Nonlinear Analysis

In addition to basic linear analysis, Dr. Frame can model problems involving nonlinear geometric effects. The solution and interaction are still real-time and interactive, so there is not necessarily any significant increase in analysis complexity or overhead beyond what is inherent in the phenomenon itself.

Geometric Nonlinearities

Dr. Frame3D's modeling of geometric nonlinearities includes the effects of (compressive) axial force/bending moment interactions within individual members, and the overall influence of loading applied to a deformed structure (P-Delta effects).

To enable geometrically nonlinear analysis, choose "2nd Order Analysis" from the "Modeling" menu, or more conveniently, type the ',' (comma) key. To disable this analysis and return to standard linear analysis, choose the command again, or type the comma key again.

For any analysis, a few taps of the comma provides a quick and informative perspective on the relative significance of geometric nonlinearities.

The screen shot below shows a geometrically nonlinear analysis in progress:

There are several things to note about this figure:

• The stability ratio thermometer at the bottom of the screen provides a visual cue that second order analysis is turned on, and a qualitative view of the relative stability of the structure (The "phi's off" label indicates that the analysis is being done without strength reduction factors). The stability ratio is calculated by computing the ratio of the determinant of the stiffness of the nonlinear system to the determinant of the stiffness of the linear system. As will be illustrated below, when the structure goes unstable, the system will stop updating, the moment diagram will not be drawn, and the thermometer will change color.
  

  To emphasize: the stability ratio is a relative measure for a given structure with a given loading. It should not be interpreted as an absolute quantity that can be used to compare the stability of two different frames, for example.

• The moment diagrams for the columns show clearly the influence of the axial/bending interaction, as the plots are not simple straight lines as they would be for the linear case.

2nd Order Algorithm

Dr. Frame3D performs 2nd-order analysis in the following manner:

1. A standard linear solution is computed, and the resulting axial loads, P, in each member are determined.
2. The system is analyzed again using modified member stiffnesses based on the solution to the following differential equation and boundary conditions: The solution to this equation is easily obtained (see e.g., Timoshenko & Gere's Theory of Elastic Stability), and for compressive axial loads leads to member stiffnesses including terms containing sin(kL) and cos(kL) with k = sqrt(P/EI).

Manual Iteration

In the general case there will be some redistribution of axial load within the structure due to the modified stiffness, i.e., the axial loads, P, used in computing the modified stiffnesses in each member will change, but this is not accounted for in the two-step (2nd-order) algorithm described above. Additional iteration can be performed by repeatedly selecting the "nth order step" command form the "Modeling" menu, or more directly by typing the "i" (for "iterate") key.

It is atypical in practical circumstances that additional iteration makes any observable difference, but for structures very near their stability limit, the additional iterations can be important. Since it only takes a few taps of the "i" key to do the extra iterations, it is not a bad idea to get in the habit of doing this as a final check.

The most sensitive measure of the state of the convergence is the change in the stability ratio with each iteration. When the stability ratio stabilizes, the solution has converged.

If during an iteration step the structure goes unstable (or sometimes just very nearly so), the iteration is unlikely to converge. Back off the load a bit and try again.

It is best not to think of stability as a unique, static value of loading: numerical quirks can occasionally make an overloaded structure appear stable. For design, always vary load magnitude and directions in the vicinity of stability-controlled states. Better yet, change the design if possible so stability is not the controlling failure state.

Load-Dependent EI

To approximately model more realistic axial capacities and member behavior than those predicted by a strictly elastic analysis, one can choose to have the effective EI of each member reduced as a function of its axial load. In particular, with the "Load-Dependent EI" option selected (available via the "Modeling" menu or by using the '/' key), each member's effective EI is computed as follows:

in which P_y represents the simple yield capacity of the section, P is the axial load present in the member, and phi_c is a strength reductions factor. This relation is simply a rearrangement of the AISC LRFD column buckling criterion, and accounts for residual stress effects, partial yielding, and initial out of straightness. A plot of this relation is shown below:

Just as in a real structure, load-dependent bending stiffness significantly alters the member's ability to carry load and to contribute stiffness to the rest of the structure. For most practical analyses involving steel members, it is recomended that this option be used whenever geometrically nonlinear analysis is performed. See the Design Checks section for further discussion of the application and modeling accuracy of this approach.

Accuracy

With load-dependent EI disabled, Dr. Frame3D's analysis is identical to an Euler -style linear stability analysis, and the structure will go unstable at the Euler buckling load. Since the exact solution to the linear stability equation is used, the predicted buckling loads will be exact to machine precision (assuming you have the patience to home in on a particular load so accurately). Bending moment amplification in each of the member's principal planes similarly will be computed "exactly", to the degree one's member's behave linearly without bounds and linear stability is valid.

The differential equation given above assumes small curvature and no coupling between principal planes, and therefore cannot give accurate results for arbitrarily large deformations. With displacement plotting enabled, it is unlikely you will be able to ignore the very large displacements that can occur in the vicinity of a stability limit, but it should be emphasized that these results decrease in validity as the displacement magnitude increases. Again, it is very rare that you will find this to be an issue with practical designs.

When load-dependent EI is enabled and the "Resistance Factors On" option has been selected from the "Modeling" menu, Dr. Frame3D will predict member capacities for isolated members identical to those determined on the basis of the AISC LRFD column design equation. This is illustrated in the following figure:

The 382.2 k value matches the AISC result to 4 significant figures (the match would be to machine accuracy were it not for the fact that Dr. Frame3D computes its own radii of gyration as sqrt(I/A) rather than using tabulated values).

Bending moment amplification will be computed more accurately than can be achieved using approximate design code approaches, and there is no need to deal separately with amplification and sidesway effects. Again, see the Design Check section for further discussion and examples concerning these issues.

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