Nonlinear Analysis

In addition to basic linear analysis, Dr. Frame2D 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. Frame2D'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. Frame2D performs 2nd-order analysis in the following manner:

A standard linear solution is computed, and the resulting axial loads, P, in each member are determined.
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. Frame2D'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. Frame2D 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. Frame2D 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.

Plastic Hinges

Dr. Frame2D can model discrete material nonlinearity by allowing the addition of plastic hinges to a structure. There are two important caveats to be aware of when using plastic hinges:

Plasticity introduces inherent history dependence into an analysis. Dr. Frame2D itself currently has limited facilities to manage and represent load histories, and it is possible to do a variety of things that would not be physically realizable on an actual structure, such as relocating a plastic hinge arbitrarily, or making overly large, instantaneous changes in loading, support conditions, and so on (Note that his implies that load increments cannot be applied by changing load factors globally). Therefore, it is up to the user to be careful in subjecting a structure to reasonable sequences of events.
Just like in a physical laboratory, it is generally a good idea to use displacement-controlled loading when dealing with yielding systems. In the context of Dr. Frame, this means that using supports with imposed displacements is a preferred mechanism for applying yield-inducing loads. For cases in which multiple plastic hinges form simultaneously, this is the only way such multiple hinges can be added interactively.

One further caveat with this particular release is that there can be occasional quirky behavior when both 2nd order analysis is being performed and plastic hinges are present. It is usually sufficient to use the manual iteration facility to clean up these anomalies.

To create a plastic hinge, use the standard Internal Hinge Tool but hold down the ctrl key while clicking on a member at the desired location. (You will note that the cursor changes while the control key is held down.)

To remove a plastic hinge, select the hinge and type the "Delete" or "Backspace" key.

Plastic hinges cannot be moved using the mouse or cursor keys. To modify a plastic hinge select it to bring up the following inspector table:

By default, plastic hinges are perfectly plastic: the post-yield stiffness setting can be used to model more general bilinear behavior.

The plastic rotation setting is primarily useful for reinitializing plastic hinges or for observing the value of the current plastic residual. As the message shown in the figure above indicates, it is possible to apply modifications via this inspector that could not be performed physically. Broadly speaking, if the residual plastic rotation is non-zero, then changes made with this inspector are likely to be fishy from a physical perspective.

The plastic moment setting defines the moment at which the hinge is activated (the sign of the moment does not matter). As will be seen below, plastic hinges automatically set their own limit moment when they are introduced on a structure, which is assumed to be at a point of activation (i.e., yielding): using this inspector allows one to pre-install unactivated hinges, which will then automatically yield later.

The primary use for this inspector is to reinitialize and possibly relocate plastic hinges, or to adjust some of their parameters at the onset of yielding.

Example

It is perhaps easiest to illustrate plastic hinge functionality in the context of an example. The beam shown below has been loaded to its bending capacity by means of an imposed end displacement:

The moment diagram above is normalized, but in this case the value of M_p for the beam is 79 k-ft. To continue loading, a plastic hinge is introduced in the structure by control-clicking with the hinge tool on the beam at its left end (when multiple members frame into a joint, it is best to have displacements on to make sure one gets the hinge on the intended member). This results in a plastic hinge appearing as follows:

The plastic hinge automatically assigns itself the current moment at the point it is placed as its plastic moment. In this case, the hinge's M_p is set at 79 k-ft.

The applied displacement can now be increased (by turning on the Show Reactions option (type '8') one could monitor the corresponding force applied at the pin support):

In the figure above, the moment diagram is showing absolute rather than relative moments, but there has been no change in the actual moment distribution itself. The end displacement has increased, and a label has been added to the plastic hinge (using the Label Tool) for purposes of monitoring the associated plastic rotation, which can be seen now to be non-zero. Increased displacement with no increase in moments and reactions is typical plastic behavior.

If the imposed displacement is decreased, the moment will decrease linearly while the plastic rotation will remain locked in. If the displacement is increased again to a sufficent magnitude (in either direction), the plastic hinge will yield again and plastic rotation will continue to accrue.

The figure below shows the state of the beam after the 5.9" displacement shown above is removed (most easily accomplished by selecting the support and typing the '0' (zero) key):

The residual moment, displacement, and plastic rotation can all be seen directly.

Algorithm

To model plastic hinges, Dr. Frame2D exploits the piecewise-linear nature of bilinear models. In particular, during loading a plastic hinge is simply a standard hinge with an associated applied moment, and during unloading it is simply an imposed internal rotation discontinuity. Following each user action, a solution is generated assuming each hinges load/unload state is as it was, the hinges are checked for state changes, and then a new solution is generated, if necessary. This continues until all hinges have a state consistent with the user action, or until a fixed number of iterations has been exceeded.

Dr. Frame2D only does piecewise linear analysis. Imposing large load steps, making arbitrary geometry changes, or introducing modified member properties can all lead to inaccurate or inconsistent results. In general, actions that causes any plastic hinge to change state between loading and unloading should be performed carefully, i.e., using small increments. Away from these transition points, one can proceed normally.

In the example in the previous subsection, most actions could be performed routinely (such as removing the imposed support displacement in one step at the end). Unloading/reloading past a yield point, however, would need to be done slowly if tracking plastic rotation accumulation were an important part of the analysis. Note also that the example above has only a single plastic hinge--with multiple hinges, additional care is required.

Axial Plasticity

Dr. Frame2D does not yet support axial plasticity directly, but to a degree it is possible to model such behavior manually. The frame below is loaded laterally such that the diagonal brace is carrying an axial load of 294 kips:

Let us assume the diagonal member's yield capacity is 290 kips. To model this, we can select the diagonal member to bring up the Member Info inspector. By interactively adjusting the member's Misfit/Prestrain value until the load in the member reaches the desired yield capacity, we obtain the following configuration:

In effect we applied our own plastic deformation manually, and the structure is now in a state consistent with the applied loading and the member capacities. If we remove the load, we get the following residual state:

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