DESIGN OF PROTOTYPE BUILDINGS

4.1 Introduction

The purpose of this chapter is to show the arguments and procedures used to define the dimensions of the prototype buildings (PB).

National building codes, like the ASCE 7-05 code, and other national codes implement the philosophy of Capacity Design to dimension buildings capable of withstanding the Maximum Considered Earthquake (MCE) in a particular location. Park (2006) refers that this philosophy of design is the brainchild of John Hollings, a New Zealander design engineer. Holling's original proposal was extended and refined by Robert Park and Tom Paulay (2006). Paulay and Priestley (1992) did a further extension of the explanation of this approach in their book.

In the spirit of the philosophy of Capacity Design, the designer decides the locations where ductility is allowed to develop, and the magnitude of seismic induced lateral force that the building must take. Under the designer's control, the building will respond to the future seismic events in a mode planned by the designer. With this approach, the maximum lateral load that the building can take is limited by design. Although it is expected that seismic events larger than the MCE will produce larger demands of ductility at the designated locations, and that the developed PH will have enough ductility capacity. The last statement requires careful analysis, because it does not define how much larger should the expected capacity be. Dissertation Methodology

A very important requisite for this approach to work is that all the remaining elements of the building, which are not permitted to develop ductility, stay in the elastic range. Park and Paulay (2006) present strong arguments to support the idea of not let the columns in a framed building to develop ductility. They argue that column sidesway plastic mechanisms (weak stories) must be avoided, and beam sidesway plastic mechanisms enforced. Nowadays, there are widely recognized the limitations to develop curvature ductility by the columns, and their reparation after the earthquake. Therefore, the plastic mechanism commonly known as strong column-weak beam (SCWB) is enforced for the design of the prototype buildings in this research. Dissertation Methodology

The SCWB plastic mechanism is expected to develop rotational ductility at some, or all beams. And the columns, to stay in the elastic range of strains. Design codes (American Society of Civil Engineers, 2006) prescribe some minimum values of the ratio of ∑ M_p columns/ ∑ M_p beams for columns and beams interconnected at a frame node. The intention of these rules is to avoid the possibility of the columns to reach their plastic moment and, after that, the developing of ductility demands. Paulay (1992) evaluates these code rules and gives arguments to support his conclusion about the limited effectivity of these rules. He emphasizes that there are diverse situations where these rules fail, and lead to designs where columns will have unexpected ductility demands.

In the section 4.2, the one-story PB is designed, and in section 4.3 the two story PB. An alternative procedure is proposed in these sections to design the beams and columns of these buildings. The intention of this procedure is to propose designs, which can develop the SCWB plastic mechanism, before reaching the ultimate base shear force of the building. Dissertation Methodology

4.2 One Story Building

Target Backbone Shape.

Figure 4 shows the backbone of the lateral force versus roof displacement for a single story frame. As it is known, this shape is determined by the type of plastic mechanism developed by the frame and by how much ductility capacity it has. The over strength factor, Ω0, and the deflection amplification factor, C_d, have a definitive influence on the total ductility demand that potentially can be developed in the beam, before reaching the plastic moment in the columns. If the beam can supply the demanded curvature ductility, the frame withstands the MCE by developing a SCWB plastic mechanism, as it is desired by the designer.

Figure 4: Story plan view and location of the reference nodes

It is not the aim of this research to study buildings that comply strictly with any design code, but to study prototype buildings whose behavior can be approximately compared with buildings designed following recommendations of the design code.

These four parameters define completely the backbone curve for the one story prototype frame. The dimensioned frame is warranted to develop a SCWB plastic mechanism when required by any extreme seismic event. The prototype frame is calibrated to develop an arbitrary ultimate lateral load, and Fu is tied to the used Mpb (an arbitrarily chosen value). The Fu value is adjusted in section 4.2, such that the prototype building can take the actual maximum demand of lateral load, induced by the earthquake data set. Herein is introduced an alternative method to evaluate the backbone curve, including the values that define the kinks. This method is named the Hysteretic Displacement Method (HDM). The HDM evaluates the relation between the applied lateral displacement and the lateral load response. The intention of this procedure is to identify the plastic mechanism developed by the frame. The HDM purpose is equivalent to the purpose of the nonlinear static method of analysis, best known as the Pushover method (PM) (Krawinkler, 1998; American Society of Civil Engineers, 2000). A fundamental difference between both methods is that the PM evaluates the nonlinear response to a fixed pattern of lateral load, applied at the story level, and the HDM evaluates the nonlinear response to a fixed pattern of lateral displacement. Another important difference is that the HDM lets to evaluate the response for reversible cyclic displacements. The HDM has the steps described below: Dissertation Methodology

Step 1. A sequence of discrete displacement steps is created using the equation 1.

Equation 1

D_s = D_max sin (2π/Steps - s )

Where:

Steps = Total number of displacement steps

s = displacement step => s G {0,1, , Steps}

Ds = Displacement corresponding to the s-th displacement step

D_max = Maximum lateral displacement

Figure 5 shows the displacement step history. The frame is under the action of a forced roof lateral displacement of magnitude Ds. The total number of displacement steps, required to complete a cycle, is selected to minimize the errors due to local changes in stiffness of beams and columns, during calculation in a displacement step. Typically, this is a large integer number. Though easy to handle by the computer. Using a trial and error procedure, D_max is increased up to the value where the frame reaches its ultimate lateral load and starts yielding in a plastic collapse mechanism. Dissertation Methodology

Figure 5: Displacement step

Step 2. The mathematical model is used herein to calculate the response of the prototype frame. The F function is used to calculate the lateral response.

The rotation demand at ends of columns and beams is shown in figure 5. The reported values are in radians. The histories of bending moments at these elements are shown in figure 6. The reported values are normalized dividing them by their corresponding plastic moment. It is evident that the beam starts to yield plastically under smaller lateral displacement at the roof level. After the lateral displacement has reached a larger magnitude, the lower end of the column starts yielding. In between these two events, the frame is developing a SCWB plastic mechanism, as it is planned. If the seismic event continues demanding ductility, eventually the columns yield and the frame reach its ultimate lateral load capacity. Dissertation Methodology