Miscellaneous Information

Iteration Control - Improvement of the Convergence Concrete Law

Usually a tolerance of 0.002 is sufficient for the concrete law (record SYST

...TOL). This tolerance is also needed for the energy convergence. With nega- tive TOL -1.50 a fixed absolute tolerance of 1.5 kN can be defined, if necessary not before 40% of the iterations: TOL4 -1.50

concrete law, could initially be solved by increasing the number of iterations (SYST...ITER). If after, for example, 50 iteration the energy still increases, the load can not be taken up anymore, this is because:

– a lack of reinforcement (tensile forces can not be compensated anymore)

– the plate is to thin (compression failure)

– the elements next to the singularity are to small (shear problem)

All these three problems lead to a point where the load can not be compensated anymore, due to an increase in strain. So the product of load·strain = energy will increase constantly.

This can only be resolved by altering the system: – increase reinforcement

– adjust the dimensions

– decrease the load (try a lower load step)

If the program does not reach a residual force equilibrium, even if the energy converges (the energy seems to remain close to the limiting value), then generally it could be attributed to the following reason: The program does not reach equilibrium in the normal direction of the plate - small changes in the strain plane generate large normal forces. Although this phenomena is usually insignificant for plate calculations and only has a local influence on the result. This becomes apparent when a check is performed on the residual forces (WinGRAF...nodes...residual forces).

Often a damping of the iteration is successful with SYST...FMAX 0.90 (FMAX smaller than 1.0 or FMAX 1.10).

If no convergence is found, the intermediate results of the iterations are saved with the load case numbers from 90001. They can be checked in the ANIMATOR with displacements and in WinGRAF with residual forces in order to find out the cause of the lacked convergence. With ECHO RESI 7 this can be enforced also for a convergent run.

Often the convergence can be improved by the lowering of the concrete tension stiffness e.g. to 0.5 N/mm2_{. This is because the negative stiffness, on the de-} creasing curve of the concrete stress-strain curve, is not that big and it can be equilibrated by the positive stiffness of the reinforcement including the tension s- tiffening. On the other hand it is possible that a bigger concrete tension stiffness

could instantaneously release a large amount of concrete failure energy. This energy would then spread like a chain reaction through the system and conver- gence would be prevented. An increase in the minimum reinforcement would also improve the convergence, because the reinforcement would counteract the negative stiffness of the decreasing stress-strain curve.

Steel Law

Steel plates or shells do not encounter instantaneous tension failure, as is the case for the concrete law. Besides being able to increase the number of itera- tions (SYST...ITER), one also has the possibility to generate a trilinear instead of a bilinear stress-strain curve, which has its advantages. The tangential stiffness in a bilinear curve is equal to zero, i.e. a strain correction of the program would not alter the stress. Another advantage is the slowly increasing curve, which is favourable for the Newton-Raphson-method.

In addition the steel law allows a step-wise increase of the load, as described in chapter "bearing load iteration".

Tangential Stiffness

For non-linear material calculations one always works with linear initial stiffness and compensation of the unabsorbed residual forces. An experimental material stiffness was implemented for improving the convergence of the iterations, but it did not achieve the desired effect and was thus deactivated.

Non-linear material calculations, according to the first-order theory, utilize the Crisfield standard (CTRL ITER 0) in the iteration control. The line-search with the geometric-tangential stiffness matrix is only utilized in case of second-order theory are crack springs (CTRL ITER 3). The material matrix is always substi- tuted with the linear initial stiffness and is not tangentially updated.

You can always try both variants CTRL ITER 0 or CTRL ITER 3 but please start with the default (no input to CTRL ITER).

Bearing Load Iteration

In a lot of cases it is of interest to establish the maximum bearing load of a given system. To do this the bearing load iteration is applied. Here the load is increased step-wise until the point of failure is reached. The point of failure is interpreted as a lack in energy convergence, i.e. the system starts to fail if the energy is steadily increased during a bearing load iteration. A simple bearing load iteration can be found in example ase9_quad_euler_beam.dat, for cracked concrete also in the verification examples nonlinear_quad_concrete_beam.dat

For non-linear material calculations it often happens that this automatic method does not reach adequate equilibrium, due to a lack in normal force convergence, even if the bearing load has not been reached yet. This can be overcome by en- tering a negative input for STEP under the record ULTI. Now the load is contin- uously increased, even if no convergence is reached after every individual load step. The user has to be assessed then the systems bearing capacity according to the energy convergence, the remaining residual forces and the load deflec- tion curve. Alternatively the procedure could gain stability through the dynamic calculation.

2.13

Membrane Structures: Formfinding and Static Analysis

2.13.1 Overview

Membrane structures are characterized by transferring of loads only with normal forces. Bending moments and shear forces are not available. The analysis with real membrane elements is more comfortable and more exactly unlike the simplified processing with a truss model, because the geometry and the stress state can be generated any exactly. An orientation of the truss elements in defined directions is not necessary.

The first task is the formfinding during the analysis of membrane structures. A corresponding form is searched for a desired stress state in the membrane. A soap skin is only result here for the isotropic prestress. Forms which are different to the soap skin need a normal force distribution which modifies itself about the structure.

If the membrane form is found, real load cases can be calculated with this new form as initial system. The membrane must be omitted here for compression. Further textile properties are realized mostly by a simplified linear- elastic or- thotropic material law.

Edge stiffenings with edge cables, inside cables or compression arches have to be considered in real structures.