The Arc-Length method (Riks method) is currently one of the most stable, efficient, and reliable iterative control methods for nonlinear analysis of structures. It effectively analyzes the pre- and post-buckling of nonlinear structures and their buckling path tracking, making it well-known in the “structural field”. Large general finite element software such as ANSYS and Abaqus have specialized calculation modules. Taking ANSYS as an example, some precautions when using the Arc-Length method for calculations are briefly described.
The APDL command for ANSYS Arc-Length method is as follows:
ARCLEN, Key, MAXARC, MINARC
MAXARC: Maximum multiplier of the reference arc-length radius (default = 25).
MINARC: Minimum multiplier of the reference arc-length radius (default = 1/1000).
This command is used to activate the Arc-Length method and set the minimum and maximum multipliers of the arc-length radius. The reference arc-length radius can be calculated from the load or displacement increment obtained in the first iteration of the first substep. It is calculated by the following equation:
Reference Arc-Length Radius = Total Load (or Displacement) / NSBSTP
Where the Reference Arc-Length Radius is the reference value of the arc-length radius, and NSBSTP is the number of substeps set in the NSUBST command.
The multipliers MAXARC and MINARC can then be used to define the limits of the arc-length radius, as shown in the following equation:
lower limit = MINARC * (Reference Arc-Length Radius)
upper limit = MAXARC * (Reference Arc-Length Radius)
In each subsequent substep calculation, a new arc-length radius is first calculated based on the previous substep’s arc-length radius and the solution condition. Then, this newly calculated arc-length radius is further modified to ensure that it is within the upper and lower limits. When even the minimum radius cannot converge, the Arc-Length method will automatically stop.
Some Notes to Keep in Mind When Using the Arc-Length Method in ANSYS:
- The Arc-Length Method cannot be run simultaneously with AUTOTS, LNSRCH, and PRED commands.
- To obtain the structural ultimate load, it is recommended to load the structure in /solu with a load about 20% higher than the predicted buckling load. The Arc-Length Method is only suitable for static analysis with asymptotic loading.
- When choosing the number of sub-steps, consider that too many sub-steps can result in long solution times. Ideally, select the minimum number of sub-steps required for an effective solution. Sometimes, the number of sub-steps may need to be adjusted and re-solved as needed.
- When using the Arc-Length Method to reduce solution time, the maximum number of balance iterations in a single sub-step should be less than or equal to 15.
- Generally, it is advisable to avoid using JCG or PCG solvers (EQSLV) with the Arc-Length Method since it may result in a negative definite stiffness matrix (negative main diagonal) that could cause solution failure.
- The Arc-Length Method stops solving under three conditions: (1) when the limit defined by the ARCTRM or NCNV command is reached; (2) when convergence is achieved within the applied load range; (3) when the solution is abandoned.
- An unsuccessful Arc-Length analysis may be due to the Arc-Length radius being too large or too small. Study the load offset curve to understand the problem, and then use the NSUBST and ARCLEN commands to adjust the size and range of the Arc-Length radius to appropriate values.
- The overall Arc-Length load factor (ALLF item in the SOLU command) may be positive or negative. Similarly, TIME, which is related to the overall Arc-Length load factor in Arc-Length analysis, may be positive or negative. A negative value of ALLF or TIME indicates that the Arc-Length characteristic is loading in the opposite direction to maintain stability in the structure. Negative ALLF or TIME values are generally encountered in various sudden transition analyses. If TIME is negative, remember to define a suitable range of change (IXRANGE or IYRANGE) before generating any POST26 graphics.
- When reading basic data for POST1 post-processing (SET), it should be based on the load step and sub-step number (LSTEP and SBSTEP) or the data setting number. Do not reference the results of TIME values because TIME values do not always increase monotonically in Arc-Length analysis. A single TIME value may involve more than one solution. Additionally, the program cannot interpret negative TIME values correctly (which may be encountered in a sudden transition analysis).
In summary, the Arc-Length Method is highly versatile, with convergence and stability that are significantly better than other methods for dealing with negative stiffness problems. It can be used for both softening and hardening structures and should be considered first for systems with a high degree of nonlinearity. However, the method is computationally expensive, and for general nonlinear problems, it is recommended to choose simpler methods.