The analysis is performed using the Finite Element Method (FEM) for plane trusses. During the analysis process, the program forms the model effective Stiffness Matrix [K] as well as the vector of the effective External Forces {F}. The supports are also taken into account to form [K] and {F}. Then, the linear equations system {F}=[K]{D} is solved in order for the effective Displacement Vector {D} to be calculated:
{D}=[K]-1·{F}
If the model is statically well-defined then the effective Stiffness Matrix [K] is reversible (Matrix Determinant greater than zero) and no problems will occur during the analysis process.
Errors during the analysis process
Computational errors may occur during the analysis process. In that case, the program warns the user with the following message:
1. 'Finite Element Analysis failed. Please check the Model.'
This error occurs due to the model stiffness matrix being non-reversible. There may be a zero-element appearing at the matrix diagonal. In this case the determinant of the model effective stiffness matrix is equal to zero, the matrix is non-reversible and the linear system of equations cannot be solved for the displacements. The structure is essentially a mechanism that cannot be analyzed. In this case, check the Model and make sure it is not a mechanism.
Example: An under-constrained Model, such as a member with a roller support, as shown below. The Model can freely rotate and move and equilibrium cannot be reached, by any means. The Model cannot be analyzed and the programs gives us the error message, as shown below.
2. 'Finite Element Analysis verification failed: Model not in equilibrium state. Please check the Model, it appears to be a mechanism.'
This case does not have to do with a computational error occurring during the Analysis process. The Analysis has been carried out "successfully" by the computer, and no numerical errors occurred. Yet, after the Analysis is finished, EngiLab Truss.2D performs several verification tests to make sure that the Analysis results are valid and trustworthy. In particular, the program calculates the following, for the entire Model (taking also into account the calculated support reactions):
•Sum of Forces in the X- direction (ΣFX)
•Sum of Forces in the Y- direction (ΣFY)
•Sum of Moments about the (0, 0) point on the plane (ΣM(0,0)).
These calculations take into account the external forces and the calculated (from the Analysis) support reactions. In a well-defined Model that has been successfully analyzed, all three values must be equal to zero, i.e. ΣFX=0, ΣFY=0, ΣM(0,0)=0 so that the Model is in equilibrium state after the external loads have been applied.
If, after the analysis, any of these values is not equal to zero (given a tolerance value of 10-5, due to rounding errors), then this is a clear indication that something is wrong with the Model and the program will show the following error message.
This means that most probably the structure is a mechanism. In this case the model stiffness matrix is in fact non-reversible but during the analysis process instead of the right zero value, a very small non-zero value appears at the matrix diagonal due to computational rounding errors. As a result, the Analysis solver finds a solution (with no practical interest) to the linear equations system, yet the resultant displacements have excessive values and the Model is NOT in equilibrium state.
This, for example, can happen in the Model shown below. The Model is obviously a mechanism (only two X-Rollers as supports), as was the one above, but this time the computer gives us the second error message, instead of the first.
Possible reasons for such error messages:
•There are Nodes that are not connected to the model via Elements.
•The Constraints (supports) are inadequate (under-constrained Model), for example the two models shown above.
•Some parts of the structure are not connected to each other and as a result the forces cannot be transferred from one part of the structure to another.
•Due to some other reason, the structure is a mechanism.