ABSTRACT
This work which is titled the application of differential equations in the study of beams will be limited to the study of torsion.The researcher commenced the study by a derivational of the torsional equilibrium equation by the application of differential equations. The existing equation due to St. Venant’s classical torsion theory which was based on the in dependent summation of the combined torsional stresses was compared with the derived equation by Osade be which took into consideration the interactive effect of pure and warping torsion. Values obtained from the derivation were used to obtain closed form expressions that explicitly described the torsional stresses along the continuous beam under various loading and support conditions using the initial value approach to obtain deflections, rotations and bi-moments. The results of the comparative analysis based on the numerical study further revealed that the present equation based on the interactive effect of the combined torsional state shows a significant effect on torsional rigidity, giving rise to higher stresses, thereby offering safe design.
CHAPTER ONE
1.1 INTRODUCTION
In a thin walled beam, the shear stresses and strains are much larger relative to those in solid rectangular beams. When thin wall structures are twisted, there is a phenomenon called warping of the cross section. The cross section warps and Bernoulli’s hypothesis is violated. Warping is defined as the out-of-plane distortion of the beam’s cross section in the direction of its longitudinal axis due to the twisting of its cross section [5]. This has led to the development of this research work which concentrates specifically on single cell and continuous single cell thin wall structures in which problems arise due to the torsion response of thin walled box beams. Structural codes during the first half of the century were silent regarding torsion design. Thin wall open sections usually have a very low torsional rigidity, combined bending and twisting occurs in axially loaded members such as angles and channels, whose shear center axis and centroidal axis does not coincide. According to Holgate,thin wall structures aregenerally assumedtoinclude:structures made of thin steel plates,such as boxcolumnsand beams,cranes,pressurevessels,pipelines,air-craft wings and steel silos, Also a limited range of concrete structures such as shells, folded plate, cooling towers and some bridge elements. In complex structures such as curved beams, helical stairways and eccentrically loaded box beams, torsional response dominates the structural behavior. Torsional moments tend to twist the structural member around its longitudinal axis, inducing shear stresses. However, structural members are subjected to pure torsion moment. In most cases, torsion moment act concurrently with bending moment and shearforces.
In basic engineering structures such as bridges and buildings are subjected to dynamical loads, these structural elements set up vibrations and interacting forces (bending moment stresses and torsion) hinders its stability under load. Analytical formulations in structural engineering helps to understand the behavior of complex structural systems .The subject of torsion is widely avoided in most analysis due to the shear complication of solving a plethora of differential equations and then correctly combining them with any additional planar shear and bending forces on the structure[6-9]. According to Osadebe (2005), the recent multitude of research efforts invested in thin-walled structures are justified by:
