Keywords: dynamic stiffness matrix, helicoidal beams, transport matrix formulation, structural analysis.

Starting from the solution of differential equations governing stress and deformation states using the transport (transfer) matrix method, this paper outlines matrix formulations used to numerically construct the dynamic structural stiffness matrix for a circular helicoidal beam within the framework of linear elasticity [1]. Specifically the paper emphasizes the main features of a Fortran computer code developed for this purpose and compares it to the general finite element method [2].

With reference to a helicoidal bar [1] with helix angle and opening angle , the position vector in rectangular coordinates is used with Fernet's formulas [3] to yield the following matrix relationship

in which a state sub-vector in local l-coordinate is related to state vector in global o-coordinate using the transformation matrix that accounts for the true curvature and torsion of the helicoidal segment.

The governing scaled differential matrix equation for a helicoid differential segment of mass per unit length and mass moment of inetia , is written [4] as

and the Cayley-Hamilton Theorem [3,5] is used to write the solution of the equation in terms of powers of a (12x12) matrix . The matrix operations result into a transport matrix equation that is also extended to handle generalized load cases. The extended form is written in the following generic matrix form

from which the (12x1) scaled transport load vector at the R-end is determined for specified load distributions after the transport (12x12) matrix is constructed. Then noting that an equation similar to equation (9), but at the left end of the bar, is also obtained by exchanging indices R and L, and replacing angle a with -a in all transport functions involved, the vectors of stress resultants at the L-end and the R-end of the helicoidal beam for the transport matrix formulation, and for the stiffness matrix formulation are related by comparing the two sign conventions of the two methods [5]. Then after some algebraic-manipulations including matrix equations (10) for the L and R ends of the bar, the vectors of stress resultants and the vectors of deformations are related by the following partitioned matrix equations

The above condensed matrix form is identified as the stiffness matrix equation for a helicoid beam segment. The two sub-vectors of the column vector on the right hand-side of equation (10) represent stiffness load vectors and in the local l-coordinate systems as all operations of angular transformation are built within the sub-matrices through the transformation matrix  [6].

The computer code in its present form has been designed to construct and solve the scaled matrix equations (10) for a stand-alone helicoidal beam instead of using less specialized isoparametric finite elements [2]. It requires only few parameters to define the geometry and material characteristics of the beam. The structural geometry is defined by radius R, helix angle , and the cross section properties (i.e.: A, I_x, I_y and I_z), while the material properties are defined by material constants E and . As such, the code is user-friendly and reduces the utilization-hardship that is invariably present in almost all general purpose computer programs. More-over, and due to the low number of design variables, the code will also make the search process for optimal designs simpler and short. The code is, therefore, strongly recommended for implementation within a general purpose finite element code to solve space structural analysis and/or design problems that involve helicoidal beams.

1

A. Abdul-Baki, D. Bartel, "Analysis of Helicoidal Girders", Eng. J., ASCE, 84-99, 1969.

2

K.J. Bathe, Finite Element Procedures, New York, Prentice-Hall, 1996.

3

F.B. Hildebrand, Methods of Applied Mathematics, 2nd edition, Englewood Cliffs, New Jersey: Prentice Hall, 259-279, 1965.

4

S.A. Alghamdi, J.J. Tuma, "Static Analysis of Helicoidal Bars", Proceedings, Fourth International Conference on Civil & Structural Engineering Computing, College University, 247-258, London, 19-21 September, 1989.

5

S.A. Alghamdi, B.O. Elbedoor, "Dynamic Stiffness Matrix of a Circular Helicoidal Bar", CE-Graduate Report (unpublished), Department of Civil Engineering, King Fahd University of Petroleum & Minerals, Dhahran, 1992.

6

S.A. Alghamdi, A.A. Boumenir, "Numerical Solution for the Differential Equations Governing the Free Vibrations of Space Helicoidal Bars", The Arabian Journal for Science and Engineering, 23, 183-193, 1998.

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