Structures, statically determinate indeterminate

Not all structures can be fully analyzed by the methods of statics. If the number of discrete equilibrium equations is equal to the number of unknown loads, then the structure is said to be statically determinate and rigid. If there are more unknowns than equations, then the structure is statically indeterminate. If there are more equations than unknowns, then the structure is said to be statically indeterminate and nonrigid. 

In the analysis of statically Indeterminate structures, static Indeterminate forces Bj, (1=1,...,n) are Introduced In order to satisfy n boundary conditions which cannot be satisfied from equilibrium only. Given the bending moment Mo(x) of the associated statically determinate beam, the bending moment M(x) is... 

In which M (x) are the bending moments within the associted statically determinate beam due to statically Indeterminate forces B < equal to unity. Since the Bj are determined from deflection boundary conditions and deflections depend on the elastic properties of the structure, the constants B will be random. Let the prescribed boundary conditions be... 

The present study shows that It is possible to evaluate the variability of statically determinate and statically indeterminate structures due to spatial variation of elastic properties without resort to finite element analysis. If a Green s function formulation is used, the mean square statistics of the indeterminate forces are obtained in a simple Integral form which is evaluated by numerical methods in negligible computer time. It was shown that the response variability problem becomes a problem Involving only few random variables, even if the material property is considered to constitute stochastic fields. The response variability was estimated using two methods, the First-Order Second Moment method, and the Monte Carlo simulation technique. 

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