Dynamic Response of 3D Frame-Plate Subjected to Wind Load using Finite Element Method

This paper studied the dynamic response of a 3D frame-plate system subjected to wind load. The government equations were established using finite element method. The numerical results were carried out using the Newmark direct integration method. The influence of some parameters on the dynamic response of the system was also investigated and discussed. The numerical results of this work can be useful for calculating, design and examine in practical. Keywords—3D frame-plate, dynamic response, wind load, finite element method.


INTRODUCTION
The tall building is vulnerable to wind loads due to their structural and aerodynamic characteristics, and their designs are frequently governed by wind loads over design loads. Due to the importance of wind load in tall building design, researchers have made attempts to analyze the dynamic response of the tall building as a complex system of frame and plate. Gu [1] analyzed the effect of a cross-wind on a typical tall building. Holmes [2] studied the dynamic response of a lattice tower subjected to aerodynamic load with aerodynamic damping. Vibration and motion of a rectangular high-rise building under wind load were investigated by Katagiri [3]. Jeong [4] investigated dynamic response of a tall building subjected to strong wind load using finite element method (FEM) with a frame-plate system model. In [5], Park studied the dynamic response of a high-rise building under wind load with the support of GPS. Kijewski [6] used a full-scale to study the behavior of tall building under wind load with a frame-plate model. Aly [7] studied vibration and vibration control a high-rise building under wind load with passive and active tuned mass damper. Kim [8] and Rosa et al. [9] applied the frame-plate model to investigate the wind-induced excitation control of a tall building with tuned mass dampers. Belloli et al. [10] studied the effects of wind load on a high slender tower using numerical and experimental method. Lin et al. [11] using FEM to investigate characteristics of wind forces acting on a tall building. Mendis et al.
[12] studied the dynamic behavior of a tall building subjected to wind load. This paper aims to investigate the dynamic response of a 3D frame-plate system subjected to wind load using FEM.

II. MODEL AND FINITE ELEMENT FORMULATION
In this paper, an 8-story 3D frame-plate system under wind load is considered as shown in Fig. 1. Hypothesis: the materials are linear elasticity, the deformation of the system is small, the plate satisfies Reissner-Mindlin plate theory.

A. Space beam element
The frame was described by space beam element (3D beam element). The space beam element has 2 nodes and 6 degrees of freedom per node, including , , , , , . The strain fields are obtained as following formulas The nodal displacement vector is     , , , , , , , , , , , x y z axes at node The nodal load vector of the beam element is expressed as     1 2 3 10 11 12 , , ,..., , , bT e In the global coordinate where   e T is the coordinate axes transition matrix.

B. Plate element
According to Reissner-Mindlin plate theory, the displacement fields are written as following formulas The transverse shear deformation of the plate is obtained as The linear elastic stress-strain relationship of plate is defined as − is shear modulus of material.
In this paper, the plate is discretized by quadrilateral 4node plate element, each node contains 6 degrees of freedom as in Fig. 3. The displacement and rotations in the element are interpolated from the nodal values as  The train energy of a plate element is given by where  is the shear correction factor, in general 5 / 6.

 =
The element stiffness matrix is given as The equation of motion of 3D frame-plate system is represented as follows are the mass, damping and stiffness matrices of the system, respectively. They are obtained by assembling all its element matrix through the direct stiffness method and imposing the prescribed boundary conditions. The damping matrix is constructed by using the Rayleigh damping theory in following form: with a and b are calculated by using damping ratio  and natural frequencies 12 ,  as following formula 12 12 2 1 This is a secondary linear differential equation with timedependent coefficients. This equation is integrated using the Newmark direct integration method. The average constant acceleration method is 0.25  = and 0.5

 =
which ensures numerically unconditional stability is used.

E. Finite element algorithm
Step 1: Calculate the mass, damping, stiffness matrices and force vector.
Step 2: Set initial conditions.
Step 3: Calculate initial parameters. Step Step 5: Export and display results.
III. RESULTS AND DISCUSSION The 3D frame-plate system including 25 vertical columns, 80 beams and 8 floors as shown in Fig. 1   According to Fig. 4 -Fig. 6, when the system subjected to wind load, it deforms and vibrates. The horizontal acceleration, velocity and displacement of point A is larger than point B.

A. Influence of the system height
The relationship between the system height H and the displacement of point A and B are shown in Fig. 7. It shows that the displacement of the system has a nonlinear dependence on the height of the system. The displacement of point A and B increase when increasing the height of the system. The displacement of point A increases faster than the displacement of point B, so when increasing the system height, the upper points will fluctuate greatly.

B. Influence of the wind velocity
Influence of the wind velocity is shown in Figure 6. The velocity of wind has a great effect on the deflection of the system. It shows that the displacement of point A and B increase when increasing the wind velocity. When the wind velocity is slow, the displacement increases slowly then increases quickly when the wind velocity is large. Therefore, it can be seen that the strong wind greatly affects the behavior of the frame-plate system. IV. CONCLUSION In this study, a 3D model of a frame-plate system was developed and investigated. The algorithm and dynamic analysis program were established based on finite element method and Newmark direct integration method. The numerical results show that the wind load has a great effect on the dynamic response of model and can cause the dangerous status of the system. The results of this paper can be useful for calculation, design and selection of reasonable solutions for new design and reinforcement of modern tall buildings.