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The Hinge Unloading Method option appears on the Nonlinear Parameters form. Access the Nonlinear Parameters form as follows:
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When a hinge unloads, the program must find a way to remove the load that the hinge was carrying and possibly redistribute it to the remainder of the structure. Hinge unloading occurs whenever the stress-strain (force-deformation or moment-rotation) curve shows a drop in capacity, such as is often assumed from point C to point D, or from point E to point F (complete rupture).
Such unloading along a negative slope may be unstable in a static analysis, and a unique solution is not always mathematically guaranteed. In dynamic analysis (and the real world) inertia provides stability and a unique solution.
For static analysis, special methods are needed to solve this unstable problem. Different methods may work better with different problems. Different methods may produce different results with the same problem. SAP2000 provides three different methods to solve this problem of hinge unloading, which are described next.
If all stress-strain slopes are positive or zero, these methods are not used unless the hinge passes point E and ruptures. Instability caused by geometric effects is not handled by these methods.
Note: If needed during a nonlinear direct-integration time-history analysis, SAP2000 will use Method 2: Apply Local Redistribution.
Hinge Unloading Method
Unload Entire Structure option. When a hinge reaches a negative-sloped portion of the stress-strain curve, the program continues to try to increase the applied load. If this results in increased strain (decreased stress), the analysis proceeds. If the strain tries to reverse, the program instead reverses the load on the whole structure until the hinge is fully unloaded to the next segment on the stress-strain curve. At this point the program reverts to increasing the load on the structure. Other parts of the structure may now pick up the load that was removed from the unloading hinge.
Whether the load must be reversed or not to unload the hinge depends on the relative flexibility of the unloading hinge compared with other parts of the structure that act in series with the hinge. This is very problem-dependent, but it is automatically detected by the program.
This method is the most efficient of the three methods available, and is usually the first method you should try. It generally works well if hinge unloading does not require large reductions in the load applied to the structure. It will fail if two hinges compete to unload, i.e., where one hinge requires the applied load to increase while the other requires the load to decrease. In that case, the analysis will stop with the message "UNABLE TO FIND A SOLUTION," in which case you should try one of the other two methods.
This method uses a moderate number of null steps.
Apply Local Redistribution option. This method is similar to the first method, except that instead of unloading the entire structure, only the element containing the hinge is unloaded. When a hinge is on a negative-sloped portion of the stress-strain curve and the applied load causes the strain to reverse, the program applies a temporary, localized, self-equilibrating, internal load that unloads the element. This causes the hinge to unload. After the hinge has unloaded, the temporary load is reversed, transferring the removed load to neighboring elements. This process is intended to imitate how local inertia forces might stabilize a rapidly unloading element.
This method is often the most effective of the three methods available, but usually requires more steps than the first method, including a lot of very small steps and a lot of null steps. The limit on null steps should usually be set between 40% and 70% of the total steps allowed.
This method will fail if two hinges in the same element compete to unload, i.e., where one hinge requires the temporary load to increase while the other requires the load to decrease. In that case, the analysis will stop with the message "UNABLE TO FIND A SOLUTION," after which you should divide the frame object so the hinges are separated and try again. Check the .LOG file to see which objects are having problems. Caution: The object length may affect default hinge properties that are automatically calculated by the program, so fixed hinge properties should be assigned to any objects that are to be divided.
Restart Using Secant Stiffness option. This method is quite different from the first two. Whenever any hinge reaches a negative-sloped portion of the stress-strain curve, all hinges that have become nonlinear are reformed using secant stiffness properties, and the analysis is restarted. The secant stiffness for each hinge is determined as the secant from point O to point X on the stress strain curve, where: Point O is the stress-strain point at the beginning of the pushover case (which usually includes the stress due to gravity load); and Point X is the current point on the stress-strain curve if the slope is zero or positive, or else it is the point at the bottom end of a negatively sloping segment of the stress-strain curve.
When the load is re-applied from the beginning of the analysis, each hinge moves along the secant until it reaches point X, after which the hinge resumes using the given stress-strain curve.
This method is similar to the approach suggested by the FEMA 273 guidelines, and makes sense when viewing pushover analysis as a cyclic loading of increasing amplitude rather than as a monotonic static push.
This method is the least efficient of the three, with the number of steps required increasing as the square of the target displacement. It is also the most robust (least likely to fail) provided that the gravity load is not too large. This method may fail when the stress in a hinge under gravity load is large enough that the secant from O to X is negative. On the other hand, this method may also give solutions where the other two fail due to hinges with small (nearly horizontal) negative slopes.
Example
Consider the frame with two pushover hinges shown below:
The force deflection characteristics of the two hinges are illustrated below:
When the pushover analysis is run, it turns out that pushover hinge 2 passes point E (i.e., unloads twice) before pushover hinge 1 reaches point C and begins to do any unloading.
The pushover curve using the Restart Using Secant Stiffness option is shown in the figure below. The pushover curve is made up of five separate curves. Curve 1 ends when pushover hinge 2 reaches point C and the structure must drop load. Curve 2 ends when pushover hinge 2 reaches point E and the structure must drop additional load. Curve 3 ends when pushover hinge 1 reaches point C and Curve 4 ends when pushover hinge 2 reaches point E. Curve 5 ends when the displacement target is reached. Note that since a mechanism has occurred, curve 5 is just a flat line along the X-axis.
The following ten figures show the loading and unloading paths taken by pushover hinge 1 and pushover hinge 2 for each of the five components of the pushover curve.
