Direct Strength Design for Cold-Formed Steel Members with Perforations Progress Report 1 C. Moen and B.W. Schafer AISI-COS Meeting February 21, 2006

Outline Objective and challenges Project overview FE stability studies –fundamentals, plates and members with holes Modal identification and c FSM Existing experimental column data –elastic buckling studies: hole effect, boundary conditions –strength prediction by preliminary DSM stub columns, long columns Conclusions

Objective Development of a general design method for cold-formed steel members with perforations.

Perforation patterns in CFS

Direct strength prediction P n = f (P y, P cre, P crd, P cr )? Input –Squash load, P y –Euler buckling load, P cre –Distortional buckling load, P crd –Local buckling load, P cr Output –Strength, P n

Direct strength for members with holes P n = f (P y, P cre, P crd, P cr )? Does f stay the same? Gross or net, or some combination? Explicitly model hole(s)? Accuracy? Efficiency? Identification? Just these modes?

DSM for columns without holes 267 columns,  = 2.5,  = 0.84

Outline Objective and challenges Project overview FE stability studies –fundamentals, plates and members with holes Modal identification and c FSM Existing experimental column data –elastic buckling studies: hole effect, boundary conditions –strength prediction by preliminary DSM stub columns, long columns Conclusions

Project Update Originally proposed as a three year project. Year 1 funding was provided, we are currently ½ way through year 1. Project years 1: Benefiting from existing data 2: Identifying modes and extending data 3: Experimental validation & software

Project year 1 Focus has primarily been on compression members with isolated holes in the first 6 mos.

Project year 2

Project year 3

Outline Objective and challenges Project overview FE stability studies –fundamentals, plates and members with holes Modal identification and c FSM Existing experimental column data –elastic buckling studies: hole effect, boundary conditions –strength prediction by preliminary DSM stub columns, long columns Conclusions

ABAQUS Element Accuracy Motivation –For the student to learn and understand sensitivity of elastic (eigen) stability response to FE shell element solutions –In particular, to explore FE sensitivity in members with holes –To take the first tentative steps towards providing practicing engineers real guidance when using high level FE software for elastic stability solutions of unusual situations

Stiffened element in uniform compression (benchmark: stiffened plate in compression)

Linear vs. quadratic elements S4/S4R S9R5 models compared at equal numbers of DOF

Number of elements along the length 2.5 elements per half-wave shown

S9R5 sensitivity to modeling corners 1 element in corner 3 elements in corner Use of quadratic shell elements that can have an initially curved geometry shown to be highly beneficial/accurate here.

FE vs FSM comparisons SSMA 362S in pure compression FE = ABAQUS FSM = CUFSM model length = half-wavelength in ABAQUS (ABAQUS boundary conditions = “pinned ends”)

Exploring local buckling difference number of local buckling half-waves in ABAQUS model (physical length of ABAQUS model is increased)

Outline Objective and challenges Project overview FE stability studies –fundamentals, plates and members with holes Modal identification and c FSM Existing experimental column data –elastic buckling studies: hole effect, boundary conditions –strength prediction by preliminary DSM stub columns, long columns Conclusions

Mesh sensitivity around holes 4 layers of elements shown SS

Mesh sensitivity around holes

Do holes decrease local buckling this much??

The square plate problem Much of the fundamental research on plates with holes has been conducted on square plates. The idea being that one local buckle evenly fits into a square plate. So, examining the impact of the hole in a square plate examines the impact in a localized fashion? = ?

l =4w w =92.075mm Local buckling in an a/b = 4 plate Conclusion? Lots of wonderful theoretical studies are not really relevant...

“SSMA” hole and varied plate width 4w

Local plate stability with a hole Observed loss of local stability much less than in a square plate. We will revisit this basic plot for member local buckling as well.

Outline Objective and challenges Project overview FE stability studies –fundamentals, plates and members with holes Modal identification and c FSM Existing experimental column data –elastic buckling studies: hole effect, boundary conditions –strength prediction by preliminary DSM stub columns, long columns Conclusions

SSMAS w/ hole Member Study L = 1220mm = 48 in.

CUFSM elastic buckling (no hole) half-wavelength (mm) P cr /P y

ABAQUS model Classical FSM style boundary conditions are employed, i.e., pinned free-to-warp end conditions.

Local (L) buckling P cr no hole = 0.28P y, with hole = 0.28P y

Distortional (D) buckling P crd no hole = 0.64P y, with hole = 0.65P y

Distortional (DH) buckling around the hole P crd no hole = 0.64P y, with hole = 0.307P y

Antisymm. dist. buckling (DH2) at the hole P crd no hole = 0.64P y, with hole = 0.514P y

Global flexural torsional (GFT) buckling P crd no hole = 0.61P y, with hole = 0.61P y

Impact of hole location on buckling values

Hole location impact on GFT GFT mode with hole at midspan Mixed GFT-L-D mode observed with hole near end. BC influence near the ends, under further study..

SSMAS w/ hole Member Study 2 L = 1220mm = 48 in. b

Hole size and member buckling modes

Observed buckling modes LDH D GFT

“DH” mode 0.62P y 0.38P y 0.35P y 0.31P y 0.30P y

Outline Objective and challenges Project overview FE stability studies –fundamentals, plates and members with holes Modal identification and c FSM Existing experimental column data –elastic buckling studies: hole effect, boundary conditions –strength prediction by preliminary DSM stub columns, long columns Conclusions

Modal identification Mixing of modes (a) complicates the engineers/analysts job (b) may point to post-buckling complications We need an unambiguous way to identify the buckling modes A significant future goal of this research is the extension of newly developed modal identification tools to members with holes

We can’t effectively use FEM We “need” FEM methods to solve the type of general stability problems people want to solve today –tool of first choice –general boundary conditions –handles changes along the length, e.g., holes in the section 30 nodes in a cross-section 100 nodes along the length 5 DOF elements 15,000 DOF 15,000 buckling modes, oy! Modal identification in FEM is a disaster

Special purpose finite strip can fail too

c FSM c FSM = constrained finite strip method The “constraints” restrict the FSM model to deformations within a selected mode – for instance, only distortional buckling c FSM adopts the basic definitions of buckling modes developed by GBT researchers My research group has been developing this method as a means to provide modal decomposition and modal identification Extension of modal identification to general purpose FE results has a potentially huge impact on our problem

modal decomposition

modal identification

at the heart of cFSM are the mechanics-based modal buckling definitions

Global modes are those deformation patterns that satisfy all three criteria. now let us examine these three criteria... #1 #2 #3

#1 membrane strains:  xy = 0, membrane shear strains are zero,  x = 0, membrane transverse strains are zero, and  v = f(x), long. displacements are linear in x within an element. #1 #2 #3

#2 warping:  y  0, longitudinal membrane strains/displacements are non-zero along the length. #1 #2 #3

#3 transverse flexure:  y = 0, no flexure in the transverse direction. (cross-section remains rigid!) #1 #2 #3

Distortional modes are those deformation patterns that satisfy criteria #1 and #2, but do not satisfy criterion #3 (i.e., transverse flexure occurs). #1 #2 #3

Local modes are those deformation patterns that satisfy criterion #1, but do not satisfy criterion #2 (i.e., no longitudinal warping occurs) while criterion #3 is irrelevant. #1 #2 #3

Other modes (membrane modes ) do not satisfy criterion #1. Note, other modes typically do not exist in GBT, but must exist in FSM or FEM due to the inclusion of DOF for the membrane. #1 #2 #3

#1 #2 #3 G D L O all deformations

lipped channel column example 80mm 200mm 15mm t=2mm E=210000MPa, =0.3 FSM DOF: 4 per node, total of 24 u v w  G:4 D:2 L:8 O:10 all deformations

G and D deformation modes

L deformation modes

O deformation modes

Modal decomposition Begin with our standard stability (eigen) problem Now introduce a set of constraints consistent with a desired modal definition, this is embodied in R Pre-multiply by R T and we create a new, reduced stability problem that is in a space with restricted degree of freedom, if we choose R appropriately we can reduce down to as little as one “modal” DOF

modal decomposition

modal identification

Years 2 and 3 of this project... extending modal identification to FE is one of the keys to creating a general method that all can agree upon. We need to remove the ambiguity in visual modal identification (a small problem for members without holes, but a much more important one for members with holes!)

Outline Objective and challenges Project overview FE stability studies –fundamentals, plates and members with holes Modal identification and c FSM Existing experimental column data –elastic buckling studies: hole effect, boundary conditions –strength prediction by preliminary DSM stub columns, long columns Conclusions

Study of experimentally tested members Collection of experimental column data Estimation of elastic buckling P cr, P crd, P cre using FE to capture influence of hole and reflect test boundary conditions Examination of initial DSM strength predictions for tested sections

Geometry of available specimens Stub columns Total of 51 specimens Boundary conditions... Remember the square plate lesson in local buckling Distortional restrained

Geometry of available specimens Long columns Total of 15 specimens Member geometry not varied significantly, but hole size range is fairly large

Length of available tested specimens stub columns more testing needed here to understand what is going on with holes and distortional buckling..

Histogram of normalized hole size enough specimens with big holes? ?

Outline Objective and challenges Project overview FE stability studies –fundamentals, plates and members with holes Modal identification and c FSM Existing experimental column data –elastic buckling studies: hole effect, boundary conditions –strength prediction by preliminary DSM stub columns, long columns Conclusions

Elastic local buckling in stub columns increasing hole size P cr,no hole = pin free- to-warp boundary conditions

Boundary conditions of stub test matter The bigger the hole and the shorter the specimen the more important the BC Local buckling

Boundary condition effect: local buckling

Effect of isolated holes on local buckling in long columns? remember...

Distortional buckling (effect of holes) (stub column data) identification of D modes can be challenging, minimum D mode = “DH” mode

Distortional buckling (effect of holes) (long column data) identification of D modes can be challenging, minimum D mode = “DH” mode

Stub column testing restrains distortional buckling

Ends free to warp vs. fixed Remember! D modes are defined by the warping (longitudinal deformations) warping distribution defined by cFSM

Global buckling in long columns (effect of holes) Effect of holes on global buckling modes greater than anticipated, still under study...

Outline Objective and challenges Project overview FE stability studies –fundamentals, plates and members with holes Modal identification and c FSM Existing experimental column data –elastic buckling studies: hole effect, boundary conditions –strength prediction by preliminary DSM stub columns, long columns Conclusions

Preliminary DSM for stub columns Local strength P ne set to P y ? P y,net P y,gross ? P y,net P y,gross lowest local mode in an FE model with hole and test boundary conditions

Gross vs. Net Area

First cut of DSM on Stub Columns (NET YIELD: P y =P y,net Local slenderness plotted for all data)

First cut of DSM on Stub Columns (GROSS YIELD: P y =P y,g Local slenderness plotted for all data)

Preliminary DSM for stub columns Local strength Distortional strength P ne set to P y ? P y,net P y,gross ? P y,net P y,gross lowest local mode in an FE model with hole and test boundary conditions lowest distortional mode (includes DH) in an FE model w/ hole and test bc’s ? P y,net P y,gross ? P y,net P y,gross

DSM prediction for stub columns mean test-to-predicted = 1.18 standard deviation = 0.16 *P cr by FE reflects test boundary conditions, minimum D mode selected, P y =P y,net NET

DSM prediction* for stub columns mean test-to-predicted = 1.04 standard deviation = 0.16 *P cr by FE reflects test boundary conditions, minimum D mode selected, P y =P y,g GROSS

Preliminary DSM for Stub Columns member length/web depth (L/H)

Preliminary DSM for long columns Global buckling Local buckling Distortional buckling

Global buckling in long columns

Local-global in long columns

Preliminary DSM for long columns member length/web depth (L/H)

Outline Objective and challenges Project overview FE stability studies –fundamentals, plates and members with holes Modal identification and c FSM Existing experimental column data –elastic buckling studies: hole effect, boundary conditions –strength prediction by preliminary DSM stub columns, long columns Conclusions

We are off and running on columns with holes Local buckling (a) doesn’t really follow unstiffened element approximation (at least for elastic buckling) (b) should be modeled consistent with application, i.e., stub column boundary conditions, no square plates Distortional buckling is even more of a mess than usual as it appears to get mixed with local buckling, particularly around hole locations. What does P crd  P cr imply? We need better modal identification tools! Global buckling needs further study, P cre sensitivity to isolated holes here is a bit surprising DSM (preliminary) based on gross section yield instead of net section yield has the best accuracy, what does this imply? The boundary conditions of the test and the hole should be explicitly modeled for finding P cr. Existing data does not cover distortional buckling well. We need additional experimental work and nonlinear FE modeling!

“DH” mode as hole location moves

Beginning of BC study Pinned free-to-warp ends, midspan warping restrained Pinned fixed warping ends, far end is torsion free