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3.4 pushover analysis

NASRIN AFROZ, profile picture
NASRIN AFROZ

This document discusses pushover analysis, which is an inelastic static analysis method used to evaluate seismic performance of structures. It begins by outlining the target performance levels dictated by codes, then provides an overview of current analysis methods and their limitations. Next, it describes the steps of a pushover analysis in detail, including defining member behavior, applying loads, specifying the load pattern, and incrementally forming plastic hinges. An example application to a 3-story frame structure is presented to demonstrate the process. The document concludes by emphasizing pushover analysis as a practical alternative to time history analysis for estimating seismic response.

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Pushover Analysis an Inelastic Static Analysis Methods courtesy of Barış Binici
Target Performance Dictated by codes (DBYBHY 2007, Section 1.2.1): “....The objective of seismic resistant design is to have no structural/nonstructural damage in low magnitude earthquakes, limited and repairable damage in moderate earthquakes and life safety for extreme earthquakes...”
Current Status )( )( 1 1 TR TAW V a t  • Equivalent Lateral Force Procedure - Assume global ductility (Ra) - Detail accordingly • Modal Superposition Procedure - Include higher mode effects • Time History Analysis - Rarely used - Tedious and requires hysteretic models
Critique of Current Practice Advantages : - Simple to use - Have proven to work - Became a tradition all over the world - Uncertainty is lumped and easier to deal with Disadvantages : - No clear connection between capacity and demand - No option for interfering with the target performance - No possibility of having the owner involved in the decision process - Not easily applicable to seismic assessment of existing structures
DBYBHY 2007 (Chapter 7) - Evaluation and Strengthening of Existing Buildings is based on structural performances. - Steps: • Collect information from an existing structure • Assess whether info is dependable and penalize accordingly • Conduct structural analysis - Linear static analysis - Nonlinear static analysis (Pushover analysis) - Incremental pushover analysis - Time history analysis • Identify for each member the damage level • Decision based on number of elements at certain damage levels
Time History? - Actual earthquake response is hard to predict anyways. - Closest estimate can be found using inelastic time-history analysis. - Difficulties with inelastic time history analysis: - Suitable set of ground motion (Description of demand) - hysteretic behavior models (Description of capacity) - Computation time (Time) - Post processing (Time and understanding) Alternative approach is pushover analysis. Düzce Ground Motion -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0 5 10 15 20 25 30 Sec. Acceleration(g)
Pushover Analysis • Definition: Inelastic static analysis of a structure using a specified (constant or variable) force pattern from zero load to a prescribed ultimate displacement. • Use of it dates back to 1960s to1970s to investigate stability of steel frames. • Many computer programs were developed since then with many features and limitations.
Available Computer Programs • Design Oriented: SAP 2000, GTSTRUDL, RAM etc. • Research Oriented: Opensees, IDARC, SeismoStrut etc. What is different? • User interface capabilities • Analysis options • Member behavior options
Section Damage Levels Damage levels are established based on concrete outermost compressive fiber strain and steel strain (for nonlinear analysis procedure).
Section Damage Levels How should these values be decided? - Construction practice - Experience of engineers - Input of academicians
Curvature demand at target curvatures Φp = θp / Lp Φt = Φy + Φp 0 100 200 300 400 500 600 0.0000 0.0200 0.0400 0.0600 0.0800 0.1000 0.1200 Eğrilik (rad/m) Moment (kN.m) AK GV GÇ (Φt)(Φy)
How do we estimate strains from a structural analysis? Strain Moment Curvature Moment My øy øu Moment Plastic Rotations My θpu θpu =(øu – øy) Lp OR θp =(ø – øy) Lp Where Lp = 0.5h Utilize this idealized moment-rotation response in inelastic structural analysis
Definition of Potential Plastic Hinges • End regions of columns and beams (center for gravity loads) are the potential plastic hinges • Plastic hinges are hinges capable of resisting My (not significantly more, hardening allowed) undergoing plastic rotations h Lp Elastic Beam- Column Element Plastic Hinges Rigid End zones
Elastic Parts For regions other than plastic hinging occurs, cracking is expected therefore use of cracked stiffness is customary (0.4-0.8) EIo Eğrilik Moment EIo 0.4-0.8EIo Curvature
Pushover Analysis
Steps of Pushover Analysis: A Simple Incremental Procedure 1. Build a computational model of the structure
Steps of Pushover Analysis 2. Define member behavior – Beams: Moment-rotation relations – Columns: Moment-rotation and Interaction Diagrams – Beam-column joints: Assume rigid (DBYBHY 2007 ) – Walls: Model as beam columns but introduce a shear spring to model shear deformations – Use cracked rigidities for elastic portions
Steps of Pushover Analysis 3. Apply gravity loads 1.0 G + n Q n=0.3 (live load reduction factor) (if the interaction diagrams will not be used a good estimate of the moment capacity of column hinges needs to be made) Possibilities: - Based on initial gravity load analysis - Based on a beam hinging mechanism - Based on elastic lateral force analysis with an assumed reasonable Ra value.
Steps of Pushover Analysis 4. Specify a Lateral Load Profile: (Inverted triangular, constant, first mode shape are some of the possibilities) It is a good idea to have a spreadsheet page ready indicating all members, current load increment 5. Lateral Load Incrementing: Step 1: Elastic analysis is valid up to the formation of the first hinge, i.e. when the first critical location reaches its moment capacity. • Find the lateral loads that cause first hinge formation (V1). • Record all member forces and deformations (F1, d1).
Steps of Pushover Analysis Step 2: Beyond Step 1, yielded element’s critical location cannot take any further moment. Therefore place an actual hinge at that location. Conduct an analysis increment for this modified structure. This load increment should be selected such that upon summing the force resultant from this incremental step and previous step, second hinge formation is reached. V2 = V1 + ΔV F2 = F1 + ΔF d2 = d1 + Δd Results from Step 1 + Results from an incremental analysis with a hinge placed at first yield location = Second Hinge formation
Steps of Pushover Analysis . . Step i: Similar to step 2 but additional hinges form and incremental analysis steps are conducted for systems with more hinges. Results are added to those from the previous step Vi = Vi-1 + ΔV Fi = Fi-1 + ΔF di = di-1 + Δd Results from Step i-1 + Results from an incremental analysis with a hinge placed at i-1th yield location = ith hinge formation
Steps of Pushover Analysis Step n: Sufficient number of plastic hinges have formed and system has reached a plastic mechanism. Note that this could be a partial collapse mechanism as well. Beyond this point system rotates as a rigid body. ANALYSIS DONE - Plot Base Shear- Roof Displacement - Check member rotations and identify performance levels
Example Application: 3 Story- 2 Bay RC Frame (Courtesy of Ahmet Yakut) M O D E L 3m 3m 3m 1 2 3 10 11 12 13 14 15 4 5 6 7 8 9 6m 6m J1 J2 J3 J4 J8 J7 J6 J5 J9 J10 J11 J12
Assumptions Assume • Constant Axial Load on Columns for Analysis Steps • Rigid-plastic with no hardening or softening moment-rotation behavior for columns and beams • plastic hinging occurs when moment capacity is within 5% tolerance • Load combinations 1.0 DL + 0.3 LL and 1.0 DL + 0.3 LL+1.0EQ to compute axial load levels DL=10kN/m DL=15kN/m DL=15kN/m LL=2kN/m LL=2kN/m LL=2kN/m EQ=60kN EQ=40kN EQ=20kN SABİT YÜK HAREKETLİ YÜK YATAY YÜK
DATA 10-f10 60cm 60cm Columns 3-f10 3-f10 25cm 50cm Beams Steel (fyd=495 Mpa) Concrete (fcd=25 Mpa) Clear cover=5 cm E=2.779E+4 MPa M+ is the same as M- Note that if this is a seismic evaluation problem strength values obtained at site should be used!
Section Capacities Eğrilik Moment fy My fult Eleman N My Φy Φ u l t kN kNm rad/m rad/m 1 -83,786 124 0,0055 0,111 2 -51,347 115,5 0,0056 0,115 3 -19,872 107,5 0,0056 0,119 4 -253,392 166 0,0059 0,085 5 -158,905 143 0,0060 0,099 6 -64,797 119 0,0060 0,113 7 -124,104 133,5 0,0056 0,105 8 -77,747 122 0,0057 0,112 9 -31,201 110 0,0054 0,118 10 5,606 49 0,0073 0,103 11 1,421 50 0,0069 0,102 12 -17,233 53 0,0069 0,099 13 5,606 49 0,0073 0,103 14 1,421 50 0,0069 0,102 15 -17,233 53 0,0069 0,099 Elemnaların Moment-eğrilik ilişkileri elasto-plastik, pekleşmesiz To be conservative smaller axial load from two load combinations can be selected (as long as N<Nb) Idealized member moment curvature relations for estimated axial load level Member
Effect of Axial Force • Compute the moment capacity by accounting for axial force variation • Always remain on the yield surface
Step 1 DL=10kN/m DL=15kN/m DL=15kN/m LL=2kN/m LL=2kN/m LL=2kN/m EQ=3kN EQ=2kN EQ=1kN COMBO2: 1.0 DL + 0.3 LL + 1.0 EQ Detection of first yield (moment reaches My±5%My ) 6 Frame Joint Myield M Element Label kNm kNm J1 124.0 -4.33 J2 124.0 20.60 J2 115.5 -22.14 J3 115.5 21.00 J3 107.5 -22.23 J4 107.5 27.35 J5 166.0 6.23 J6 166.0 -0.60 J6 143.0 3.50 J7 143.0 -2.94 J7 119.0 1.52 J8 119.0 -3.29 J9 133.5 16.03 J10 133.5 -20.07 J10 122.0 26.88 J11 122.0 -24.83 J11 110.0 22.95 J12 110.0 -30.82 J2 49.0 -42.74 J6 49.0 -49.58 YIELDED J3 50.0 -43.24 J7 50.0 -49.28 J4 53.0 -27.35 J8 53.0 -34.34 J6 49.0 -45.48 J10 49.0 -46.95 J7 50.0 -44.83 J11 50.0 -47.79 J8 53.0 -31.05 J12 53.0 -30.82 0.2947 11 12 6 7 4 14 15 Condition 13 5 8 9 3 10 1 2 First yielding stage Total Base Shear (kN)= Lateral Disp. at J4 (mm)= J4 (monitored node )
Step 2 (Incremental) ΔEQ=3kN ΔEQ=2kN ΔEQ=1kN Actual hinge at previously yielded location for the incremental analysis New locations at which yield moments within tolerance are reached 6 12 0.2865 Total Lateral Disp. at J4 (mm)= 0.5812 Frame M ΔM M + ∆M Element kNm kNm (kNm) -4.33 6.39 2.06 20.60 0.76 21.36 -22.14 2.05 -20.10 21.00 -2.18 18.82 -22.23 0.24 -21.99 27.35 -1.82 25.53 6.23 6.47 12.71 -0.60 0.39 -0.21 3.50 2.79 6.29 -2.94 -3.15 -6.09 1.52 1.56 3.08 -3.29 -3.43 -6.72 16.03 6.48 22.51 -20.07 0.20 -19.87 26.88 2.57 29.45 -24.83 -2.26 -27.09 22.95 0.15 23.10 -30.82 -1.80 -32.62 -42.74 1.29 -41.46 -49.58 0.00 -49.58 YIELDED -43.24 2.42 -40.82 -49.28 -2.36 -51.64 YIELDED -27.35 1.82 -25.53 -34.34 -1.73 -36.07 -45.48 2.40 -43.08 -46.95 -2.38 -49.33 YIELDED -44.83 2.35 -42.48 -47.79 -2.41 -50.19 YIELDED -31.05 1.71 -29.34 -30.82 -1.80 -32.62 13 14 15 9 10 11 12 5 6 7 8 1 2 3 4 Inc. Lateral Disp. at J4 (mm)= Total Base Shear (kN) = Total Incremental Load (kN)= Condition
Step 3 (Incremental) Actual hinges at previously yielded location for the incremental analysis New location at which yield moment within tolerance are reached ΔEQ=21kN ΔEQ=14kN ΔEQ=7kN 42 54 2.94 Total Lateral Disp. at J4 (mm)= 3.5212 Frame M ΔM M + ∆M Element kNm kNm (kNm) 2.06 57.79 59.85 21.36 12.12 33.48 -20.10 24.68 4.58 18.82 -16.19 2.64 -21.99 -2.12 -24.11 25.53 -18.94 6.58 12.71 56.85 69.56 -0.21 12.18 11.97 6.29 24.58 30.87 -6.09 -13.41 -19.49 3.08 0.99 4.07 -6.72 -34.94 -41.67 22.51 53.65 76.16 -19.87 18.00 -1.88 29.45 18.00 47.45 -27.09 -8.15 -35.24 23.10 -8.15 14.95 -32.62 -18.38 -51.00 -41.46 12.56 -28.90 -49.58 0.00 -49.58 YIELDED -40.82 14.07 -26.75 -51.64 0.00 -51.64 YIELDED -25.53 18.94 -6.58 -36.07 -17.61 -53.68 YIELDED -43.08 12.40 -30.68 -49.33 0.00 -49.33 YIELDED -42.48 14.40 -28.08 -50.19 0.00 -50.19 YIELDED -29.34 17.33 -12.01 -32.62 -18.38 -51.00 12 13 14 15 8 9 10 11 1 2 3 4 5 6 7 Inc. Lateral Disp. at J4 (mm)= Total Base Shear (kN) = Condition Total Incremental Load (kN)=
ΔEQ=3kN ΔEQ=2kN ΔEQ=1kN Step 4 (Incremental) Actual hinges at previously yielded location for the incremental analysis New location at which yield moment within tolerance are reached 6 60 0.4692 Total Lateral Disp. at J4 (mm)= 3.9904 Frame M ΔM M + ∆M Element kNm kNm (kNm) 59.85 8.59 68.44 33.48 2.00 35.48 4.58 3.91 8.49 2.64 -1.96 0.67 -24.11 0.29 -23.82 6.58 -1.96 4.63 69.56 8.43 77.99 11.97 2.07 14.04 30.87 3.95 34.82 -19.49 -1.77 -21.26 4.07 0.50 4.57 -41.67 -3.40 -45.07 76.16 7.95 84.12 -1.88 2.90 1.02 47.45 2.90 50.35 -35.24 -0.50 -35.74 14.95 -0.50 14.45 -51.00 -3.35 -54.36 -28.90 1.91 -26.99 -49.58 0.00 -49.58 YIELDED -26.75 2.26 -24.49 -51.64 0.00 -51.64 YIELDED -6.58 1.96 -4.63 -53.68 0.00 -53.68 YIELDED -30.68 1.88 -28.79 -49.33 0.00 -49.33 YIELDED -28.08 2.27 -25.81 -50.19 0.00 -50.19 YIELDED -12.01 3.40 -8.61 -51.00 -3.35 -54.36 YIELDED 13 14 15 9 10 11 12 5 6 7 8 1 2 3 4 Condition Inc. Lateral Disp. at J4 (mm)= Total Base Shear (kN) = Total Incremental Load (kN)=
ΔEQ=18kN ΔEQ=12kN ΔEQ=6kN Step 5 (Incremental) 36 96 3.41 Total Lateral Disp. at J4 (mm)= 7.4004 Frame M ΔM M + ∆M Element kNm kNm (kNm) 68.44 55.34 123.78 35.48 15.86 51.34 8.49 28.66 37.15 0.67 -6.38 -5.71 -23.82 10.42 -13.40 4.63 -15.82 -11.19 77.99 54.50 132.49 14.04 16.03 30.06 34.82 28.70 63.52 -21.26 -6.00 -27.26 4.57 10.75 15.33 -45.07 -15.83 -60.90 84.12 51.48 135.60 YIELDED 1.02 21.43 22.45 50.35 21.43 71.78 -35.74 1.18 -34.57 14.45 1.18 15.62 -54.36 0.00 -54.36 -26.99 12.80 -14.19 -49.58 0.00 -49.58 YIELDED -24.49 16.80 -7.69 -51.64 0.00 -51.64 YIELDED -4.63 15.82 11.19 -53.68 0.00 -53.68 YIELDED -28.79 12.68 -16.12 -49.33 0.00 -49.33 YIELDED -25.81 16.75 -9.05 -50.19 0.00 -50.19 YIELDED -8.61 15.83 7.22 -54.36 0.00 -54.36 YIELDED 12 13 14 15 8 9 10 11 1 2 3 4 5 6 7 Condition Inc. Lateral Disp. at J4 (mm)= Total Base Shear (kN) = Total Incremental Load (kN)=
Step 6 (Incremental) ΔEQ=0.06kN ΔEQ=0.04kN ΔEQ=0.02kN 0.12 96.12 0.01277 Total Lateral Disp. at J4 (mm)= 7.41317 Frame M ΔM M + ∆M Element kNm kNm (kNm) 123.78 0.25 124.03 YIELDED 51.34 0.03 51.38 37.15 0.08 37.23 -5.71 -0.03 -5.74 -13.40 0.03 -13.37 -11.19 -0.06 -11.25 132.49 0.26 132.75 30.06 0.02 30.09 63.52 0.07 63.60 -27.26 -0.02 -27.29 15.33 0.04 15.36 -60.90 -0.06 -60.96 135.60 0.00 135.60 YIELDED 22.45 0.09 22.54 71.78 0.09 71.87 -34.57 0.00 -34.57 15.62 0.00 15.63 -54.36 0.00 -54.36 -14.19 0.05 -14.14 -49.58 0.00 -49.58 YIELDED -7.69 0.06 -7.63 -51.64 0.00 -51.64 YIELDED 11.19 0.06 11.25 -53.68 0.00 -53.68 YIELDED -16.12 0.05 -16.07 -49.33 0.00 -49.33 YIELDED -9.05 0.06 -8.99 -50.19 0.00 -50.19 YIELDED 7.22 0.06 7.28 -54.36 0.00 -54.36 YIELDED 13 14 15 9 10 11 12 5 6 7 8 1 2 3 4 Condition Inc. Lateral Disp. at J4 (mm)= Total Base Shear (kN) = Total Incremental Load (kN)=
Step 7 (Incremental) ΔEQ=4.8kN ΔEQ=3.2kN ΔEQ=1.6kN 9.6 105.72 1.3 Total Lateral Disp. at J4 (mm)= 8.71317 Frame M ΔM M + ∆M Element kNm kNm (kNm) 124.03 0.00 124.03 YIELDED 51.38 4.04 55.42 37.23 8.81 46.05 -5.74 -3.63 -9.37 -13.37 2.07 -11.30 -11.25 -5.15 -16.40 132.75 35.16 167.90 YIELDED 30.09 -3.63 26.45 63.60 2.03 65.63 -27.29 -2.56 -29.84 15.36 3.01 18.38 -60.96 -5.18 -66.14 135.60 0.00 135.60 YIELDED 22.54 5.95 28.49 71.87 5.95 77.82 -34.57 -1.02 -35.58 15.63 -1.02 14.61 -54.36 0.00 -54.36 -14.14 4.77 -9.37 -49.58 0.00 -49.58 YIELDED -7.63 5.70 -1.93 -51.64 0.00 -51.64 YIELDED 11.25 5.15 16.40 -53.68 0.00 -53.68 YIELDED -16.07 5.67 -10.40 -49.33 0.00 -49.33 YIELDED -8.99 5.57 -3.42 -50.19 0.00 -50.19 YIELDED 7.28 5.18 12.46 -54.36 0.00 -54.36 YIELDED 12 13 14 15 8 9 10 11 1 2 3 4 5 6 7 Total Base Shear (kN) = Total Incremental Load (kN)= Condition Inc. Lateral Disp. at J4 (mm)=
Step 9 (Incremental) 39 144.72 12.69 Total Lateral Disp. at J4 (mm)= 21.40317 M ΔM M + ∆M kNm kNm (kNm) 124.03 0.00 124.03 YIELDED 55.42 -46.64 8.78 46.05 5.74 51.79 -9.37 -44.15 -53.51 -11.30 1.29 -10.01 -16.40 -38.69 -55.09 167.90 0.00 167.90 YIELDED 26.45 -46.22 -19.76 65.63 6.05 71.68 -29.84 -43.74 -73.58 18.38 1.72 20.10 -66.14 -38.78 -104.91 135.60 0.00 135.60 YIELDED 28.49 -24.15 4.35 77.82 -24.15 53.68 -35.58 -21.98 -57.57 14.61 -21.98 -7.37 -54.36 0.00 -54.36 -9.37 52.37 43.00 -49.58 0.00 -49.58 YIELDED -1.93 45.43 43.51 -51.64 0.00 -51.64 YIELDED 16.40 38.69 55.09 YIELDED -53.68 0.00 -53.68 YIELDED -10.40 52.27 41.87 -49.33 0.00 -49.33 YIELDED -3.42 45.46 42.03 -50.19 0.00 -50.19 YIELDED 12.46 38.78 51.24 -54.36 0.00 -54.36 YIELDED Condition Total Incremental Load (kN)= Total Base Shear (kN) = Inc. Lateral Disp. at J4 (mm)= ΔEQ=19.5kN ΔEQ=13kN ΔEQ=6.5kN
Step 9 (Incremental) Frame M ΔM M + ∆M Element kNm kNm (kNm) 124.03 0.00 124.03 YIELDED 8.78 -1.83 6.95 51.79 0.44 52.22 -53.51 -1.74 -55.25 -10.01 0.30 -9.71 -55.09 0.00 -55.09 167.90 0.00 167.90 YIELDED -19.76 -1.82 -21.59 71.68 0.44 72.12 -73.58 -1.44 -75.02 20.10 0.64 20.74 -104.91 -1.86 -106.77 135.60 0.00 135.60 YIELDED 4.35 -0.84 3.50 53.68 -0.84 52.83 -57.57 -0.54 -58.11 -7.37 -0.54 -7.91 -54.36 0.00 -54.36 43.00 2.27 45.27 -49.58 0.00 -49.58 YIELDED 43.51 2.03 45.54 -51.64 0.00 -51.64 YIELDED 55.09 0.00 55.09 YIELDED -53.68 0.00 -53.68 YIELDED 41.87 2.26 44.13 -49.33 0.00 -49.33 YIELDED 42.03 2.08 44.11 -50.19 0.00 -50.19 YIELDED 51.24 1.86 53.10 YIELDED -54.36 0.00 -54.36 YIELDED 12 13 14 15 8 9 10 11 1 2 3 4 5 6 7 Condition ΔEQ=0.75kN ΔEQ=0.50kN ΔEQ=0.25kN
Step 10 (Incremental) 4.2 150.42 1.94 Total Lateral Disp. at J4 (mm)= 23.90917 Frame M ΔM M + ∆M Element kNm kNm (kNm) 124.03 0.00 124.03 YIELDED 6.95 -5.34 1.61 52.22 2.18 54.40 -55.25 -4.04 -59.29 -9.71 3.14 -6.57 -55.09 0.00 -55.09 167.90 0.00 167.90 YIELDED -21.59 -5.17 -26.76 72.12 2.35 74.47 -75.02 -4.19 -79.21 20.74 3.00 23.73 -106.77 0.00 -106.77 135.60 0.00 135.60 YIELDED 3.50 -2.09 1.41 52.83 -2.09 50.74 -58.11 0.16 -57.95 -7.91 0.16 -7.75 -54.36 0.00 -54.36 45.27 7.52 52.79 YIELDED -49.58 0.00 -49.58 YIELDED 45.54 7.18 52.72 YIELDED -51.64 0.00 -51.64 YIELDED 55.09 0.00 55.09 YIELDED -53.68 0.00 -53.68 YIELDED 44.13 7.52 51.65 YIELDED -49.33 0.00 -49.33 YIELDED 44.11 7.18 51.30 YIELDED -50.19 0.00 -50.19 YIELDED 53.10 0.00 53.10 YIELDED -54.36 0.00 -54.36 YIELDED 13 14 15 9 10 11 12 5 6 7 8 1 2 3 4 Total Incremental Load (kN)= Total Base Shear (kN) = Inc. Lateral Disp. at J4 (mm)= Condition ΔEQ=2.1kN ΔEQ=1.4kN ΔEQ=0.7kN
Collapse Mechanism S Y S T E M I S U N S T A B L E Beam sway mechanism is observed No further lateral load incrementing possible (only rigid body motion) 0 20 40 60 80 100 120 140 160 0 5 10 15 20 25 30 Roof Displacement (mm) BaseShear(kN)
What did we obtain? • A simple representation of the capacity curve • Plastic mechanism and sequence of hinge formation • Lateral load and displacement capacity • Ductility and plastic rotation demand 0 20 40 60 80 100 120 140 160 0 5 10 15 20 25 30 Top Displacement (mm) TotalBaseShear(kN) Incremental SAP2000 SAP 2000 built in pushover analysis options include: • hardening/loss of strength • P-M interaction • Systematic stiffness approach
Concluding Remarks • Nonlinear analysis is becoming a part of the profession • It gives us information on displacements which are indicators of damage • Never forget that estimating deformations is harder compared to estimating strength • Never replace engineering judgment with any analysis procedure

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3.4 pushover analysis

  • 1. Pushover Analysis an Inelastic Static Analysis Methods courtesy of Barış Binici
  • 2. Target Performance Dictated by codes (DBYBHY 2007, Section 1.2.1): “....The objective of seismic resistant design is to have no structural/nonstructural damage in low magnitude earthquakes, limited and repairable damage in moderate earthquakes and life safety for extreme earthquakes...”
  • 3. Current Status )( )( 1 1 TR TAW V a t  • Equivalent Lateral Force Procedure - Assume global ductility (Ra) - Detail accordingly • Modal Superposition Procedure - Include higher mode effects • Time History Analysis - Rarely used - Tedious and requires hysteretic models
  • 4. Critique of Current Practice Advantages : - Simple to use - Have proven to work - Became a tradition all over the world - Uncertainty is lumped and easier to deal with Disadvantages : - No clear connection between capacity and demand - No option for interfering with the target performance - No possibility of having the owner involved in the decision process - Not easily applicable to seismic assessment of existing structures
  • 5. DBYBHY 2007 (Chapter 7) - Evaluation and Strengthening of Existing Buildings is based on structural performances. - Steps: • Collect information from an existing structure • Assess whether info is dependable and penalize accordingly • Conduct structural analysis - Linear static analysis - Nonlinear static analysis (Pushover analysis) - Incremental pushover analysis - Time history analysis • Identify for each member the damage level • Decision based on number of elements at certain damage levels
  • 6. Time History? - Actual earthquake response is hard to predict anyways. - Closest estimate can be found using inelastic time-history analysis. - Difficulties with inelastic time history analysis: - Suitable set of ground motion (Description of demand) - hysteretic behavior models (Description of capacity) - Computation time (Time) - Post processing (Time and understanding) Alternative approach is pushover analysis. Düzce Ground Motion -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0 5 10 15 20 25 30 Sec. Acceleration(g)
  • 7. Pushover Analysis • Definition: Inelastic static analysis of a structure using a specified (constant or variable) force pattern from zero load to a prescribed ultimate displacement. • Use of it dates back to 1960s to1970s to investigate stability of steel frames. • Many computer programs were developed since then with many features and limitations.
  • 8. Available Computer Programs • Design Oriented: SAP 2000, GTSTRUDL, RAM etc. • Research Oriented: Opensees, IDARC, SeismoStrut etc. What is different? • User interface capabilities • Analysis options • Member behavior options
  • 9. Section Damage Levels Damage levels are established based on concrete outermost compressive fiber strain and steel strain (for nonlinear analysis procedure).
  • 10. Section Damage Levels How should these values be decided? - Construction practice - Experience of engineers - Input of academicians
  • 11. Curvature demand at target curvatures Φp = θp / Lp Φt = Φy + Φp 0 100 200 300 400 500 600 0.0000 0.0200 0.0400 0.0600 0.0800 0.1000 0.1200 Eğrilik (rad/m) Moment (kN.m) AK GV GÇ (Φt)(Φy)
  • 12. How do we estimate strains from a structural analysis? Strain Moment Curvature Moment My øy øu Moment Plastic Rotations My θpu θpu =(øu – øy) Lp OR θp =(ø – øy) Lp Where Lp = 0.5h Utilize this idealized moment-rotation response in inelastic structural analysis
  • 13. Definition of Potential Plastic Hinges • End regions of columns and beams (center for gravity loads) are the potential plastic hinges • Plastic hinges are hinges capable of resisting My (not significantly more, hardening allowed) undergoing plastic rotations h Lp Elastic Beam- Column Element Plastic Hinges Rigid End zones
  • 14. Elastic Parts For regions other than plastic hinging occurs, cracking is expected therefore use of cracked stiffness is customary (0.4-0.8) EIo Eğrilik Moment EIo 0.4-0.8EIo Curvature
  • 16. Steps of Pushover Analysis: A Simple Incremental Procedure 1. Build a computational model of the structure
  • 17. Steps of Pushover Analysis 2. Define member behavior – Beams: Moment-rotation relations – Columns: Moment-rotation and Interaction Diagrams – Beam-column joints: Assume rigid (DBYBHY 2007 ) – Walls: Model as beam columns but introduce a shear spring to model shear deformations – Use cracked rigidities for elastic portions
  • 18. Steps of Pushover Analysis 3. Apply gravity loads 1.0 G + n Q n=0.3 (live load reduction factor) (if the interaction diagrams will not be used a good estimate of the moment capacity of column hinges needs to be made) Possibilities: - Based on initial gravity load analysis - Based on a beam hinging mechanism - Based on elastic lateral force analysis with an assumed reasonable Ra value.
  • 19. Steps of Pushover Analysis 4. Specify a Lateral Load Profile: (Inverted triangular, constant, first mode shape are some of the possibilities) It is a good idea to have a spreadsheet page ready indicating all members, current load increment 5. Lateral Load Incrementing: Step 1: Elastic analysis is valid up to the formation of the first hinge, i.e. when the first critical location reaches its moment capacity. • Find the lateral loads that cause first hinge formation (V1). • Record all member forces and deformations (F1, d1).
  • 20. Steps of Pushover Analysis Step 2: Beyond Step 1, yielded element’s critical location cannot take any further moment. Therefore place an actual hinge at that location. Conduct an analysis increment for this modified structure. This load increment should be selected such that upon summing the force resultant from this incremental step and previous step, second hinge formation is reached. V2 = V1 + ΔV F2 = F1 + ΔF d2 = d1 + Δd Results from Step 1 + Results from an incremental analysis with a hinge placed at first yield location = Second Hinge formation
  • 21. Steps of Pushover Analysis . . Step i: Similar to step 2 but additional hinges form and incremental analysis steps are conducted for systems with more hinges. Results are added to those from the previous step Vi = Vi-1 + ΔV Fi = Fi-1 + ΔF di = di-1 + Δd Results from Step i-1 + Results from an incremental analysis with a hinge placed at i-1th yield location = ith hinge formation
  • 22. Steps of Pushover Analysis Step n: Sufficient number of plastic hinges have formed and system has reached a plastic mechanism. Note that this could be a partial collapse mechanism as well. Beyond this point system rotates as a rigid body. ANALYSIS DONE - Plot Base Shear- Roof Displacement - Check member rotations and identify performance levels
  • 23. Example Application: 3 Story- 2 Bay RC Frame (Courtesy of Ahmet Yakut) M O D E L 3m 3m 3m 1 2 3 10 11 12 13 14 15 4 5 6 7 8 9 6m 6m J1 J2 J3 J4 J8 J7 J6 J5 J9 J10 J11 J12
  • 24. Assumptions Assume • Constant Axial Load on Columns for Analysis Steps • Rigid-plastic with no hardening or softening moment-rotation behavior for columns and beams • plastic hinging occurs when moment capacity is within 5% tolerance • Load combinations 1.0 DL + 0.3 LL and 1.0 DL + 0.3 LL+1.0EQ to compute axial load levels DL=10kN/m DL=15kN/m DL=15kN/m LL=2kN/m LL=2kN/m LL=2kN/m EQ=60kN EQ=40kN EQ=20kN SABİT YÜK HAREKETLİ YÜK YATAY YÜK
  • 25. DATA 10-f10 60cm 60cm Columns 3-f10 3-f10 25cm 50cm Beams Steel (fyd=495 Mpa) Concrete (fcd=25 Mpa) Clear cover=5 cm E=2.779E+4 MPa M+ is the same as M- Note that if this is a seismic evaluation problem strength values obtained at site should be used!
  • 26. Section Capacities Eğrilik Moment fy My fult Eleman N My Φy Φ u l t kN kNm rad/m rad/m 1 -83,786 124 0,0055 0,111 2 -51,347 115,5 0,0056 0,115 3 -19,872 107,5 0,0056 0,119 4 -253,392 166 0,0059 0,085 5 -158,905 143 0,0060 0,099 6 -64,797 119 0,0060 0,113 7 -124,104 133,5 0,0056 0,105 8 -77,747 122 0,0057 0,112 9 -31,201 110 0,0054 0,118 10 5,606 49 0,0073 0,103 11 1,421 50 0,0069 0,102 12 -17,233 53 0,0069 0,099 13 5,606 49 0,0073 0,103 14 1,421 50 0,0069 0,102 15 -17,233 53 0,0069 0,099 Elemnaların Moment-eğrilik ilişkileri elasto-plastik, pekleşmesiz To be conservative smaller axial load from two load combinations can be selected (as long as N<Nb) Idealized member moment curvature relations for estimated axial load level Member
  • 27. Effect of Axial Force • Compute the moment capacity by accounting for axial force variation • Always remain on the yield surface
  • 28. Step 1 DL=10kN/m DL=15kN/m DL=15kN/m LL=2kN/m LL=2kN/m LL=2kN/m EQ=3kN EQ=2kN EQ=1kN COMBO2: 1.0 DL + 0.3 LL + 1.0 EQ Detection of first yield (moment reaches My±5%My ) 6 Frame Joint Myield M Element Label kNm kNm J1 124.0 -4.33 J2 124.0 20.60 J2 115.5 -22.14 J3 115.5 21.00 J3 107.5 -22.23 J4 107.5 27.35 J5 166.0 6.23 J6 166.0 -0.60 J6 143.0 3.50 J7 143.0 -2.94 J7 119.0 1.52 J8 119.0 -3.29 J9 133.5 16.03 J10 133.5 -20.07 J10 122.0 26.88 J11 122.0 -24.83 J11 110.0 22.95 J12 110.0 -30.82 J2 49.0 -42.74 J6 49.0 -49.58 YIELDED J3 50.0 -43.24 J7 50.0 -49.28 J4 53.0 -27.35 J8 53.0 -34.34 J6 49.0 -45.48 J10 49.0 -46.95 J7 50.0 -44.83 J11 50.0 -47.79 J8 53.0 -31.05 J12 53.0 -30.82 0.2947 11 12 6 7 4 14 15 Condition 13 5 8 9 3 10 1 2 First yielding stage Total Base Shear (kN)= Lateral Disp. at J4 (mm)= J4 (monitored node )
  • 29. Step 2 (Incremental) ΔEQ=3kN ΔEQ=2kN ΔEQ=1kN Actual hinge at previously yielded location for the incremental analysis New locations at which yield moments within tolerance are reached 6 12 0.2865 Total Lateral Disp. at J4 (mm)= 0.5812 Frame M ΔM M + ∆M Element kNm kNm (kNm) -4.33 6.39 2.06 20.60 0.76 21.36 -22.14 2.05 -20.10 21.00 -2.18 18.82 -22.23 0.24 -21.99 27.35 -1.82 25.53 6.23 6.47 12.71 -0.60 0.39 -0.21 3.50 2.79 6.29 -2.94 -3.15 -6.09 1.52 1.56 3.08 -3.29 -3.43 -6.72 16.03 6.48 22.51 -20.07 0.20 -19.87 26.88 2.57 29.45 -24.83 -2.26 -27.09 22.95 0.15 23.10 -30.82 -1.80 -32.62 -42.74 1.29 -41.46 -49.58 0.00 -49.58 YIELDED -43.24 2.42 -40.82 -49.28 -2.36 -51.64 YIELDED -27.35 1.82 -25.53 -34.34 -1.73 -36.07 -45.48 2.40 -43.08 -46.95 -2.38 -49.33 YIELDED -44.83 2.35 -42.48 -47.79 -2.41 -50.19 YIELDED -31.05 1.71 -29.34 -30.82 -1.80 -32.62 13 14 15 9 10 11 12 5 6 7 8 1 2 3 4 Inc. Lateral Disp. at J4 (mm)= Total Base Shear (kN) = Total Incremental Load (kN)= Condition
  • 30. Step 3 (Incremental) Actual hinges at previously yielded location for the incremental analysis New location at which yield moment within tolerance are reached ΔEQ=21kN ΔEQ=14kN ΔEQ=7kN 42 54 2.94 Total Lateral Disp. at J4 (mm)= 3.5212 Frame M ΔM M + ∆M Element kNm kNm (kNm) 2.06 57.79 59.85 21.36 12.12 33.48 -20.10 24.68 4.58 18.82 -16.19 2.64 -21.99 -2.12 -24.11 25.53 -18.94 6.58 12.71 56.85 69.56 -0.21 12.18 11.97 6.29 24.58 30.87 -6.09 -13.41 -19.49 3.08 0.99 4.07 -6.72 -34.94 -41.67 22.51 53.65 76.16 -19.87 18.00 -1.88 29.45 18.00 47.45 -27.09 -8.15 -35.24 23.10 -8.15 14.95 -32.62 -18.38 -51.00 -41.46 12.56 -28.90 -49.58 0.00 -49.58 YIELDED -40.82 14.07 -26.75 -51.64 0.00 -51.64 YIELDED -25.53 18.94 -6.58 -36.07 -17.61 -53.68 YIELDED -43.08 12.40 -30.68 -49.33 0.00 -49.33 YIELDED -42.48 14.40 -28.08 -50.19 0.00 -50.19 YIELDED -29.34 17.33 -12.01 -32.62 -18.38 -51.00 12 13 14 15 8 9 10 11 1 2 3 4 5 6 7 Inc. Lateral Disp. at J4 (mm)= Total Base Shear (kN) = Condition Total Incremental Load (kN)=
  • 31. ΔEQ=3kN ΔEQ=2kN ΔEQ=1kN Step 4 (Incremental) Actual hinges at previously yielded location for the incremental analysis New location at which yield moment within tolerance are reached 6 60 0.4692 Total Lateral Disp. at J4 (mm)= 3.9904 Frame M ΔM M + ∆M Element kNm kNm (kNm) 59.85 8.59 68.44 33.48 2.00 35.48 4.58 3.91 8.49 2.64 -1.96 0.67 -24.11 0.29 -23.82 6.58 -1.96 4.63 69.56 8.43 77.99 11.97 2.07 14.04 30.87 3.95 34.82 -19.49 -1.77 -21.26 4.07 0.50 4.57 -41.67 -3.40 -45.07 76.16 7.95 84.12 -1.88 2.90 1.02 47.45 2.90 50.35 -35.24 -0.50 -35.74 14.95 -0.50 14.45 -51.00 -3.35 -54.36 -28.90 1.91 -26.99 -49.58 0.00 -49.58 YIELDED -26.75 2.26 -24.49 -51.64 0.00 -51.64 YIELDED -6.58 1.96 -4.63 -53.68 0.00 -53.68 YIELDED -30.68 1.88 -28.79 -49.33 0.00 -49.33 YIELDED -28.08 2.27 -25.81 -50.19 0.00 -50.19 YIELDED -12.01 3.40 -8.61 -51.00 -3.35 -54.36 YIELDED 13 14 15 9 10 11 12 5 6 7 8 1 2 3 4 Condition Inc. Lateral Disp. at J4 (mm)= Total Base Shear (kN) = Total Incremental Load (kN)=
  • 32. ΔEQ=18kN ΔEQ=12kN ΔEQ=6kN Step 5 (Incremental) 36 96 3.41 Total Lateral Disp. at J4 (mm)= 7.4004 Frame M ΔM M + ∆M Element kNm kNm (kNm) 68.44 55.34 123.78 35.48 15.86 51.34 8.49 28.66 37.15 0.67 -6.38 -5.71 -23.82 10.42 -13.40 4.63 -15.82 -11.19 77.99 54.50 132.49 14.04 16.03 30.06 34.82 28.70 63.52 -21.26 -6.00 -27.26 4.57 10.75 15.33 -45.07 -15.83 -60.90 84.12 51.48 135.60 YIELDED 1.02 21.43 22.45 50.35 21.43 71.78 -35.74 1.18 -34.57 14.45 1.18 15.62 -54.36 0.00 -54.36 -26.99 12.80 -14.19 -49.58 0.00 -49.58 YIELDED -24.49 16.80 -7.69 -51.64 0.00 -51.64 YIELDED -4.63 15.82 11.19 -53.68 0.00 -53.68 YIELDED -28.79 12.68 -16.12 -49.33 0.00 -49.33 YIELDED -25.81 16.75 -9.05 -50.19 0.00 -50.19 YIELDED -8.61 15.83 7.22 -54.36 0.00 -54.36 YIELDED 12 13 14 15 8 9 10 11 1 2 3 4 5 6 7 Condition Inc. Lateral Disp. at J4 (mm)= Total Base Shear (kN) = Total Incremental Load (kN)=
  • 33. Step 6 (Incremental) ΔEQ=0.06kN ΔEQ=0.04kN ΔEQ=0.02kN 0.12 96.12 0.01277 Total Lateral Disp. at J4 (mm)= 7.41317 Frame M ΔM M + ∆M Element kNm kNm (kNm) 123.78 0.25 124.03 YIELDED 51.34 0.03 51.38 37.15 0.08 37.23 -5.71 -0.03 -5.74 -13.40 0.03 -13.37 -11.19 -0.06 -11.25 132.49 0.26 132.75 30.06 0.02 30.09 63.52 0.07 63.60 -27.26 -0.02 -27.29 15.33 0.04 15.36 -60.90 -0.06 -60.96 135.60 0.00 135.60 YIELDED 22.45 0.09 22.54 71.78 0.09 71.87 -34.57 0.00 -34.57 15.62 0.00 15.63 -54.36 0.00 -54.36 -14.19 0.05 -14.14 -49.58 0.00 -49.58 YIELDED -7.69 0.06 -7.63 -51.64 0.00 -51.64 YIELDED 11.19 0.06 11.25 -53.68 0.00 -53.68 YIELDED -16.12 0.05 -16.07 -49.33 0.00 -49.33 YIELDED -9.05 0.06 -8.99 -50.19 0.00 -50.19 YIELDED 7.22 0.06 7.28 -54.36 0.00 -54.36 YIELDED 13 14 15 9 10 11 12 5 6 7 8 1 2 3 4 Condition Inc. Lateral Disp. at J4 (mm)= Total Base Shear (kN) = Total Incremental Load (kN)=
  • 34. Step 7 (Incremental) ΔEQ=4.8kN ΔEQ=3.2kN ΔEQ=1.6kN 9.6 105.72 1.3 Total Lateral Disp. at J4 (mm)= 8.71317 Frame M ΔM M + ∆M Element kNm kNm (kNm) 124.03 0.00 124.03 YIELDED 51.38 4.04 55.42 37.23 8.81 46.05 -5.74 -3.63 -9.37 -13.37 2.07 -11.30 -11.25 -5.15 -16.40 132.75 35.16 167.90 YIELDED 30.09 -3.63 26.45 63.60 2.03 65.63 -27.29 -2.56 -29.84 15.36 3.01 18.38 -60.96 -5.18 -66.14 135.60 0.00 135.60 YIELDED 22.54 5.95 28.49 71.87 5.95 77.82 -34.57 -1.02 -35.58 15.63 -1.02 14.61 -54.36 0.00 -54.36 -14.14 4.77 -9.37 -49.58 0.00 -49.58 YIELDED -7.63 5.70 -1.93 -51.64 0.00 -51.64 YIELDED 11.25 5.15 16.40 -53.68 0.00 -53.68 YIELDED -16.07 5.67 -10.40 -49.33 0.00 -49.33 YIELDED -8.99 5.57 -3.42 -50.19 0.00 -50.19 YIELDED 7.28 5.18 12.46 -54.36 0.00 -54.36 YIELDED 12 13 14 15 8 9 10 11 1 2 3 4 5 6 7 Total Base Shear (kN) = Total Incremental Load (kN)= Condition Inc. Lateral Disp. at J4 (mm)=
  • 35. Step 9 (Incremental) 39 144.72 12.69 Total Lateral Disp. at J4 (mm)= 21.40317 M ΔM M + ∆M kNm kNm (kNm) 124.03 0.00 124.03 YIELDED 55.42 -46.64 8.78 46.05 5.74 51.79 -9.37 -44.15 -53.51 -11.30 1.29 -10.01 -16.40 -38.69 -55.09 167.90 0.00 167.90 YIELDED 26.45 -46.22 -19.76 65.63 6.05 71.68 -29.84 -43.74 -73.58 18.38 1.72 20.10 -66.14 -38.78 -104.91 135.60 0.00 135.60 YIELDED 28.49 -24.15 4.35 77.82 -24.15 53.68 -35.58 -21.98 -57.57 14.61 -21.98 -7.37 -54.36 0.00 -54.36 -9.37 52.37 43.00 -49.58 0.00 -49.58 YIELDED -1.93 45.43 43.51 -51.64 0.00 -51.64 YIELDED 16.40 38.69 55.09 YIELDED -53.68 0.00 -53.68 YIELDED -10.40 52.27 41.87 -49.33 0.00 -49.33 YIELDED -3.42 45.46 42.03 -50.19 0.00 -50.19 YIELDED 12.46 38.78 51.24 -54.36 0.00 -54.36 YIELDED Condition Total Incremental Load (kN)= Total Base Shear (kN) = Inc. Lateral Disp. at J4 (mm)= ΔEQ=19.5kN ΔEQ=13kN ΔEQ=6.5kN
  • 36. Step 9 (Incremental) Frame M ΔM M + ∆M Element kNm kNm (kNm) 124.03 0.00 124.03 YIELDED 8.78 -1.83 6.95 51.79 0.44 52.22 -53.51 -1.74 -55.25 -10.01 0.30 -9.71 -55.09 0.00 -55.09 167.90 0.00 167.90 YIELDED -19.76 -1.82 -21.59 71.68 0.44 72.12 -73.58 -1.44 -75.02 20.10 0.64 20.74 -104.91 -1.86 -106.77 135.60 0.00 135.60 YIELDED 4.35 -0.84 3.50 53.68 -0.84 52.83 -57.57 -0.54 -58.11 -7.37 -0.54 -7.91 -54.36 0.00 -54.36 43.00 2.27 45.27 -49.58 0.00 -49.58 YIELDED 43.51 2.03 45.54 -51.64 0.00 -51.64 YIELDED 55.09 0.00 55.09 YIELDED -53.68 0.00 -53.68 YIELDED 41.87 2.26 44.13 -49.33 0.00 -49.33 YIELDED 42.03 2.08 44.11 -50.19 0.00 -50.19 YIELDED 51.24 1.86 53.10 YIELDED -54.36 0.00 -54.36 YIELDED 12 13 14 15 8 9 10 11 1 2 3 4 5 6 7 Condition ΔEQ=0.75kN ΔEQ=0.50kN ΔEQ=0.25kN
  • 37. Step 10 (Incremental) 4.2 150.42 1.94 Total Lateral Disp. at J4 (mm)= 23.90917 Frame M ΔM M + ∆M Element kNm kNm (kNm) 124.03 0.00 124.03 YIELDED 6.95 -5.34 1.61 52.22 2.18 54.40 -55.25 -4.04 -59.29 -9.71 3.14 -6.57 -55.09 0.00 -55.09 167.90 0.00 167.90 YIELDED -21.59 -5.17 -26.76 72.12 2.35 74.47 -75.02 -4.19 -79.21 20.74 3.00 23.73 -106.77 0.00 -106.77 135.60 0.00 135.60 YIELDED 3.50 -2.09 1.41 52.83 -2.09 50.74 -58.11 0.16 -57.95 -7.91 0.16 -7.75 -54.36 0.00 -54.36 45.27 7.52 52.79 YIELDED -49.58 0.00 -49.58 YIELDED 45.54 7.18 52.72 YIELDED -51.64 0.00 -51.64 YIELDED 55.09 0.00 55.09 YIELDED -53.68 0.00 -53.68 YIELDED 44.13 7.52 51.65 YIELDED -49.33 0.00 -49.33 YIELDED 44.11 7.18 51.30 YIELDED -50.19 0.00 -50.19 YIELDED 53.10 0.00 53.10 YIELDED -54.36 0.00 -54.36 YIELDED 13 14 15 9 10 11 12 5 6 7 8 1 2 3 4 Total Incremental Load (kN)= Total Base Shear (kN) = Inc. Lateral Disp. at J4 (mm)= Condition ΔEQ=2.1kN ΔEQ=1.4kN ΔEQ=0.7kN
  • 38. Collapse Mechanism S Y S T E M I S U N S T A B L E Beam sway mechanism is observed No further lateral load incrementing possible (only rigid body motion) 0 20 40 60 80 100 120 140 160 0 5 10 15 20 25 30 Roof Displacement (mm) BaseShear(kN)
  • 39. What did we obtain? • A simple representation of the capacity curve • Plastic mechanism and sequence of hinge formation • Lateral load and displacement capacity • Ductility and plastic rotation demand 0 20 40 60 80 100 120 140 160 0 5 10 15 20 25 30 Top Displacement (mm) TotalBaseShear(kN) Incremental SAP2000 SAP 2000 built in pushover analysis options include: • hardening/loss of strength • P-M interaction • Systematic stiffness approach
  • 40. Concluding Remarks • Nonlinear analysis is becoming a part of the profession • It gives us information on displacements which are indicators of damage • Never forget that estimating deformations is harder compared to estimating strength • Never replace engineering judgment with any analysis procedure