Nonlinear methods are best applied when inelastic material behavior is considered in structural modeling and analysis. Nonlinear methods include static-pushover and dynamic time-history analyses. If only elastic material behavior is to be considered, linear methods should be suitable for design. Linear methods include strength-based and response-spectrum analyses. These methods, and the relations between their attributes, are presented in Figure 1:

Unable to render embedded object: File (Analysis type.png) not found.

Each of these analysis methods has benefits and limitations. An overview of each method is as follows:

• Strength-based analysis is a static-linear procedure where structural components are selected such that their capacity exceeds the demand from loading. Strength-based demand-capacity (D-C) ratio indicate the adequacy of each component. If capacity does not meet demand, redesign is necessary. Modeling and analysis is fairly simple and straight-forward in that only component elastic-stiffness properties are considered. Strength-based analysis is also the least time consuming method.

• Response-spectrum analysis is a dynamic-linear method where maximum structural response is plotted as a function of structural period and damping. Response may be the peak acceleration, velocity, or displacement which results from a given time-history record. Response-spectrum analysis is a linear application dependent upon the superposition of gravity and lateral effects. As a result, structures must remain essentially elastic during response-spectrum analysis. A series of time-history records may be analyzed and enveloped to form a smooth design spectrum.

• Static-pushover analysis is a static-nonlinear procedure which indicates the elastic and inelastic performance of a component or system subjected to monotonic loading which continually increases through an ultimate condition. This method produces a nonlinear force-deformation (F-D) relationship which provides insight into the ductility and limit-state behavior of a structure. Deformation parameters may be translational or rotational. Response is a function of both strength and deformation.

• Time-history analysis is a dynamic-nonlinear method which characterizes the dynamic response and inelastic behavior of a structure subjected to the time-history acceleration record of a ground motion (earthquake). The nonlinear material properties of ductile components, designed to yield under substantial loading, are modeled such step-by-step integration may capture inelastic effect. Simultaneously, P-delta effect take into account the influence of gravity loading on deformed structural configuration. Output may provide data, plots, and animations indicating structural response and component behavior.

Engineers may use any of these four analysis methods to suit the following purposes:

• To characterize and gain insight into structural behavior; and

• To generate information useful to the design decision-making process.

Capacity Design

During nonlinear analysis, ductility contributes an additional level of complexity to modeling and design. Ductile elements, either designed to specifically or possibly achieve inelastic behavior under certain loading scenario, may satisfy demand-capacity criteria through either of two parameters: strength or deformation.

It is best to select which elements will be permitted to yield (ductile components) and which will remain elastic (brittle components) during preliminary design. This way the designer determines how a structure will perform, and not the computational tools and analysis procedures. This is critical in that many sources of uncertainty are inherent to analytical modeling. A computer model may indicate behavior which will not actually occur in a real structure. Analysis models are only a simulation of physical phenomena that is impossible to predict exactly. Therefore it is best to select which elements will be permitted to yield, and implement those ductile systems from the beginning. This allows the engineer to create a more reliable model, and should lead to better structural design.

This frames the basic tenants of Capacity Design; structures should be designed with ductile systems predetermined. This way, systems permitted to yield may be designed with sufficient deformation capacity, and systems chosen to remain elastic are designed with sufficient strength capacity. To reiterate this point:

• Ductile components predetermined to undergo yielding shall be designed with sufficient deformation capacity such that they satisfy displacement-based demand-capacity ratio; and

• Brittle components which will remain elastic are designed to achieve sufficient strength such that they satisfy strength-based demand-capacity ratio.

A benefit to Capacity Design is that only the elastic material properties of brittle components need to be modeled. If demand on any brittle component is found to exceed its capacity, redesign involves simply selecting an element with greater strength such that it remains elastic. Modeling only the linear projection of initial stiffness for all elastic elements simplifies analysis and reduces computational time and quantity of output data.

Sources of nonlinearity

There are two types of nonlinearity, including geometric and material, described as follows:

1. Geometric nonlinearity

Geometric nonlinearity, also known as P-delta effect, becomes critical when gravity load acts on a structural system which has displaced laterally. Secondary moments are then induced which contribute significantly to structural behavior. It is important to account for P-delta effect during nonlinear analysis. It doesn't increase computational time very much, and is accurate for drift levels up to 10%.

Large deformation analysis is another P-delta effect, except that it is denoted by the upper-case delta symbol. This type of geometric nonlinearity is assessed on the local level where equilibrium about column deformation is considered. This however only becomes significant in especially slender columns or at unreasonably large displacements. Further, it increases computational time significantly. An easier way to model this type of P-delta effect is to model additional nodes along a column length, which transfers the behavior (large displacement effect) into P-triangle.

2. Material nonlinearity

Material nonlinearity is the characterization of inelastic behavior within a component or system. Material nonlinearity is characterized by a force-deformation relationship. Two basic relationship types exist, including monotonic and hysteretic.

Monotonic curve

Static-pushover analysis utilizes monotonic loading. Here, a component or system is subjected to a load condition where the independent variable, a deformation parameter, increases from zero to an ultimate value. The corresponding dependent variable, a force-based parameter, is plotted against deformation to produce a nonlinear, monotonic curve. Some examples of monotonic force-deformation relationships (and their mechanism) may include stress-strain (axial), moment-curvature (flexure), plastic-hinging (rotation), etc.

Figure 2 presents a static-pushover curve. Under monotonic loading, the force-deformation relationship begins linear-elastically, following the initial-stiffness tangent to a yield point. Inelastic behavior then begins, advancing through a series of limit states until an ultimate condition is achieved. Any strength-gain represents hardening, and strength-loss represents softening. After softening, a residual value may be achieved, which may sustain through unrealistically large displacements before reaching an ultimate condition. This nonlinear force-deformation relationship may then be simplified with little compromise to analysis accuracy through idealization as a series of linear segments, as shown in Figure 3. Please notice that immediate-occupancy (IO), life-safety (LS), and collapse-prevention (CP) limit states are denoted on the curve. While these parameters relate to structural serviceability, limit states may also be specific to plastic thresholds, as shown in Figure 4:

Unable to render embedded object: File (General monotonic curve.png) not found.

Unable to render embedded object: File (Idealized general curves.png) not found.

Unable to render embedded object: File (Serviceability curves.png) not found.

Hysteretic cycle

Dynamic time-history analysis tracks the hysteretic behavior of a component or system subjected to cyclic loading. Here, material nonlinearity is plotted in a series of hysteresis loops. Rather than following a single monotonic curve to an ultimate condition, hysteresis repeatedly reverses the orientation of loading. Once some degree of inelasticity is achieved, behavior will begin to deviate from that of the monotonic curve with each unloading and reloading in the opposite direction. As shown in Figure 5, both stiffness and strength will deviate from their initial relationships as hysteretic cycles progress. Stiffness typically degrades, which is indicated by a decrease in slope upon load reversal. Strength levels may increase initially, but typically also degrade with cyclic behavior. A ductile system succeeds in maintaining its post-peak strength through hysteretic behavior and increasing levels of deformation.

Unable to render embedded object: File (Hysteresis.png) not found.

Characterizing the development of strength and stiffness relationships, as they progress through dynamic time-history analysis, is an important feature of accurate nonlinear modeling. PERFORM-3D is a computational tool which provides this capability.

Depending on structural configuration and material, a hysteretic cycle may be one of many different types. Figures 6-10 illustrate some of the possible behaviors.

Conclusion

While accurate prediction of structural behavior is desirable, analysis models can only idealize the performance of real structures. Those using software tools should note that exact prediction of behavior is not possible. The objective of structural analysis is to generate information useful to the design decision-making process. Nonlinear methods enable greater insight into dynamic and inelastic structural behavior.