Theoretical Basis of Pushover Analysis
This article summarizes nonlinear analysis and the pushover method, first through their physical concepts and then through the relevant provisions of TBDY 2018 and their implementation in StatiCAD. For the usage steps, see the article How to Perform a Nonlinear Pushover Analysis.
1. Why nonlinear analysis?
Linear analysis assumes that every member stays elastic throughout the earthquake and reduces the seismic load with the response modification coefficient; it accepts that members yield but does not show where, how much and in what order the yielding occurs. In the assessment of an existing building, this is exactly the information that is needed: how much plastic deformation do which beams and columns undergo, and are these deformations within safe limits?
The approach that answers this question is Deformation-Based Assessment and Design (ŞGDT) (TBDY 2018 Chapter 5). Force-based calculation asks “how much force can the section carry?”, deformation-based calculation asks “how much can the section rotate and elongate?”.
2. Material models (Annex 5A)
- Reinforcing steel: Elastic up to the yield strength fy; then a yield plateau and strain hardening (increase of strength towards fu); it ends at the rupture strain εsu.
- Confined and unconfined concrete (Mander model): Stirrups and cross-ties compress the core concrete laterally; this confinement effect increases both the strength (fco → fcc) and the ductility. The effective lateral confining pressure is computed as fe = 0.5 · ke · (ρx + ρy) · fyw (ρx, ρy: volumetric ratios of transverse reinforcement, ke: confinement effectiveness coefficient).
3. Section level: moment–curvature relationship
If the material models are distributed over the fibres of the section and a step-by-step increasing curvature is applied to the section, with internal force equilibrium established at each step, the moment–curvature (M–φ) curve is obtained. From the curve, the yield moment and curvature (My, φy), the maximum moment reached through strain hardening (MMax), the ultimate curvature (φu) and the effective flexural stiffness (EIe) are read.
In columns and shear walls the M–φ curve depends on the axial force; therefore, in the assessment these members are analysed with N = G + nQ coming from the vertical loads.
4. Member level: plastic hinge
Under earthquake, the moment is largest at the member ends and yielding starts first at the ends. In the lumped plasticity model the member body remains elastic, and plastic hinges carrying the M–φ behaviour of the section are placed at its ends. The plastic rotation is θp = φp · Lp, and the plastic hinge length is Lp = 0.5·h (TBDY 5.4.5.2).
StatiCAD has two hinge models: elastic–perfectly plastic (EPP) (default, conservative side) and strain-hardening (bilinear); in the latter, the hinge rises up to MMax with a stiffness derived from the actual M–φ strain-hardening slope of the section.
5. System level: pushover analysis
Pushover analysis is a nonlinear static calculation performed by increasing a proportional lateral load pattern on the building step by step while the vertical loads (G + nQ) are kept constant. In single-mode pushover, TBDY prescribes that the load pattern be chosen proportional to the first mode shape.
As the analysis proceeds, hinges form, the stiffness of the system decreases, and eventually the target displacement is reached or a mechanism forms. The StatiCAD solver uses the event-to-event method: at each step it finds exactly the next end that will yield, increases the load exactly up to that event, opens the hinge and continues by updating the stiffness. In this way the pushover curve is produced without increment error, with its break points coinciding with the real events.
6. Conditions of single-mode pushover
Fixed single-mode pushover (TBDY 5.6.3, Annex 5B) cannot be used in every building; according to TBDY 5.6.2.2:
| Condition | Limit | Meaning |
|---|---|---|
| Torsional irregularity coefficient ηbi | ≤ 1.4 (at every storey) | Plan torsion must be limited. |
| First-mode effective mass ratio | ≥ 70% | Behaviour must be dominated by a single mode. |
If the conditions are not satisfied, multi-mode pushover or time-history analysis is required. StatiCAD checks both conditions automatically and shows them in the report. For an existing building, a knowledge level factor is applied according to the information collected: 1.0 for the comprehensive knowledge level and 0.75 for the limited knowledge level.
7. Annex 5B: finding the seismic demand
Annex 5B converts the multi-degree-of-freedom building into an equivalent single-degree-of-freedom system of the first mode: the pushover curve is converted to a capacity diagram in modal acceleration–modal displacement axes and compared with the earthquake demand spectrum in the same axes.
In medium- and long-period structures the equal displacement rule applies (CR = 1). In short-period structures the demand is larger than the elastic value and is amplified with the CR coefficient (Annex 5B, iteratively). The roof displacement demand found marks the performance point on the pushover curve; the assessment is made according to the state of the building at that point.
8. Damage limits and member damage regions
At the performance point, the plastic rotation of each hinge and the section strains are known. TBDY compares them with three limits:
| Limit | Concrete (εc) | Reinforcement (εs) |
|---|---|---|
| GÖ (Collapse Prevention) | 0.0035 + 0.04·√ωwe ≤ 0.018 | 0.4·εsu |
| KH (Controlled Damage) | 0.75 · εc(GÖ) | 0.75 · εs(GÖ) |
| SH (Limited Damage) | 0.0025 | 0.0075 |
ωwe is the effective confinement reinforcement ratio: the better the confinement, the higher the GÖ limit. The plastic rotation limits are computed with TBDY Eq. 5.6 and 5.7b; θp(KH) = 0.75 · θp(GÖ).
Comparing the demand with the limits places each member in one of four regions: Limited Damage, Significant Damage, Advanced Damage and Collapse region.
9. Building performance level decision (TBDY 15.8)
Once the member regions are known, the decision is made at the building level. In summary: at the Limited Damage level, at most 20% of the beams at each storey may pass into the Significant Damage region; at the Controlled Damage level, at most 35% of the beams may pass into the Advanced Damage region and the contribution of vertical members in the Advanced Damage region to the storey shear is limited; at the Collapse Prevention level, severe damage is permitted to a limited extent. If these conditions cannot be satisfied either, the building is in a collapse state.
StatiCAD applies these checks storey by storey and states the justification of the decision explicitly in the report (e.g. “SH sağlanamadı: kat 1’de Belirgin kiriş oranı %20’yi aşıyor → bina KH” (SH not satisfied: the Significant Damage beam ratio exceeds 20% at storey 1 → building KH)).
10. Multi-mode pushover and IRSA
For buildings that do not satisfy the conditions of single-mode pushover, TBDY 5.6.6 allows multi-mode pushover. In StatiCAD each mode is pushed separately with its own mass × mode shape pattern, the demand of each mode is found from Annex 5B with its own period (CR > 1 in short-period modes), and the member end plastic rotations are combined with the CQC rule. In accordance with TBDY 5.6.6.2, modal internal forces are not combined statistically: since the moment in a yielded section cannot exceed the yield moment, the combination is made at the deformation level. The fact that the method gives the same result as linear analysis with mode combination when elastic stiffnesses are used (5.6.6.1) is shown numerically in every analysis.
IRSA (incremental mode combination) advances the pushover in the linear segments between hinge formations together with all modes; in each segment the modal analysis is renewed with the current stiffness and the combination is made in every segment.
11. Implementation and assumptions in StatiCAD
| Step | Counterpart in StatiCAD |
|---|---|
| Material models | Annex 5A: Mander confined and unconfined concrete, strain-hardening steel |
| Section analysis | M–φ with the fibre model; two axes in columns; flange (T-section) effect and separate left/right, positive/negative capacities in beams |
| Hinge | Lumped plasticity, Lp = 0.5h; EPP (default) or strain-hardening |
| Solver | Event-to-event incremental pushover; rigid diaphragm; eight direction combinations ±X ± 0.3Y; shear deformations included |
| Demand | Annex 5B single-mode transformation; modal quantities from modal analysis; CR iteration |
| Checks | ηbi ≤ 1.4, mass participation ≥ 70%, knowledge level factor |
| Damage | Plastic rotation limits and section strain checks; GÖ limit with ωwe at shear wall end regions |
| Second order | P-Δ effects are optionally taken into account with geometric stiffness |
| Verification | The same model can be solved in SAP2000 and the curve, hinge states and decisions compared automatically |
Assumptions: The axial force of columns and shear walls is constant throughout the pushover (N = G + nQ; the axial force–moment interaction is not updated step by step). There is no strength drop (descending branch) in the elastic–perfectly plastic model; this difference is deliberate in SAP2000 comparisons.