Based on Hamilton’s principle, this study has developed a continuous treatment for the steelconcrete composite truss beam (SCCTB). It has also deduced the SCCTB element stiffness matrix and mass matrix, which include the effects of interface slip, shear deformation, moment of inertia, and many other influencing factors. A finite beam element method (FBEM) program for SCCTB’s natural vibration frequency has been developed and used to calculate the natural vibration frequencies of several SCCTBs with different spans and different degrees of shear connections. The FBEM’s calculation results of several SCCTBs agree well with the results obtained from ANSYS. Based on the results of this study, the following conclusions can be drawn. For the SCCTB with highorder natural vibration frequency and with short span, the effect of the shear deformation is greater. Hence, the effect of the shear deformation on the SCCTB’s highorder natural vibration frequency cannot be ignored. On the other hand, the effect of the interface slip on the SCCTB’s highorder natural vibration frequency is insignificant. However, the effect of the interface slip on the SCCTB’s loworder natural vibration frequency cannot be ignored.
The SCCTB is a newtype structural member. It is developed on the basis of ordinary composite beams and uses shear connectors to connect the concrete slab to the steel truss into one entire joint work. Comparing with ordinary truss beams, SCCTB is more effective because it uses both the compressive strength of concrete and the tensile strength of steel. Hence, it is characterized by high utilization rate of indoor space, high economic efficiency, high bending stiffness, and high bearing capacity [
Even though a beam is designed as a full composite section, due to the deformation of shear studs, it cannot respond with perfect composite action without slip. Further, unlike Euler–Bernoulli beams in which there is infinite shear stiffness, it may not be appropriate to design other beams by neglecting the shear deformation of the concrete slab and steel truss beam, for example, composite beams with a small spantodepth ratio [
Hitherto, there are many studies on the shear behavior of hybrid steeltrussedconcrete beams, which are constructed by embedding the prefabricated steel trusses into a concrete core cast in situ. On the other hand, there are few studies on the mechanical properties of SCCTB. Using the software ABAQUS to carry out the finite element (FE) numerical simulations, Monaco [
Giltner and Kassimali [
The natural vibration characteristic analysis is the basis to investigate the dynamic characteristics of SCCTB. So, it is necessary to investigate its natural vibration characteristics. As shown by the earlier studies, the mechanical properties of SCCTB are affected by the interface slip, shear deformation, moment of inertia, and many other factors. However, there are only a few studies on the SCCTB’s natural vibration characteristics which have taken these factors into account. In this study, a continuous treatment for SCCTB has been developed. It has also deduced the SCCTB element stiffness matrix and mass matrix with the cubic Hermite polynomial shape function, which includes the effects of the interface slip, shear deformation, moment of inertia, and many other influencing factors. Hence, even with fewer degrees of freedom, the precision in the calculation is satisfactory. Based on the developed SCCTB element, this study has developed a FBEM program which can calculate the natural vibration frequency of SCCTBs commonly used in the engineering practice. Finally, it calculates the natural vibration frequencies of several SCCTBs with different spans and different degrees of shear connections. Based on the results of the analyses, some conclusions which are related to the engineering design are drawn.
In Figure
Sectional dimension and the coordinate system of the SCCTB.
The strain of oblique bracing member can be expressed as
Further, the axial force of the oblique bracing member is given by
The longitudinal component of the axial force of the oblique bracing member is given by
Therefore, the longitudinal equivalent area of the oblique bracing member can be expressed as follows:
The longitudinal displacement of SCCTB can be expressed as follows [
The axial displacement of the oblique web member is given by
According to the displacement model of SCCTB in Section
Derived from (
Based on the strain models of SCCTB in Section
The shearing force per unit length of the interface is given by
In order to satisfy the consistency requirements, for the different displacement functions to contribute to the same strain function, the degree of polynomial has to be the same after the finite element approximation. Otherwise, there are unreal geometric constraint conditions, which can significantly reduce the accuracy of the finite element method and result in shear locking [
Nodal degrees of freedom of the SCCTB element.
The Hermite polynomial shape function to satisfy the continuity condition of
Substituting (
Substituting (
The kinetic energy of the SCCTB can be expressed as
The strain energy of SCCTB can be expressed as
According to (
By substituting (
By substituting (
Based on the SCCTB element stiffness matrix
The transformational relationship between the SCCTB overall degree of freedom vector before and after applying the boundary constraint can be expressed as follows:
The free vibration function for a structure with multiple degrees of freedom can be expressed as follows:
Let
Substituting (
Substituting (
Using (
The software MATLAB has been used to develop the FBEM program of the abovementioned SCCTB element. Then, the natural vibration frequency of SCCTB has been solved by including the effects of both the interface slip and shear deformation.
If the shear deformation is not considered, then
Using (
Using (
After obtaining the SCCTB element stiffness matrix and mass matrix without the effect of the shear deformation, the same method in Section
To verify the accuracy of the developed FBEM (Section
The calculations by the finite element method have been carried out using the finite element program, ANSYS. The upper chord and the lower chord have been simulated using SHELL43 shell element. The vertical web member, oblique web member, lower horizontal connection member, and oblique bracing member have been simulated using BEAM188 elements. The concrete slab has been simulated using SOLID65 solid elements. The studs have been simulated using COMBIN14 spring elements. The elastic modulus
Tables
Comparison of calculation results between the FBEM and ANSYS model (SCCTB1).

Computation methods  Natural frequencies (Hz)  

1st  2nd  3rd  4th  5th  6th  
0.40 

9.443  24.192  44.764  70.425  100.160  122.580 

9.313  24.049  44.384  68.993  96.811  126.960  

9.706  26.119  50.630  83.145  123.639  172.038  


0.60 

9.580  24.359  44.930  70.570  100.280  122.610 

9.480  24.274  44.629  69.225  97.022  127.146  

9.895  26.404  50.977  83.524  124.037  172.447  


0.80 

9.701  24.512  45.087  70.709  100.390  122.630 

9.628  24.482  44.861  69.448  97.227  127.329  

10.064  26.670  51.311  83.892  124.427  172.850  


1.00 

9.809  24.653  45.236  70.844  100.510  122.660 

9.760  24.676  45.083  69.665  97.428  127.508  

10.216  26.920  51.631  84.251  124.810  173.248  


2.00 

10.214  25.236  45.884  71.454  101.020  122.770 

10.257  25.484  46.056  70.653  98.362  128.356  

10.790  27.980  53.066  85.920  126.630  175.165  



9.206  5.633  3.629  2.350  1.577  1.087 
Comparison of calculation results between the FBEM and ANSYS model (SCCTB2).

Computation methods  Natural frequencies (Hz)  

1st  2nd  3rd  4th  5th  6th  
0.40 

6.717  17.314  32.190  50.824  72.887  97.708 

6.598  17.177  32.065  50.470  71.700  95.122  

6.795  18.224  35.278  57.903  86.098  119.830  


0.60 

6.833  17.465  32.347  50.966  73.013  97.813 

6.734  17.370  32.284  50.686  71.903  95.307  

6.944  18.454  35.562  58.214  86.427  120.170  


0.80 

6.935  17.602  32.494  51.103  73.134  97.915 

6.853  17.548  32.491  50.893  72.100  95.488  

7.075  18.666  35.832  58.516  86.750  120.504  


1.00 

7.025  17.729  32.634  51.235  73.252  98.014 

6.957  17.712  32.688  51.094  72.292  95.665  

7.190  18.864  36.090  58.810  87.065  120.834  


2.00 

7.352  18.242  33.234  51.826  73.796  98.481 

7.338  18.384  33.539  52.001  73.182  96.501  

7.612  19.683  37.228  60.157  88.549  122.409  



10.079  6.564  4.396  2.944  2.026  1.429 
From Tables
Relationship between the calculation errors of FBEM and the mode orders of natural vibration frequency.
Relationship between the shear deformation effect and the mode orders of natural vibration frequency.
Relationship between the interface slip effect and the mode orders of natural vibration frequency.
(
(
(
(
(
(
Figures
Comparison between the first six flexural modes of ANSYS and FBEM models of SCCTB1.
Comparison between the first six flexural modes of ANSYS and FBEM models of SCCTB2.
By including the effects of the interface slip, shear deformation, moment of inertia, and many other influencing factors on SCCTB, and based on Hamilton’s principle, the SCCTB element stiffness matrix and mass matrix with cubic Hermite polynomial shape function have been developed. Based on the developed SCCTB element, this study has also developed a FBEM program which can calculate the natural vibration frequency of SCCTBs that are common in the engineering practice. The program has been used to calculate the natural vibration frequencies of several SCCTBs with different spans and different degrees of shear connections. The conclusions are as follows:
The FBEM’s calculation results of the natural vibration frequencies of several SCCTBs agree well with the results of ANSYS’s finite element calculations. This is an indication that the SCCTB element stiffness matrix and mass matrix developed in this study are effective. Hence, this is a basis for the further applications of the FBEM to the dynamic calculations of SCCTB.
The SCCTB’s bending vibrationtype of loworder is dominated by the bending deformation caused by the section rotation, and the shear deformation effect of SCCTB’s loworder natural vibration frequency is insignificant.
For the SCCTB with highorder natural vibration frequency, the effect of the shear deformation is greater. Hence, the shear deformation effect on the SCCTB’s highorder natural vibration frequency cannot be ignored.
Under different degrees of shear connections, the curves for the shear deformation effect and natural vibration frequency order of SCCTBs are overlapped with each other. This is an indication that the effect of the degree of shear connections on SCCTB’s shear deformation is not significant.
While the interface slip effect on the SCCTB’s highorder natural vibration frequency is insignificant, the effect on the SCCTB’s loworder natural vibration frequency is dominated by the section rotation deformation which cannot be ignored.
The authors declare that they have no conflicts of interest.
The research described in this paper was financially supported by the National Natural Science Foundation of China (51778630, 51408449 and 51378502), the Special Fund of Strategic Leader in Central South University of China (under Grant 2016CSU001), and the Fundamental Research Funds for the Central Universities of Central South University of China (2016zzts078).