# Solutions

The UDEC calculation cycle repeats over-and-over until a steady-state solution has been achieved (i.e, the maximum nodal force vector or the maximum unbalanced force is zero). For a numerical analysis, the out-of-balance force will never reach exactly zero. Generally it is sufficient to say that the model is in equilibrium when the maximum unbalanced force is small compared to the total applied forces in the problem. For example, if the maximum unbalanced force is initially 1 MN and drops to approximately 100 N, then the model can be considered at equilibrium, within 0.01% of the initial maximum unbalanced force. In this way, the user can assess when equilibrium has been reached.

In addition to cycling or stepping a specified number of timestep calculations, UDEC features a number of solve tool that enables the automatic detection of the steady-state solution for mechanical problems. Modeling continues to be performed until a limiting condition is met, including the following.

### Solve Ratio

The ratio of the maximum unbalanced force to the total applied forces in the model is small. The default limit is 0.00001, but this can be specified by the user as appropriate.

### Solve Age

The model will cycle until the total problem time equals a specified value. If creep is active, creep time is used for the limit. Otherwise, mechanical time is used.

### Solve Elastic

Sets joints and zone constitutive models to infinite strength for initial model equilibrium. This prevents artificial plastic deformations due to numerical shock. Afterwards the model is solved again with the strength values specified.

### Solve Relax

Used to slowly reduce the forces on the inside of an excavation to avoid a large tensile stress wave. This tensile stress wave may cause dynamic failure in zones that would not normally fail under the static stresses caused by the excavation. SOLVE relax can also be used to reduce the boundary forces on the internal boundary of an excavation down to a prescribed level, to simulate the 3D effect of a tunnel advance. A ground reaction curve can also be generated automatically if a table number and a history are supplied.

### Solve FOS

UDEC provides a manual or automatic factor-of-safety solution using the Strength Reduction Method (SRM) that can be used for stability analyzes of models (e.g., slopes, retaining walls, tunnels, etc.). The SRM progressively reduces the shear strength of the material to bring the slope to a state of limiting equilibrium. The automated method can be applied to the Mohr-Coulomb, ubiquitous-joint and Hoek-Brown material models. The Coulomb joint model and certain strength properties for structural elements can also be incorporated into the automated factor-of-safety calculation. Manually, the SRM approach can be applied to essentially any material failure model to evaluate a factor of safety based upon the reduction of a specified strength property or property group.

## Energy Calculation

When either a stress (force) or boundary-element outer boundary being used (i.e., the outer boundary of the model be free to deform), energy changes can be measured for a UDEC model (intact rock, the joints and for the work done on boundaries). Energy components calculated include:

• Total boundary loading work;
• Potential energy;
• Kinetic energy;
• Damped energy;
• Strain energy (stored in the rock mass);
• Viscous boundary work (dynamic analysis); and
• Plastic work.