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Introduction The demand of the large space and using diversity, cantilever slabs are commonly used in many structural systems, such as bridge deck slabs, balcony, awning, and balcony window. Which are usually damping decreases and the lower natural frequencies, it attached more and more attention. Zihai Shi[Reference.
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The model and parameters The model of the cantilever slab was developed under construction part in ABAQUS. One side of the cantilever slab with dimensions of 5. 7m is fixed. The cantilever length was 2m, the root thickness of the cantilever slab is 200mm, end thickness was 180mm, as shown in Fig 1. The three-dimensional entity unit with eight nodes hexahedron reduced integral (C3D8R) were adopted to simulate the concrete. Three-dimensional linear truss elements with two nodes were adopted to simulate the steel. The section area was changed with the diameters. The connected was carried out by EMBEDDED ELEMENDT[] ABAQUS，Inc．ABAQUS user manual，version 6. 7[M]．2007． ] between with steel and concrete. The boundary condition was fixed one side and the others were free. The uniformly distributed load was applied. The FEM was shown in Fig 2. Typical material and geometrical properties[] Zheng Hong-yu, Su Yi-sheng, Engineering mechanics[J]. 2007(2); Vol. 24 No. 2: 120-128. ] for the cantilever slabs that had been considered in this study were given in Table 1. Fig. 2. The section size and reinforcement of the cantilever slab Table 1 Material and geometrical properties for the cantilever slab E（N/m2） n (kg/m3) ft（N/m2） Fc(N/m2) Reinforcing Steel.
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Analysis on Vibrate frequency and displacements of cantilever slabs (i) vibration of the structure （a）Mode 1: f=4. 91Hz； （b）Mode 2: f=9. 22Hz （c）Mode 3: f=23. 30Hz； （d）Mode 4: f=38. 64Hz Fig. 3. The first four modes of the cantilever slabs Recently years, because of new structural material, construction method and structural analysis method, the modern engineering structures become lighter, more soft, span bigger. It caused more easy vibration under load. People in the building would feel discomfort, tension, even fear, it will decrease the comfort of the structure directly. As well as increase the floor quality will reduce the natural frequency of vibration of structure. It is possible to reduce the vibration of the cantilever structure, but the effect is limited. So it is not recommend by increasing the quality of structure. The stiffness of the structure is enhanced and its natural frequency is increased by improving the material elastic modulus[] Xi Ling lu, GuoFang Jin, etc, Non-liner theory and apply of steel reinforcement structure [M]. Shanghai, Tongji Univerisity publishing company, 1999. ]. The bearing capacity of the structure is decreased by 26. 3% with the vibration frequency increased by 10%. Because of the damping ratio of reinforced concrete structures change range is small, and there are lots of factors affect the structural damping ratios[] WANG Suguo, The Effect of Slabs on the Failure Mode of Reinforced Concrete Frame Structures Journal of CiviI，Architectural&Environmental Engineering. 2009(2); V01. 31 No. 1: 66-70. ], such as material damping, the surrounding medium, support style etc. The vibration and displacement behaviors of cantilever slabs with different overhang length or diameters of the reinforced steel were discussed using the nonlinear finite element software in the paper. The vibration properties of the cantilever slab can be changed with slab quality, stiffness, overhang length, and damping etc. The non-linear FEM(Fig. 2. ) was built in Abaqus. The first four order vibration frequencies amplifying diagram were shown in figure 3. Fig. 3 presents an overview of the vibration behavior obtained from the cantilever slab. It was found that the first mode of vibration in Fig. 3(a) indicates bend motion. The second, third and fourth mode indicated high amplitude of motion at a specific location i. e. at the middle part of the first bay of the cantilever slab, as shown in Fig. 3(b, c, d). （a）d=12mm （b）d=14mm Fig. 4. The maximum displacement plot of cantilever slab with different diameters (ii) Displacement Under the same assumption (overhang length is 2m, as shown in Fig. 2. ), the uniformly pressure of 3. 5KN/m2 were applied to the cantilever slabs. The reinforced steel was respectively setting as 12@100 and 14@100. The magnified displacement plots of deformed and undeformed cantilever slabs considered different diameters were shown in Fig. 4. It was found that the maximum displacement in the load direction was 8. 495mm and 7. 194mm, respectively. So the maximum displacement was decreased with increasing the diameters of the reinforcement. The overhang length was 1. 8m, 1. 5m and 1. 2m, respectively. The maximum displacement and first fourth mode of vibration at the load of 3. 5KN/m2 were calculated in Table 2. As it can be observed in Table 2, the maximum displacement of cantilever slab was decreasing with reducing of the overhang length. It would be speed up when the overhang length less than 1. 5m. The stiffness of the structure is enhanced and its natural frequency is increased by reducing the overhang length. Table 2 The maximum displacement and vibration frequencies with different overhang length Overhang length(m) Reinforce The maximum displacement(mm) Mode 1 (hz) Mode2(hz) Mode 3 (hz) Mode 4(hz).
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47 The stiffness, mass and overhang length of the structure had influence on the dynamic characteristics of the cantilever slabs. By computing and comparing displacement and vibration frequencies it was found that overhang length and reinforcement diameters had much greater influence on structural properties. The material elastic modulus and mass had limited influence on the dynamic characteristics of the structure; on the other hand may it have reversed affection. Of course, there are complex factors influencing the displacement and vibration frequencies of the cantilever slabs, for examples, overhang length, reinforcing, elastic modulus, damping and mass, etc.
DOI: 10.1016/0022-460x(75)90006-1
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Coclusions A three-dimensional FEM of the cantilever slabs was developed using Abaqus finite element software. The stiffness, mass, damping and overhang length have influence on the displacement and vibration. As shown in these figures and tables, the following relation is introduced in this paper: (1) The overhang length of cantilever slabs is reduce, the small the maximum displacement will be; (2) The overhang length, damping and the reinforcement have large influence on the vibration of cantilever slabs.
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