Finite Strip Analysis 1 local distortional later-torsional length of a half sine wave buckling multiplier (stress, load, or moment) Finite Strip Analysis and the Beginnings of the Direct Strength Method Toronto, July 2000 AISI Committee on Specifications

Finite Strip Analysis 2 Overview Introduction Background A simple verification problem Analyze a typical section Interpreting results (k, f cr, P cr, M cr ) Direct Strength Prediction Improve a typical section Individual analysis - “do it yourself”

Finite Strip Analysis 3 Introduction Understanding elastic buckling (stress, modes, etc.) is fundamental to understanding the behavior and design of thin-walled structures. Thorough treatment of plate buckling separates design of cold-formed steel structures from typical structures. Hand solutions for plate buckling have taken us a great distance but more modern approaches may be utilized now. Finite strip analysis is one efficient method for calculating elastic buckling behavior.

Finite Strip Analysis 4 Introduction No new theory: Finite strip analysis uses the same “thin plate” theory employed in classical plate buckling solutions (e.g., k = 4) already in current use. Organized: The nuts and bolts of the analysis is organized in a manner similar to the stiffness method for frames and thus familiar to a growing group of engineers. Efficient: Single solutions and parameter studies can be performed on PCs Free: Source code and programs for the finite strip analysis is free

Finite Strip Analysis 5 Introduction

Finite Strip Analysis 6 What has to be defined? nodes give node number give coordinates indicate if any additional support exist along the longitudinal edge give applied stress on node elements give element number give nodes that form the strip give thickness of the strip property give E, G, and v lengths give all the lengths that elastic buckling should be examined at

Finite Strip Analysis 7 Finite Strip Software CU-FSM –Matlab based full graphical version –DOS engine only (execufsm.exe) Helen Chen has written a Windows front end “procefsm.exe” which uses the CU-FSM DOS engine (Thanks Helen!) Other programs with finite strip capability –THINWALL from University of Sydney –CFS available from Bob Glauz

Finite Strip Analysis 8 Overview Introduction Background A simple verification problem Analyze a typical section Interpreting results (k, f cr, P cr, M cr ) Direct Strength Prediction Improve a typical section Individual analysis - “do it yourself”

Finite Strip Analysis 9 Theoretical Background Elastic buckling in matrix form Initial elastic stiffness [K] –specialized shape functions Geometric stiffness [K g ] Forming the solution Elastic buckling solution

Finite Strip Analysis 10 Elastic Buckling (Matrix form) standard initial elastic stiffness form consider effect of stress on stiffness consider linear multiples of constant stress (f 1 ) eigenvalue problem gives solution

Finite Strip Analysis 11 What is [K]? [K] is the initial elastic stiffness

Finite Strip Analysis 12 Shape Functions these shape functions are also known as [N]

Finite Strip Analysis 13 Strain-Displacement and [K] Plane stress K uv comes from these strain- displacement relations Bending K w  comes from these strain- displacement relations

Finite Strip Analysis 14 Plane Stress Initial Stiffness

Finite Strip Analysis 15 Bending Initial Stiffness

Finite Strip Analysis 16 [K g ] is the stress dependent geometric stiffness, (compressive stresses erode stiffness) The terms may be derived through –consideration of the total potential energy due to in-plane forces, or –equivalently consider equilibrium in the deformed geometry, (i.e., consider the moments that develop in the deformed geometry due to forces which are in-plane in the undeformed geometry), also –one can consider K g as a direct manifestation of higher order strain terms. What is [K g ]?

Finite Strip Analysis 17 Developing [K g ] where {d} is the nodal displacements the same shape functions as before are used, therefore [G] is determined through partial differentiation of [N].

Finite Strip Analysis 18 Geometric Stiffness

Finite Strip Analysis 19 The stiffness matrix for the member is formed by summing the element stiffness matrices (this is done in exactly the same manner as the stiffness solution for frame analysis) –generate stiffness matrices in local coordinates –transform to global coordinates ([k] n =  T k’  –add contribution of each strip to global stiffness, symbolically: Forming the Complete Solution

Finite Strip Analysis 20 Eigenvalue Solution The solution yields – the multiplier which gives the buckling stress –{d} the buckling mode shapes The solution is performed for all lengths of interest to develop a complete picture of the elastic buckling behavior

Finite Strip Analysis 21 Overview Introduction Background A simple verification problem Analyze a typical section Interpreting results (k, f cr, P cr, M cr ) Direct Strength Prediction Improve a typical section Individual analysis - “do it yourself”

Finite Strip Analysis 22 A Simple Verification Problem Find the elastic buckling stress of a simply supported plate using finite strip analysis. SS width = 6 in. (152 mm) thickness = 0.06 in. (1.52 mm) E = ksi ( MPa) v = 0.3 Hand Solution: = ksi

Finite Strip Analysis 23 Finite Strip Analysis Notes (analysis of SS plate) double click procefsm.exe enter elastic properties into the box enter the node number, x coordinate, z coordinate, and applied stress –1,0.0,0.0,1.0 which means node 1 at 0,0 with a stress of 1.0 –2,6.0,0.0,1.0 which means node 2 at 6,0 with a stress of 1.0 enter the element number, starting node number, ending node number, number of strips between the nodes (at least 2 typically 4 or more) and the thickness –1,1,2,2,0.06 which means element 1 goes from node 1 to 2, put 2 strips in there and t=0.06 select plot cross-section to see the plate enter a member length (say 6) and number of different half wavelengths (say 10) do File - Save As - plate.inp now select view/revise raw data file

Finite Strip Analysis 24 Finite Strip Analysis Notes (analysis of SS plate) continued View/Revise Raw Data File shows the actual text file that is used by the finite strip analysis program. All detailed modifications must be made here before completing the analysis. The format of the file is summarized as: The x, y, z,  degrees of freedom are shown in the strip to the right. Supported degrees of freedom are supported along the entire length (edge) of the strip. The ends of the strip are simply supported (due to the selected shape functions). Set a DOF variable to 0 to support that DOF along the edge

Finite Strip Analysis 25 Finite Strip Analysis Notes (analysis of SS plate) continued First modify degrees of freedom so the plate is simply supported along the long edges (the loaded edges are always simply supported). Put a pin along the left edge and a roller along the right edge. – becomes – stays the same – becomes Now delete the last line and replace it with the specific lengths that you want to use, say for instance “ ” Now change the #lengths listed in the thrid column of the first line of the file to match the selected number, in this example we have 6 different lengths Now select Save for Finite Strip Analysis and save under the name plate.txt Select Analysis - Open Then type ‘plate.txt’ for the input file and ‘plate.out’ for the output file Select Output - then plate.out - and open Select Plot curve and plot mode, the result of this example is ksi (vs ksi hand solution - repeat using 4 strips - then result is ksi)

Finite Strip Analysis 26 Overview Introduction Background A simple verification problem Analyze a typical section Interpreting results (k, f cr, P cr, M cr ) Direct Strength Prediction Improve a typical section Individual analysis - “do it yourself”

Finite Strip Analysis 27 Analyze a typical section Pure bending of a C –Quickie hand analysis –Finite strip analysis using procefsm.exe –Discussion Pure compression of a C –Quickie hand analysis –Finite strip analysis –Discussion Comparisons and Further Discussion

Finite Strip Analysis 28 Strong Axis Bending of a C Approximate the buckling stress for pure bending comp. lip comp. flange web

Finite Strip Analysis 29 Strong Axis Bending of a C Approximate the buckling stress for pure bending lip flange web * this k value would be fine-tuned by AISI B4.2

Finite Strip Analysis 30 Finite Strip Analysis Steps (strong axis bending of a C) Double click on procefsm.exe Select File - Open - C.inp Plot Cross Section View/Revise Raw Data File Go to bottom of text file and change lengths to “ ” Go to top of file (1st line 3rd entry) change the number of lengths from 20 to 19 Save for finite strip analysis as C.txt Select Analysis - Open - execufsm Enter in DOS window ‘C.txt’ return then ‘C.out’ Select output - ‘C.out’ - open Check 2D, Check undef, push plot mode button Push plot curve, set half wave-length to 5 rehit plot mode, set to 30 and plot local buckling at 40ksi (~5 in. 1/2wvlngth), dist buckling at 52ksi (~30 in. 1/2wvlngth)

Finite Strip Analysis 31 local buckling half-wavelength buckling multiplier local distortional

Finite Strip Analysis 32 distortional

Finite Strip Analysis 33 Discussion Finite strip analysis identifies three distinct modes: local, distortional, lateral-torsional The lowest multiplier for “each mode” is of interest. The mode will “repeat itself” at this half-wavelength in longer members Higher multipliers of the same mode are not of interest. The meaning of the “half-wavelength” can be readily understood from the 3D plot. For example:

Finite Strip Analysis 34 Discussion How do I tell different modes? –wavelength: local buckling should occur at wavelengths near or below the width of the elements, longer wavelengths indicate a different mode of behavior –mode shape: in local buckling, nodes at fold lines should rotate only, if they are translating then the local mode is breaking down What if more minimums occur? –as you add stiffeners and other details more minima may occur, every fold line in the plate adds the possibility of new modes. Definitions of local and distortional buckling are not as well defined in these situations. Use wavelength of the mode to help you decide.

Finite Strip Analysis 35 Compression of a C Approximate the buckling stress for pure compression lip flange web * this k value would be fine-tuned by AISI B4.2

Finite Strip Analysis 36 Finite Strip Analysis Steps (compression of a C) Double click on procefsm.exe Select File - Open - C.inp Change all applied stress to compression +1.0 Plot Cross Section View/Revise Raw Data File Go to bottom of text file and change lengths to “ ” Go to top of file (1st line 3rd entry) change the number of lengths from 20 to 14 Save for finite strip analysis as C.txt Select Analysis - Open - execufsm Enter in DOS window ‘C.txt’ return then ‘C.out’ Select output - ‘C.out’ - open Check 2D, Check undef, push plot mode button Push plot curve, set half wave-length to 6 and rehit plot mode local buckling at 7.5ksi (pure compression)

Finite Strip Analysis 37 Finite Strip Analysis Compression of a C f cr local = 7.5 ksi f cr distortional ~ 20 ksi (this value may be fine tuned by selecting more lengths and re-analyzing) f cr overall at 80 in. = 29 ksi

Finite Strip Analysis 38 Comparision of Elastic Results Hand Analysis –compression lip=56.6 flange=62.4 web=5.2 ksi –bending lip=56.6 flange=62.4 web=31.3 ksi Finite strip analysis –compression local=7.5 distortional~20ksi –bending local=40 distortional=52 ksi

Finite Strip Analysis 39 Comparision of Results for Buckling Stress of a C Hand Analysis –Compression Lip = 56.6 ksi Flange = 62.4 Web = 5.2 –Bending Lip = 56.6 Flange = 62.4 Web = 31.3 Finite strip analysis –Compression Local = 7.5 ksi Distortional ~ 20 –Bending Local = 40 Distortional = 52

Finite Strip Analysis 40 Overview Introduction Background A simple verification problem Analyze a typical section Interpreting results (k, f cr, P cr, M cr ) Direct Strength Prediction Improve a typical section Individual analysis - “do it yourself”

Finite Strip Analysis 41 Converting the results If f 1 is the applied stress in the finite strip analysis and the multiplier that results from the elastic buckling then f cr = f 1 is known. How do we get k? P cr ? M cr ? k is found via where: b = element width of interest (flange, web, lip etc.) P cr = A g f cr M cr = S g f cr (as long as f 1 is the extreme fiber stress of interest)

Finite Strip Analysis 42 Converting the results - Example For the C in pure compression what does the finite strip analysis yield for the local buckling k of the web? For the flange? Solutions are different when you recognize the interaction!

Finite Strip Analysis 43 Converting the results - Example For the C in compression what is the elastic critical local buckling load? distortional buckling load? overall? –From CU-FSM or hand calculation get the section properties –(P cr ) local = A g f cr = 0.885in 2 *7.5ksi = 6.6 kips –(P cr ) distortional = A g f cr = 0.885in 2 *20 ksi = 17.7 kips –(P cr ) overall at 80 in. = A g f cr = 0.885in 2 *29 ksi = 25.7 kips

Finite Strip Analysis 44 Converting the results - Example For the C in bending what is the elastic critical local buckling moment? distortional buckling moment? –From CU-FSM or hand calculation get the section properties –(M cr ) local = S g f cr = 2.256in 3 *40 ksi = 90 in-kips –(M cr ) distortional = S g f cr = 2.256in 3 *52 ksi = 117 in-kips

Finite Strip Analysis 45 How can I use this information? Known –local buckling load (P cr ) local from finite strip analysis –distortional buckling load (P cr ) distortional from finite strip analysis –overall or Euler buckling load (P cr ) Euler may be flexural, torsional, or flexural-torsional in the special case of K x =K y =K t then we may use finite strip analysis results, in other cases hand calculations for overall buckling of a column are used –yield load (P y ) from hand calculation Unknown –design capacity P n Methodology: Direct Strength Prediction

Finite Strip Analysis 46 Overview Introduction Background A simple verification problem Analyze a typical section Interpreting results (k, f cr, P cr, M cr ) Direct Strength Prediction Improve a typical section Individual analysis - “do it yourself”

Finite Strip Analysis 47 Direct Strength Prediction The idea behind Direct Strength prediction is that with (P cr ) local, (P cr ) distortional and (P cr ) Euler known an engineer should be able to calculate the capacity reliably and directly without effective width. Current work suggests the following approach for columns –Find the inelastic long column buckling load (P ne ) using the AISC column curves already in the AISI Specification –Check for local buckling using new curve (less conservative than Winter) on the entire member with the max load limited to P ne –Check for distortional buckling using Hancock’s curve (more conservative than Winter) with the max load limited to P ne –Design strength is the minimum

Finite Strip Analysis 48 Direct Strength for Columns

Finite Strip Analysis 49 Direct Strength for Columns (cont.) *these calculations include long column interaction, to ignore this interaction replace P ne with P y

Finite Strip Analysis 50 Column Example Consider the lipped C we have been analyzing. Assume 50 ksi yield, L=80 in. and K x =K y =K t =1.0 From finite strip analysis we know: –P crl = 6.6 kips –P crd = 17.7 kips –P cre = 25.7 kips –also P y = A g f y = 0.885*50 = kips

Finite Strip Analysis 51 Column Example (cont.) P crl = 6.6 kips P crd = 17.7 kips P cre = 25.7 kips P y = kips

Finite Strip Analysis 52 Column Example (cont.) *these calculations include long column interaction, to ignore this interaction replace P ne with P y P crl = 6.6 kips P crd = 17.7 kips P cre = 25.7 kips P y = kips

Finite Strip Analysis 53 Column Example (cont.) P crl = 6.6 kips P crd = 17.7 kips P cre = 25.7 kips P y = kips

Finite Strip Analysis 54 Column Example (cont.) *these calculations include long column interaction, to ignore this interaction replace P ne with P y P crl = 6.6 kips P crd = 17.7 kips P cre = 25.7 kips P y = kips P ne = kips

Finite Strip Analysis 55 Direct Strength for Beams Direct Strength prediction for beams follows the same format as for columns. –Find the inelastic lateral buckling load (M ne ) using the strength curves already in the AISI Specification –Check for local buckling using new curve (less conservative than Winter) on the entire member with the max moment limited to M ne –Check for distortional buckling using Hancock’s curve (more conservative than Winter) with the max moment limited to M ne –Design strength is the minimum Note, all the beams studied at this time have been laterally braced - therefore the interaction between local and lateral buckling and distortional and lateral buckling has not been examined. For now, it is conservatively assumed that these modes can interact in the same manner as completed for columns. (This is what we do now when we use the effective section modulus)

Finite Strip Analysis 56 Direct Strength for Beams

Finite Strip Analysis 57 Direct Strength for Beams (cont.) *these calculations include long column interaction, to ignore this interaction replace P ne with P y

Finite Strip Analysis 58 Beam Example Consider the lipped C we have been analyzing. Assume 50 ksi yield, assume the member is laterally braced, but distortional buckling is still free to form and thus a concern. Find the nominal capacity. From finite strip analysis we know: –M crl = 90 in-kips –M crd = 117 in-kips –M cre = braced –also M y = S g f y = 2.256*50 = 113 in-kips

Finite Strip Analysis 59 Beam Example (cont.) M crl = 90 in-kips M crd = 117 in-kips M cre = braced M y = 113 in-kips M ne = M y since “braced” = 113 in-kips M nl = 89 in-kips

Finite Strip Analysis 60 Beam Example (cont.) *these calculations include long column interaction, to ignore this interaction replace P ne with P y M crl = 90 in-kips M crd = 117 in-kips M cre = braced M y = 113 in-kips M nd = 86 in-kips M n = 86 in-kips, distortional controls even though elastic critical is 30% higher

Finite Strip Analysis 61 Direct Strength Verification Existing experimental data on laterally braced beams and centrally loaded columns (unbraced) have been collected and evaluated –Laterally Braced Beams: Experimental data includes lipped and unlipped C’s, Z’s, rectangular and trapezoidal decks w/ and w/o int. stiffener(s) in the web and flange for a total of 574 members –Columns: Experimental data includes lipped C's, Z's, lipped C's with int. web stiffeners, racks, racks with compound lips for a total of 227 members Finite strip analysis was conducted on each member, then combined with the experimental results to compare vs. the strength curves suggested for Direct Strength prediction

Finite Strip Analysis 62 Laterally Braced Beams

Finite Strip Analysis 63 Columns

Finite Strip Analysis 64 Direct Strength Verification (cont.) As the data shows, trends are clear, but large scatter can exist for a particular member. Using separate strength curves given herein for local and distortional limits –Laterally braced beams have test to predicted ratio of 1.14 for all data, 1.05 for local limits and 1.16 for distortional limits –Columns have test to predicted ratio of 1.01 for all data Preliminary calibration using the strength curves suggested herein appear to support the use of traditional  =0.9 for beams and  =0.85 for columns (presumably ASD factors would also remain unchanged)

Finite Strip Analysis 65 Deflection Calculations Example strength & deflection calc. completed at f y Example strength & deflection calc. completed at f a It is anticipated that degradation of gross properties (i.e, A g I g ) due to local/distortional/overall buckling may be approximated in the same manner as the degradation in the strength, e.g.,

Finite Strip Analysis 66 Limitations of Direct Strength Conservative solution when one element is extraordinarily slender (f cr approaches zero and the strength with it) Interaction of local-distortional-overall buckling not studied thoroughly for beams and deserves further study for columns Calibration of methods incomplete Impact of proposed changes incomplete Integration into existing design methods incomplete...

Finite Strip Analysis 67 Overview Introduction Background A simple verification problem Analyze a typical section Interpreting results (k, f cr, P cr, M cr ) Direct Strength Prediction Improve a typical section Individual analysis - “do it yourself”

Finite Strip Analysis 68 Improve a Section Examine the results for the C section in compression that we previously evaluated Suggest alternative design Implement the design change Evaluate the finite strip results Recalculate the strength

Finite Strip Analysis 69 Previous Results

Finite Strip Analysis 70 Alternative Design Suggestions Intermediate stiffener in the web Multiple intermediate stiffeners in the web Decrease width of one flange in order to incorporate a small fold in the web so that sections could be nested together others... –________________________

Finite Strip Analysis 71 Suggested Alternative OldNew (2.44,0.84) (2.44,0.0) (0.50,0.00) (0.50,1.50) (0.00,2.00) (0.00,8.44) (2.44,8.44) (2.44,7.60)

Finite Strip Analysis 72

Finite Strip Analysis 73 Convert stress to load For the C in compression what is the elastic critical local buckling load? distortional buckling load? overall? –From CU-FSM or hand calculation get the section properties –(P cr ) local = A g f cr = 0.868in 2 *13.2ksi = 11.4 kips –(P cr ) distortional = A g f cr = 0.868in 2 *28 ksi = 24.3 kips –(P cr ) overall at 80 in. = A g f cr = 0.868in 2 *26 ksi = 22.6 kips –P y for 50 ksi yield = A g f y = 0.868in 2 *50 ksi = 43.4 kips

Finite Strip Analysis 74 Direct Strength P crl = 11.4 kips P crd = 24.3 kips P cre = 22.6 kips P y = 43.4 kips

Finite Strip Analysis 75 Direct Strength (cont.) *these calculations include long column interaction, to ignore this interaction replace P ne with P y P crl = 11.4 kips P crd = 24.3 kips P cre = 22.6 kips P y = 43.4 kips P ne = 19.4 kips

Finite Strip Analysis 76 Comparison “typical” C P crl = 6.6 kips P crd = 17.7 kips P cre = 25.7 kips P y = 44.2 kips P ne = 21.6 kips P nl = 12.2 kips P nd = 14.9 kips P n = 12.2 kips “nestable” C P crl = 11.4 kips P crd = 24.3 kips P cre = 22.6 kips P y = 43.4 kips P ne = 19.4 kips P nl = 13.7 kips P nd = 15.8 kips P n = 13.7 kips

Finite Strip Analysis 77 Changing boundary conditions –eliminate a mode you are not interested in by temporarily supporting a DOF –use symmetry and anti-symmetry to examine different modes and behavior –bound solutions by looking at fix vs. free conditions Extras

Finite Strip Analysis 78 Extras Adding an elastic support –you want to add elastic springs to the cross-section to model external (continuous) support. This can be done only indirectly by adding an unloaded strip to your model –The elastic stiffness k x, k y, k z and k  of a single strip is given previously. Note selection of strip width, t, and boundary conditions will generate all 4 k’s currently no method is available for adding springs for only one DOF

Finite Strip Analysis 79 Extras defining a single unloaded strip fixed at the 1 edge and attached at the 2 edge will add the above elastic stiffness to the solution wherever the 2 edge is attached to the member. Remember, this stiffness is along the length of the member (the length of the strip) in order to make an unloaded strip you will have to create a very short dummy element near the 2 edge because loading is defined at the nodes not the elements

Finite Strip Analysis 80 Extras Assemblage –2 C’s linked together vs. a single C section results are in bending. CU-FSM used for analysis

Finite Strip Analysis 81 Extras Beam-columns –load with the expected stress distribution instead of doing separate beam and column analysis –Make your own elastic buckling interaction curve for a particular shape

Finite Strip Analysis 82 Extras Matlab version (free, but you need Matlab) –Robust graphical interface –Calculates section properties –Add P and M, or combinations thereof directly instead of adding stress –Compare two analyses directly –Perform parametric studies –Completely free and available source, modify as you wish –Write your own programs and software that call any of the finite strip routines

Finite Strip Analysis 83 Limitations of Finite Strip Elastic only: elastic buckling stress is useful, but it is –not an allowable stress, –not necessarily the stress at which buckling “ensues”, –not a conservative design ultimate stress. Optimum design –Do not design so that the elastic buckling stress of all modes is at or near the same stress - this invariably leads to coupled instabilities and should be avoided - note that different strength curves are used for different modes. Identification of minima –Certain members and certain loadings blur the distinctions between modes. Exercise judgment and remain conservative when in doubt, i.e., the distortional buckling strength curve is more conservative than the local buckling curve.

Finite Strip Analysis 84 Limitations of Finite Strip Mixed wavelength modes –Finite strip analysis can not identify situations when the wavelength in the flange differs from that in the web. Finite element analysis indicates this can happen in certain circumstances. Analysis to date shows that the finite strip formulation does not lead to undo errors, but further work needed. Point-wise bracing, punch-outs... –Any item that discretely varies along the length must be “smeared” in the current approach. The benefit of individual braces is difficult to quantify. This is a practical limit not a theoretical limit of the current approach. For now, engineering judgment is required when evaluating bracing, punch-outs etc.. Shear interaction

Finite Strip Analysis 85 Overview Introduction Background A simple verification problem Analyze a typical section Interpreting results (k, f cr, P cr, M cr ) Direct Strength Prediction Improve a typical section Individual analysis - “do it yourself”

Finite Strip Analysis 86 Geometry of Examples C_nolip C C_deep C_rack L L_lip H o Z o Z_deep plate 6 drawings not to scale, all dim. in inches

Finite Strip Analysis 87 END

Finite Strip Analysis 88 Classical Derivation for k Classical methods tend to use an energy approach (following Timoshenko) –assume a series for deflected shape (must match boundary conditions) –determine the internal strain energy (independent of loading) –determine the external work (dependent on loading) –form the total potential energy –take the variation of the total potential energy with respect to the amplitude coefficients of the series and set to zero –determine the desired number of terms to be used in the series –calculate the buckling stress and k

Finite Strip Analysis 89 Classical derivation for k (cont.) consider solution for a simply supported plate with a stress gradient displaced shape internal energy external work total potential energy variation and solution