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General Description
Section 1
1-72
5) Two types of coordinate systems are used in the generation of
the required matrices and are referred to as local and global
systems.
Local coordinate axes are assigned to each individual element and
are oriented such that computing effort for element stiffness
matrices are generalized and minimized. Global coordinate axes
are a common datum established for all idealized elements so that
element forces and displacements may be related to a common
frame of reference.
Basic Equation
The complete stiffness matrix of the structure is obtained by
systematically summing the contributions of the various member
and element stiffness. The external loads on the structure are
represented as discrete concentrated loads acting only at the nodal
points of the structure.
The stiffness matrix relates these loads to the displacements of the
nodes by the equation:
A
j
= a
j
+ S
j
x D
j
This formulation includes all the joints of the structure, whether
they are free to displace or are restrained by supports. Those
components of joint displacements that are free to move are called
degrees of freedom. The total number of degrees of freedom
represent the number of unknowns in the analysis.
Method to Solve for Displacements
There are many methods to solve the unknowns from a series of
simultaneous equations. An approach which is particularly suited
for structural analysis is called the method of decomposition. This
method has been selected for use in STAAD. Since the stiffness
matrices of all linearly elastic structures are always symmetric, an
especially efficient form of the decomposition called Modified
Cholesky's method may be applied to these problems. This method
is reasonably accurate and well suited for the Gaussian elimination
process in solving the simultaneous equations.