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Civil-Comp Proceedings
ISSN 1759-3433 CCP: 81
PROCEEDINGS OF THE TENTH INTERNATIONAL CONFERENCE ON CIVIL, STRUCTURAL AND ENVIRONMENTAL ENGINEERING COMPUTING Edited by: B.H.V. Topping
Paper 98
Transport Matrix Based Dynamic Stiffness Matrix of a Circular Helicoidal Bar S.A. Alghamdi
Department of Civil Engineering, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia S.A. Alghamdi, "Transport Matrix Based Dynamic Stiffness Matrix of a Circular Helicoidal Bar", in B.H.V. Topping, (Editor), "Proceedings of the Tenth International Conference on Civil, Structural and Environmental Engineering Computing", Civil-Comp Press, Stirlingshire, UK, Paper 98, 2005. doi:10.4203/ccp.81.98
Keywords: dynamic stiffness matrix, helicoidal beams, transport matrix formulation, structural analysis.Summary
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 I and _{y}I),
while the material properties are defined by material constants _{z}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.
References
- 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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