bajoria 2010

Journal of Constructional Steel Research 66 (2010) 428–441

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Journal of Constructional Steel Research

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Modal analysis of cold-formed pallet rack structures with semi-rigid connectionsKamal M. Bajoria ∗, Keshav K. Sangle, Rajshekar S. TalicottiDepartment of Civil Engineering, Indian Institute of Technology Bombay, Powai, Mumbai-400076, India

a r t i c l e i n f o

Article history:Received 23 December 2008Accepted 10 October 2009

Keywords:Finite element modal analysisCold-formed steelSemi-rigid connections

a b s t r a c t

The three dimensional (3D) model of conventional pallet racking systems were prepared using the finiteelement program ANSYS and free vibration modal analysis carried out on conventional pallet racks withthe 18 types of column sections developed alongwith semi-rigid connection. The stiffness of the connectorwas tested using the conventional cantilever method and also using a double cantilever method. Non-linear finite element analysis of both the testswas carried out. From the experimental study on connectionand finite elementmodal analysis, a simple analytical model that captures the seismic behavior of storageracks in their down aisle direction is proposed. The model is aimed at developing simplified equation forthe fundamental period of storage racks in their down aisle direction. A parametric study was carried outto find out fundamental mode shape and time period. Finite element method is used for the accuracy andappropriateness of cold-formed steel frame.

© 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Themost important parameter in the analysis and design of anystructure subjected to seismic load is the fundamental time pe-riod of the structure. In addition to the fundamental time period,understanding of the fundamental and other mode shapes of thestructure under seismic load is also equally important. For steelstructuresmade up fromhot rolled sections and for reinforced con-crete structures, significant research has been done on fundamen-tal time period and mode shapes of the structure. For these typesof structures analysis and design for seismic load is well set andalmost in all the codes this procedure is given. Rack structures arevery similar to the framed steelworks traditionally used for civiland commercial buildings, but great differences exist in membergeometry and in connection systems. The structural behavior ofindustrial storage racks under seismic load depends on how theindividual components like beam to column connections, columnbases andmembers perform interactively with each other. The be-havior of 3D frames under seismic load is very complex because ofmany parameters such as semi-rigid nature of connections, pres-ence of significant perforations in uprights and susceptibility to lo-cal buckling and torsional–flexural buckling. As towhichmethodofanalysis ismost suitable to solve this problemwill certainly dependon the tools available with the designer. The analysis model can beas simple as using a sub-structure model such as isolating the col-umn and using the alignment chart, or as sophisticated as using

∗ Corresponding author. Tel.: +91 22 9821129187.E-mail addresses: [email protected] (K.M. Bajoria), [email protected]

(K.K. Sangle).

0143-974X/$ – see front matter© 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.jcsr.2009.10.005

numerical methods to analyse the entire frame. With the avail-ability of powerful computers and software, the latter approachhas become more attractive, allowing more complex and efficientanalysis.The dynamic analysis and design of thin walled cold-formed

steel pallet racking structure with perforated open upright sectionand semi-rigid joints presents several challenges to the structuralengineers. Therefore understanding of the structural behavior ofrack structures is very important.Carlos Aguirre [1] performed non-linear analyses of the rack

structure under different seismic conditions, considering the mea-sured moment–rotation curves. Results showed that non-linearcalculated displacements weremore than twice the displacementspredicted with the classical linear analysis. Beale and Godley [2]performed sway analysis of spliced rack structures. The structureswere analyzed by considering an equivalent free-sway column andusing computer algebra generated modified stability functions toincorporate the non-linear P-∆ effects. The effect of semi-rigidbeam to upright, splice to upright connections are fully includedin the analysis. Each section of upright between successive beamlevels in the pallet rack is considered to be a single column ele-ment. The results of the analysis have been compared with a tra-ditional finite element solution of the problem. Godley et al. [6]performed analysis and design of un-braced pallet rack structuressubjected to horizontal and vertical loads. The structures are an-alyzed by considering an equivalent free-sway column and solv-ing the differential equations of flexure, including P-∆ effect. Initialimperfections within the frame are allowed. Results of the analysisare compared with a traditional non-linear finite element solutionof the same problem. Danny and Raymond [4] have carried out an-alytical work by modeling the pallet rack and merchandise in the

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K.M. Bajoria et al. / Journal of Constructional Steel Research 66 (2010) 428–441 429

Fig. 1. Medium weight section 1.6, 1.8 and 2.0 mm.

Fig. 2. Heavy weight section 2.0, 2.25 and 2.5 mm.

analysis software and run this model for the elcentro earthquakedata’s. The objective of this work is to perform a preliminary studyof dynamic behavior of a typical storage rack loadedwithmerchan-dise subjected to earthquake ground motion using finite elementsimulation. ABAQUS finite element code was used for this studyand the structural improvement was recommended based on thesimulation outcome. Blume et al. (Fema-460-2005) [3] performedstatic and response spectrum analyses to investigate the applica-bility of the eccentric braced frame concept to storage racks in or-der to improve their seismic behavior in the cross aisle direction.The results of the study indicated that aside from a considerablesavings in steel material, the eccentric bracing system could un-dergo significantly more inelastic deformations without structuralinstability than conventional bracing systems. Although the ana-lytical results were promising, the authors recommended also thatexperimental investigations needed to be conducted before imple-menting the eccentric bracing system in storage racks. Such exper-imental results are not available to date.Lewis [5] worked on the down aisle stability of rack structures.

In his analysis, a single internal upright column carrying both

vertical and horizontal loads was used. The model allowed forsemi-rigid connections between beams and uprights and betweenthe bases of uprights and the floors. However, the model onlyallowed for column flexibility below the level of the second beam,the rest of the column being treated as rigid. This assumptionbecomes increasingly unsafe as the number of storey levelsincreases.This paper deals with the free vibration finite element modal

analysis of 3D frame of a cold-form steel storage rack structures,with semi-rigid connections. Results are presented from the 3Danalysis carried out on 3D frameswith 18 types of column sectionsdeveloped. Based on these results simplified mathematical modelis proposed to find out the fundamental time period of the cold-formed steel conventional pallet rack structure.

2. Column sections used in the study

In this paper open sections and torsionally strengthened sec-tions were used. Original open sections were strengthened by pro-viding channel and hat stiffeners to avoid the local buckling of

430 K.M. Bajoria et al. / Journal of Constructional Steel Research 66 (2010) 428–441

Fig. 3. Torsionally strengthened MW and HW sections with channel and hat stiffeners.

125

50

Fig. 4. Details of box beam section (Thickness of web of the beam = 2 mm andthickness of flange of beam= 4 mm).

uprights. These sections areMW (MediumWeight) column sectionhaving three thicknesses 1.6 mm, 1.8 mm and 2.0 mm each withhat and channel stiffener and HW (Heavy Weight) column sectionhaving three thicknesses 2.0 mm, 2.25 mm and 2.5 mm each withhat and channel stiffener. Their cross sectional geometry is givenin Figs. 1–3. Purpose of choosing three different thicknesses is toknow the change in behavior when the sections are made locallystable by having higher thickness.

3. Calculations of sectional properties

For the above sections, sectional properties are calculated basedon weighted average section. A weighted average section is a sec-tion that uses an average thickness in the web portion to accountfor the absence of the material due to the holes along the lengthof the section and additional thickness for the additional materialof channel and hat stiffener. Excel program is developed to calcu-late the sectional properties of sections used in this study. Sectionalproperties of the sections are given in Table 1 and material prop-erties of the same sections are given in Table 2.

4. Stiffness of the connection

Stiffness of the connections for semi-rigid racks is consideredas shown in Table 3 in proposed analytical model. This stiffness is

Fig. 5. Connection details.

based on the experimental and non-linear finite element analysisstudy. The stiffness of the connector developed was tested usingthe conventional cantilever method and then also using a doublecantilever method. To verify the results obtained from both thetests, a full scale frame test was carried out. Non-linear finiteelement analysis of both the tests and also of the full scale test wascarried out using ANSYS software(Fig. 6).


Table 1Properties of column section.

Type of section Section propertiesA (mm2) Ixx (mm4) Iy (mm4) J (mm4) C.G (mm)

(x, y)

MWS 1.6 389.53 269028 302208 311.6 0, 46.31MWS 1.8 438.21 302626 339983 443.58 0, 46.32MWS 2.0 487.00 336369 377784 608.774 0, 46.31MWCS 1.6 512.80 426500 330300 463300 0, 39.12MWCS 1.8 561.70 463000 369200 512000 0, 40.01MWCS 2.0 610.60 498800 408000 559300 0, 40.75MWHS 1.6 611.60 432400 405600 521200 0, 38.21MWHS 1.8 660.40 469800 444400 570700 0, 39.03MWHS 2.0 709.30 506500 483200 618300 0, 39.74HWS 2.0 593.02 514270 854484 744.669 0, 54.66HWS 2.25 667.06 578437 961214 1060.02 0, 54.66HWS 2.5 741.21 642731 1068050 1454.09 0, 54.67HWCS 2.0 783.80 825550 990900 1065000 0, 45.03HWCS 2.25 856.90 891300 1099000 1163000 0, 46.09HWCS 2.05 929.90 955400 1208000 1255000 0, 46.98HWHS 2.0 887.90 830300 1175000 1156000 0, 44.85HWHS 2.25 960.90 896500 1283000 1252000 0, 45.81HWHS 2.5 1030.0 961100 1392000 1343000 0, 45.13

Table 2Material properties used in FEA.

Yield stress (σy) (MPa) Ultimate stress (σu) (MPa) Modulus of elasticity (E) (GPa) Density (Kg/m3) Percentage elongation. (%)

365 569 212 7860 29

Table 3Stiffness of the connections for semi-rigid rack.

Type of section HWS 2.0 HWS 2.25 HWS 2.5 MWS 2.0 MWS 1.8 MWS 1.6

Stiffness of connection (KN m) 50 70 90 80 60 40

(a) Finite element model of double cantilever test. (b) Four nodes monitored to determine the rotation (Double cantilevertest).

Fig. 6.

Rotational stiffness of the type of connection used in the palletrack solely depends upon the geometry and engagement lengthof the joint. Further design of rack structures mainly governedby stability analysis therefore actual stress level present in the

members is much lower than yield stress. In light of the aboveexplanation, results obtained from the double cantilever test onsemi-rigid frame for static load are used for the time dependentanalysis (Fig. 7).


Table 4Properties of the finite elements used in the analysis in brief.

Element name Shell 63 Solid 45 Conta 173

Position of connectorelement

Upright, beam, beam connector Connector hook, spacer bar,bracing

Contact between connector and upright

Description Plastic shell element 3D structural solid element 3D surface-to-surface contact elementNumber of nodes 4 8 4Degree of freedom x, y and z translation and rotational displacements x, y and z translation displacements x, y and z translation displacements

Fig. 7. Finite element simulation Von Misses stress (Double cantilever test).

5. Development of a single degree of freedom analytical model

For these types of frames, the stiffness in the down aisle direc-tion is almost always less than cross aisle direction. Therefore the

fundamental mode of vibration is always seen to be in down aisledirection. Therefore it is proposed here to develop an equivalentsingle degree of freedom analytical model. Analytical model pre-sented in this paper is developed on the same line which is givenin FEMA-460 [6] with a little modification. Analytical model usedin the Fema-460 is based on the experimental study conducted ontwo bay 3D pallet racks made up of hot rolled sections by Filia-trault and Wanitorkul. In the proposed single degree of freedomanalytical model of this paper, stiffness of connection is taken fromexperimental and non-linear finite element study conducted oncold-formed section. Free vibration analysis result of this analyt-ical model is compared with the results obtained by free vibrationfinite element modal analysis carried out on total 192 3D frameswith different configurations by using Ansys software (Fig. 8(a, b,c, d)).In order to derive simplified expressions for the down aisle

fundamental period, of storage racks, the following assumptionsare made.• Uniform beam to upright connection is used throughout theframe.• The beams are spaced uniformly along the height of frame withheight except that bottom beam.• All connections of the racks experience simultaneously similarrotations at all times. This assumption implies that the connec-tion rotational stiffness is smaller than the rotational stiffnessof the beams and uprights.

(a)M–θ curve by finite element analysis (Load on left side ofupright).

(b)M–θ curve by experiment (Load on left side of upright).

(c)M–θ curve by finite element analysis (Load on right side ofupright).

(d)M–θ curve by experiment (Load on right side ofupright).

Fig. 8.


(a) Kinematics assumption for storage racks indown aisle direction.

(b) Idealization ofsingle degree offreedom system.

Fig. 9.

a b

Fig. 10. (a) Single bay without mass. (b) Single bay with mass.

Fig. 11. Double bays with mass.


Fig. 12. Details of 3D frame in study.

Table 5HWS semi-rigid without mass, 6.05 m height (Single bay).

Type of frame Length of bay Time period frommathematical model (s)

Time period from FE freevibration analysis (s)

Ratio=MM/FEFVA Actual difference

HWS 2 mm Tk 2.4 0.377 0.411 0.917 0.033HWCS 2 mm Tk 2.4 0.371 0.417 0.890 0.045HWHS 2 mm Tk 2.4 0.358 0.406 0.881 0.048HWS 2 mm Tk 2.0 0.358 0.390 0.917 0.032HWCS 2 mm Tk 2.0 0.353 0.397 0.889 0.044HWHS 2 mm Tk 2.0 0.342 0.387 0.883 0.045HWS 2 mm Tk 1.6 0.338 0.368 0.920 0.029HWCS 2 mm Tk 1.6 0.337 0.376 0.896 0.038HWHS 2 mm Tk 1.6 0.326 0.367 0.888 0.040HWS 2.25 mm Tk 2.4 0.352 0.370 0.952 0.017HWCS 2.25 mm Tk 2.4 0.349 0.376 0.929 0.026HWHS 2.25 mm Tk 2.4 0.339 0.368 0.920 0.029HWS 2.25 mm Tk 2.0 0.334 0.351 0.949 0.017HWCS 2.25 mm Tk 2.0 0.333 0.359 0.928 0.025HWHS 2.25 mm Tk 2.0 0.324 0.351 0.921 0.027HWS 2.25 mm Tk 1.6 0.282 0.332 0.850 0.049HWCS 2.25 mm Tk 1.6 0.284 0.340 0.834 0.056HWHS 2.25 mm Tk 1.6 0.276 0.334 0.828 0.057HWS 2.5 mm Tk 2.4 0.336 0.337 0.995 0.001HWCS 2.5 mm Tk 2.4 0.334 0.343 0.973 0.009HWHS 2.5 mm Tk 2.4 0.326 0.338 0.966 0.011HWS 2.5 mm Tk 2.0 0.318 0.321 0.990 0.003HWCS 2.5 mm Tk 2.0 0.318 0.328 0.970 0.009HWHS 2.5 mm Tk 2.0 0.311 0.323 0.963 0.011HWS 2.5 mm Tk 1.6 0.299 0.309 0.968 0.009HWCS 2.5 mm Tk 1.6 0.302 0.311 0.968 0.009HWHS 2.5 mm Tk 1.6 0.295 0.307 0.963 0.011

• The fundamental period of vibration therefore be calculated byan equivalent single degree of freedom system correspondingto an assumed first down aisle mode of deformation of the rack.• In the present analytical model stiffness of the base plate is as-sumed as stiffness of the upright connected to the base plate.• It is assumed that the semi-rigid beam to upright connectionhas been designed with sufficient ductility so that the staticjoint properties can be assumed to remain same during timedependent analysis (Table 4).

Fig. 9(a) illustrates the assumed lateral first mode deformationof a three level storage rack in its down aisle direction accordingto the assumption listed above. It is assumed that the rotationalstiffness (kc) of the beam to upright connections is known at the

target displacement ‘u’. The total rotational stiffness between thebeam and uprights (kbu) indicated in Fig. 9, is the sum in series ofthe rotational stiffness of the connection (kc) and of the flexuralrotational stiffness of the beam end (kbu).

kbu =kckbekc + kbe

. (1)

Similarly, the total stiffness at the base of each upright (ku)indicated in Fig. 9(a) is the sum in series of the rotational stiffnessof the base plate (kb) and of the flexural stiffness of the base uprightend (kce).

ku =kbkcekb + kce

. (2)


Table 6MWS semi-rigid without mass, 6.05 m (Single bay).




MWS 1.6 mm Tk 2.4 0.408 0.417 0.977 0.009MWCS 1.6 mm Tk 2.4 0.420 0.430 0.976 0.010MWHS 1.6 mm Tk 2.4 0.412 0.423 0.975 0.010MWS 1.6 mm Tk 2.0 0.383 0.395 0.971 0.011MWCS 1.6 mm Tk 2.0 0.397 0.408 0.971 0.011MWHS 1.6 mm Tk 2.0 0.391 0.402 0.972 0.011MWS 1.6 mm Tk 1.6 0.358 0.370 0.967 0.012MWCS 1.6 mm Tk 1.6 0.373 0.385 0.968 0.012MWHS 1.6 mm Tk 1.6 0.370 0.381 0.970 0.011MWS 1.8 mm Tk 2.4 0.350 0.340 1.027 0.009MWCS 1.8 mm Tk 2.4 0.361 0.351 1.030 0.010MWHS 1.8 mm Tk 2.4 0.359 0.348 1.033 0.011MWS 1.8 mm Tk 2.0 0.326 0.322 1.011 0.003MWCS 1.8 mm Tk 2.0 0.339 0.334 1.016 0.005MWHS 1.8 mm Tk 2.0 0.338 0.331 1.020 0.006MWS 1.8 mm Tk 1.6 0.302 0.304 0.995 0.001MWCS 1.8 mm Tk 1.6 0.316 0.316 1.001 0.000MWHS 1.8 mm Tk 1.6 0.317 0.314 1.007 0.002MWS 2 mm Tk 2.4 0.350 0.340 1.027 0.009MWCS 2 mm Tk 2.4 0.361 0.351 1.030 0.010MWHS 2 mm Tk 2.4 0.359 0.348 1.033 0.011MWS 2 mm Tk 2.0 0.326 0.322 1.011 0.003MWCS 2 mm Tk 2.0 0.339 0.334 1.016 0.005MWHS 2 mm Tk 2.0 0.338 0.331 1.020 0.006MWS 2 mm Tk 1.6 0.302 0.304 0.995 0.001MWCS 2 mm Tk 1.6 0.316 0.316 1.001 0.000MWHS 2 mm Tk 1.6 0.317 0.314 1.007 0.002

Table 7HWS semi-rigid without mass, 4.5 m height (Single bay).




HWS 2 mm Tk 2.4 0.256 0.268 0.955 0.012HWCS 2 mm Tk 2.4 0.249 0.269 0.927 0.019HWHS 2 mm Tk 2.4 0.239 0.260 0.919 0.021HWS 2 mm Tk 2.0 0.243 0.254 0.957 0.010HWCS 2 mm Tk 2.0 0.238 0.256 0.931 0.017HWHS 2 mm Tk 2.0 0.229 0.248 0.923 0.018HWS 2 mm Tk 1.6 0.230 0.239 0.962 0.008HWCS 2 mm Tk 1.6 0.227 0.242 0.938 0.014HWHS 2 mm Tk 1.6 0.218 0.235 0.930 0.016HWS 2.25 mm Tk 2.4 0.240 0.242 0.989 0.002HWCS 2.25 mm Tk 2.4 0.236 0.244 0.965 0.008HWHS2.25 mm Tk 2.4 0.228 0.238 0.957 0.010HWS 2.25 mm Tk 2.0 0.228 0.230 0.989 0.002HWCS 2.25 mm Tk 2.0 0.225 0.233 0.967 0.007HWHS2.25 mm Tk 2.0 0.218 0.227 0.959 0.009HWS 2.25 mm Tk 1.6 0.215 0.217 0.991 0.001HWCS 2.25 mm Tk 1.6 0.214 0.220 0.970 0.006HWHS2.25 mm Tk 1.6 0.207 0.215 0.964 0.007HWS 2.5 mm Tk 2.4 0.229 0.222 1.031 0.006HWCS 2.5 mm Tk 2.4 0.226 0.225 1.006 0.001HWHS 2.5 mm Tk 2.4 0.220 0.220 0.999 0.000HWS 2.5 mm Tk 2.0 0.217 0.211 1.028 0.006HWCS 2.5 mm Tk 2.0 0.216 0.214 1.007 0.001HWHS 2.5 mm Tk 2.0 0.210 0.210 1.000 0.000HWS 2.5 mm Tk 1.6 0.205 0.199 1.027 0.005HWCS 2.5 mm Tk 1.6 0.205 0.203 1.009 0.001HWHS 2.5 mm Tk 1.6 0.200 0.199 1.002 0.0009

5.1. Simplified equation for fundamental period of vibration

The applied moment about the base (Mbi) caused by the lateralinertia forces is given by

Mbi =NL∑i=1

Wpiguhpi =

1g

NL∑i=1

Wpih2piθ (3)

whereWpi= the weight of ith pallet supported by the storage rack.

hpi= the elevation of the center of gravity of the ith pallet withrespect to the base of the storage rack.g= the acceleration due to gravity.NL= the number of loaded level.

The resisting moment about the base (Mbr ) is given by

Mbr = −(Nckbu + NbKu

)θ (4)

whereNc= the number of beam to upright connections.


Table 8MWS semi-rigid without mass, 4.5 m height (Single bay).




MWS 1.6 mm Tk 2.4 0.303 0.296 1.022 0.006MWCS 1.6 mm Tk 2.4 0.308 0.302 1.022 0.006MWHS 1.6 mm Tk 2.4 0.307 0.300 1.023 0.007MWS 1.6 mm Tk 2.0 0.268 0.265 1.013 0.003MWCS 1.6 mm Tk 2.0 0.276 0.272 1.012 0.003MWHS 1.6 mm Tk 2.0 0.270 0.266 1.013 0.003MWS 1.6 mm Tk 1.6 0.250 0.248 1.008 0.002MWCS 1.6 mm Tk 1.6 0.260 0.256 1.012 0.003MWHS 1.6 mm Tk 1.6 0.255 0.252 1.014 0.003MWS 1.8 mm Tk 2.4 0.271 0.266 1.016 0.004MWCS 1.8 mm Tk 2.4 0.277 0.271 1.020 0.005MWHS 1.8 mm Tk 2.4 0.277 0.271 1.020 0.005MWS 1.8 mm Tk 2.0 0.244 0.239 1.020 0.004MWCS 1.8 mm Tk 2.0 0.252 0.246 1.025 0.006MWHS 1.8 mm Tk 2.0 0.249 0.242 1.029 0.007MWS 1.8 mm Tk 1.6 0.227 0.224 1.012 0.002MWCS 1.8 mm Tk 1.6 0.236 0.232 1.019 0.004MWHS 1.8 mm Tk 1.6 0.234 0.229 1.024 0.005MWS 2 mm Tk 2.4 0.254 0.243 1.046 0.011MWCS 2 mm Tk 2.4 0.260 0.247 1.052 0.012MWHS 2 mm Tk 2.4 0.261 0.248 1.050 0.012MWS 2 mm Tk 2.0 0.237 0.229 1.034 0.007MWCS 2 mm Tk 2.0 0.244 0.234 1.042 0.009MWHS 2 mm Tk 2.0 0.245 0.235 1.039 0.009MWS 2 mm Tk 1.6 0.219 0.214 1.023 0.005MWCS 2 mm Tk 1.6 0.227 0.220 1.033 0.007MWHS 2 mm Tk 1.6 0.228 0.220 1.036 0.007

Table 9HWS and MWS semi-rigid with mass, 4.5 and 6.05 m height (Single bay).




HWS and MWS semi-rigid with mass, 6.05 m height

HWS 2 mm Tk 2.4 1.722 1.931 0.891 0.208HWCS 2 mm Tk 2.4 1.614 1.889 0.854 0.275HWHS 2 mm Tk 2.4 1.531 1.815 0.843 0.284HWS 2 mm Tk 2.0 1.551 1.756 0.883 0.205HWCS 2 mm Tk 2.0 1.458 1.709 0.853 0.250HWHS 2 mm Tk 2.0 1.385 1.643 0.843 0.257HWS 2 mm Tk 1.6 1.368 1.550 0.882 0.181HWCS 2 mm Tk 1.6 1.289 1.509 0.854 0.219HWHS 2 mm Tk 1.6 1.226 1.451 0.845 0.224MWS 2 mm Tk 2.4 1.910 1.910 0.999 0.000MWCS 2 mm Tk 2.4 1.873 1.875 0.998 0.002MWHS 2 mm Tk 2.4 1.819 1.822 0.998 0.002MWS 2 mm Tk 2.0 1.697 1.728 0.982 0.030MWCS 2 mm Tk 2.0 1.667 1.696 0.982 0.029MWHS 2 mm Tk 2.0 1.621 1.648 0.983 0.026MWS 2 mm Tk 1.6 1.472 1.528 0.963 0.055MWCS 2 mm Tk 1.6 1.449 1.500 0.966 0.050MWHS 2 mm Tk 1.6 1.411 1.457 0.968 0.045

HWS and MWS semi-rigid with mass, 4.5 m height

HWS 2 mm Tk 2.4 1.163 1.290 0.901 0.127HWCS 2 mm Tk 2.4 1.083 1.248 0.868 0.164HWHS 2 mm Tk 2.4 1.022 1.191 0.858 0.169HWS 2 mm Tk 2.0 1.051 1.167 0.900 0.116HWCS 2 mm Tk 2.0 0.980 1.129 0.868 0.148HWHS 2 mm Tk 2.0 0.926 1.079 0.858 0.152HWS 2 mm Tk 1.6 0.929 1.031 0.900 0.102HWCS 2 mm Tk 1.6 0.868 0.998 0.870 0.129HWHS 2 mm Tk 1.6 0.821 0.953 0.861 0.132MWS 2 mm Tk 2.4 1.307 1.300 1.005 0.007MWCS 2 mm Tk 2.4 1.278 1.272 1.004 0.006MWHS 2 mm Tk 2.4 1.225 1.220 1.004 0.005MWS 2 mm Tk 2.0 1.167 1.176 0.991 0.009MWCS 2 mm Tk 2.0 1.142 1.151 0.992 0.008MWHS 2 mm Tk 2.0 1.097 1.112 0.987 0.014MWS 2 mm Tk 1.6 1.017 1.046 0.973 0.028MWCS 2 mm Tk 1.6 0.998 1.018 0.980 0.020MWHS 2 mm Tk 1.6 0.9616 0.984 0.976 0.022


(a) First mode shape—sway in down aisledirection.

(b) Second mode shape—torsion. (c) Third mode shape.

(d) Fourth mode shape—sway in cross aisledirection.

(e) Fifth mode shape—torsion. (f) Sixth mode shape—torsion, buckling ofbracing and sway in cross aisle direction.

Fig. 13. Mode shape of 3D frame in modal analysis.

Nb= the number of base plate connections.

Equating the applied moment to the resisting moment yields theequation of motion for the equivalent single degree of freedomsystem.

1g

NL∑i=1

Wpih2piθ + (Nckbu + Nbku) θ = 0. (5)

The fundamental period of vibration (T ) is expressed as

T = 2π

√√√√√ NL∑i=1Wpih2pi

g (Nckbu + Nbku). (6)

Substituting Eqs. (1) and (2) into Eq. (6) yields

T=2π

√√√√√√√NL∑i=1Wpih2pi

g(Nc(kckbekc+kbe

)+ Nb

(kbkcekb+kce

)) (7)


(g) Seventh to seventeenth modeshape—buckling of beam.

(h) Eighteenth mode shape—third sway indown aisle direction.

(i) First mode shape—sway in down aisle direction.

(j) Second mode shape—torsion.

Fig. 13. (continued)


(k) Third mode shape—second sway in down aisle direction.

(l) Fourth mode shape—sway in cross aisle direction.

(m) Fifth mode shape—second sway in cross aisle direction and torsion.

Fig. 13. (continued)

whereWpi= the effective seismic weight of the Ith pallet supported by

the storage rack.hpi= elevation of center of gravity of the Ith pallet with respect

to the base of the storage rack.g= acceleration of gravity.NL= number of loaded level.kc= rotational stiffness of each beam to upright connection.

kb= rotational stiffness of each base plate connection assumingkb = kc .

Nc= the number of beam to upright connection.

Nb= the number of base plate connection.

The beam end and the base upright end rotational stiffness aregiven by


Table 10HWS and MWS semi-rigid without mass, 4.5 m and 6.05 m height (Double bay).




HWS and MWS semi-rigid without mass, 6.05 m height (Double bay)

HWS 2 mm Tk 2.4 0.344 0.373 0.921 0.029HWCS 2 mm Tk 2.4 0.341 0.383 0.889 0.042HWHS 2 mm Tk 2.4 0.333 0.385 0.866 0.051MWS 2 mm Tk 2.4 0.319 0.321 0.993 0.001MWCS 2 mm Tk 2.4 0.328 0.329 0.998 0.000MWHS 2 mm Tk 2.4 0.333 0.339 0.983 0.005

HWS and MWS semi-rigid without mass, 4.5 m height (Double bay)


HWS and MWS semi-rigid with mass, 6.05 m height (Double bay)


HWS and MWS semi-rigid with mass, 4.5 m height (Double bay)


kbe =6EIbL

kce =4EIcH

whereE= Young’s modulus of the beams and columns.Ib=moment of inertia about the bending axis of each beam.L= clear span of the beam.Ic=moment of inertia of each base upright.H= clear height of the upright.

6. Parameters used in the study for conventional pallet rackstructure

Finite element analysis was done for 192 frames braced withcombination of horizontal and diagonal bracing. Parameters usedin the study are

(a) Height of the frame= 4.55 and 6.05 m.(b) 18 types of column sections.(c) One type of bracing systems i.e. Horizontal with inclined.(d) Hollow stringer beam = 150 mm deep × 50 mm wide and2.0, 2.5, 2.75 and 3 mm thick.

(e) Center to center distance between beams= 0.83 and 0.9 m.(f) Depth of frame= 1 m.(g) Three length of bay= 1.6, 2.0 and 2.4 m.(h) Vertical distance between the horizontal brace axis= 0.6 m.(i) Cross sectional area of horizontal brace = 0.00316 m2.(j) Cross sectional area of diagonal(inclined) brace = 0.00316m2.(k) Horizontal distance betweenneutral axis of the columns=1m.(l) Angle between horizontal and diagonal braces = 32◦.

7. Finite element modeling and analysis

Finite element analysis of the 3D frameswasdevelopedwith thehelp of APDL (Ansys parametric design language) to run the samein Ansys and get the results of the desired analysis. The structureof the program is developed in APDL.Finite element analysis was done for 192 frames. Uprights

(Fig. 3) of the frames and flange and web of box beam (Fig. 4) aremodeled in shell 63 element with mesh size 5 mm. The spacersused for connecting the upright and bracing are modeled usingsolid 45 elements. Spacer bar are created at the hole of flangepart of the upright at a distance 600 mm center to center. Fourcorner of holes on both the flange of upright and four nodes ofthe spacer bar on each side of flange are connected to each other.Bracings are connected to spacer bar of each upright. At the levelof beam, beam connectors in the form of angle section are createdin shell 63 element. One leg of the beam connector is directlyconnected to the beam end and on other leg 3 hooks (Fig. 5) inconnectors are created in solid 45 element and these hooks inconnector are connected to the four corner of the holes onweb partof upright. These connections represent actual connection betweenbeamand uprights. These connections are considered as semi-rigidconnection (Figs. 10–12).

8. Results and observations

8.1. Results

Results of 3D frames are tabulated in Appendix. Mode shapes ofthe frame are shown in Fig. 13. Most of the frame has same type ofmode shape.


Table 11HWS and MWS Semi-rigid with mass, 4.5 m and 6.05 m height (Six bay).




HWS and MWS semi-rigid without mass, 6.05 m height (Six bay)


HWS and MWS semi-rigid without mass, 4.5 m height (Six bay)


HWS and MWS semi-rigid with mass, 6.05 m height (Six bay)


HWS and MWS semi-rigid with mass, 4.5 m height (Six bay)


8.2. Observations

1. Fundamental time period of rack structures of semi-rigidconnections from finite element free vibration modal analysisare very close to fundamental time period from proposedanalytical model.

2. Though the percentage variation of fundamental time period is1 to 10 the actual difference between time periods is very less.

3. With the help of simulation of double cantilever test ofconnection, stiffness of the connection can be found. There is noneed to always conduct the experiments to find out the stiffnessof the connection.

4. Percentage difference between times period of frames madeup from original open section is less as compared to framesmade up from torsionally strengthened sections (i.e. Sectionwith channel and hat stiffener).

5. Proposedmathematical model has been checked for single, twoand six bays of frameswith andwithoutmass. For all the framesresults from mathematical model are very close to the freevibration modal analysis result. That means this model can beused for any number of bays.

9. Conclusions

To study the various mode shapes and to find the fundamental

time period, finite element free vibration model analysis of 3Dconventional pallet rack structures was carried out. Simplifiedmathematical model is proposed to find outthe fundamental timeperiod of semi-rigid conventional pallet rack structure. Result ofsimplified mathematical model and finite element free vibrationmodel analysis was compared so that this simplifiedmathematicalmodel can be implemented in the design of these frames.

Appendix

See Tables 5–11.

References

[1] Carlos Aguirre. Seismic behavior of rack structures. Journal of Construction SteelResearch 2005;61:607–24.

[2] Beale RG, Godley MHR. Sway analysis of spliced pallet rack structures. Journalof Computer and Structures 2004;83:2145–6.

[3] FEMA-460. Seismic considerations for steel storage racks located in areasaccessible to the public; September 2005.

[4] ChanDannyH, Yee RaymondK. Structural behavior of storage rack under seismicground motion. CA: San Jose State University; 2003.

[5] Lewis GM. Stability of rack structures. Journal of Thin-Walled Structures 1991;12:163–74.

[6] Godley MHR, Beale RG, Feng X. Analysis and design of down aisle pallet rackstructures. Journal of Computer Structures 2000;77(4):391–401.

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