COLD-FORMED STEEL FRAME AND BEAM-COLUMN DESIGN.pdf

SCHOOL OF CIVIL AND ENVIRONMENTAL ENGINEERING

Cornell UniversityHOLLISTER HALL, ITHACA NY 14853-3501

October 6, 2000

TO: Prof. T. Pekz

FROM: Rao Kotha. S., Ph. D.

SUBJECT: COLD-FORMED STEEL FRAME AND BEAM-COLUMN DESIGN Final Report

1 INTRODUCTION

Design of cold-formed steel pallet rack frames has traditionally been empirical due totheir complex behavior and lack of information with regard to the influence of variousparameters on the overall system strength. The behavior and ultimate strength of a typicalpallet rack frame are characterized by many parameters such as flexibility of beam tocolumn joints, column base fixity, perforations in their column members, local bucklingof member components, geometric and material imperfections and complex bucklingbehavior of column members.

The recommendations of the current American pallet rack frame design specification(RMI 1997) are not very different from its earlier versions which were formulated basedon limited experimental studies carried out at Cornell University (Pekz 1975). Since itsearliest edition, many developments have taken place in the manufacturing and use of therack frames. However, very little efforts are made to improve their design procedurereflecting the changes in the technology. A review of the current specification (RMI1997) reveals that its design recommendations are empirical and conservative with regardto the estimation of frame strength under static loading condition. With the help of newanalytical methods available today such as FEM, it is possible to study the behavior ofrack frames more easily and with greater reliability. This would result in more accurateand economical designs of the rack frames.

As an attempt towards achieving this objective, the RMI and the AISI have jointlysponsored a project on Cold-Formed Steel Frame and Beam-Column Design with Prof.Pekz of Cornell University as principal investigator. This report presents the results ofthe research carried out during October 1998 September 2000 on the above project.

22 OBJECTIVES AND SCOPE

The following are the objectives of the present work.

1. To quantify the beam to column joint flexibility of commonly used pallet rack framesand develop a general q-M relationship for the same

2. To study the column base stiffness characteristics of pallet rack frames and itsinfluence on overall system strength

3. To quantify the approximations in the current RMI (1997) design procedure

4. To verify the beam-column interaction equation of the current American cold-formedsteel design specification (AISI 1996) in the context of pallet rack frame design andmake modifications, if required.

Experimental and Numerical (FE) procedures are used to achieve the above objectives.

The scope of the present study is restricted to analysis and design of 2D pallet rackframes. Only lipped channel columns are considered in the study. Various parametersconsidered in the study include: the type and configuration of pallet rack frames,perforations in the columns, beam-to-column joint flexibility, column base fixity,warping in the column members and global imperfections. The local buckling of beamand column members - another important parameter is ignored in the present study.

3 DETAILS OF THE WORK

This section presents the details and results of the present work in two parts (i) Analysisand (ii) Design.

3.1 Analysis

Experimental methods and Finite element technique are the two procedures available tostudy the complex behavior of pallet rack frames. In the present study it is proposed thatthe experimental methods be used to understand the behavior of components of rackframes (e.g. beam-to-column joint, column base etc.) and finite element method to studythe overall system behavior. The details of the work are presented below.

3.1.1 Behavior of frame components

Behavior of beam-to-column joints and column bases of different types of pallet rackframes are studied in the present work as described below.

33.1.1.1 Beam-to-Column Joint Flexibility

The connection between the shelf beam and column members of pallet rack frames isgenerally flexible. The influence of joint flexibility on the overall frame behavior andstrength is significant. However, due to a very large variety of connection types used inpractice no attempt has been made so far to standardize these joint types nor to develop ageneral expression for the joint stiffness which could be used for all types of connections.

The current RMI specification accounts for the effect of joint flexibility on columnstrength by modifying the pallet beam stiffness and in turn modifying the column endrestraint offered by the beam member. The joint stiffness required to do so is to bedetermined experimentally by individual manufacturer. The specification suggests usingthe secant stiffness corresponding to 0.85 times the ultimate moment capacity of the jointas determined from physical tests. While the specification procedure is simple to use,there may be cases where the above assumption/simplification does not hold true. Hence,it is always rational and safe to use the correct joint stiffness value in the frame analysisby adopting joint q-M relationship valid through the entire load history.

In the present study such an expression for joint stiffness which can be used in the frameanalysis is developed. For this purpose, the experimental data available in terms of jointmoment-rotation history of a variety of joints as provided by different manufacturers isused. The following six types of joints are considered in the study.

Type of Frame/Joint Column Beam depth (in)

ALDR 3.0 x 1.57 x 0.71 x 0.091 3.75 Type B 2.92 x 2.36 x 0.71 x 0.077 4.09 Type C 2.93 x 1.61 x 0.71 x 0.073 5.22 Type D

U 3.0 3.0 x 3.0 x 0.71 x 0.091 3.00 U 3.5 3.0 x 3.0 x 0.71 x 0.091 3.50 U 5.0 3.0 x 3.0 x 0.71 x 0.091 5.00

In each of the Type D joints, again two different shapes are considered. They only differin the configuration. Their geometric dimensions are same as given above.

By trail and error, a tri-linear q-M equation has been established to fit the experimentaldata of the above joint types. The corresponding joint stiffness is given by the followingequations.

bxDK 588.= for uMM 75000 ..

4The joint stiffness as given by Eq 3.1 is expressed in terms of the beam depth andultimate moment capacity of the joint as it was understood from previous experimentaland numerical studies that the joint stiffness is primarily due to distortion of a smallportion of the column member confined to the region of magnitude equal to beam depthas shown in Fig 3.1.1.1.1 (This figure is taken from study on FE models presented insection 3.1.2.1).

Fig 3.1.1.1.1

Eq 3.1 is compared with the experimental results in Fig 3.2.

(a) (b)

Finite Elemnt Model ofBeam-column Joint

As shown, Joint flexibility is mainly due todistortion of small portion of the columnflange connected to pallet beam.

Joint Deformation at P = 0.75 Pult

Beam-column joint stiffness

0

5

10

15

20

25

0 0.05 0.1 0.15 0.2

Joint Rotation (rad)

Mo

men

t (ki

p-in

)

ALDR AHDR Proposed


0

4

8

12

16

20

0 0.05 0.1 0.15 0.2


Mom

ent (

kip-

in)

Type B Proposed

5 (c) (d)

(e) (f)

Fig 3.1.1.1.2

It should be noted that the proposed expression for joint stiffness is not a general equationapplicable for all types of joints. This can be used only after validating with moreexperimental data of wider range of joint types. It may also be noted that the aboveequation requires the ultimate moment capacity known a priory. The joint stiffness asgiven by the above expression also needs to be validated against the effect of moment toshear ratio on the joint.


0

4

8

12

16

20

24

0 0.05 0.1 0.15 0.2


Mo

men

t (ki

p-i

n)

Type C Proposed


0

2

4

6

8

10

12

0 0.02 0.04 0.06 0.08 0.1Rotation (rad)

MO

men

t (k

ips-

in)

U 3.0 (Type 1) U 3.0 (Type 2) Proposed


0

2

4

6

8

10

12

0 0.02 0.04 0.06 0.08 0.1

Rotation (rad)

MO

men

t (ki

ps-

in)

U 3.5 (Type1) U 3.5 (Type 2) ProposedBeam-column joint stiffness

0

2

4

6

8

10

12

14

0 0.02 0.04 0.06 0.08 0.1

Rotation (rad)

MO

men

t (k

ips-

in)

U 5.0 (Type 1) U 5.0 (Type 2) Proposed

63.1.1.2 Column Base Fixity

The column base stiffness of pallet rack frames is characterized by the base platedimensions, number, dimensions and layout of bolts, ratio of moment to axial load at thecolumn base and foundation characteristics. The degree to which each parameter effectsthe base stiffness depends on the way the column is connected to the foundation. Thetotal base rotation can be divided into the following three components.

1. Base rotation due to deformation of foundation : Until no tension is developed at thecolumn base, the entire base rotation is due to deformation of foundation. The effectof deformation of bolts is negligible in this case as they dont resist any external load.An exception to this case is separation of base plate (due to lifting of plate ends along the periphery of plate) from the foundation under very high axial loadespecially when thick base plate is used. Such cases need to be analyzed carefully.

2. Base rotation due to bending of base plate : When the lateral load on the column (oreccentricity of axial load) is large enough to cause tension at the column base, thebase plate separates from the foundation in the tension region. This happens soirrespective of the number and arrangement of bolts. In such cases, the joint rotationwill primarily be due to the bending of plates. The effect of deformation of bolts onbase rotation could be neglected for all practical purposes under the assumption thatthey are overdesigned. In such case, perfect contact between the base plate andfoundation at the locations of the bolts may be assumed for analysis purpose.However, if the bolts fail in tension no contact between the base plate and foundationin the tension zone can be assumed.

3. Base rotation due to deformation of bolts : The effect of extension of bolts on thebase stiffness is considerable when some portion of the column base experiencestension. However, it is always accompanied by bending of base plate and as themoment to axial load ratio increases, this effect becomes negligible. In order to studythe effect of bolt extension on base stiffness, the problem can be divided into thefollowing two cases. When the bolt tension is smaller than the bolt capacity : In this case both the bolt

deformation and bending of base plate contribute to the base rotation. The lay outof bolts also influences the base flexibility.

When the bolt tension exceeds the bolt capacity : This is a special case of category2 as discussed above

In the present work, the base stiffness characteristics due to bending of base plate(category 2) is studied. The following assumptions are made in the study.

Bolts are overdesigned against strength and bond Deformation of bolts is negligible Base plate is in contact with the foundation at discrete points (i.e. at the

location of bolts) in the tension zone until failure.

7The details and results of the work are described below.

Objectives : Following are the objectives of the study on column base flexibility due to bending of base plate.

To study the effect of various parameters on column base stiffness To quantify the stiffness characteristics of various types of commonly used column

bases and establish the range of variation for the same in terms of the base stiffnessvalue recommended by the current RMI specification

For this purpose, a parametric study involving a large number of nonlinear finite elementanalyses of a 3x3x0.091 lipped channel cantilever column has been carried out. Thelength of the column is taken as 30``. All the influencing parameters are considered in thestudy. They include : Axial and lateral loads on the column, base plate configuration andnumber of bolts. Four types of base plates (8x5, 8x8, 6x6 and 7x7) with 3thickness values (0.25, 0.375 and 0.5) in each category are considered in the study.Each column was analyzed for 4 different lateral load to axial load ratios, ranging from1.5% to pure moment case. Fig 3.1.1.2.1 shows the schematic diagram of the model usedin the parametric study.

Fig 3.1.1.2.1

C.G

P

aPTop End Plate

Foundation block(Fixed at bottom)

Base Plate

Spring (Bolt)

Contact Elements(To model gap)

a = 0.0, 1/3, 1/6, 1/60

3.0``x3.0``x0.7`` Column

8For the purpose of analysis, ABAQUS, a general purpose FE commercial software hasbeen used. The features of the FE model used in the study are described below.

Four noded shell element (S4R5) : Used to model the cantilever column and base and topplates.

3D solid elements : To model the foundation. The foundation was assumed made of steeland 1.0`` deep. The base of the foundation was assumed to be fixed in the study. (A laterstudy of one column case (wherein the base was modeled as 6`` deep concrete block withfixed base) showed that the influence of the depth and material of the foundation isinsignificant. However this needs to be verified by further study).

Nonlinear spring elements : To model the anchor bolts. Two springs are used to modeleach bolt one to model shear and one to model the stiffness in the axial direction.However, the stiffnesses of the springs in both the directions are taken to be very high inthe present study assuming that the connections are over designed as discussed before.

Contact elements : To model the gap between the base plate and the foundation.

The moment-rotation relationships for various base plate configurations as obtained fromthe finite element analysis are plotted in Figs 3.1.1.2.2-5. Each of Fig. 3.1.1.2.2(a)-5 (a)contains 12 graphs showing the effects of axial load and plate thickness on column baseflexibility for a given plan area of base plate. Graphs corresponding to very high axialload (Plat=1.6% of Pax) case are shown in larger scale separately in Fig 3.1.1.2.2(b)-5 (b).The base moment used to develop these graphs includes the P-D nonlinear effect. Alsoshown in the above plots is the column base stiffness value as recommended by thecurrent American pallet rack design specification (RMI 1997).

The following are a few observations made based on the study.

The moment-rotation relationship of a typical pallet rack frame base is generally nonlinear

Higher the axial load on the column stiffer is the column base.

While the axial load increased the base stiffness by about 20% when the base plate is thin(0.25``, thinnest of plates studied), its effect is found to be insignificant in the case of thickerplates (Figs 3.1.1.2.2(a)-5 (a)).

when the axial load on the column is accompanied by only a small amount of lateral load(1.6% of axial load), the column base stiffness is found to be very close to that of RMIspecification value (Figs 3.1.1.2.2(b)-5 (b)).

For lateral load to axial load ratios other than 0.016, the initial stiffness of column base isfound to vary from 0.3-0.7 times the RMI specification value, depending on the base plateconfiguration and amount of axial load on the column (Figs 3.1.1.2.2(a)-5 (a)).

The effects of base plate configuration (plan dimensions) and number of bolts on the basestiffness are found to be negligible. However in the case of thinner plates, smaller plateconfigurations seem to help increase the base stiffness slightly.

Further study aimed to verify the above observations over wider range of parameters andto quantify the base flexibility caused by the deformation of foundation and extension ofbolts is underway. The future study will consist of both experimental and finite elementstudies.

9Fig 3.1.1.2.2a

Fig 3.1.1.2.2b

Base Fixity RMI vs FEM (6 x 6 plate)

0

10

20

30

40

50

0 0.01 0.02 0.03 0.04 0.05

Support Rotation (rad)

Mo

men

t (k

ips-

in)

(P_ax=0.0) (P_lat=0.33P_ax)(P_lat=0.016P_ax) (P_lat=0.16P_ax)K_RMI

Base Fixity RMI vs FEM (6 x 6)

0

2

4

6

8

10

0 0.001 0.002 0.003 0.004


Mo

men

t (k

ips-

in)

K_RMI (P_ax=60P_lat)

10

Fig 3.1.1.2.3a

Fig 3.1.1.2.3b

Base Fixity RMI vs FEM (7x7 plate)

0

10

20

30

40

50

0 0.01 0.02 0.03 0.04 0.05Support Rotation (rad)

Mo

men

t (k

ips-

in)

(P_ax=0.0) (P_lat=0.33P_ax)(P_lat=0.016P_ax) (P_ax=6P_lat)K_RMI

Base Fixity RMI vs FEM (7x7)

0

2

4

6

8

10

0 0.001 0.002 0.003 0.004


Mo

men

t (k

ips-

in)


11

Fig 3.1.1.2.4a

Fig 3.1.1.2.4b


0

10

20

30

40

50

0 0.01 0.02 0.03 0.04 0.05


Mo

men

t (k

ips-

in)



0

2

4

6

8

10

0 0.001 0.002 0.003 0.004


Mo

men

t (ki

ps-

in)


12

Fig 3.1.1.2.5a

Fig 3.1.1.2.5b


0

10

20

30

40

50

0 0.01 0.02 0.03 0.04 0.05Support Rotation (rad)

Mo

men

t (k

ips-

in)



0

2

4

6

8

10

0 0.001 0.002 0.003 0.004 0.005 0.006


Mom

ent

(kip

s-in

)


13

3.1.2 Frame Behavior

The objective is to establish guide lines to prepare FE models of CFS pallet rack framesto study their behavior considering various influencing parameters. The following twomodels are proposed in the present study. ABAQUS a commercial finite elementsoftware is used to validate the proposed models.

3.1.2.1 Shell-contact element model

In this model the entire frame is modeled using thin shell elements. All the importantparameters that effect the frame behavior are automatically accounted for in the model resulting in an accurate numerical model very similar to physical model of the frame. Thefeatures and validation of this model as developed in the present study are discussedbelow in detail.

Modeling features

Column and Beam members : General thin shell elements with 6 dof are used to modelthe columns, beams and base plates. Holes in the columns are modeled by defining thethickness of the corresponding elements very small (~ 0.0000001 inch).

Beam-to-column joint : This is the most critical part of the modeling. The 3D nature ofthe connection behavior makes the exact joint modeling very complex, computationallyexpensive and practically prohibitive. Moreover, in the case of cold-formed steelmembers, very small or negligible out of plane stiffness of the plate element of thecolumn (like flange or web) to which the beam is connected is the primary source of thejoint flexibility. Hence, modeling of tabs/keys and other accessories could be ignoredwithout any loss of accuracy. The following two joint models are experimented in thepresent study.

Joint model-1: In this model the end of the beam is connected directly to the flange orweb of the column and the connection is treated as continuous. The end plates andtabs/keys are ignored completely. This is also termed as continuous model. Thismodel is simple to use and will be of great help when modeling large scale frames.

Joint model-2 : In this model the beam ends are connected to end connecting plateswhich in turn are connected to columns at a few points. The gap between the columnand endplate is modeled using contact elements. This model is also calleddiscontinuous model. Tabs are not included in this model.

Column base : It is suggested to use contact elements to model the discontinuity betweenthe base plate and the foundation. The foundation bloc and the anchor bolts can bemodeled by using solid and beam elements respectively.

14

Validation

The following pallet rack components and sub-assemblies are chosen for the validationstudy. The results of the analytical study obtained based on complete nonlinear(geometric + material nonlinearities) FE analysis are compared with their experimentalresults (Pekz 1975).

List of analyses carried out :

(i) Stub column - 1 case(ii) Cantilever beam test - 2 cases

(a) Direct connection between the beam end and the column flange(b) Using Contact element between the beam end and the column

flange(iii) ALDR type frame under gravity load - 3 cases


flange(c) Using Contact element between the beam end and the column

flange with 1 inch in 10 ft out of plumb of columns(iv) ALDR type frame under (gravity load + Lateral load) - 2 cases


flange(v) AHDR type frame under (gravity load + Lateral load) - 1 case

Fig 3.1.2.1.1 shows the details of the column and beam members of ALDR and AHDRframes. The corner radius and the effect of cold forming are neglected in the presentstudy.

Column post-ALDR frame Column post-AHDR frame

0.71"

1.57"

3.00"

0.534"

0.687"

0.091"

0.71"

3.0"

2. 91"

0.72"

0.531"

0. 091"

15

Beam (all pallet racks)

Column perforations *

Fig 3.1.2.1.1

* Shape of the perforations is assumed to be rectangular in the present FE study.

2.75"

3.75"

1.675"

1.875"

2.0"

1.50"

0.063"

0.070"0.133"

16

Discussion of results

Stub column

Analysis on the stub column of an ALDR (Fig. 3.1.2.1.1) column was carried out underpure compression. The objective is to test the models performance against estimating thesection strength of columns accounting for the effect of local buckling and holes.Residual stresses and imperfections are not included in the study.

The ultimate load of the stub column as obtained by the FE analysis is found to be 25.7kips as against to its experimental value of 22.8 kips with a difference of +10%.

The deformed shape and von-Mises stress (equivalent uni-axial stress for a combinedstate of stress; determined from the maximum distortion energy theory of failure,according to which the failure by yielding occurs when, at any point, the distortionenergy for unit volume in a state of combined stress becomes equal to that associatedwith yielding in simple tension) of the stub column at failure are shown below in Fig3.1.2.1.2 and Fig 3.1.2.1.3 respectively.

Fig 3.1.2.1.2 Stub column Deformed shape at failure

17

Fig 3.1.2.1.3 Stub column von-Mises stress at failure

To study the effect shear locking due to use of negligibly thin elements (0.0000001) tomodel the holes, the analysis was repeated by removing the elements at the holes. Thebehavior of the stub column as predicted by the two models is compared in Fig 3.1.2.1.4.

Fig 3.1.2.1.4 COMPARISON OF STUB COLUMN BEHAVIOR( Continuous Vs Discontinuous* )

* Continuous Thickness of elements corresponding, Discontinues No elements at holes

0

0.5

1

1.5

2

2.5

3

0 0.01 0.02 0.03 0.04 0.05

with elements

with out elements

18

The difference in the load estimates of the two models is found to be very small (3.5%).Hence, the difference in the experimental and analytical strengths is attributed toignorance of the geometric imperfections and the residual in the analytical study. It mayalso be noted from the plot that on contrary to the suspicion, the behavior is softer whenthe holes are modeled with very thin elements. This may be due to greater loss of sectionstiffness caused by out of plane bending of the negligibly thin elements in the first model.Hence, it can be concluded that the holes could be modeled successfully with negligiblythin elements with out appreciable error to predict the behavior and ultimate load.However it may be advisable to do a similar study in more complicated situations as wellbefore proceeding to the actual analysis. This is further explained in the followingsections.

Cantilever beam test

The objective is to study the performances of different shell element models in predictingthe beam-to-column joint behavior of pallet rack frames. The details of the model used(joint between ALDR column and beam Fig 3.1.2.1.1) for this purpose are shown Fig3.1.2.1.5.

Fig 3.1.2.1.5 FE model of beam-to-column joint

The following are a few observations made with regard to the models performance inpredicting the joint behavior.

P

30.0"

24.0"

19

The load versus deflection behavior is non-linear as shown in Fig 3.1.2.1.6.

Fig 3.1.2.1.6 Comparison of D-P behavior Experimental Vs Analytical

A very good correlation is found between the Analytical and Experimental resultswith respect to the deflection at the tip of the cantilever beam as shown in Fig3.1.2.1.6.

The difference between the strength estimates of the two joint models is negligible.

However, the D-P behavior as predicted by the joint model-2 is found to be softeras compared to Model-1. This is expected because, unlike in Model-1, in the case ofModel-2 the load is transferred from the beam to column at far fewer points (throughtwo points on the web and through the contact between the compression flange ofbeam and the column flange) resulting in flexible behavior.

Figs 3.1.2.1.7 and 3.1.2.1.8 show the deformation configurations of the joint aspredicted by the two models in subultimate and near failure.

20

(Joint Model-1)

(Joint Model-2)

Fig 3.1.2.1.7 Joint deformation in sub-ultimate range

21

Fig 3.1.2.1.8 Deformed shape of the joint at failure

As shown in Figs 3.1.2.1.7 and 3.1.2.1.8, irrespective of the joint model adopted, thetwo flanges of the column distorted to different amounts reflecting the semi-rigidnature of the joint. This is due to small out of plane stiffness of the column flanges.

The failure was initiated by the yielding of the material near the beam-to-column jointas expected. The distribution of von-Mises shear stress at failure is shown in Fig3.1.2.1.9.

The model is also able to predict the failure load with sufficient accuracy as comparedto the experimental load. The ratios between the analytical load and the experimentalload are found to be 1.09 (corresponding to vertical deflection=4.5 inch, which is verylarge) and 0.98 for the Joint model-1 and Joint model-2 respectively. The failuremode as predicted by the finite element analysis (yielding of the beam-column joint)is also found to be in agreement with the experimental observation.

From the results discussed above it is clear that the proposed shell element model is ableto predict the behavior and strength of pallet rack frame components and sub-assemblages accurately. This model could be used as an alternative to physical tests infuture studies.

22

Fig 3.1.2.1.9 Von-Mises stress distribution at failure

Fig 3.1.2.1.10 Deformed shape of beam-to-column joint model at failure

23

Pallet rack frames

A total of 6 analyses as discussed in the beginning of the section in section were carriedout on two different frames (ALDR and AHDR). The cross section details of thecomponents of frames are as shown in Fig 3.1.2.1.1. The schematic diagram of overallframe is given in Fig 3.1.2.1.11.

Fig 3.1.2.1.11 Details of Pallet Rack Frames

Fig 3.1.2.1.12 shows the FE mesh used to model the pallet rack frames and beam-to-column joints. As shown in fig., the frame was restrained with respect to the out of planedeflection at each story level.

93.0"93.0"

62.0"

56.0"

62.0"

W

WW

W W

W

24

Fig 3.1.2.1.12a FE model of type A-LDR frame

Fig 3.1.2.1.12b FE model of a typical interior joint (Model-1)

25

The ultimate loads and failure modes of all the frames as obtained from nonlinear finiteelement analysis are summarized in Table 3.1.2.1.1 along with their experimentalobservations.

Table 3.1.2.1.1

Failure ModeFrame Loading Joint Model

tult

anaultP

Pexp,

,Analytical Exptl

Model-1 1.58 Yd of jointsModel-2 1.47 Yd of joints

ALDR Gravity

Model-2** 1.10 Yd of joints

Yd+LBof loweststory beam

Model-1 1.12*ALDR Gravity +Horizontal Model-2 1.09*

Permanentsway

Permanent sway

AHDR Gravity +Horizontal

Model-1 1.13* Permanentsway

Permanent sway

* with respect to the lateral load ** (1/120) out of plumbLB Local Buckling

These results are discussed below in greater detail.

ALDR Frame under Gravity Load

First, the frame was analyzed using Joint model-1. As shown in Table 3.1.2.1.1, theratio between the analytical and experimental failure load was found to be very high(1.58). Interestingly, the failure modes as predicted by the FEM analysis and thatobserved in the test are also noted to be different.

The frame was re-analyzed using the Joint model-2 (though the earlier experience withthe cantilever beam problem (section2.3) showed only 10% difference in the ultimateloads of the two joint models). The ratio between the analytical and experimental failureloads with joint model-2 is found to be 1.47, with an improvement of about 10% ascompared to the joint model-1. While it is consistent with the previous observation madeat component level (cantilever problem), the reasons for a large difference inexperimental and analytical values remained undetermined at this time

While no convincing reasons could be seen for poor performance of the model, it wasdecided to re-analyze the frame considering the out of plumb of the column uprights,which was ignored, in the previous analyses. The initial configuration of the frame wasremodeled with columns 1 inch out of plumb for 10 ft of height as recognized in the RMIspecification. The Joint model-2 was used to model the connection. The ratio between theanalytical and experimental failure loads was found to be 1.10. The local imperfectionsand the residual stresses are attributed for the conservative analytical estimates (by 10%).It is learned from the above 3 analyses that the frame is very sensitive to theimperfections, which must be included in the analytical study.

The vertical deflections at mid section of the bottom story beam as estimated by theFE analysis is compared with the experimental observations in Fig 3.1.2.1.13.

26

Fig 3.1.2.1.13 Load-Deflection behavior Analytical Vs Experimental

The von-Mises stress contour (at failure) in the bottom story beam and in the interiorcolumn joint is shown below in Fig 3.1.2.1.14. It may be noted that while the stressesin the beam were below the yield stress (49 kips), the stress in the joint exceeded thematerial stress indicating the joint failure which is in contrary to the experimentalobservation. This is different from the failure mode observed in physical test (Table2.4.2.1).

27

Fig 3.1.2.1.14a von-Mises stress contour in the bottom story beam (at failure)

Fig 3.1.2.1.14b von-Mises stress contour in the interior column joint(at failure)

The deformed shape of the frame bottom interior joint at failure are shown in Fig 3.1.2.1.15.

28

Fig 3.1.2.1.15a Deflected shape at failure

Fig 3.1.2.1.15b Deflected shape at failure

29

ALDR Frame under ( Gravity + Lateral ) Load

The frame was analyzed for gravity and lateral load combination. The loading wasapplied in the same sequence as it was applied in physical test. The connection wasmodeled using both the joint models and the results are compared with experimentalvalues in Table 3.1.2.1.1.

As reported in Table 3.1.2.1.1, the difference between the failure loads of the twojoint models is negligibly small, both are found to be close to the experimental value.The conservative estimates of FE analysis could be attributed to the ignorance ofimperfections.

As shown in Fig 3.1.2.1.16-17, the performance of the model in predicting the load-deflection behavior is very satisfactory.

The failure of the frame was found to be due to large lateral deformations associatedwith yielding of interior joint of the lowest storey. This agrees with the experimentalfailure mode.

The deformed shapes of the frame at failure as observed in the analytical andexperimental studies are shown in Fig 3.1.2.1.18-20.

Fig 3.1.2.1.16 Load Vs Vertical deflection at the midsection of bottom story beam

30

Fig 3.1.2.1.17 Load Vs Lateral displacement

Fig 3.1.2.1.18 Deflected shape at failure (FEM)

31

Fig 3.1.2.1.19 Deflected shape at failure (Experimental)

32

Fig 3.1.2.1.20 Deflected shape at failure (Experimental)

33

AHDR Frame under ( Gravity + Lateral ) Load

The frame was analyzed for gravity and lateral load combination as in the previousexample. However, in view of negligible difference in the performance of the twojoint models as noted in the previous examples, joint model-1, which is simple to use,was adopted in this analysis.

As noted in Table 3.1.2.1.1, the comparison between the analytical and experimentalestimates of frame strengths is very good.

The lateral displacement at the top story beam level and the vertical deflections atthe mid section of bottom story beam are plotted against the load in Fig 3.1.2.1.21. Asshown in the figure, the model is able to predict the frame behavior through the entireload history.

Fig 3.1.2.1.21a Load vs Lateral displacement

34

Fig 3.1.2.1.21b Load vs Vertical deflection at the midsection of bottom story beam

Conclusion

The foregoing results demonstrate clearly the accuracy of the proposed shell elementmodel to predict the behavior and strength of pallet rack frames accounting for variousparameters. However, it was also observed that modeling of full frame by shell elementsis tedious, time expensive and requires experienced analysts. These drawbacks limit theuse of shell element model to a reference/benchmark numerical model. This model bestserves in validation studies (as an alternative to experiments) to evaluate the performanceof other simpler numerical or analytical techniques.

3.1.2.2 Beam-spring element model

While the shell element model discussed above is versatile, rational and accurate, it is notsuitable for routine design works and in situations where large number of frames arerequired to be analyzed. Hence, there is a need of a simpler analytical model. Such amodel is explored below.

The objective of the study presented in this section is to develop a simple finite elementmodel capable of estimating the behavior and ultimate load capacity of pallet rack framesaccurately. This helps to generate large amount of data of Pu and Mu of pallet rackframes, more efficiently, required to quantify the approximations in the current RMIspecification and suggest modifications if required.

35

Modeling Features

The influencing parameters and how they are incorporated in the present FE modeldeveloped in ABAQUS are described in the following table.

Table 3.1.2.2.1 Details of Beam element model

Parameter Modeling features RemarksColumn members B31OS-thin walled beam element

with 7 dof/nodeThe seventh dof models thewarping of the section important property of theopen section columns ofrack frames.

Pallet beams B31-general beam element with 6dof/node.

As the shelf beams of palletrack frames are generallyclosed section type, thewarping effect could beneglected.

Joint Flexibility Flexible joint model : Beamends are connected to columnsthrough nonlinear springs withstiffness characteristics defined byEq 3.1, discussed in Sec 3.1.1.1.

The joint stiffnesscharacteristics are assumedto be same in both loadingand unloading.

Column Base Fixity Flexible base :Linear* spring element with astiffness equal to 0.5 times theRMI specification value.* Effect of nonlinearcharacteristics on frame strengthis discussed in Appendix

This is average value of theupper and lower bound basestiffnesses of the columnbase types studied in thepresent study (Please refer tosection 3.1.1.2 for details).

Holes in the column(Section`)

Equivalent thickness calculatedbased on I_avg is used to definethe geometry of the columnmembers.

Simple average principle isused in the present study.

Warping d.o.f.(at column base)

1. Fixed2. Free

The two extreme conditionsare used to quantify theeffect of warping on thefailure strength.

Local Buckling Ignored completely

The study is restricted to, at thisstage, the frames locally stablemembers.

Justified since the membersare found to be stable withrespect to local bucklingfrom FE analysis with shellelement model and theory.

Imperfections 1/120 out of plumb in the columns As specified by RMI (Alsoused in shell elementmodels).

36

Validation

The two pallet rack frames (ALDR and AHDR) used before in the validation study ofshell element model are analyzed using the beam element model with the above featuresand the results are summarized in Table 3.1.2.2.2. In the case of ALDR frame the framecapacities as evaluated by the current RMI design rules are also compared. The resultspresented for AHDR frame correspond to warping fixed case. (It was observed later thatin the case of frames made of AHDR type columns, the end condition wrto warping dofdoes not have significant effect on the frame strength).

Table 3.1.2.2.2a Comparison of Analytical, specification and Experimental results ALDR Frame

Ratio = (Pu,cal /Pu,exptl )Beam Element Model

Warping (at base)Load

combinationFree Fixed

Shellelementmodel

RMISpecification

Gravity 0.75 1.08 1.10 0.43 0.52

Gravity +Lateral

0.80 0.91*

0.98**1.09*

1.06**0.42 0.50

* with respect to lateral load ** with respect to gravity load

Table 3.1.2.2.2b Comparison of Beam and Shell element models - AHDR Frame

Load Combination (Pu,beam /Pu,shell)

Gravity + Lateral (0.75%gravity) 1.10

Gravity + Lateral (1.5% of gravity) 1.05

The following are a few observations made in this regard.

The effect of warping constraint on the frame strength depends on the type of columnmembers. The difference between the strengths of warping free and warping fixedcases may vary from 4% (AHDR type columns) to 25% (ALDR columns).

The limited parametric study shows that the warping fixed case simulates theexperimental condition better (Table 3.1.2.2.2a).

The beam element model with the proposed features (Table 3.1.2.2.1) is capable ofestimating the frame strengths accurately as compared to experimental and rationalshell element based FE results.

The current RMI specification underestimates the strengths of pallet rack frames byabout 50% (more results on RMI procedure are presented in section 3.2.1). Thisestablishes the need to review and improve the current design procedure (RMI 1997).

The foregoing discussion qualifies the proposed use of beam element model in futurestudies on development of design procedures for rack frames. The following sectiondiscusses the details of the work done in this direction.

37

3.2 Design

Work carried out under this heading consists of two parts :

1. Study of the current RMI design procedure

2. Development of a new design procedure

3.2.1 Current pallet rack design procedure (RMI 1997)

As noted before, the current American pallet rack design procedure (RMI 1997) is veryconservative. This is attributed to the following inconsistencies in the procedure. Thespecification suggests linear (1st order elastic) frame analysis to obtain the memberforces. It uses effective length procedure to include the frame effect. The effective lengthfactors are calculated either from rational buckling analysis or by using the alignmentcharts. It also suggests a set of fixed effective length factors in lieu of the aboveprocedures. The effect of beam-to-column joint flexibility is accounted for indirectly byreducing the beam stiffness in the calculation of column effective factor. Thespecification recommends the linear beam-column interaction equation of the currentcold-formed steel design specification (AISI 1996) to check the member safety. Toaccount for moment magnification due to the presence of axial load on the column it usesthe simple amplification factor derived from linear elastic buckling analysis. The effectsof local buckling and holes in the columns are considered using effective section andQ-factor approaches respectively. While the specification procedure has been madeconservative in the absence of knowledge of the effect of various parameters on theoverall frame behavior, it is not known to what magnitude each of these parameterseffects the accuracy of the design procedure.

To quantify the conservatism in the current procedure caused by the use of approximateeffective length factors, linear beam-column interaction equation and linear momentamplification factor, a large parametric study involving nonlinear finite element analysisof five types of frames has been carried out. The details of the analyses are given below.

Frame Type Column Beam

AHDR 2.87 x 2.91 x 0.091 3.75Type B 3.0 x 2.36 x 0.077 4.09Type D

U 3.0 3 x 3 x 0.091 3.00 U 3.5 3 x 3 x 0.091 3.50 U 5.0 3 x 3 x 0.091 5.00

In each category, 6 frame configurations (BxS : 2x3,4x3,6x3,4x4,4x6 and 6x6) subjectedto three load combinations: Gravity only, (Gravity+lateral equal to 1.5% of Gravity) and(Gravity+lateral equal to 15% of Gravity) are analyzed for two column base stiffnessvalues (Kbase=KRMI, KRMI/2). The beam element model described in previous section withnonlinear beam-to-column joint stiffness as defined by Eq 3.1 has been used in the finiteelement study.

38

The frame capacities as estimated by the RMI specification are compared with the finiteelement results in Tables 3.2.1.1-6.

Table 3.2.1.1a Evaluation of RMI approach ( Kb = KRMI )

R = (Ucal / UFEM)Product DFrame

AHDR B-type U 3.0 U 3.5 U 5.0Remarks

2Bx3S 0.75 0.75 0.75 0.72 0.70

4Bx3S 0.66 0.66 0.70 0.68 0.66

6Bx3S 0.72 0.72 0.74 0.72 0.67

4Bx4S 0.81 0.80 0.82 0.75 0.67

4Bx6S 0.62 0.56 0.74 0.74 0.72

6Bx6S 0.63 0.56 0.69 0.86 0.85

AISI Beam-column Interaction equation

Loading

Gravity only (all butone span case)

Notional load = 0.0

Base fixity

Kb = KRMI

Effective length factors

Kx = 1.7Kphi = 0.8Ky = 1.0

Table 3.2.1.1b Evaluation of RMI approach ( Kb = KRMI )



2Bx3S 0.55 0.65 0.63 0.62 0.63

4Bx3S 0.49 0.57 0.59 0.59 0.59

6Bx3S 0.53 0.63 0.62 0.62 0.60

4Bx4S 0.59 0.69 0.69 0.64 0.60

4Bx6S 0.46 0.49 0.62 0.64 0.65

6Bx6S 0.46 0.49 0.58 0.74 0.76


Loading

Gravity only (all butone span case)

Notional load = 0.0

Base fixity

Kb = KRMI


Kx = Align. chartsKphi = 0.8Ky = 1.0

39




2Bx3S 0.76 0.70 0.93 0.84 0.69

4Bx3S 0.80 0.72 0.96 0.85 0.72

6Bx3S 0.88 0.84 0.91 - 0.80

4Bx4S 0.84 0.76 0.96 0.88 0.75

4Bx6S 0.90 0.75 1.00 0.95 0.82

6Bx6S 0.88 0.86 0.95 0.90 0.88


Loading

Gravity+ 1.5% Lateral

Base fixity

Kb = KRMI


Kx = 1.7Kphi = 0.8Ky = 1.0




2Bx3S 0.57 0.61 0.80 0.74 0.63

4Bx3S 0.60 0.62 0.82 0.74 0.66

6Bx3S 0.68 0.75 0.80 - 0.73

4Bx4S 0.63 0.67 0.82 0.77 0.69

4Bx6S 0.68 0.66 0.86 0.83 0.74

6Bx6S 0.68 0.77 0.82 0.79 0.81


Loading


Base fixity

Kb = KRMI



40




2Bx3S 0.86 0.80 0.78 0.84 0.85

4Bx3S 0.84 0.70 0.87 0.72 0.87

6Bx3S 1.17 0.80 0.83 0.93 0.93

4Bx4S 0.92 0.69 0.92 0.90 0.97

4Bx6S 0.87 0.87 0.92 1.04 0.95

6Bx6S - - 0.89 0.90 0.90


Loading

Gravity+ 15% Lateral

Base fixity

Kb = KRMI


Kx = 1.7Kphi = 0.8Ky = 1.0




2Bx3S 0.72 0.73 0.72 0.78 0.81

4Bx3S 0.71 0.65 0.80 0.67 0.83

6Bx3S 1.03 0.75 0.78 0.88 0.89

4Bx4S 0.78 0.64 0.84 0.83 0.92

4Bx6S 0.75 0.81 0.85 0.97 0.90

6Bx6S - - 0.82 0.84 0.85


Loading


Base fixity

Kb = KRMI



41

Table 3.2.1.4a Evaluation of RMI approach ( Kb = KRMI /2)


AHDR U 3.0 U 3.5 U 5.0Remarks

2Bx3S 0.64 0.76 0.76 0.71

4Bx3S 0.71 0.73 0.70 0.65

6Bx3S 0.78 0.76 0.73 0.67

4Bx4S 0.85 0.66 0.78 0.71

4Bx6S 0.92 1.04 0.94 0.71

6Bx6S 0.95 1.04 0.93 0.93


Loading

Gravity only(All spans)

Notional load = 0.0

Base fixity

Kb = KRMI/2


Kx = 1.7Kphi = 0.8Ky = 1.0

Table 3.2.1.4b Evaluation of RMI approach ( Kb = KRMI /2)



2Bx3S 0.50 0.67 0.66 0.64

4Bx3S 0.52 0.61 0.61 0.59

6Bx3S 0.57 0.64 0.63 0.60

4Bx4S 0.62 0.55 0.67 0.64

4Bx6S 0.68 0.88 0.80 0.64

6Bx6S 0.70 0.88 0.80 0.84


Loading

Gravity only(All spans)

Notional load = 0.0

Base fixity

Kb = KRMI/2



42




2Bx3S 0.79 0.79 0.83 0.69

4Bx3S 0.79 0.79 0.87 0.71

6Bx3S 0.82 0.84 0.89 0.76

4Bx4S 0.85 0.84 0.88 0.84

4Bx6S 0.94 0.97 0.98 0.91

6Bx6S 0.92 0.98 0.99 0.96


Loading


Notional load = 0.0

Base fixity

Kb = KRMI/2


Kx = 1.7Kphi = 0.8Ky = 1.0




2Bx3S 0.59 0.67 0.72 0.63

4Bx3S 0.59 0.67 0.76 0.65

6Bx3S 0.63 0.73 0.78 0.68

4Bx4S 0.65 0.72 0.77 0.76

4Bx6S 0.72 0.83 0.85 0.82

6Bx6S 0.70 0.84 0.87 0.86


Loading


Notional load = 0.0

Base fixity

Kb = KRMI/2



43




2Bx3S 0.90 0.91 0.93 0.74

4Bx3S 0.81 1.08 0.90 0.85

6Bx3S 1.12 0.92 1.11 1.64

4Bx4S 0.92 0.98 1.33 1.47

4Bx6S 0.95 0.97 1.01 1.92

6Bx6S 1.10 - 1.01 2.00


Loading


Notional load = 0.0

Base fixity

Kb = KRMI/2


Kx = 1.7Kphi = 0.8Ky = 1.0




2Bx3S 0.76 0.82 0.86 0.69

4Bx3S 0.70 0.99 0.83 0.81

6Bx3S 0.93 0.85 1.02 1.56

4Bx4S 0.79 0.89 1.24 1.40

4Bx6S 0.81 0.89 0.94 1.83

6Bx6S 0.95 1.30 0.94 1.89


Loading


Notional load = 0.0

Base fixity

Kb = KRMI/2



44

From the above comparative study, it may be noted that :

(For the purpose of discussion, the RMI design procedure is deviled into two methods based onthe effective length factors used. They are referred below as Procedure I (Kphi = 0.8 and Kx =1.7) and Procedure II (Kphi = 0.8 and Kx = from alignment charts)).

If the column base fixity is equal to the RMI stiffness value

The strength estimates of the RMI by procedure I are generally goodfor the Gravity+lateral load cases. The specification capacities inthese cases are conservative by 10%-15% (except in a few frames) ascompared to finite element results. However, in the Gravity loadonly case, the specification is found to be conservative by as much as40%.

Procedure II of RMI is found to be 20%-50% conservative when theload on the frame is either gravity or gravity+small lateral load andabout 15%-25% for large lateral load (+gravity) case.

If the column base is equal to half the RMI stiffness value

Procedure I is found to be less conservative in this case as to the caseof frames with stiffer column bases (case discussed above). The RMIstrengths are about 10%-35% conservative as compared to FE results.However, in the case of a few frames subjected to gravity+largelateral load (Table 3.2.2.6a) the specification results are found to beunsafe. The FE results corresponding to these frames may need to bereviewed to verify the abnormality.

Strengths estimated by Procedure II are 20%-40% conservative whenthe load on the frame is either gravity or gravity+small lateral loadand about 15%-25% for large lateral load (+gravity) case. However, afew frame strengths are overestimated under large later loads.

In order to quantify the net effect of various approximations in the RMI procedure (i.e.other parameters such as local buckling, holes in the column etc. as well), the results ofALDR frame presented in previous section (Table 3.1.2.2.2) could be used. As notedbefore, the net effect of all the approximations is to underestimate the frame strength by40%-50%. However, as this observation is based on single frame analysis, a moredetailed study in this direction covering wider range of products is required to quantifythe actual effect of the approximations.

From the foregoing discussion it may be said that the RMI specification rules are notadequate to model various parameters accurately. Though the numbers shown above as tothe unconsrvatism of the RMI are based on the preliminary study and need verification,they do give approximate idea and it could be said that the net effect of approximations ofRMI will not be less than 30% with regard to strength estimates. This suggests the needof a better design procedure which is developed in the following section.

45

3.2.2 Development of Design Procedure

The objective is to develop a more rational and accurate design procedure for CFS palletrack frames by making use of the existing design equations. The beam-columninteraction equations and moment amplification factors of the current AISI and AISCspecifications are used for this purpose. A total of five possible methods devised bydifferent combinations of these equations are experimented to arrive at the mostacceptable procedure. The details of the procedures are presented in table 3.2.2.1. It maybe noted that the effect of local buckling is not included in the study.

Table 3.2.2.1 Evolution of possible design procedure

Method Interaction Equation Remarks

Method 1

(RMI-Proc.I)

Kx = 1.7; Kphi =0.8 & Ky = 0.85

M = a*Mlin = a*Elin*P a = 1 / (1-P/Pex)

Elin = Linear Eccentricity = (Mlin/Plin) from linear analysis

Method 2(RMI- Proc.IIwith modified Ks)

01.=+nn M

MPP

Kx = 1.0; Kphi =0.8 & Ky = 0.85

All other parameters as inMethod 1.

Method 3

(AISC proc. I)

Kx = 1.0; Kphi =0.8 & Ky = 0.85

M = a*Mlin = a*Elin*P

a = 1 / ( 1-(SP/SH)(D/L) )

D = Inter-story drift at firstfloor level determined from linearFE analysis with initial Beam-column joint stiffness

All other parameters as inMethod 1

Method 4

(AISC proc.II)

012

.=+nn M

MPP

for 20 .

nPP

0198

.=+nn M

MPP

for 20 .>

nPP

D = Inter-story drift at firstfloor level determined from linearFE analysis with secant Beam-column joint stiffness correspondingto 0.99 Mu (of joint moment capacity)

All other parameters as inMethod 3

Method 5 : M-P equation and effective length factors as in method 2 and a

46

the moment amplification factor as in method 4.Comparative study

The axial load and moment capacities of a large number of rack frames of different typesas estimated by these methods and nonlinear FE analysis are compared in tables 3.2.2.2-3.2.2.7. The details of the frames are as given in section 3.2.1. The simple beam elementmodel described in sec.3.1.2.2 was used to obtain the reference FE strengths. However,the following linear beam-to-column joint stiffness values were used to model jointflexibility instead of nonlinear stiffness characteristics (Eq 3.1) originally suggested inthe model. In all the FE analyses, the column base stiffness was assumed to be half theRMI value and the warping at the column base was restrained.

Beam-to-column joint stiffness characteristics

The beam-to-column joints are modeled using linear spring elements with two differentstiffness characteristics : (i) Initial joint stiffness until Mu and (ii) Secant stiffnesscorresponding to 0.99Muas described in the following Fig.

Ks

Beam-to-column joint Stiffness Characteristics

ROTATION

0.99Mu

Mu

0.75 Mu

Ki

Ki = Initial stiffness (methods 1,2,3)

Ks = Secant stiffness (methods : 4,5)

MOMENT

47

Table 3.2.2.2 Comparison Of Design procedures (Gravity+0.0075% lateral to trigger failure )

Method 1 : AISI M_P Eqn. & AISI Amplification factorMethod 2 : AISI M_P Eq. & AISI Amplification factorMethod - 3 : AISC M_P En. & AISC B2 factor (deflection based) - initial stiffnessMethod 4 : AISC M_P Eq. & AISC B2 factor (deflection based) secant stiffnessMethod 5 : AISI M_P Eq. & AISC B2 factor secant stiffness(Kx = 1.7; Kphi

=0.8) in Method 1 & (Kx = 1.0; Kphi =0.8) in Methods 2,3,4,5.

(P,M)r = [ (P,M)cal / (P,M)fem ]

Frame Method - 1 Method -2 Method - 3 Method - 4 Method - 5

Type Confg( BxS )

Pr Mr Pr Mr Pr Mr Pr Mr Pr Mr

2 x 3 0.68 0.12 0.89 0.11 0.88 0.46 0.85 1.42 0.84 1.16

4 x 3 0.71 0.13 0.94 0.12 0.90 0.77 0.86 2.36 0.85 2.14

6 x 3 0.78 0.14 1.03 0.13 0.99 1.28 0.90 3.07 0.90 3.07

4 x 4 0.85 0.15 1.11 0.14 0.98 1.32 0.87 2.42 0.86 2.19

4 x 6 0.92 0.16 1.21 0.15 0.91 1.32 0.75 1.85 0.75 1.69

AHDR

6 x 6 0.95 0.17 1.25 0.16 0.90 1.73 0.74 2.42 0.74 2.17

2 x 3 0.80 0.15 1.06 0.13 1.05 0.43 1.04 0.78 1.04 0.78

4 x 3 0.72 0.13 0.96 0.13 0.96 0.73 0.94 6.26 0.93 5.6

6 x 3 0.76 0.14 1.02 0.13 1.00 0.99 0.93 4.14 0.93 3.55

4 x 4 0.66 0.12 0.87 0.11 0.83 1.69 0.72 6.7 0.72 5.62

4 x 6 1.05 0.19 1.40 0.18 1.03 3.66 0.79 5.77 0.79 5.37

U3.0

6 x 6 1.04 0.19 1.40 0.12 0.97 2.09 0.74 5.19 0.74 2.91

2 x 3 Data Needs verification

4 x 3 0.70 0.07 0.93 0.06 0.93 0.34 0.93 1.12 0.92 0.86

6 x 3 0.73 0.13 0.97 0.12 0.97 1.04 0.93 3.85 0.93 3.71

4 x 4 0.78 0.14 1.03 0.13 0.96 0.89 0.89 1.50 0.88 1.41

4 x 6 0.94 0.17 1.26 0.16

U3.5

6 x 6 0.94 0.17 1.25 0.16Data Needs verification

2 x 3 0.72 0.13 0.95 0.12 0.95 0.25 0.95 0.39 0.95 0.39

4 x 3 0.66 0.12 0.87 0.11 0.87 0.29 0.87 0.62 0.86 0.56

6 x 3 0.67 0.12 0.89 0.11 0.89 0.33 0.89 0.91 0.88 0.81

4 x 4 0.71 0.13 0.95 0.12 0.90 3.42 0.91 2.03 0.91 2.03

4 x 6 0.71 0.13 0.95 0.12 0.89 0.83 0.71 3.13 0.71 2.91

U5.0

6 x 6 0.93 0.16 1.25 0.16 0.95 2.33 0.79 3.35 0.79 2.89

48

Table 3.2.2.3 Comparison Of Design procedures ( Gravity+0.75% lateral load)

Method 1 : AISI M_P Eq. & AISI Amplification factorMethod 2 : AISI M_P Eq. & AISI Amplification factorMethod - 3 : AISC M_P Eq. & AISC B2 factor (deflection based) - initial stiffnessMethod 4 : AISC M_P Eq. & AISC B2 factor (deflection based) secant stiffnessMethod 5 : AISI M_P Eq. & AISC B2 factor secant stiffness(Kx = 1.7; Kphi




Type Confg( BxS )


2 x 3 0.73 0.32 0.95 0.32 0.84 0.60 0.78 0.71 0.77 0.68

4 x 3 0.83 0.35 1.10 0.35 0.93 0.72 0.87 0.79 0.86 0.76

6 x 3 0.81 0.34 1.06 0.34 0.88 0.71 0.83 0.79 0.81 0.72

4 x 4 0.85 0.34 1.12 0.34 0.89 0.73 0.82 0.85 0.81 0.80

4 x 6 0.96 0.37 1.26 0.37 0.91 0.92 0.80 1.11 0.79 1.01

AHDR

6 x 6 0.95 0.32 1.25 0.32 0.87 0.84 0.76 0.99 0.74 0.86

2 x 3 0.72 0.27 0.96 0.27 0.86 0.49 0.82 0.54 0.81 0.52

4 x 3 0.81 0.32 1.08 0.32 0.94 0.62 0.88 0.70 0.86 0.67

6 x 3 0.81 0.30 1.08 0.30 0.92 0.59 0.86 0.68 0.83 0.62

4 x 4 0.87 0.33 1.15 0.33 0.94 0.70 0.86 0.82 0.84 0.77

4 x 6 0.91 0.34 1.21 0.35 0.85 0.87 0.71 1.10 0.70 0.99

U3.0

6 x 6 0.91 0.34 1.21 0.34 0.83 0.92 0.68 1.09 0.66 1.00

2 x 3 0.90 0.25 1.21 0.25 1.09 0.45 1.06 0.50 1.03 0.47

4 x 3 0.93 0.27 1.24 0.27 1.09 0.54 1.05 0.60 1.02 0.55

6 x 3 0.94 0.27 1.26 0.28 1.07 0.54 1.02 0.61 1.00 0.56

4 x 4 0.92 0.31 1.23 0.32 1.02 0.67 0.96 0.78 0.94 0.71

4 x 6 0.98 0.36 1.30 0.36 0.95 0.92 0.84 1.06 0.83 0.99

U3.5

6 x 6 1.01 0.37 1.34 0.38 0.93 0.96 0.83 1.16 0.81 1.05

2 x 3 0.70 0.30 0.94 0.30 0.88 0.47 0.85 0.56 0.83 0.52

4 x 3 0.74 0.30 0.99 0.30 0.90 0.51 0.86 0.61 0.84 0.57

6 x 3 0.72 0.33 0.95 0.33 0.87 0.57 0.82 0.68 0.79 0.63

4 x 4 0.75 0.30 1.00 0.30 0.89 0.54 0.83 0.68 0.80 0.62

4 x 6 0.82 0.32 1.09 0.32 0.92 0.69 0.72 0.92 0.76 0.87

U5.0

6 x 6 0.94 0.32 1.25 0.32 0.91 0.86 0.80 0.99 0.79 0.90

49

Table 3.2.2.4 Comparison Of Design procedures ( Gravity+1.5% lateral load)





Type Confg( BxS )


2 x 3 0.78 0.49 1.03 0.50 0.89 0.79 0.84 0.88 0.82 0.82

4 x 3 0.79 0.46 1.03 0.47 0.87 0.77 0.83 0.83 0.79 0.75

6 x 3 0.83 0.48 1.09 0.50 0.90 0.82 0.84 0.86 0.82 0.80

4 x 4 0.85 0.47 1.11 0.49 0.88 0.82 0.82 0.90 0.80 0.84

4 x 6 0.95 0.49 1.22 0.51 0.89 0.93 0.80 1.03 0.78 0.94

AHDR

6 x 6 0.92 0.46 1.20 0.47 0.85 0.87 0.76 0.97 0.74 0.88

2 x 3 0.78 0.43 1.04 0.44 0.92 0.67 0.88 0.73 0.86 0.69

4 x 3 0.79 0.40 1.04 0.41 0.89 0.65 0.84 0.72 0.81 0.65

6 x 3 0.85 0.48 1.11 0.50 0.94 0.80 0.88 0.86 0.86 0.80

4 x 4 0.84 0.40 1.10 0.41 0.89 0.69 0.83 0.86 0.81 0.71

4 x 6 0.96 0.48 1.26 0.50 0.91 0.92 0.78 1.07 0.75 0.93

U3.0

6 x 6 0.98 0.49 1.29 0.51 0.91 0.97 0.76 1.07 0.74 0.97

2 x 3 0.83 0.36 1.11 0.38 0.99 0.57 0.95 0.59 0.92 0.56

4 x 3 0.88 0.44 1.15 0.45 0.98 0.72 0.95 0.75 0.92 0.71

6 x 3 0.90 0.38 1.17 0.40 0.99 0.64 0.96 0.68 0.92 0.62

4 x 4 0.89 0.38 1.17 0.40 0.96 0.67 0.91 0.72 0.89 0.67

4 x 6 0.98 0.50 1.29 0.52 0.95 0.96 0.85 1.04 0.83 0.95

U3.5

6 x 6 0.98 0.48 1.29 0.50 0.92 0.92 0.83 1.02 0.81 0.93

2 x 3 0.70 0.45 0.92 0.47 0.86 0.65 0.82 0.73 0.79 0.68

4 x 3 0.72 0.40 0.95 0.41 0.86 0.60 0.81 0.67 0.79 0.62

6 x 3 0.75 0.41 1.00 0.43 0.87 0.63 0.84 0.72 0.81 0.66

4 x 4 0.83 0.44 1.11 0.46 0.97 0.70 0.90 0.80 0.88 0.75

4 x 6 0.91 0.42 1.20 0.44 0.99 0.73 0.86 0.89 0.84 0.81

U5.0

6 x 6 0.95 0.47 1.25 0.49 0.92 0.91 0.82 1.02 0.80 0.93

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Table 3.2.2.5 Comparison Of Design procedures ( Gravity+15% lateral load)





Type Confg( BxS )


2 x 3 0.90 0.85 1.03 0.89 0.99 1.01 0.96 1.03 0.89 0.93

4 x 3 0.81 0.86 0.89 0.89 0.87 0.99 0.86 1.00 0.80 0.92

6 x 3 1.13 0.81 1.30 0.84 1.22 0.99 1.17 0.99 1.10 0.90

4 x 4 0.93 0.83 1.04 0.88 0.98 1.00 0.94 0.99 0.88 0.91

4 x 6 0.95 0.86 1.07 0.89 0.96 1.02 0.92 1.02 0.86 0.94

AHDR

6 x 6 1.11 0.82 1.25 0.86 1.12 0.98 1.07 0.93 1.00 0.91

2 x 3 0.91 0.83 1.05 0.88 1.02 1.00 1.00 1.00 0.92 0.91

4 x 3 1.09 0.79 1.24 0.83 1.21 0.96 1.18 0.97 1.09 0.87

6 x 3 0.92 0.85 1.05 0.90 1.01 1.02 0.99 1.03 0.91 0.94

4 x 4 0.99 0.88 1.12 0.92 1.05 1.04 1.03 1.07 0.95 0.96

4 x 6 0.98 0.88 1.10 0.92 0.99 1.05 0.92 1.05 0.87 0.97

U3.0

6 x 6 1.42 0.84 1.59 0.88 1.42 0.99 1.32 1.00 1.25 0.93

2 x 3 0.94 0.85 1.08 0.89 1.05 1.02 1.04 1.03 0.96 0.93

4 x 3 0.90 0.82 1.03 0.86 0.99 0.98 0.98 0.99 0.90 0.89

6 x 3 1.12 0.79 1.28 0.83 1.23 0.95 1.20 0.95 1.11 0.86

4 x 4 1.34 0.76 1.55 0.81 1.45 0.92 1.43 0.94 1.32 0.85

4 x 6 1.02 0.86 1.14 0.89 1.04 1.04 0.99 1.03 0.92 0.95

U3.5

6 x 6 1.01 0.86 1.14 0.89 1.02 1.06 0.97 1.03 0.91 0.94

2 x 3 0.74 0.83 0.86 0.88 0.85 0.98 0.83 1.00 0.77 0.91

4 x 3 0.86 0.81 0.99 0.86 0.97 0.97 0.95 0.99 0.88 0.89

6 x 3 1.65 0.73 1.90 0.77 1.86 0.87 1.81 0.89 1.67 0.80

4 x 4 1.49 0.73 1.71 0.78 1.67 0.88 1.58 0.88 1.48 0.81

4 x 6 1.93 0.76 2.23 0.80 2.09 0.91 1.96 0.93 1.81 0.84

U5.0

6 x 6 2.01 0.78 2.28 0.82 2.06 0.94 1.98 0.96 1.83 0.86

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Table 3.2.2.6 Summary of Comparitive study - Axial load capacitiesMethod 1 : AISI M_P Eq. & AISI Amplification factor

Method 2 : AISI M_P Eq. & AISI Amplification factorMethod - 3 : AISC M_P Eq. & AISC B2 factor (deflection based) - initial stiffnessMethod 4 : AISC M_P Eq. & AISC B2 factor (deflection based) secant stiffnessMethod 5 : AISI M_P Eq. & AISC B2 factor secant stiffness(Kx = 1.7; Kphi


The ratios reported are with respect to the axial load capacity : (Pr = Pcal / Pfem )Loading Statistics Method - 1 Method -2 Method - 3 Method - 4 Method - 5

Min 0.66 0.87 0.87 0.71 0.71Max 1.05 1.40 1.05 1.04 1.04GravityAvg 0.80 1.06 0.93 0.86 0.85Min 0.72 0.94 0.83 0.68 0.66Max 1.01 1.34 1.09 1.06 1.03Gravity+0.75% lat.Avg 0.85 1.14 0.92 0.85 0.83Min 0.70 0.92 0.85 0.76 0.74Max 0.98 1.29 0.99 0.96 0.92Gravity+1.5% lat.Avg 0.86 1.13 0.92 0.85 0.82Min 0.74 0.86 0.87 0.86 0.80Max 2.01 2.28 2.09 1.98 1.83Gravity+15% lat.Avg 1.13 1.29 1.21 1.17 1.09

Table 3.2.2.7 Summary of Comparitive study - Moment capacityMethod 1 : AISI M_P Eq. & AISI Amplification factorMethod 2 : AISI M_P Eq. & AISI Amplification factorMethod - 3 : AISC M_P Eq. & AISC B2 factor (deflection based) - initial stiffnessMethod 4 : AISC M_P Eq. & AISC B2 factor (deflection based) secant stiffnessMethod 5 : AISI M_P Eq. & AISC B2 factor secant stiffness(Kx = 1.7; Kphi


The ratios reported are with respect to the moment capacity : (Mr = Mcal / Mfem )Loading Statistics Method - 1 Method -2 Method - 3 Method - 4 Method - 5

Min 0.12 0.06 0.25 0.39 0.39Max 0.19 0.18 3.66 6.26 5.62GravityAvg 0.14 0.13 1.25 2.82 2.47Min 0.25 0.25 0.49 0.54 0.47Max 0.37 0.37 0.96 1.16 1.05Gravity+0.75% lat.Avg 0.32 0.32 0.69 0.81 0.74Min 0.36 0.38 0.57 0.59 0.56Max 0.50 0.51 0.96 1.07 0.97Gravity+1.5% lat.Avg 0.44 0.46 0.77 0.85 0.78Min 0.73 0.77 0.87 0.88 0.80Max 0.88 0.92 1.06 1.07 0.97Gravity+15% lat.Avg 0.82 0.86 0.98 0.99 0.90

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While the results presented in the above tables are based on preliminary investigation, itappears at this stage that the AISC procedure with Kx = 1.0; Kphi =0.8 in conjunction witheither initial joint stiffness (method 3) or with secant joint stiffness (method 4) may beused to predict the pallet rack frame capacities with sufficient accuracy.

4 PLANS FOR FURTHER STUDY

The following objectives have been set for the future study.

The effect of beam end shear and the ratio of the shear to moment on the beam-to-column joint stiffness by modeling the joints more accurately. A more refined finiteelement modeling including the endplates, tabs and actual hole shape is required for thispurpose. Such a study will shed light on several important topics as

the correlation of the results from a cantilever test and a portal test behavior under reversal of loading on the joint (such as the case when the

beams are subjected to reverse curvature).

The study of column base flexibility by means of experimental investigation andmore refined finite element analysis to include the contribution of foundation and anchorbolt deformations

The rotation restraint offered by the adjacent members to the columns against twistingat shelf beam level by 3D finite element analysis using shell-contact and simple beam-spring element models.

The work done so far has been empirical in nature. Having developed reasonablygood empirical design procedure (AISI M-P+AISC B2 factor or AISC M-P+AISC B2factor) to estimate the frame capacities, in the future work, the behavior of pallet rackframes will be studied in more detail enabling to explain the working of the aboveprocedures more rationally and convincingly.

The future work will also account for the effects of other parameters such as localbuckling etc., which were not considered in the present study.

5 SUMMARY and CONCLUSION

The behavior of pallet rack frames with semi-rigid beam-column joints and flexiblecolumn bases was studied by experimental and numerical (FEM) investigations. Ageneral q-M relationship was established to model the beam-to-column joint stiffnessof pallet rack frames. The column base flexibility of rack frames caused by base platebending was studied in detail and quantified in terms of the current specification value.Guidelines were framed to carry out nonlinear FE analysis of rack frames accounting forvarious influencing parameters.

53

A critical review of the current American rack frame design specification (RMI 1997)was carried out and found to be conservative by about 30% with regard to strengthestimates. The sources of conservatism in the specification were identified and an attemptwas made to improve the same. A few suggestions were made to deal with the bucklingand nonlinear behavior of frames in a more rational way. These include modifiedeffective length factors, beam-column interaction equation and moment amplificationfactor. The use of Kx = 1.0; Kphi =0.8 as effective length factors in conjunction with theAISC (or AISI) beam-column design procedure and AISC B2-fator was found to give themost acceptable frame strengths. Plans for the future work were discussed.

54

References

1. AISC (1998), LRFD Manual of steel construction - second edition, AmericanInstitute of Steel Construction Inc., Chicago, IL, USA.

2. AISI (1996), Specification or the Design of Cold-formed steel structural members,American Iron and Steel Institute, NW, Washington D.C., USA.

3. Pekz, T. (1975) Pallet rack tests, Department of Structural Engineering Report,Cornell University, September 1975.

4. RMI (1997), Specification for the design testing and utilization of industrial steelstorage racks, Rack Manufacturers Institute, Charlotte NC, USA.

Bibliography

1. ASCE (1997) Effective length and Notional load approaches for Assessing FrameStability : Implications for American Steel Design, Special publication of ASCE(Task Committee on Effective length factor).

2. Chen, W. F. (1985) Connection Flexibility and Steel Frames, Proceedings of asession sponsored by structural Division of the ASCE in conjunction with the ASCEConvention in Detroit, Michigan.

3. Chen, W. F. and Atsuta, T. (1975) Theory of Beam-Columns Volume 1: In-planeBehavior and Design, Advanced Book Program, Mc Graw Hill Book Company.

4. Chen, W. F. and Atsuta, T. (1976) Theory of Beam-Columns Volume 2: SpaceBehavior and Design, Advanced Book Program, Mc Graw Hill Book Company.

5. Chen, W. F. and Lui, E. M. (1991) Stability Design of Steel Frames, CRC Press.

6. Christopher, J. E., and Bjorhovde, R. (1999) Semi-rigid Frame Design Methods forPracticing Engineers, Engineering Journal AISC, Vol.36, No. 1, 12-28.

7. Galambos, T. V. (1960) Influence of Partial Base Fixity on Frame Stability, Journalof Structural Engineering ASCE, Vol. 86, No. ST3, May 1960.

8. Lau, S. M., Kirby, P. A. and Davison, J. B. (1999) Semi-rigid Design of PartiallyRestrained Columns in Non-sway Steel Frames, Journal of Constructional SteelResearch, Vol.50, 305-328.

9. Narayanan, R. (1985) Steel Framed Structures : Stability and Strength, ElsevierApplied Science Publishers.

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10. Ronald D. Zieman, William McGuire and Gregory G. Dierlein (1992) Inelastic LimitStates Design. Part I: Planar Frame Studies, Journal of Structural EngineeringASCE, Vol. 118, No.9, 2532-2549.

11. Ronald D. Zieman, William McGuire and Gregory G. Dierlein (1992) Inelastic LimitStates Design. Part II: Three-Dimensional Frame Study, Journal of StructuralEngineering ASCE, Vol. 118, No.9, 2550-2568.

12. Salmon, C. G., Schenker, L. and Jhonston, B. G. (1955) Moment-RotationCharacteristics of Column Anchorages, Proceedings of ASCE, Vol. 81, No. ST3,April 1955.

13. White, D. W., and Clarke, M. J. (1997) Design of Beam-columns. I: Philosophiesand Procedures, Journal of Structural Engineering ASCE, Vol. 123, No.12, 1556-1564.

14. White, D. W., and Clarke, M. J. (1997) Design of Beam-columns. II: Comparison ofStandards, Journal of Structural Engineering ASCE, Vol. 123, No.12, 1565-1575.

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APPENDIX

Effect of Nonlinear base stiffness on frame strength

As shown in the report (sec 3.1.1.2), the column base stiffness of a typical rack frame isnonlinear in nature. Hence, the corresponding moment-rotation relationship throughcomplete load history is required for proper consideration of its effect on the behaviorand failure strength of a frame. However, before developing such an expression, it isdeemed to quantify the effect of nonlinear base stiffness on the frame strength ascompared to the capacity of the frame with linear base stiffness. This study enables toassess the actual need of a nonlinear expression for base stiffness.

For this purpose, a large number of pallet rack frames of different types are analyzed byFEM for two base stiffness characteristics: Linear and Nonlinear. The details of the studyare discussed below.

Parameters

Types of Frames - 4 types (AHDR, U3.0, U3.5, U5.0)Frame configuration - 6 types (Bays : 2-6 ; Stories : 3-6)Loading - 4 types (Gravity load + Lateral load)

(Lateral load = 0.0%, 0.75%, 1.5% and 15% of axial load)

Column base stiffness - Linear / Nonlinear as per the case

This resulted in FE analyses of 192 rack frames (96 in each of the two base stiffness cases).

FE modeling features : Used beam element model with the following features.

Shelf Beams - General beam element with 6 dof/nodeColumns - Special beam element with 7 dof/nodeColumn perforations - Average section propertiesBeam-column joints - Nonlinear Spring elements with stiffness characteristics as defined in Section 2.2.Column base fixity - Linear column base : Stiffness equal to half the RMI specification value.

i.e. a stiffness value equivalent to (I/L = bd2/2880). This is also the average base stiffness value of a typical rack frame whose stiffness may range from 0.35 to 0.7 times the RMI value, as discussed in section 2.3. This is an approximation only.

Nonlinear column base : The following tri-linear moment-rotation relationship, which approximates the base stiffness of a column with 0.35`` thick base plate, is used in the analysis.

K = (6*E*I/L) for 0.0 < M < 0.75 Mu = (3*E*I/L) for 0.75 Mu < M < 0.95 Mu ~ 0.0 for 0.95 Mu < M

57

The nonlinear base stiffness as expressed by these expressions is compared with theactual moment-rotation relationship (as obtained by FEA) in the following figure.

Results and Discussion

Comparison of the FE results of the two cases showed that the frame strengths are noteffected significantly by nonlinear characteristics of the column base stiffness. Themaximum difference between the frame strengths corresponding to the linear andnonlinear base stiffness cases was found to be about 5%. The column base moments atfailure are found to be smaller than 0.75 Mu in both cases and this is understood to be thereason for the negligible effect of the nonlinear base stiffness characteristics.

Conclusion

From the above results, it may be said that the column base stiffness of pallet rack framescould be assumed as linear in nature without any loss of accuracy in estimating the framestrengths.

Base Fixity RMI vs FEM (7x7x0.375)

0

10

20

30

40

50

0 0.01 0.02 0.03 0.04


Mo

men

t (ki

ps-

in)

K_RMI FEMApprox. (nonlinear) Approx. (linear)

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APPENDIX

Notional load Approach

Isolated member

Objective: Possible application of notional load method as an alternative toeffective length method to estimate the design strength of cold-formed steel beam-column members.

Result : A parametric study was carried out on isolated cold-formed steel singlysymmetric lipped channel beam-column members with different end conditions to verifythe application of notional load approach. This approach involves the use of the actualunsupported length of the member, instead of effective length, to calculate the strength ofthe member.

While the method was found to be promising in predicting the strength of members thatmay fail in flexural buckling mode, the method failed to be encouraging in the case ofmembers sensitive to torsional-flexural bucking failure. However, it was decided tocontinue the work to check the validity of the method in the context of full frames andsub-assemblies. This was taken up in the following period and the results are brieflydescribed in section 2.2.

System level

Objectives

Feasibility study of the application of Notional load approach to estimate the failurestrength of cold-formed steel frames with semi-rigid beam column joints and flexiblecolumn bases.

Results

A parametric study on 5 types of unbraced (sway allowed) frames with 5 types of flexiblebeam-column joints and two column base stiffnesses (hinged and RMI value) subjectedto two different load combinations (Gravity+Lateral (1.5% and 15% of Gravity) load)was carried out to study the possible application of Notional load approach. Thefollowing observations had been made based on the comparative study between theNotional load approach and Nonlinear finite element analysis of frames.

The failure strength of cold-formed steel frames with semi-rigid joints could be estimatedby using the AISI (1996) member strength interaction equation with effective lengthfactors Kx = 1.0, Kphi = 1.0 when the column base fixity is equal to that of RMI value (I/L

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= bd2/1440) and Kx = 1.5, Kphi = 1.5 for hinged base. The results were found to beaccurate enough for all practical purposes.

It was also found, based on parametric study (the base stiffness as the variable) on AHDRframes of two configurations (2Bx3S and 4Bx4S), that the effective length factors Kx =1.0, Kphi = 1.0 could be used to predict the frame strength accurately when the basestiffness is in the range of fixed to 1/10th of the RMI value. As the base stiffnessapproaches hinged case the effective length factors are found to be increased to Kx = 1.5,Kphi = 1.5 in order to estimate the frame strengths accurately.

For column base fixities between hinged and 1/10 th of RMI value, a relationshipbetween the column base stiffness and the effective length factors needs to be established.However, it was deemed to quantify the practical range of column base stiffnesses beforetaking up further work in this direction.

Objectives

To evaluate the performance of the Notional load approach (considering it as a requiredor an acceptable alternative procedure) in the case of frames with practical column basefixities and other load combinations that have not been covered so far.

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INTERIM REPORT

COLD-FORMED STEEL FRAME AND BEAM-COLUMN DESIGN

By

Kotha S. Rao, Ph.D.

Professor Teoman Pekz(Proncipal investigator)

October 6, 2000

Department of Civil and Environmental EngineeringCornell University

Ithaca, NY 14853, USA

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Department of Civil and Environmental EngineeringCornell University

Ithaca, NY 14853, USA

INTERIM REPORT

COLD-FORMED STEEL FRAME AND BEAM-COLUMN DESIGN

By

Kotha S. Rao, Ph.D.

Professor Teoman Pekz(Proncipal investigator)

October 6, 2000

A Research Project sponsored by

The American Iron and Steel Instituteand Rack Manufacturers Institute

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CONTENTS

1 INTRODUCTION 1

2 OBJECTIVES AND SCOPE 2

3 DETAILS OF THE WORK 2

3.1 Analysis 2

3.2.1 Behavior of Frame Components 23.2.1.1 Beam-to-Column joint flexibility 33.2.1.2 Column base fixity 6

3.2.2 Frame Behavior 133.2.2.1 Shell-contact element model 13

Modeling features 13 Validation study 14

Stub column 16

Cantilever beam test 18

Pallet rack frames 233.2.2.2 Beam-spring element model 34

Modeling features 35 Validation study 36

3.2 Design 37

3.2.1 Current pallet rack design procedure (RMI 1997) 373.2.2 Development of Design Procedure 45

4 PLANS FOR FURTHER STUDY 52

5 SUMMARY AND CONCLUSION 52

REFERENCES / BIBLIOGRAPHY 54

APPENDIX 55

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