NCCI: Simple methods for second order effects in portal frames

Transcription

1 NCC: Simple metods for second order effects in portal frames NCC: Simple metods for second order effects in portal frames NCC: Simple metods for second order effects in portal frames Tis NCC presents information concerning second-order effects in portal frames. Simple metods to take second-order effects in portal frames into account are presented. Contents. General. Elastic frame analysis 3. odified first order metod, for plastic frame analysis 6 4. References Page

2 NCC: Simple metods for second order effects in portal frames NCC: Simple metods for second order effects in portal frames. General Second-order effects occur due to sway of te frame. Te sway causes eccentricity of vertical loading tat generates second-order moments in te columns. Te effects of te deformed geometry (second-order effects) sall be considered if tey increase te action effects significantly or modify significantly te structural beaviour, see EN 993--, section 5.. Wen a frame is analysed using an elastic metod, te in-plane second-order effects can be allowed for by using: a) first-order analysis, amplified sway moment metod b) first-order analysis, terative metod c) first-order analysis, wit sway-mode buckling lengt. Te amplified sway moment metod and te iterative metod (as a global approac and as a practical example) are presented witin tis NCC. Wen plastic metods are used for frame analysis, second-order effects can be considered by te use of modified first-order analysis. See Section 3. A calculation can also be done by using an applicable software performing a nd order analysis wit relevant imperfections acc. to EN section Elastic frame analysis. Amplified sway moment metod Te amplified sway moment metod is te simplest one for introducing nd order effects for elastic frame analysis, te principle is given in EN A first-order linear elastic analysis is first carried out; ten te orizontal loads H Ed (e.g. wind) and equivalent loads V Ed φ due to imperfections are amplified by a sway factor so as to ascertain for second-order effects. For portal frames wit sallow roof slopes, provided tat te axial compression in te beams or rafters is not significant and α cr 3,0 te sway factor can be calculated according to: αcr were α cr may be calculated in accordance wit EN (4) as sown in SN004. (.) Page

3 NCC: Simple metods for second order effects in portal frames NCC: Simple metods for second order effects in portal frames. terative metod (global approac) Te goal of tis metod is, tat te effects due to te presence of bot a compression force N Ed and a global rotation φ of te member are replaced by opposite sway forces tat act perpendicular to te compressed member. Te sway forces cause te same second order effects in te member (see Figure.). Were te rotation φ may be te initial sway imperfection, but can also be te sway angle under US combination (including te initial sway imperfection if necessary). Figure. Forces equivalent to nd order effects due to sway Te calculation steps of te iterative metod can be summarized as follows. ) perform te st order elastic analysis of te portal frame wit all applied loads (V + H) and initial sway imperfections (φ init V) if relevant. Results to be considered from tis analysis for te metod: compression force N Ed,i and sway angle φ i for eac compressed member. ) Determine te opposite sway forces φ i N Ed,i for eac compressed member, to be applied in te directions wic amplifies te sway angles (see figure above). 3) Perform a new st order elastic analysis of te portal frame wit all applied loads (V + H), initial sway imperfections (φ init V) if relevant, and all sway forces φ i N Ed,i applied at ends of eac compressed members. Results to be considered from tis new analysis for te metod: new values of compression force N Ed,i and sway angle φ i for eac compressed member. Ten return to ) until convergence is reaced on displacements (e.g. φ i ) or bending moments. Page 3

4 NCC: Simple metods for second order effects in portal frames NCC: Simple metods for second order effects in portal frames teration H + φ init V V N Ed, φ N Ed, φ teration H + φ init V V φ.n Ed, φ.n Ed, teration 3 H + φ init V φ ().N Ed, () φ.n Ed, φ ().N Ed, () V φ.n Ed, φ ().N Ed, () φ ().N Ed, () and so fort till convergence is reaced : φ i (n) φ i (n-). N Ed, () φ () N Ed, (3) φ (3) N Ed, () φ () N Ed, (3) Generally 3 iterations are sufficient for reacing a quite acceptable convergence. At te end of te process, te internal forces and moments, and te displacements as well, can be considered as tose obtained from a second order analysis. φ (3) Page 4

5 NCC: Simple metods for second order effects in portal frames NCC: Simple metods for second order effects in portal frames.3 terative metod (practical example) Te second-order effects are commonly referred to as being due to P-Δ effects, i.e. an axial load P applied at an eccentricity Δ (see Figure.). Δ 0 P Figure. P-Δ effects in a portal frame Te P-Δ procedure can be separated into te following steps: ) Performing a first order elastic analysis for te portal frame wit all te applied loads, including self weigt (and imperfection loads, were required) and determination of internal forces and moments ( ). ) Calculation of orizontal deflection Δ 0 (see Figure.) due to te applied loads. 3) Determination of additional internal moments due to structural deformations (according to te orizontal deflections Δ 0 ). A first approximation of te results could be = +. 4) Determination of additional orizontal deflections Δ due to Tis can be done troug a orizontal load H (in analogy to te imperfection loads). Te orizontal deflection due to tis orizontal load H as to be determined. 5) Te orizontal deflection Δ causes an additional internal moment, tat again causes an additional deflection Δ and so on. Because of te fact tat te additional deflection becomes smaller and smaller additional iterative processes can be neglected. Te calculation procedure can be simplified troug use of te geometrical series. Tus according to te procedure mentioned above, can be calculated troug: Page 5

6 NCC: Simple metods for second order effects in portal frames NCC: Simple metods for second order effects in portal frames = = = q + q + q (.) Δ wit: = = = q and because te sway stiffness of te frame is constant and terefore: = q Te last line of equation (.) is equivalent to te geometrical series. t converges to te value: = q (.3) To obtain te geometrical series te additional moments for te following steps are estimated by te first additional moment. Tus te geometrical series approac is (except for some special cases) just an approximation. Te accuracy of tis approximation can be proved troug an additional calculation step. Terefore te additional moment due to te orizontal deflections Δ needs to be calculated and te geometrical series approac gives a proved approximation: q + = + (.4) A comparison of tese results provides additional information concerning te quality of te results, were q i is a significant reference value. f q is constant te result of te calculation will not cange anymore. Te deflections accounting for second-order effects may be calculated by analogy to te internal forces and moments by te geometrical series approac: Δ Δ 0 = Δ Δ 0 3. odified first order metod, for plastic frame analysis 3. Design pilosopy n te absence of elastic-plastic second order analysis software, te design pilosopy is to derive loads tat are amplified to account for te effects of deformed geometry (second-order (.5) Page 6

7 NCC: Simple metods for second order effects in portal frames NCC: Simple metods for second order effects in portal frames effects). Application of tese amplified loads troug a first-order analysis gives bending moments, axial forces and sear forces tat include te second-order effects approximately. Te amplification is calculated by a metod tat is sometimes known as te ercant- Rankine metod. Tis provides an equivalent metod for plastic analysis to te metod for elastic frames in EN (4), For frames were te first sway buckling mode is predominant, first order elastic analysis sould be carried out wit subsequent amplification of relevant action effects (e.g. bending moments) by appropriate factors. Because, in plastic analysis, te plastic inges limit te moments resisted by te frame, te amplification is performed on te actions tat are applied to te first-order analysis instead of te action effects tat are calculated by te analysis. Te metod places frames into one of two categories: Category A: Regular, symmetric and mono-pitced frames (Section 3..) Category B: Frames tat fall outside of Category A but excluding tied portals (Section 3..) For eac of tese two categories of frame, a different load amplification factor sould be applied. Te metod as been verified [4, 5] for frames tat satisfy te following criteria:. Frames in wic 8 for any span. Frames in wic α cr 3 were is span of frame (see Figure 3.) α cr is te eigt of te lower column at eiter end of te span being considered (see Figure 3.) is te elastic critical buckling load factor (calculated eiter exactly using software or estimated from te first sway mode (see Section 3.3)) Oter frames sould be designed using second-order elastic-plastic analysis software. 3. Amplification factors 3.. Category A: Regular, symmetric and asymmetric pitced and mono-pitced frames Regular, symmetric and mono-pitced frames (Figure 3.) are eiter single-span frames or multi-span frames in wic tere is only a small variation in eigt () and span () between te different spans; variations in eigt and span of te order of 0% may be considered as being sufficiently small. n te traditional industrial application of tis approac, for suc frames first-order analysis may be used if all forces and moments are amplified for simplicity and safety by αcr, even toug tis is conservative for te column axial forces. Page 7

8 NCC: Simple metods for second order effects in portal frames NCC: Simple metods for second order effects in portal frames 3.. Category B: Frames tat fall outside of Category A and excluding tied portals For frames tat fall outside of Category A, first-order analysis may be used if all te applied, loads are amplified by α cr (a) ono-pitc (b) Single-span (c) ulti-span Note: is measured from te intersection of te rafter centerline and te column centerline, ignoring any aunc Figure 3. Examples of Category A 3.3 Estimate of α cr For frames witin te limits of Notes and of EN , α cr may be calculated from (5.) in tat clause, as sown in SN004. For frames outside te limits of Notes and, but satisfying te criteria and in Section 3. above, te following metod may be used. For eac load case, an estimate of te elastic critical buckling load factor may be obtained as follows. For frames in wic te rafters are straigt between te columns, as in Figure 3.(a): α cr,est = α cr,s, est For frames wit pitced rafters, suc as in Figure 3.(b) and Figure 3.(c): α cr,est = min ( α ) were cr,s, est;αcr,r,est α cr,s,est is te estimate of α cr for sway buckling mode (see Section 3.3.) α cr,r,est is te estimate of α cr for rafter snap-troug buckling mode (see Section 3.3.) Page 8

9 NCC: Simple metods for second order effects in portal frames NCC: Simple metods for second order effects in portal frames 3.3. Sway buckling load factor Te parameters required to calculate α cr,s,est for a portal frame are sown in Figure 3.. As can be seen, δ HEF is te lateral deflection at te top of eac column wen subjected to an arbitrary lateral load H EHF. (Te magnitude of te total lateral load is arbitrary, as it is simply used to calculate te sway stiffness H EHF /δ EHF.) Te orizontal load applied at te top of eac column sould be proportional to te vertical reaction. Tus, for an individual column: H V EHF, i US, i H = V EHF US were is te sum of all te equivalent orizontal forces at column tops (see Figure 3.(a)) is te sum of all factored vertical reactions at US calculated from first-order plastic analysis is equivalent orizontal force at top of column i (tere are two columns in a singlespan portal, tree in a two-span portal, etc.) V US,i is factored vertical reaction at US at column i, calculated from first-order plastic analysis H EHF V US H EHF,i w US N R,US H US,A V US, A H US, B VUS, B (a) Frame under load at US (b) Reactions and axial force in rafter at US δ EHF,A δ EHF,B H EHF,A H EHF,B (c) Horizontal deflection at top of columns Figure 3. Diagram sowing parameters required to estimate α cr An estimate of α cr can ten be obtained from N αcr,s,est = 0, 8 N R, US R,cr max i V US,i H δ EHF,i EHF,i min Page 9

10 NCC: Simple metods for second order effects in portal frames NCC: Simple metods for second order effects in portal frames were N N R,US R,cr max is maximum ratio in any rafter N R,US is axial force in rafter (see Figure 3.(b)) N π Er R,cr= is Euler load of te rafter on full span (assumed pinned) r δ EHF,i is in-plane second moment of area of rafter is orizontal deflection of column top (see Figure 3.(c)) i V US, i H δ EHF, i EHF, i min is te minimum value for columns to n (n = te number of columns) 3.3. Rafter snap-troug buckling load factor For frames wit rafter slopes not steeper tan : (6 ), α cr,r may be taken as: α D 55,7 = ( 4 + ) + 75 ( tan θ ) c r cr, r,est Ω f r yr Tis as to be cecked because it is possible to design portals of 3 spans or more wit stiff outer bays tat provide orizontal support to te rafters of te inner spans. Ten te rafters of te inner spans can act as arces wit te orizontal reaction provided by te outer bays. Were tis arcing action works, te rafters will support more vertical load tan if tey were acting only as beams. Tis ceck is used to ensure tat te rafters are not so flexible tat tey snap troug. But were Ω, α cr,r = were D c r is cross-section dept of rafter is span of bay is mean eigt of column from base to eaves or valley is in-plane second moment of area of te column (taken as zero if te column is not rigidly connected to te rafter, or if te rafter is supported on a valley beam) is in-plane second moment of area of te rafter f yr is nominal yield strengt of te rafters in N/mm r Page 0

11 NCC: Simple metods for second order effects in portal frames NCC: Simple metods for second order effects in portal frames θ r r is roof slope if roof is symmetrical, or else θ r = tan - ( r /) is eigt of apex of roof above a straigt line between te tops of columns Ω is arcing ratio, given by Ω = W r /W 0 W 0 W r is value of W r for plastic failure of rafters as a fixed ended beam of span is total factored vertical load on rafters of bay f te two columns or two rafters of a bay differ, te mean value of c sould be used. 4. References Te rules in tis NCC are based on: EN 993--: Eurocode 3: Design of Steel Structures Part -: General rules and rules for Buildings () Horne,.R. Safeguards against frame instability in te plastic design of single-storey pitced roof frames, paper presented at te Conference on te beaviour of slender structures, City University, ondon, 977 (3) Davies, J.. Te stability of multi-bay portal frames, Te Structural Engineer, Vol 69 No., June 99 (4) BS paper B/55/3/04_5505, Final report on Pi project 38/9/4 cc796, SC 004 (5) im, J.B.P., King, C.., Ratbone, A.J., Davies, J.. and Edmondson, V.: Eurocode 3 and te in-plane stability of portal frames, Te Structural Engineer, 83, No., 005, p43. Page

12 NCC: Simple metods for second order effects in portal frames NCC: Simple metods for second order effects in portal frames Quality Record RESOURCE TTE NCC: Simple metods for second order effects in portal frames Reference(s) ORGNA DOCUENT Name Company Date Created by attias Oppe RWTH Aacen Tecnical content cecked by Cristian üller RWTH Aacen Editorial content cecked by Tecnical content endorsed by te following STEE Partners:. UK G W Owens SC 30/3/06. France A Bureau CTC 8/3/06 3. Sweden B Uppfeldt SB 3/3/06 4. Germany C üller RWTH 0/3/06 5. Spain J A Cica abein 8/3/06 Resource approved by Tecnical Coordinator TRANSATED DOCUENT Tis Translation made and cecked by: Translated resource approved by: G W Owens SC 08/7/06 Page