Keywords: body attached load, critical speed of wind, follower forces, flutter instability, space attached load.

In engineering the effects of follower forces on structures are obviously, such as wind blows through bridges, or the propulsive force of a rocket, or pouring water into a storage tank. A follower force is a kind of load where the magnitude and direction are dependent on the deformation of the structures. Besides above mentioned, hydraulic pressure, gas flow or current magnetic field interactive forces acting on micro-structures, they all have the similar effects. According to the definition of the follower forces given by Schweizerhof and Ramm [1], they can be classified as two kinds; body attached loads or space attached loads. In the literature a big event had occurred that was the collapse of the Tacoma Narrows Bridge as a result of the wind blowing the bridge through a period of time in a steady way on November 7, 1940.

Wind follower forces may cause structures becoming flutter instability from a dynamics point of view if the wind speed exceeds the critical speed of the wind that the structure can withstand. Because the forces acting on the structures are deformation dependent, they are typically called follower forces. Engineers and scholars have done a lot of work on this topic. One of most famous scholars is Botolin [2]. He had solved many engineering problems related to follower force effects on structures. He set up the differential equations for the different kinds of structures and imposed appropriate boundary conditions such as cantilever beams under follower forces, wing of airplane under wind flow and their corresponding boundary conditions, and so on. But for a complex structure using the differential equation has its limitations in the analysis of the structure as a result of imposing the boundary conditions. This causes difficulties in the solution of the complicated differential equations with specific boundaries.

Owing to invention of the computers, and the enhanced computational ability, the use of the stiffness matrix method has become a worldwide trend for the solution of the complex and large structures under the action of follower forces like cooling towers for nuclear power plants under wind loads or long span bridges under wind loads. One can use variational principles to conquer the difficulties of solving differential equations with boundary conditions, by solving algebraic equations systematically with the aids of computers instead of differential equations, (Belytschko et al, [3]), or using natural formulation, (Argyris and Symeondis [4]). Recently many researchers have used finite element methods to solve the problems such as Vitaliani and Gasparini [5] and Lazzari [6]. Using finite element methods one has to derive the tangential stiffness matrix that will contained the so called load stiffness matrix, most researchers take the variation of non-conservative work to derive the so called the load stiffness matrix, (Hibbitt [7], Schweizerhof and Ramm [1]). They have made great advances in the analysis of the structures under follower forces. But for beginners it is not an easy thing to understand the variation principles. In this paper it is of interest to provide a physical intuitive and graphical interpretation method to enable students and novices understand the effects and to derive the load stiffness matrix of a simple two node planar truss element based on simple mechanics and a preliminary knowledge of mathematics of vectors. The load stiffness matrix once derived, can easily be incorporated into finite element programs, the analysis of the structures under follower forces. In this paper a body attached load definition direct concept will be proposed for the forming of the load stiffness matrix of planar truss elements. A simple two-truss member under different load cases for follower forces will be presented.

1

Schweizerhof, K., Ramm, E., "Displacement dependent pressure loads in nonlinear nonlinear finite element analyses," Comput. Strut., Vol. 18(6), 1099-148, 1984. doi:10.1016/0045-7949(84)90154-8

2

Bolotin, V.V., "Nonconservative Problems of the Theroy of Elastic Stability", Pergamon Press, London, 1963.

3

Belytschko ,T., Moran, B. Liu, W.K. "Nonlinear Finite Elements for Continua and Structures", John Wiley & Sons, Third Avenue, N.Y., 2000.

4

Argyris, J.H., Symeonidis S., "Nonlinear finite element analysis of elastic systems under non-conservative forces loading - natural formulation - I Quasistatic problems", Comput. Meth. Appl. Mech. Engng, Vol. 26, 75-123, 1981. doi:10.1016/0045-7825(81)90131-6

5

Vataliani, R.V., Gasparini, A.M., "Finite element solution of the stability problem for nonlinear undamped and damped systems under nonconservative loading," Int. J. Solids Structures., Vol. 34(19), 2497-2516, 1997. doi:10.1016/S0020-7683(96)00115-1

6

Lazzari, M., Vitaliani, R.V., Majowiecki, M., Saetta V.A.S., "Dynamic behaviour of a tensegrity system subjected to follower wind loadinge," Comput. Strut., Vol. 81(6), 2199-217, 2003. doi:10.1016/S0045-7949(03)00291-8

7

Hibbitt, H.D., "Somer follower forces and load stiffness," Int. J. Numer. Meth. Eng., Vol. 14, . 937-941. 1979. doi:10.1002/nme.1620140613

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