2902.0  290.20  29.026  14.513  2385.2  238.51  23.802  11.824 
III mode  4371.6  437.15  43.559  21.518  12223  1222.3  121.65  51.888 
I mode ( 
2385.2  238.52  23.858  11.939  1037.9  104.09  12.602  7.7410 
II mode ( 
2902.0  290.20  29.030  14.529  2385.2  238.52  23.856  11.934 
III mode ( 
4371.6  437.15  43.623  21.667  12223  1222.3  122.04  51.743 





I mode  1037.9  104.08  12.583  7.7554  2385.2  238.51  23.802  11.825 
II mode  2385.2  238.51  23.802  11.824  2902.0  290.20  29.026  14.513 
III mode  12223  1222.3  121.65  51.888  4371.6  437.15  43.559  21.518 
I mode ( 
1037.9  104.09  12.602  7.7410  2385.2  238.52  23.858  11.939 
II mode ( 
2385.2  238.52  23.856  11.934  2902.0  290.20  29.030  14.529 
III mode ( 
12223  1222.3  122.04  51.743  4371.6  437.15  43.623  21.667 





I mode  1209.7  121.32  14.754  9.1755  1209.7  121.32  14.754  9.1755 
II mode  4016.5  401.63  40.041  19.819  4016.5  401.63  40.041  19.819 
III mode  12456  1245.5  123.88  54.506  12456  1245.5  123.88  54.506 
I mode ( 
1209.7  121.33  14.810  9.2169  1209.7  121.33  14.810  9.2169 
II mode ( 
4016.5  401.64  40.137  20.022  4016.5  401.64  40.137  20.022 
III mode ( 
12456  1245.6  124.29  54.312  12456  1245.6  124.29  54.312 
The differential equations of equilibrium in orthogonal curvilinear coordinates for the free vibrations of simply supported structures have been exactly solved in threedimensional form by using the exponential matrix method. The proposed general 3D formulation uses an exact geometry for shells, and it allows results for spherical, open cylindrical, closed cylindrical and flat panels to be obtained. The first three vibration modes have been investigated for several geometries, both isotropic and orthotropic layers, various thickness ratios, and imposed halfwave numbers. The modes plotted through the thickness make it possible to recognize the most complicated cases and these results will be useful benchmarks to validate future refined 2D models. The method is simple and intuitive and it will be extended to multilayered structures (also embedding functionally graded layers). This extension will give a global threedimensional overview of the free vibration problem of onelayered and multilayered plates and shells. The present method has a fast convergence and it is not heavy from the computational point of view for each geometry (plates and shells). This feature is an important advantage with respect to other methods proposed in the literature that are valid only for a chosen geometry.
The author declares that there is no conflict of interests regarding the publication of this paper.