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The Current Stiffness Parameter

The current stiffness parameter (cstif) is output to the nonlinear log file and is a useful measure of the degree of nonlinearity being experienced by a structure. In its simplest, unscaled form it effectively measures the stiffness (k) of the system as related to the initial tangential predictor. Alternatively stated it is the ratio of the current stiffness of the structure to its initial stiffness. The unscaled stiffness is evaluated according to

k = Dq/Dp

or scaled (to account for the vectorial nature of the Dq and Dp components) as

k = (DqT Dp)/(DpT Dp)

where Dq is the incremental applied load vector and Dp, the resulting tangential displacements. LUSAS then computes the cstif parameter as follows

cstif = k/k0

Where k0 is the initial k value evaluated at the initial iteration of the first load increment.

Tangential Stiffness

The term tangential refers to the momentary stiffness value of the overall structure. For instance, consider an overall force-displacement response for a structure as follows…

Nonlinear_CSTIF_parameter_Tangential.gif (3737 bytes)

The value of stiffness at a force ft and displacement dt is the tangential stiffness at that point - the magnitude being evaluated from the gradient which is tangent to the curve at that point.

CSTIF Characteristics

Consider the following force vs displacement response:

Nonlinear_CSTIF_parameter.gif (5577 bytes)

From which the following comments made be made…

  • cstif may be negative, positive or zero and may vary between the maximum positive and negative real values for the computer
  • A positive value indicates a "hardening" stiffness whilst a negative value indicates "softening"
  • For "force" loading cstif represents the equivalent tangential stiffness of the overall structure
  • For incremental or total prescribed displacement loading, the tangential loading is zero and, hence, cstif does not give any useful information
  • cstif will be unity on the 1st iteration of the 1st increment of the solution
  • Many structures exhibit a response in which the structure softens as the load is applied. In such situations, it is very useful to force the solution procedure to automatically switch from load control to arc-length control as the limit point is reached. Because cstif is known to be zero at such a point, the introduction of a suitable value of cstif below which this switch is automatically introduced is possible. This is supported in LUSAS and the parameter can be set by specifying the "stiffness ratio to switch to arc-length". The default threshold is taken as 0.4. If the arc-length variable (isurf) equals unity then the arc-length procedure has been invoked.
  • Stress stiffening or second order strain effects associated with geometric nonlinearity may result in large values of cstif
  • The rate of convergence will normally decrease as cstif approaches a zero magnitude

How to Interpret CSTIF

A number of implications regarding the state of the analysis may be derived according to the CSTIF value, and are as follows:


CSTIF Magnitude


Approximately 1
  • The overall stiffness of the structure has remained unchanged by the effects of the nonlinearity
0 < cstif < 1
  • Typically indicates a softening behaviour, but may also be post yield and/or pre-buckling behaviour
  • A "limit" or "saddle" point has been reached
  • An element or global mechanism has been excited
  • Catastrophic material failure. This may indicate incorrect nonlinear material parameters
  • The buckling load level has been reached
  • Contact or other load reaction mechanism has been lost
cstif < 0
  • Unloading is occurring
  • Softening as a result of material nonlinearity. Significant softening may indicate incorrect nonlinear material parameters
  • Post buckling behaviour
  • Contact and any other load reaction mechanism has been lost
cstif > 1
  • Hardening behaviour
  • Post buckling behaviour
  • Stress stiffening influence becoming significant
  • Material locking under large plastic strain
Sudden changes
  • Too large a load increment
  • Too slack a convergence tolerance
  • Data error


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