A twolayer (connected by stubs) partial composite plate is a structure with outstanding advantages which can be widely applied in many fields of engineering such as construction, transportation, and mechanical. However, studies are scarce in the past to investigate this type of structure. This paper is based on the new modified firstorder shear deformation plate theory and finite element method to develop a new fournode plate element with nine degrees of freedom per node for static bending and vibration analysis of the twolayer composite plate. The numerical results are compared to published data for some special cases. The effects of some parameters such as the boundary condition, stiffness of the connector stub, heighttowidth ratio, thicknesstothickness ratio between two layers, and aspect ratio are also performed to investigate new numerical results of static bending and free vibration responses of this structure.
Twolayer beam and plate are among the most commonly used structures in many fields of engineering, such as construction, transportation, and mechanical, because of their advantages in comparison with onelayer structures; we can see these structures in practice such as steelconcrete composite beams and plates, layered wooden beams and plates, and woodconcrete and timbersteel floor structures. Some advantages of this type of composite structure can be considered such as easy manufacturing, taking advantage of the material properties of two components which made the structure. In fact, there are many different grafting methods to manufacture twolayer beams or plates; for example, two components of a structure are sandwiched together at the edges, using glue or connector stub to bond two parts at the contact surface. Because of these advantages, many researchers have focused on the mechanical analysis of these structures.
Twolayer beam has been investigated by Foraboschi [
Latham et al. [
This paper aims to use the finite element method (FEM) based on a new modified firstorder shear deformation plate theory (FSDT) to analyze static bending and free vibration of the twolayer composite plate. The proposed method shows the simple formulations and computational efficiency. The accuracy and reliability of the proposed method are validated with other published results. Several numerical examples and influence of some parameters on static bending and free vibration of the twolayer composite plate are also investigated; these new results have significant impacts on the use of this structure in practice.
The organization of this paper is as follows: Section
The geometry model of the problem is a twolayer plate including two isotropic plates which bonded together at the contact surface using connector stubs, as shown in Figure
Geometrical notation of the twolayer plate.
The assumptions of the twolayer composite plate include the following: the materials of each layer are linear, elastic, and isotropic; the displacement and rotation of the plate are small; there is no delamination phenomenon between two layers; the deflection of the stub is in the contact interface of two components; and the mass of the stub is much smaller than the mass of the plate, so we assume that it is neglected.
The basic equations of the Mindlin plate theory are [
Equilibrium of moments about
Substituting equation (
Assume that the total deflection consists of two parts which are bending deflection and transverse shear, while angles of the plate crosssectional slope are a result of rotation of pure bending and shear angles:
Substituting equation (
Equations (
The total deflection is
Using the new modified firstorder shear deformation theory, the displacement field of the plate can be rewritten in the following form [
The straindisplacement relations may be written as follows [
Equation (
The plate is discretized using the quadrilateral fournode element, as shown in Figure
A twolayer composite plate element.
The element nodal displacement vector can be given as
The nodal displacement vector of each component is
In equations (
The static bending deflection
The displacements
The bending deflection
The total static deflection of the plate is
The rotations of cross sections on the neutral plane are
Taking coordinate values
Taking equation (
Substituting equations (
By substituting equation (
According to Hooke’s law, the relationship between stresses and strains can be given as
The calculation model for the connector stub is shown in Figure
Axial displacement assuming a twolayer composite plate with connector stubs.
By using Lagrangian approximation for both axial displacement and rotation of two components to determine the axial deflection of the connector stub, the interfacial slips of two components are given by
The strain energy of two components and connector stub can be given by
The kinetic energy is
Equation (
The work done by external distribution normal force acting on the top surface of layer
The Lagrangian function of the plate element is defined by
Using Lagrange’s equations, the governing equations of motion of the plate element are given by
Substituting equations (
Note that we need not any selective reduced integration or reduced integration scheme to calculate the matrices and vectors in equations (
For the static bending problem, ignoring
Solving equation (
For free vibration analysis, neglecting the effect of external force vector
Solving equation (
We now focus on numerically studying the static bending and free vibration responses of the twolayer composite plate. Each edge of the plate can be under the simply supported boundary condition or clamped boundary condition. For the simply supported edge,
For the clamped edge,
To verify the proposed method, we compare the deflections in two cases of materials: homogeneous and composite materials. Firstly, a square homogeneous plate under contribution load
Table
Comparison of nondimensional deflection of square plates with
Source  SSSS  CCCC  




 
Ferreira [ 
0.004271  0.004060  0.001503  0.001264 
Exact solution [ 
0.004270  0.004060  —  0.001260 
Present ( 
0.004290  0.004079  0.001508  0.001272 
Present ( 
0.004300  0.004079  0.001523  0.001291 
We see that when the thickness of one layer is much higher than that of the other layer and the stiffness of the connector stub is very small, the behavior of the twolayer plate is the same as the behavior of the onelayer plate with the same thickness. In addition, when two layers have the same thickness and material, the stiffness of the connector stub gets a much higher value
In order to further confirm the accuracy of this method, we compare the nondimensional deflection of a square SSSS composite plate (
Comparison of the nondimensional deflection of a square SSSS composite plate subject to uniformly distributed load and central concentrated load.

Uniformly distributed load  Central concentrated load  

Reddy [ 
Present  Reddy [ 
Present  
1.6955  1.6538  4.6664  4.6421 
It is shown plainly that the present results have a good agreement with the results given in [
In this section, a rectangular twolayer composite plate subjected to the uniform load
Nondimensional deflection of the plate as a function of ratio


 

SSSS  CCCC  CSCS  CCSS  SSSS  CCCC  CSCS  CCSS  
0.5  6.4728  1.6252  5.4088  3.1243  12.5027  3.1584  10.4767  6.0528 
0.625  5.3098  1.4770  3.8595  2.7089  10.2589  2.8726  7.4844  5.2510 
0.75  4.2368  1.2639  2.6511  2.2449  8.1874  2.4598  5.1458  4.3531 
0.875  3.3289  1.0313  1.8013  1.7999  6.4339  2.0083  3.4991  3.4914 
1.0  2.5968  0.8135  1.2304  1.4124  5.0196  1.5849  2.3919  2.7406 
1.2  1.7421  0.5350  0.6889  0.9375  3.3683  1.0431  1.3408  1.8201 
1.4  1.1793  0.3464  0.4036  0.6196  2.2806  0.6760  0.7866  1.2035 
1.6  0.8107  0.2259  0.2475  0.4139  1.5683  0.4413  0.4830  0.8046 
1.8  0.5676  0.1501  0.1582  0.2818  1.0984  0.2936  0.3094  0.5482 
2.0  0.4050  0.1020  0.1051  0.1958  0.7839  0.1999  0.2058  0.3812 
2.2  0.2943  0.0711  0.0721  0.1389  0.5699  0.1395  0.1416  0.2707 
2.4  0.2177  0.0507  0.0510  0.1006  0.4217  0.0997  0.1003  0.1962 
2.6  0.1637  0.0370  0.0371  0.0742  0.3172  0.0729  0.0730  0.1450 
2.8  0.1249  0.0276  0.0275  0.0558  0.2423  0.0544  0.0544  0.1091 
3.0  0.0967  0.0209  0.0209  0.0426  0.1877  0.0415  0.0414  0.0835 
3.2  0.0759  0.0162  0.0162  0.0331  0.1473  0.0321  0.0322  0.0650 
3.4  0.0602  0.0127  0.0127  0.0261  0.1170  0.0253  0.0254  0.0512 
3.6  0.0484  0.0101  0.0101  0.0208  0.0940  0.0202  0.0203  0.0409 
3.8  0.0392  0.0082  0.0082  0.0168  0.0763  0.0164  0.0164  0.0331 
4.0  0.0321  0.0067  0.0067  0.0137  0.0626  0.0134  0.0134  0.0270 
The influence of ratio
Nondimensional deflection of the plate as a function of ratio


 

SSSS  CCCC  CSCS  CCSS  SSSS  CCCC  CSCS  CCSS  
1.0  2.5968  0.8135  1.2304  1.4124  0.4050  0.1020  0.1051  0.1958 
1.2  3.1576  0.9907  1.4977  1.7187  0.4926  0.1243  0.1281  0.2384 
1.4  3.6969  1.1617  1.7556  2.0138  0.5769  0.1460  0.1503  0.2795 
1.6  4.1941  1.3201  1.9940  2.2864  0.6546  0.1661  0.1710  0.3175 
1.8  4.6367  1.4618  2.2070  2.5297  0.7239  0.1841  0.1896  0.3516 
2.0  5.0196  1.5849  2.3919  2.7406  0.7839  0.1999  0.2058  0.3812 
2.2  5.3436  1.6895  2.5488  2.9194  0.8348  0.2133  0.2196  0.4063 
2.4  5.6129  1.7769  2.6798  3.0684  0.8771  0.2246  0.2312  0.4273 
2.6  5.8338  1.8489  2.7875  3.1909  0.9118  0.2340  0.2408  0.4447 
2.8  6.0131  1.9077  2.8752  3.2906  0.9400  0.2416  0.2487  0.4588 
3.0  6.1575  1.9552  2.9461  3.3710  0.9628  0.2478  0.2551  0.4702 
3.2  6.2731  1.9934  3.0030  3.4355  0.9810  0.2529  0.2603  0.4795 
3.4  6.3651  2.0239  3.0485  3.4870  0.9956  0.2569  0.2644  0.4868 
3.6  6.4379  2.0482  3.0847  3.5278  1.0071  0.2602  0.2678  0.4927 
3.8  6.4954  2.0676  3.1133  3.5602  1.0162  0.2628  0.2704  0.4974 
4.0  6.5405  2.0828  3.1359  3.5856  1.0234  0.2649  0.2726  0.5011 
Nondimensional deflection of the plate as a function of ratio


 

SSSS  CCCC  CSCS  CCSS  SSSS  CCCC  CSCS  CCSS  
0  3.1903  0.9951  1.5068  1.7316  9.4673  2.9526  4.4712  5.1381 
10^{–2}  2.6790  0.8708  1.3056  1.4862  5.5538  1.9698  2.8925  3.2279 
10^{–1}  2.6051  0.8214  1.2402  1.4215  5.0730  1.6363  2.4560  2.7995 
10^{0}  2.5968  0.8135  1.2304  1.4124  5.0196  1.5849  2.3919  2.7406 
10^{1}  2.5959  0.8126  1.2293  1.4114  5.0142  1.5792  2.3850  2.7343 
10^{2}  2.5958  0.8125  1.2292  1.4113  5.0137  1.5787  2.3843  2.7336 
10^{3}  2.5958  0.8125  1.2292  1.4113  5.0136  1.5786  2.3842  2.7336 
10^{4}  2.5958  0.8125  1.2292  1.4113  5.0136  1.5786  2.3842  2.7336 
10^{5}  2.5958  0.8125  1.2292  1.4113  5.0136  1.5786  2.3842  2.7336 
10^{6}  2.5958  0.8125  1.2292  1.4113  5.0136  1.5786  2.3842  2.7336 
The effect of ratio
The effect of ratio
To show clearly the slip between the top and bottom layers at the contact place, we depicted the ratio
The effect of ratio
The slip between the top and bottom layers at the contact place varying along the
We can see again in Figure
Nondimensional deflection of the plate along the line part
To confirm the accuracy of this work, we first present a comparison of frequencies of homogeneous CCCC and SSSS plates, with
The results are listed in Tables
Comparison of the nondimensional frequencies of an SSSS plate.

Mode  Ferreira [ 
Mindlin [ 
Present 

10  1  0.9346  0.930  0.9271 
2  2.2545  2.219  2.2055  
3  2.2545  2.219  2.2055  
4  3.4592  3.406  3.3600  
5  4.3031  4.149  4.1123  
6  4.3031  4.149  4.1123  


100  1  0.0965  0.0963  0.0952 
2  0.2430  0.2406  0.2343  
3  0.2430  0.2406  0.2343  
4  0.3890  0.3847  0.3680  
5  0.4928  0.4807  0.4580  
6  0.4928  0.4807  0.4580 
Comparison of the nondimensional frequencies of a CCCC plate.

Mode  Ferreira [ 
Liew et al. [ 
Present 

10  1  1.5955  1.5582  1.5710 
2  3.0662  3.0182  2.9927  
3  3.0662  3.0182  2.9927  
4  4.2924  4.1711  4.1829  
5  5.1232  5.1218  4.9441  
6  5.1730  5.1594  4.9893  


100  1  0.1750  0.1743  0.1683 
2  0.3635  0.3576  0.3431  
3  0.3635  0.3576  0.3432  
4  0.5358  0.5240  0.4978  
5  0.6634  0.6465  0.6014  
6  0.6665  0.6505  0.6057 
Next, we compare the nondimensional frequencies of an SSSS composite plate 0/90°. The parameters of this plate are set to be
The comparison is shown in Table
Comparison of the nondimensional frequencies
Mode 

 

Reddy [ 
Present  Reddy [ 
Present  
1  1.183  1.151  0.990  1.032 
2  3.174  3.209  2.719  2.966 
3  3.174  3.209  2.719  2.966 
4  4.733  5.057  3.959  4.101 
5  6.666  6.399  5.789  5.806 
6  6.666  6.399  5.789  5.806 
In this section, we research the natural frequency of the twolayer composite plate utilizing the proposed method. The twolayer composite plate (length
Influence of ratio

Boundary  

SSSS  CCCC  CSCS  CCSS  
0.5  0.8439  1.6701  0.9329  1.2115 
1.0  1.3505  2.4421  1.9700  1.8434 
1.2  1.6474  3.0049  2.6189  2.2592 
1.4  1.9983  3.7145  3.3955  2.7687 
1.6  2.4029  4.5647  4.2971  3.3698 
1.8  2.8614  5.5501  5.3220  4.0606 
2.0  3.3737  6.6663  6.4691  4.8395 
2.2  3.9398  7.9101  7.7374  5.7053 
2.4  4.5595  9.2791  9.1262  6.6570 
2.6  5.2330  10.7714  10.6349  7.6938 
2.8  5.9601  12.3857  12.2627  8.8151 
3.0  6.7408  14.1208  14.0093  10.0203 
3.2  7.5751  15.9757  15.8739  11.3092 
3.4  8.4628  17.9494  17.8561  12.6811 
3.6  9.4040  20.0412  19.9553  14.1359 
3.8  10.3985  22.2503  22.1709  15.6730 
4.0  11.4463  24.5760  24.5023  17.2922 
The effect of ratio
Influence of ratio

Boundary  

SSSS  CCCC  CSCS  CCSS  
1.0  1.3505  2.4421  1.9700  1.8434 
1.2  1.2757  2.3022  1.8582  1.7396 
1.4  1.2231  2.2024  1.7789  1.6661 
1.6  1.1872  2.1332  1.7241  1.6156 
1.8  1.1641  2.0873  1.6880  1.5825 
2.0  1.1506  2.0591  1.6662  1.5627 
2.2  1.1445  2.0445  1.6552  1.5530 
2.4  1.1439  2.0403  1.6526  1.5511 
2.6  1.1476  2.0439  1.6562  1.5550 
2.8  1.1544  2.0535  1.6646  1.5633 
3.0  1.1635  2.0676  1.6765  1.5749 
3.2  1.1744  2.0850  1.6911  1.5889 
3.4  1.1866  2.1049  1.7077  1.6048 
3.6  1.1997  2.1266  1.7256  1.6219 
3.8  1.2133  2.1496  1.7446  1.6399 
4.0  1.2274  2.1734  1.7642  1.6585 
The effect of ratio
Nondimensional fundamental frequencies as a function of
Influence of ratio

Boundary  

SSSS  CCCC  CSCS  CCSS  
0  1.2206  2.2205  1.7881  1.6710 
10^{–2}  1.3292  2.3636  1.9142  1.7976 
10^{–1}  1.3482  2.4308  1.9625  1.8376 
10^{0}  1.3505  2.4421  1.9700  1.8434 
10^{1}  1.3507  2.4433  1.9708  1.8441 
10^{2}  1.3508  2.4435  1.9709  1.8441 
10^{3}  1.3508  2.4435  1.9709  1.8441 
10^{4}  1.3508  2.4435  1.9709  1.8441 
10^{5}  1.3508  2.4435  1.9709  1.8441 
10^{6}  1.3506  2.4434  1.9710  1.8439 
The effect of ratio
Figures
Nondimensional fundamental frequencies as a function of
Nondimensional fundamental frequencies as a function of
The first six mode shapes of a twolayer square SSSS composite plate. (a) Mode 1. (b) Mode 2. (c) Mode 3. (d) Mode 4. (e) Mode 5. (f) Mode 6.
The first six mode shapes of a twolayer square CCCC composite plate. (a) Mode 1. (b) Mode 2. (c) Mode 3. (d) Mode 4. (e) Mode 5. (f) Mode 6.
The first six mode shapes of a twolayer square CSCS composite plate. (a) Mode 1. (b) Mode 2. (c) Mode 3. (d) Mode 4. (e) Mode 5. (f) Mode 6.
The first six mode shapes of a twolayer square CCSS composite plate. (a) Mode 1. (b) Mode 2. (c) Mode 3. (d) Mode 4. (e) Mode 5. (f) Mode 6.
In this work, new numerical results of mechanical behaviors and free vibration responses of the twolayer composite plate are explored, in which two layers are connected by stubs. We used the finite element method combined with the new modified firstorder shear deformation theory, which has the following advantages:
Simple formulations for theoretical representation
No need for reduced integration or selective reduced integration scheme for the proposed method
The computed results of static bending and free vibration obtained by this approach are also compared to other solutions, showing a good agreement. We gained insight into the responses of deflections and natural frequencies. Besides, some geometrical and physical properties of this structure are also examined. Finally, from new numerical results, several conclusions may be achieved as follows:
The stiffness of the stub has a strong effect on static bending deflections and natural frequencies of the twolayer composite plate when
Likewise, the boundary condition has great effect on the slip between two layers, the minimum slip appears at the center of the plates with the symmetric boundary condition, and the slip has the greatest value at the simply supported edge.
When the ratio
The higher the ratio
The new results of this work are useful for calculation, design, and testing, as well as for giving the optimal solution for the twolayer plate and shell in engineering and technologies. This study suggests some further works on buckling, dynamic response, and heat transfer problems of the twolayer composite plate using different plate theories.
The matrix
The data used to support the findings of this study are included within the article.
The authors declare that they have no conflicts of interest.
DVT gratefully acknowledges the support of Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant no. 107.022018.30.