Page Comparison

\\

*Nonlinear* methods are best applied when inelastic behavior is considered in the modeling and analysis process. Nonlinear methods include static-pushover analysis and dynamic time-history analysis. If only elastic material behavior is to be considered, linear methods may be the most suitable. Linear methods include static strength-based analysis and dynamic response-spectrum analysis. Engineers may use any of these four analysis methods, shown below in Figure 1, to suit the following purposes:

* To gain insight into structural behavior; and

* To generate information useful for making design decisions.

\\

{center-text}Figure 1 - Analysis methods{center-text}

\\

Each analysis method has its benefits and limitations:

* *Strength-based* analysis is simple and less time consuming. Engineers consider only linear behavior, therefore only elastic stiffness properties must be modeled for each component. Aside from possible drift limitations, structural design is based only on strength parameters. Therefore, during design and analysis, engineers need only assess strength-based demand-capacity ratio. If the demand on a component is found to exceed its capacity, redesign involves simply selecting an element with greater strength.

* *Response-spectrum* analysis is a dynamic method in that it is a function of time, and it is a linear method in that it is dependent upon superposition, which does not apply to nonlinear behavior. Nonlinear behavior necessitates the simultaneous consideration of gravity effect and lateral displacement.

* *Static-pushover* analysis reveals nonlinear structural performance under monotonic loading. As static loading increases, a component or system advances past a yield point and through a range of inelastic response. Once elasticity is surpassed, static-pushover progresses through each limit state until an ultimate limit condition is achieved.

* *Time-history* analysis characterizes the dynamic response of a structure subjected to the acceleration record of a ground motion (earthquake). Material nonlinearity is modeled on the component level for ductile elements such that a step-by-step integration procedure may capture inelastic effect. Lateral behavior is evaluated simultaneously with gravity loading and equilibrium about deformed configuration such that P-delta (P-d) effect may capture geometric nonlinearity.

\\

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.

\\

h1. Sources of nonlinearity

There are two types of nonlinearity which are integral to static-pushover and time-history analysis. These types of nonlinearity include, including geometric and material, described as follows:

h2. 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.

h2. 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.

h3h5. 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:

h3h5. 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.

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.


\\

\\

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.