ACCOUNTING FOR SECOND ORDER EFFECTS DUE TO GEOMETRIC NONLINEARITY AS PER EUROCODE 2 AND EUROCODE 3 IN ETABS 2016 AND SAP2000 V19
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1 ACCOUNTING FOR SECOND ORDER EFFECTS DUE TO GEOMETRIC NONLINEARITY AS PER EUROCODE 2 AND EUROCODE 3 IN ETABS 2016 AND SAP2000 V19 I. Overview When the load acting on a structure and the resulting deflections are small enough, the loaddeflection relationship for the structure is linear. For the most part, ETABS/SAP2000 analyses assume such linear behavior. This permits the program to form the equilibrium equations using the original (undeformed) geometry of the structure. Strictly speaking, the equilibrium equations should actually refer to the geometry of the structure after deformation. The linear equilibrium equations are independent of the applied load and the resulting deflection. Thus the results of different static and/or dynamic loads can be superposed (scaled and added), resulting in great computational efficiency. However, if the load on the structure and/or the resulting deflections are large, then the loaddeflection behavior may be come nonlinear. The effect due to nonlinear load-deflection behavior is termed as Second Order Effect. Eurocode 2 and Eurocode 3 requires second order effects due to geometric nonlinearity be accounted for in the analysis stage and/or in the design stage whenever its effect is significant. Second order effects due to geometric nonlinearity are classified as global second order effect (P-Δ) and local second order effect (P-δ). Global second order effect (P-Δ) refers to the overall sway effect of structure, while local second order effect (P-δ) refers to the effect of deformation of the member between its ends. P-Δ effect is often significant, it can be accounted for fairly accurately by considering the total vertical load at a storey level, which is due to gravity loads and is unaffected by any lateral loads. P-δ effect, on the other hand, is significant only in very slender columns or columns bent in single curvature (not the usual case). P-δ effect requires consideration of axial forces in the members due to both gravity and lateral loads. Otte Technical Note No.: TN-S02 Rev. 0 1
2 This technical note discusses on how to account for both second order effects in modeling building structures. II. Defining Simplified P-Δ Analysis with Only (1) Set of Load (Gravity) For many applications, it is adequate to consider the P-Δ effect on the structure under one set of factored loads (usually gravity), and to consider all other load case analyses as linear static using the stiffness matrix developed for this one set of P-delta loads. This enables all analysis results to be scaled and superposed (when defining design load combinations) for the purposes of design. As an example, supposed that we are using the following design load combinations as per Eq of SS EN 1990:2008: (1) 1.35 Dead Load ± EHF (2) 1.35 Dead Load Live Load ± EHF (3) 1.35 Dead Load Live Load ± 0.75 Wind Load ± EHF (4) 1.35 Dead Load Live Load ± 1.5 Wind Load ± EHF (5) 1.35 Dead Load ± 1.5 Wind Load ± EHF (6) 1.0 Dead Load ± 1.5 Wind Load ± EHF For this case, the P-Δ effect due to overall sway of the structure can usually be accounted for, conservatively, by specifying the load combination in the initial P-delta Load Case to be 1.35 times the dead load plus 1.5 times the live load. This will accurately account for this effect in load combinations 2 and 3 above, and will conservatively account for this effect in load combinations 1, 4, 5 and 6. This P-Δ effect is not generally important in load combinations that has no lateral load. This Simplified P-Δ Analysis can be done in two methods: Method A: Defining Nonlinear Static P-Delta Load Case and Using Its Stiffness Matrix as Initial Stiffness of Other Load Cases (Dead, Live, etc.) Otte Technical Note No.: TN-S02 Rev. 0 2
3 Go to Define>Load Cases then define PDELTA nonlinear static load case as below: Other parameters include the number of saved steps, the number of iterations allowed per step, and the convergence tolerance. If the P-delta effect is reasonably small, the default values are adequate. Then define or modify all other linear Load Cases so that they use the stiffness from case PDELTA: Otte Technical Note No.: TN-S02 Rev. 0 3
4 Method B (Available only in ETABS): Defining Preset P-Delta Options and Using Its Stiffness Matrix as Initial Stiffness of Other Load Cases (Dead, Live, etc.) Instead of defining nonlinear static P-delta load case, Preset P-delta Options can be set in ETABS 2016 to produce the same results. ETABS 2016 will create internally a nonlinear static P-delta load case and automatically use it as initial stiffness for the other load cases. Go to Define>Preset P-delta Options then define Preset P-delta Options as below: After setting the Preset P-delta Options, check that the other load cases are using the stiffness of Preset P-delta Options as its initial stiffness. Otte Technical Note No.: TN-S02 Rev. 0 4
5 When designing for steel columns as per EC3, steel frame design preferences must be modified to specify that P-Δ is already considered in the analysis. Otherwise, effective length method will be used for steel column designs. Go to Design>Steel Frame Design>View/Revise Preferences P-Δ analysis is usually required to be done prior to designing for EC2 in ETABS 2016 and SAP2000 v19. III. Accounting for P-δ in Design Stage Although ETABS 2016 and SAP2000 v19 can analyse both P-Δ and P-δ effects, it is generally recommended that the P-δ effect be accounted for in the design stage by using code based design procedures. P-δ effect can be accounted for in EC2 design stage by either Nominal Curvature or Nominal Stiffness method. This can be specified in the Concrete Frame Design Preferences: Click on Design>Concrete Frame Design>View/Revise Preferences and choose the preferred P-δ method. Otte Technical Note No.: TN-S02 Rev. 0 5
6 Similarly, P-δ effect can be accounted for in EC3 design stage by using the relevant buckling factors and either one of the Interaction Factor Methods (Method 1/Annex A or Method 2/Annex B). Selection of Interaction Factor Method to be used can be specified in the Steel Frame Design Preferences: Click on Design>Steel Frame Design>View/Revise Preferences Otte Technical Note No.: TN-S02 Rev. 0 6
7 IV. Direct Analysis Method for EC3 The P-δ effect due to the deformation of the member between its ends can be accurately analyzed only when separate nonlinear Load Cases are run for each load combinations. Also, at least two Frame elements per column should be used. This term is called Direct Analysis Method. Again, it is recommended that this effect be accounted for instead by using the ETABS 2016 or SAP2000 v19 design features. Direct Analysis Method in ETABS 2016 and SAP2000 v19 generally involves the following procedures: 1. Reduce element stiffness (say, by 20%) by applying Frame Property Modifiers. This will account for residual stress effect on steel members. 2. Define EHF for global imperfections. 3. Divide columns and walls into 2 or more elements. 4. Define and perform nonlinear static load cases for each load combinations. 5. Local imperfections are considered through the buckling check of individual element. Nonlinear load cases for each load combinations can be easily generated by using ETABS 2016/SAP2000 v19 feature to convert load combinations to nonlinear load cases. Go to Define>Load Combinations Select all load combinations and click on Convert Combos to Nonlinear Cases. Nonlinear load cases will be automatically generated for each of the selected load combinations. Otte Technical Note No.: TN-S02 Rev. 0 7
8 V. Performing Buckling Analysis to Check Importance of Second Order Analysis As per EC3, second order effects must be considered if: α cr < 10 ; for elastic analysis α cr < 15 ; for plastic analysis Buckling analysis can be performed in ETABS 2016 and SAP2000 to check the importance of second order analysis. In ETABS 2016 and SAP2000 buckling analysis, linear (bifurcation) buckling modes of a structure can be found under any set of loads and the buckling factor α cr will be reported for each modes. Otte Technical Note No.: TN-S02 Rev. 0 8
9 To define a buckling load case, click on Define>Load Case To display the deformed shape and buckling load factor of a buckling load case, click on Display>Deformed Shape and choose the buckling load case and mode number to be displayed. Otte Technical Note No.: TN-S02 Rev. 0 9
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