Guide on Second-order and Advanced analysis of …

1 Nonlinear Integrated design and analysis software for structures Ir Prof. SL Chan Guide on Second-order and Advanced analysis of structures Version 2 - 2009 Nonlinear Integrated design and analysis software for structures Ir Prof. SL Chan CONTENT 1 Methods of - 1 - 2 Linear analysis and the effective length method .. - 5 - Important remark .. - 5 - Non-sway frame .. - 6 - design chart method .. - 6 - Sway-sensitive frames .. - 8 - Sway ultra-sensitive frames .. - 9 - 3 design by Second-order analysis .. - 10 - Software .. - 10 - Imperfections .. - 10 - Frame imperfection .. - 10 - Using eigen-buckling mode as imperfection mode .. - 10 - 4 Examples .. - 16 - Example 1 Simple benchmark example for testing of software .. - 16 - Example 2 Portal frame .. - 17 - Example 3 design of a 3-Dimensional regular steel building .. - 21 - Example 4 Slender frame in practice.

2 - 27 - 5 Application of nonlinear finite shell element method .. - 28 - design of beams for lateral-torsional buckling .. - 28 - Large deflection nonlinear analysis for column buckling .. - 31 - Buckling analysis of hinged cylindrical shell .. - 33 - 6 References .. - 36 - CHECK LIST .. - 37 - Nonlinear Integrated design and analysis software for structures Prof. SL Chan - 1 -1 Methods of analysis In the Hong Kong steel Code, both the first- order linear and Second-order non-linear analysis methods can be used. But we need to ensure the effects of change of geometry and buckling effects are correctly considered with cr not less than 5 otherwise the Second-order analysis must be used. Also, the design of irregular and slender structures should adopt the Second-order analysis with appropriate values of initial imperfections from code (see Table of the Hong Kong Steel Code or Table of Eurocode-3).

3 Software NIDA is used for the present analysis and design . Load factor in Figure 1 represents a scalar multiplied to the set of design load in a particular combined load case. To understand the method, we must first appreciate the behaviour of a structure under an increasing load. Various methods provide an answer of the collapse load under its assumptions, such as plastic collapse load which does not consider any buckling effect and P- -only Second-order analysis does not consider member imperfection and member buckling. The results of these methods are compared with the true collapse or ultimate load of a structure , u in the Figure 1 below. DeflectionLoad factor Second-order P elastic analysisFirst- order Linear AnalysisElastic critical load=2EI/L2crPDesign resistance or collapse load by Second-order Plastic AnalysisDesign resistance by "First-plastic-hinge"p= Pwhere P=1000kNBraced & hori.

4 Members 152x152x37 Others : 203x203x60 Yield stress : 355 MPa4m4m3m4m = crpyRigid plastic load=pS= Second-order P only elastic analysis cr Figure 1 design methods Nonlinear Integrated design and analysis software for structures Prof. SL Chan - 2 -Types of analyses can be explained as follows. Elastic Critical Load Factor cr cr a factor multiplied to the design load to cause the structure to buckle elastically. The deflection before buckling, large deflection and material yielding effects are not considered here and the factor is an upper bound solution that cannot be used directly for design . cr can be used to measure the instability stage of a frame against sway and buckling. Plastic collapse load factor, p. p is a load factor multiplied to the design load to cause the structure to collapse plastically but buckling and Second-order effects are not considered.

5 Because of the ignorance of buckling effects, p cannot be used for direct design and it is an upper bound solution to the true collapse load of the structure . This load factor was widely used in the past for plastic design because of its simplicity to determine. P-delta effects : refer to the Second-order effect. There are two types, being P- and P- . P- effect : Second-order effect due to change of geometry of the structure P- effect : Second-order effect due to member curvature and change of member stiffness under load. A member under tension is stiffer than under compression. Second-order analysis and design method is a better design method than the effective length method. The method determines the P- effect and the P- effect with initial imperfections so that effective length needs not assumed. Note that use of Second-order analysis without consideration of initial imperfection should be avoided as it under-designs a structure since all structures contain imperfections.

6 Nonlinear Integrated design and analysis software for structures Prof. SL Chan - 3 - Figure 2 The P- and P- effects Linear analysis or First- order linear analysis : an analysis assuming the deflection and stress are proportional to load. It does not consider buckling nor material yielding which is then considered in the design stage using code. Notional Force : a small force applied horizontally to a structure to simulate lack of verticality and imperfection. It can also be used to measure the lateral stiffness so that the elastic critical factor can be determined. Figure 3 Simulation of out-of-plumbness by the notional force PP PPPPP Nonlinear Integrated design and analysis software for structures Prof. SL Chan - 4 - Second-order P- -only analysis for plotting bending moment: an analysis used to plot the bending moment and force diagrams based on the deformed nodal coordinates.

7 It does not consider member curvature or the P- effect. This method is commonly used in software because of its simplicity. In fact, most software can only do this P- -only analysis which is not qualified for a full Second-order analysis accounting for P- and P- effects with imperfections at frame and member levels. Second-order analysis with section capacity check : an analysis allows for P- effect and the P- effect and stops at first plastic hinge. It needs not assume an effective length for the buckling strength check, but imperfection must be allowed for. Nonlinear Integrated design and analysis software for structures Prof. SL Chan - 5 -2 Linear analysis and the effective length method In the first- order linear analysis , the analysis finds the load in the columns and the buckling strength is unknown. Unlike the Second-order P- analysis which consider the increase in stress due to the Second-order and buckling effect, the first- order linear analysis needs to reduce the resistance of the columns when taking the load without considering Second-order moment.

8 To calculate the effective length other than making an assumption, we have the following method. 1. Calculate cr by one of the following methods Application of notional force. cr can be determined in Eqt. 2. Use computer to find cr hFFVNcr= ( ) For multi-storey frames, the maximum cr among all stories should be used to obtain the minimum critical load factor. cr is defined as the factor multiplied to the design load causing the frame to buckle elastically. Notional force is (1) to simulate lack of verticality of frames and taken as of the factored dead and imposed loads applied horizontally to the structure and (2) to calculate the elastic critical load factor cr to Equation 4. This percentage of notional force may vary for other types of structures like scaffolding where imperfections are expected to be more serious. (3) to classify the frame as non-sway, sway-sensitive and sway-ultra sensitive frames.

9 Important remark Elastic Critical Load Factor cr for assessing a sway and a non-sway frames should only be used for rectilinear frames under gravitational loads and it should not be used in irregular frames or structures dominated by lateral loads such as wind. Nonlinear Integrated design and analysis software for structures Prof. SL Chan - 6 - The following section describes the method of using chart to determine the effective length of a member in a regular frame. Non-sway frame When cr 10, the frame is considered as non-sway. P- effect can be ignored here and only P- effect is needed. The effective length of members in frames can be designed by chart in Figure of Hong Kong Steel Code. design chart method For a sub-frame in a multi-storey frame in Figure 4, the distribution factors, k1 and k2, are required to be determined as, Determine k1 and k2 as joint at the members theall of stiffness Totaljoint at the columns theof stiffness Total=k 1211111 KKKKKKkcc++++= 2221222 KKKKKKkcc++++= To calculate the load resistance of the column cgcpAP=, its effective length is needed to be determined from k1and k2 above with the chart in Figure 5.

10 Nonlinear Integrated design and analysis software for structures Prof. SL Chan - 7 - Figure 4 Restraint coefficients in a sub-frame Figure 5 Effective length chart for non-sway framecolumn-length being Nonlinear Integrated design and analysis software for structures Prof. SL Chan - 8 - Sway-sensitive frames When 510 > , it is a sway sensitive frame. We need to use the following chart to find the effective length factor LLe to eqt. in HK steel Code (2005). k1 and k2 are calculated similarly to the above but the stiffness coefficients for beams should follow Table of Hong Kong Steel Code. Figure 6 Effective length chart for sway frame Member buckling check to Eqt. in HK Steel Code Using effective length greater than member length to find the Pc is first carried out and the moment is not required to be amplified here because we concentrate on member buckling check.