Chapter 4. GRILLAGE MODEL DEVELOPMENT 93
4.1 Introduction 93
4.4.1 Column Plastic Hinges 105
Since the column was designed to form a plastic hinge at both the top and bottom of the column, it was necessary to include a nonlinear link element, which represented the hinges, at the top and bottom of the column, as shown in Figure 4.5. A moment-curvature
106
analysis of both the top and bottom of the column was performed using a program developed at Iowa State University, known as VSAT (Levings, 2009). The data from the moment- curvature analysis was then converted to a moment-rotation response using Equation 4.3, which accounts for rotation due to both strain penetration and plastic deformation within the hinge. It should be noted that the rotation due to elastic deformation was taken into account via the elastic frame element used to model the column. The term L’sp represents the length
that the elastic effects of strain penetration extend into either the cap or the footing, depending on the location of the hinge being analyzed. The term Lp represents the plastic
hinge length and includes the length of the plastic effects of strain penetration as well as the length representing the plastic region of the column, as the maximum curvature over this region was assumed to be constant. The terms Φe and Φp represent the elastic and plastic
curvature components, respectively. The terms fy, db, and L represent the yield stress of the
longitudinal reinforcement, the bar diameter of the longitudinal reinforcement, and the total length of the column, respectively.
Figure 4.5: Grillage Model Column Nonlinear Link Locations
½ Column Nonlinear Link Location
Nonlinear Link Location
107
Therefore, per (Priestley, Seible, & Calvi, 1996), the total rotation within the column plastic hinge region, θ, was defined as:
(4.3)
(4.4)
€
Lp = 0.08L + 0.15 fydb ≤ 0.3 f/ ydb (4.5)
The moment-rotation response input was then directly input into the properties for the nonlinear link element and placed at the top and bottom of the column. The moment-rotation properties that were input into SAP for the nonlinear link elements representing the plastic hinges are shown below in Figures 4.6 and 4.7. It is important to note that the moment values obtained from the moment-curvature analysis were halved before being input into SAP, as only half of the column was modeled due to symmetry. Also, the responses for both the top and bottom plastic hinges were essentially the same, with the bottom hinge being a little stiffer due to a slightly higher axial load from the self-weight of the column.
Figure 4.6: Predicted Top of Column Plastic Hinge Moment vs. Rotation Monotonic Response
-‐10000 -‐8000 -‐6000 -‐4000 -‐2000 0 2000 4000 6000 8000 10000 -‐0.035 -‐0.025 -‐0.015 -‐0.005 0.005 0.015 0.025 0.035 Momen t (kip-in ) Rotation (rad)
108
Figure 4.7: Predicted Bottom of Column Plastic Hinge Moment vs. Rotation Monotonic Response
Hysteretic rules were also defined for the nonlinear link element within SAP2000, which provided three possible built-in hysteretic models: Kinematic, Takeda, and Pivot. Since the Takeda and Pivot models are the most widely used for reinforced concrete columns, they were selected as the two primary models of consideration. In order to decide between the Takeda and Pivot models, a comparative analysis was performed based on the results of various column tests provided by the University of Washington column database (University of Washington, 2004). Based on the results of said comparison, specifically column Vu NH3, it was shown that the Pivot model was able to most accurately model the overall hysteretic behavior of the comparison column, as shown in Figure 4.8. Furthermore, the Takeda model defined within SAP2000 did not allow the user to modify its rules, whereas the user was able to define more of the rules when using the Pivot model, providing a more specific set of rules applicable to the column being analyzed. Therefore, the Pivot model was selected to define the hysteretic behavior of the column nonlinear link elements.
-‐10000 -‐8000 -‐6000 -‐4000 -‐2000 0 2000 4000 6000 8000 10000 -‐0.035 -‐0.025 -‐0.015 -‐0.005 0.005 0.015 0.025 0.035 Momen t (kip-in ) Rotation (rad)
109
(a) Takeda Hysteresis (b) Pivot Hysteresis Figure 4.8: Force-Displacement Hysteresis Comparison
In order to define the Pivot model for both the top and bottom nonlinear link elements, the values for α1, α2, β1, β2, and η had to be defined and input into the SAP2000
hysteretic model. The values α1 and α2 were used to define the location of the pivot point
used to determine the unloading stiffness when removing the load from a positive and negative moment value, respectively. For the sake of comparison, it was arbitrarily assumed that these values would be approximately the same. The values β1 and β2 were used to define
the pinching points that the moment-rotation response would pass through when reversing the moment toward the positive and negative direction, respectively. Again, it was arbitrarily assumed that these values would be approximately equal. It is important to note however, that when defining the moment-rotation response within SAP2000, both the first positive and negative moment-rotation values should correspond to the yield condition. This was done because SAP2000 defines the pinching points at a moment value corresponding to βFy, in
which the program assumed that the first point entered after the origin was used to define yield. The value η was used to define the amount of elastic, or initial, strength degradation experienced after any plastic deformation (Computers and Structures, Inc., 2008), (Dowell, Seible, & Wilson, 1998). The values for α and βwere defined using the charts shown in
-‐150 -‐100 -‐50 0 50 100 150 -‐2 -‐1 0 1 2 Fo rc e (k ip s) Displacement (in) UW Measured SAP Takeda -‐150 -‐100 -‐50 0 50 100 150 -‐2 -‐1 0 1 2 Fo rc e (k ip s) Displacement (in) UW Measured SAP Pivot
110
Figure 4.9 (a) and (b) respectively, which were based on the longitudinal reinforcement ratio and the axial load ratio experienced by the given column (Dowell, Seible, & Wilson, 1998). The longitudinal reinforcement ratio, ρl, and the axial load ratio, ALR, were calculated using
Equation 4.6 and 4.7, respectively, where Asl represents the area of longitudinal steel, Ag
represents the gross area of the column, and f’c represents the concrete compressive strength.
The value for η was taken as 8 in order to reflect an arbitrarily assumed amount of elastic strength degradation, to be used solely as a basis for comparison.
Figure 4.9: Pivot Hysteresis Parameters
(4.6)
(4.7)