The limitations of linear static analysis in the seismic design gave rise to non-linear static analysis or Pushover analysis. Pushover analysis can demonstrate how progressive failure in buildings really occurs and identify the mode of final failure. Pushover Analysis can also predict potential weak areas in the structure, by tracking the sequence of damages of every member in the structure (using something called ‘hinges’).
With the advent of Finite Element software in the world of seismic engineering, we have reached a stage where seismic design based on linear elastic analysis is no longer sufficient, and a dedicated non-linear analysis of the structure is required.
The linear approach uses the concept of Response Reduction factor R. When we design a structure for say, R=3, it implies that only 1/3rd of the seismic force is taken by the Limit State Capacity of the structure and further deflection is taken by the ductile capacity of the structure. However, we rarely analyze the ductile part and only fulfill the reinforcement detailing guidelines for the part. This is due to various challenges faced while analyzing the ductile capacity, such as changes in the stiffness of elements due to cracking and yielding, P-delta effects, changes in the final estimated seismic force, etc.
The elastic analysis fails to account for the resulting redistribution of forces during the progressive yielding or even predict the failure mechanisms in the structure. A non-linear static analysis that considers the inelastic behavior of the structure can predict these more accurately.
The need for this non-linear analysis to predict the performance of the structure beyond the elastic limit gave rise to Pushover analysis (PA). Using this method, we can demonstrate how progressive failure in buildings really occurs and identify the mode of final failure. Pushover Analysis can also predict potential weak areas in the structure, by tracking the sequence of damages of every member in the structure (using something called ‘hinges’).
Pushover analysis can provide the following advantages:
Let us see some of the similarities and differences between conventional seismic analysis (SA) and Pushover Analysis (PA). We will begin with the similarities.
With that idea, let us discuss some of the differences between the two methods:
Conventional Seismic Analysis results are mostly used for the design of structure, thus the loads available in the load combinations are factored, however, PA simulates the behavior of the structure under actual loads, therefore, the loads applied are not factored.
Pushover hinges are points of high flexural (or shear) deformations in a structure. These are points where the cracking and yielding are expected to occur at a higher intensity as the members approach their ultimate strength.
Pushover hinges are of several types – flexural hinges, shear hinges, and axial hinges. At the ends of beams and columns in a structure, we provide flexural and shear hinges. Axial hinges are usually provided at the ends of diagonal struts which are modeled during Pushover analysis to simulate the infill masonry walls in a structure. Figure 1 below shows the usual position of flexural, shear, and axial hinges in a typical structural frame.
Fig 1. Typical Hinge Location in a structural model
Figure 2 represents a typical flexural hinge for which the Moment-rotation relation is shown. AB is the zone when the structure behaves as linear and elastic. Point A is the unloaded state and Point B represents the yield state. Beyond Point B to Point C, the structure behaves as linear but inelastic. In this zone, the stiffness of the structure decreases. From Point C to D, there is a sudden drop in the resistance of the structure. This is followed by a total loss of resistance from Point E to Point F.
When the member lies in zone BC, i.e. in the ductile region, the nonlinear hinges go through 3 stages: Immediate Occupancy, Life Safety, and Collapse Prevention.
Fig 2. Typical Flexural Hinge, showing IO (Immediate Occupancy), LS (Life Safety), and CP (Collapse Prevention)
There are various methods of Pushover analysis:
1. Displacement Coefficient Method:
Here, the target displacement is calculated which can be the maximum translational displacement for the global modal or it could be the displacement for a specific node (called the master node in Midas Civil).
FEMA 356 and Eurocode 8 (EN 1998-1, 2003) follows this approach for Pushover analysis.
2. Capacity Spectrum Method:
In this method, the load is incremented gradually, and the hinge condition is checked at each stage until we reach the ‘Performance Point’ condition.
This method of Pushover analysis is adopted by ATC-40.
FEMA 440 is an improvement in the procedure of Pushover analysis for both the above methods.
While codes like FEMA-356, ATC-40, and FEMA-440 provide the chart for the hinge properties for the flexural hinges, the exact hinge properties (accurate M-phi curve) require the details of reinforcement provided in the member.
Using various concrete models such as the Confined Mander Model present in Finite Element software like Midas Civil , we can generate an exact M-phi curve.
However, for an RC member, reinforcement details can only be obtained when the member is designed. This implies that Pushover analysis is a second-order analysis.
To summarize, for performing accurate seismic design, we need to perform the following steps sequentially:
1. Conventional seismic analysis like Response Spectrum analysis is performed. Results obtained during analysis are used for initial structural design and provide reinforcement required in an RC section.
2. Hinge property is calculated in Midas Civil.
3. The hinges are then inserted into beams, columns, etc. at the required positions.
4. Pushover analysis is carried out.
5. The design and reinforcement detailing of the section is changed to generate the final result.
Figure 3 below shows a typical pier/pier cap section on which Pushover analysis is performed.
Fig 3. A typical substructure (pier/pier cap) section
1. Moment Curvature Analysis with GSD
Midas Civil offers a built software, General Section Designed (GSD) to find the Moment-curvature function for the cross-section of the pier.
Fig 4. Midas GSD with section properties and reinforcement details
Fig 5. Mander model in Midas Civil for Confined concrete property
Make sure to check on the ‘Display Idealized Model’ option in order to export hinge properties to Midas Civil as shown in Figure 6.
Fig 6. M-phi curve generated in GSD to be exported to Midas Civil
2. Export Hinge Properties to Midas Civil
Hinge data is automatically exported to Midas Civil by clicking the OK button.
Fig 7. Hinge data exported to Midas Civil
3. Import Hinge Properties to Midas Civil
Fig 8. Imported Hinge properties in Midas Civil
4. Assign Hinge Properties to Columns
Pushover hinge is assigned to both the columns
Fig 9. Pushover Hinge assigned to columns
5. Define Pushover Global Control
We define the necessary conditions to be applied to Pushover analysis such as Geometric Nonlinearity type, Initial Load, Convergence criteria, stiffness reduction ratio, etc.
Fig 10. Pushover Global analysis control
6. Define Pushover Global Control
Pushover Load Case has been defined as Displacement Control in the Transverse direction.
Fig 11. Pushover Load Case
Fig 12. The capacity curve in the Transverse direction with a Target displacement of 20 mm
Fig 13. Capacity Spectrum vs Demand Spectrum
Fig 14. Chord Rotation check under Pushover Hinge Results
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