الفهرس 
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Abstract
This thesis aims to study the elastic stability of curved beams
and grid shells which are represented geometrically by single or
double free curves. The analysis of these types of structures is one
of the more difficult problems that has been attempted with the
finite element method. The notion of shells with. free surfaces are
characterized completely by the motion of a reference surface, and
a finite element analysis is complicated greatly by the need to
represent accurately the motion of this reference surface. Both the
stretching and curvature change of the surface are required in the
prediction of the strains existing throughout the shell.
This thesis presents a comprehensive study of the geometric
representation of free curves and surfaces. A comparison between
the different methods which are used to represent the curves and
surfaces is made first, from which it can be concluded that the
Quadratic B-spline polynomial is very efficient to represent the free
surfaces since it requires less number of elements and control points
as well as less computations. The strain energy equations of plane curved structures are
derived first based on the Quadratic B-Spline function with three
control points. Four parameters have a significant effect on the
derivation of the strain energy equations of plane curved structures.
These parameters are :
(i) Change of curvature of the curved structure.
(ii) Axial strain of the curved structure.
(iii) Boundary conditions
(iv) External forces.
The stability of straight struts and compressed beams with
initial imperfections as weII as circular beams subjected to different
types of loads are studied using Quadratic-B-Spline finite element.
A comparison between the results and those of finite element
method and Cubic-B-spline finite element method is made. After
developing the strain energy equations of plane curved structures,.
the strain energy equations of grid shell is derived based on BiQuadratic
B-Spline finite element with nine control points. To
derive this equation, the effect of five parameters on the strain of
grid shell has to be considered. These parameters are :
(i) Change ofNorrnal Curvature.
(ii) Change of goedesic curvature.
(iii) Torsional bending term.
(iv) Boundary conditions
(v) External loads.
The stability of hyperbolic paraboloid grid shells and elliptic
paraboloid grid shells as well as cylindircal grid vaults subjected to
venical loads are studied using quadratic B-Spline elements.This element is capable of finding to a good degree’ of
accuracy the failure load, the mode of buckling and load-deflection
curve of grid shells .