Finite Element Modeling of Bolted Connections Between Coldformed

Engineering Structures 22 (2000) 1271–1284www.elsevier.com/locate/engstruct

Finite element modeling of bolted connections between cold-formed steel strips and hot rolled steel plates under static shear

loading

K.F. Chung*, K.H. IpDepartment of Civil and Structural Engineering, The Hong Kong Polytechnic University, Hung Hom, Hong Kong, People’s Republic of China

Received 3 March 1999; received in revised form 26 July 1999; accepted 12 August 1999

Abstract

A finite element model with three-dimensional solid elements is established to investigate the structural performance of boltedconnections between cold-formed steel strips and hot rolled steel plates under shear. Non-linear material, geometrical and contactanalysis is carried out to predict the load–extension curves of typical bolted connections with cold-formed steel strips of high yieldstrength and low ductility. The failure mode of interest in the present investigation is the bearing failure of cold-formed steel striparound bolt holes; a full description of the finite element model is presented. Based on test data, a stress–strain curve for the cold-formed steel strip is proposed which allows the cold-formed steel strip to yield and degrade in strength at large strain. The load–extension curves of four sets of test specimens were successfully predicted up to an extension of 3 mm. The predicted bearingresistance and the extensional stiffness of the bolted connections compare well with test data. It is found that stress–strain curves,contact stiffness and frictional coefficient between element interfaces, and clamping force developed in bolt shanks are importantparameters for accurate prediction of the load–extension curves of bolted connections. 2000 Elsevier Science Ltd. All rightsreserved.

Keywords:Bolted connections; Cold-formed steel connections; Plasticity; Strength degradation; Non-linear contact analysis

1. Introduction

Thin steel products are used extensively in the build-ing industry, ranging from purlins and lintels, to roofsheeting and floor decking. Many different shapes andsizes of thin steel sections are available for use either asbasic building elements for assembly on site, or alterna-tively as pre-fabricated panels or frames. These thin sec-tions are cold-formed by rolling or bending from stripsteel, and are given the generic title of ‘cold-formed steelsections’. The method of manufacture is important as itdifferentiates these products from hot rolled steel sec-tions such as I sections, channel sections and hollow sec-tions.

The most common sections are C and Z sections, andthe thickness of these sections typically ranges from 1.2mm to 3.2 mm. Moreover, there are a whole range of

* Corresponding author. Tel.: + 852-2766-6063; fax: + 852-2334-6389.

0141-0296/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved.PII: S0141-0296 (99)00082-6

variants of these basic shapes, including sections withsingle and double edge lips, and sections with internalstiffeners and longitudinal bends in webs. Both steelwith yield strength of 280 N/mm2 and 350 N/mm2 arecommonly used. Recently, steel with high yield strengthof 450 N/mm2 is also used in lipped C and Z sections,while new steel cladding with yield strength of 550N/mm2 is also available. There are a number of codifieddesign recommendations [1–6] on the design of cold-formed steel structures together with complementarydesign guides [7–9] and worked examples [10] to assistpractising engineers.

2. Connections between cold-formed steel stripsand hot rolled steel plates

In building construction, hot rolled steel sections arecommonly used as primary structural frames while cold-formed steel sections are used as secondary structuralmembers to support claddings as part of the building

1272 K.F. Chung, K.H. Ip / Engineering Structures 22 (2000) 1271–1284

envelopes. Connections between hot rolled steel andcold-formed steel members are commonly achieved withbolts and web-cleats. At present, many designexpressions on the load carrying capacities of boltedconnections against bearing failure may be found in vari-ous design recommendations. However, they are semi-empirical expressions formulated according to test dataof specific ranges of material properties and geometricaldimensions. While the semi-empirical expressions areapplicable for cold-formed steel strips with high ductilityand design strengths between 280 N/mm2 and 350N/mm2, they may not be suitable for high strength cold-formed steel strips of low ductility.

Due to the advancement of steel technology, cold-for-med steel strips with design yield strength up to 550N/mm2 are now available for building applications.Although the increment in strength is highly desirable,the ductility of high strength cold-formed steel strips issignificantly reduced (to only a few percentageelongation). This may affect adversely the structural per-formance of the high strength cold-formed steel strips,particularly, at connections where local stresses andstrains are very high, leading to premature failure. Aclose examination on the strength and the deformationcharacteristics of bolted connections between cold-for-med steel strips and hot rolled steel plates is desirablein order to provide efficient and safe design recommen-dations for high strength cold-formed steel structures.

This paper presents part of the findings of a researchproject on the structural performance of bolted connec-tions of cold-formed steel structures. The research pro-ject aims to examine the strength and stiffness of boltedconnections between cold-formed steel strips and hotrolled steel plates in typical building applications. Theinvestigation may be divided into the following parts:

1. coupon tests of cold-formed steel strips with differentyield strengths, tensile strengths and elongation lim-its;

2. lap shear tests on bolted connections between cold-formed steel strips and hot rolled steel plates;

3. numerical investigation on lap shear tests using three-dimensional solid elements with non-linear material,geometrical and contact analysis. The areas of inter-est are:O the load–extension curves of bolted connections

based on ‘engineered’ stress–strain curves withstrength degradation,

O the local stress and strain distribution around boltholes,

O the load path between cold-formed steel strips andhot rolled steel plates, and relative contributionsof end bearing resistance and frictional resistancebetween contact interfaces, and

O patterns of yielding and strength degradation athigh strains.

The coupon tests provide basic data to generate appro-priate ‘engineered’ stress–strain curves with strengthdegradation for cold-formed steel strips. The proposedstress–strain curves are then incorporated into a finiteelement model to examine the structural performance ofbolted connections under shear. The load–extensioncurves of the bolted connections predicted by finiteelement models are then compared with those of the lapshear tests. Full description of the finite element modelsand also the results of the finite element analyses arepresented. The significance of the stress–strain curves,contact stiffness and frictional coefficient betweenelement interfaces, and clamping force developed in boltshanks are discussed in detail.

3. Coupons tests

A total number of six tensile tests were carried outwith two steel grades of galvanized cold-formed steelstrips, namely G300 and G550; the design yield strengthof the strips are 300 N/mm2 and 550 N/mm2, respect-ively, while the nominal thicknesses are 1.6 mm and 1.5mm, respectively. For each steel material, three couponsof 20 mm width by 120 mm length were axially elon-gated up to rupture, and both strain gauges and displace-ment transducers were used to measure the deformationof the coupons before and after yielding, respectively.Table 1 summarizes the measured material propertiesand parameters of the two steel materials; the actualthicknesses of the steel materials are measured directlyfrom the steel strips after careful removal of galvaniz-ing coating.

The stress–strain curves of the cold-formed steel stripsare then deduced from the measured load–extensioncurves of the coupons based on initial dimensions asshown in Fig. 1a. It should be noted that these curvesare only accurate before the onset of necking, thereafterlocalized deformation will introduce errors to the curves[11]. Correction to the measured stress–strain curves forlarge plastic deformation is then applied to give the‘true’ stress–strain curves, which are also plotted ontoFig. 1a for direct comparison. Refer to Appendix A fordetails of the ‘true’ stress–strain curves.

4. Lap shear tests

A total of 12 lap shear tests [3,5] with both G300 andG550 cold-formed steel strips were carried out. Table 2summarizes the material grades and the nominal thick-nesses of the cold-formed steel strips together with thedimensions of washers; the bolts are grade 8.8 and 12mm diameter. In the present study, the material gradeand the dimension of the connections are chosen in sucha way that bearing failure in the cold-formed steel strips

1273K.F. Chung, K.H. Ip / Engineering Structures 22 (2000) 1271–1284

Table 1Measured material properties of cold-formed steel strips in coupon tests

Test specimen Thickness Elastic modulus Yield strength Tensile strength Elongation limit

Nominal (mm) Actual (mm) (kN/mm2) (N/mm2) (N/mm2) (%)

G550-1 1.6 1.54 222.2 616.7 625.1 1.3G550-2 1.6 1.56 220.0 622.1 648.4 4.1G550-3 1.6 1.56 212.9 623.2 641.6 2.9G300-1 1.5 1.41 203.3 320.1 440.1 20.0G300-2 1.5 1.42 206.7 321.2 470.4 25.0G300-3 1.5 1.41 202.7 320.3 450.3 21.0

Fig. 1. (a) Stress–strain curves from coupon tests and engineeringstress–strain curves of test specimens. (b) Proposed stress–strain curvesfor G550 cold-formed steel strips under tension and compression. (c)Proposed stress–strain curves for G300 cold-formed steel strips undertension and compression.

around bolt holes is always critical. For each combi-nation of material grade and washer, three specimens arespecified for testing. Fig. 2 illustrates the overall dimen-sions of the test specimens while the test set-up is shownin Fig. 3.

In each test, two identical cold-formed steel stripswere each bolted at one end to a 25-mm-thick hot rolledsteel plate, and the bolts were installed with torquewrench to a torque of 30 Nm. The ends of the test speci-men were then attached through hinges onto a tensiletest machine in which a tensile force was applied gradu-ally to the test specimen in a displacement control modeto establish the complete load–extension curve, beforeand after peak load. Both the applied tensile force andthe separation between two specific locations of the testspecimen were measured and stored into a data logger.The prescribed extension rate was set to 1 mm/min,while data were collected at an extension of 0.05-mmintervals up to a total end extension of 5.0 mm.

In addition to the total load carrying capacities ofbolted connections, the following aspects of the behav-iour of the connections are compared and discussed:

O the resistance contribution from bearing actionbetween the bolt shanks and the cold-formed steelstrips, and

O the resistance contribution of frictional actionbetween the washers and the cold-formed steel strips.

5. Test results of lap shear tests

The measured load carrying capacities of the boltedconnections at specific extensions are presented in Table2 while the measured load–extension curves of thebolted connections are presented in Figs. 4 and 5. Itshould be noted that in order to avoid excessive defor-mation in connected members, the maximum load carry-ing capacities of the bolted connections are taken as themaximum applied load corresponding to an extensionless than or equal to 3 mm [1,2]. In general, bearing


Table 2Measured strengths and flexibilities of bolted connections under static shear loadinga

Test Cold-formed steel strips Washers Applied load (kN) under extension at Deflection at 2/3 of maximumapplied load

Grade Nominal 1 mm 2 mm 3 mm (mm)thickness(mm)

A11A-1 19.85 26.44 28.96 1.02A11A-2 G550 1.6 A 18.81 25.20 28.82 1.09A11A-3 21.77 26.78 28.92 0.70

(Averaged value) 20.14 26.14 28.90 0.94A11B-1 16.66 23.31 27.18 1.22A11B-2 G550 1.6 B 17.99 24.84 29.11 1.15A11B-3 17.79 24.65 29.77 1.18

(Averaged value) 17.48 24.27 28.69 1.18A21A-4 14.73 18.44 17.91 0.69A21A-5 G300 1.5 A 16.13 18.47 18.63 0.48A21A-6 13.61 17.26 19.04 0.82

(Averaged value) 14.83 18.06 18.53 0.66A21B-4 13.66 17.66 20.79 0.78A21B-5 G300 1.5 B 16.32 20.01 20.52 0.58A21B-6 14.32 19.14 20.36 0.77

(Averaged value) 14.77 18.94 20.56 0.71

a Washer A: external diameter 25.7 mm, internal diameter 13.0 mm, thickness 2.3 mm. Washer B: external diameter 32.1 mm, internal diameter14.6 mm, thickness 3.0 mm.

Fig. 2. Geometry of cold-formed steel strip in lap shear test(dimensions in mm).

failure of cold-formed steel strips was always critical inall the tests while large localized plastic deformationaround bolt holes was also evident. Findings of the testresults of both G550 and G300 test specimens arepresented below.

5.1. G550 test specimens

The maximum load carrying capacities of test speci-mens A11A are found to be very close to those in testspecimens A11B at respective extensions, despite the useof larger washers in test specimens A11B. As the sametorque is used for bolt installation in both sets of testspecimens, the compressive normal stresses and also thefrictional stresses between the washers and the cold-for-med steel strips will vary inversely with the sizes of thewashers. The two effects of reduced frictional stress andincreased contact area tend to cancel out each other, andthe load carrying capacities of both sets of test specimensare very close. Fig. 3. Configuration of typical test specimen in lap shear test.


Fig. 4. Theoretical and experimental load–deflection curves forbolted connections with G550 cold-formed steel strips.

It is further considered that the resistance contributionof bearing action between the bolt shanks and the cold-formed steel strips is much more important than the con-tribution of frictional action between the washers and thecold-formed steel strips. The bearing resistance isdirectly related to the tensile strength of the steel strips,and is, thus, more important in bolted connectionsbetween high strength cold-formed steel strips.

The end sections of test specimens A11A and A11Bwere found to curl up after the peak load carryingcapacities are reached after the extension exceeded 3mm.


The maximum load carrying capacities of test speci-mens A21B are found to be approximately 10% higherthan those in test specimens A21A due to the presenceof larger washers. This suggests that the resistance con-tribution of frictional action between the washers andthe cold-formed steel strips may be significant for lowstrength cold-formed steel strips. Furthermore, curling ofthe ends of the specimens was observed at an extensionof about 2 mm, before the peak load carrying capacitieswere reached.

Both the coupon tests and the lap shear tests providebasic data in the forms of stress–strain curves and load–

Fig. 5. Theoretical and experimental load–deflection curves forbolted connections with G300 cold-formed steel strips.

extension curves for the calibration of finite elementmodeling.

6. Numerical investigation

With the advent of computer hardware and software,numerical simulation has drawn the attention ofresearchers in various engineering disciplines. In thefield of structural engineering, results from finite elementsimulation may provide detailed information on thestress and the strain distributions in structures. Suchinformation is not easily available from experiments,and, therefore, numerical investigation may be used toprovide supplementary data for improved understanding.Furthermore, parametric studies on the finite elementmodels may be performed to improve the efficiency ofstructural design. In recent years, advancement in finiteelement formulation has produced robust algorithms inhandling large deformation, plasticity and contact com-patibility. This allows numerical modeling to investigatevarious large strain elasto-plastic problems [12–14].While much effort has been devoted to two-dimensionalelasto-plastic analysis with contact elements [15,16],extension of the technique to three-dimensional struc-tures is limited. Much of the research work is devotedto the analysis of metal forming [17,18].


The pioneering work in three-dimensional modelingof bolted connection is attributed to Krishnamurthy andGraddy [19]. They employed elastic linear analysis withcontact conditions simulated approximately by attachingand releasing appropriate nodes after each loadincrement. Until recently, a rigorous approach isreported [20,21] modeling beam-to-column connectionswith bolted extended end-plates in hot rolled steel con-struction. Solid elements together with contact elementsare found to produce good results compared with testdata. The following section presents the research work[22–24] using three-dimensional solid elements withnon-linear material, geometrical and contact analysis toinvestigate the structural performance of bolted connec-tions between cold-formed steel strips and hot rolledsteel plates under static shear loading.

6.1. Finite element modeling

In the present study, the ANSYS (Version 5.3) [25]finite element package is used to predict the load–exten-sion curves of the bolted connections between cold-for-med steel strips and hot rolled steel plates under shear.Three-dimensional eight-node iso-parametric solidelements SOLID45 are employed to model all thecomponents, namely, the cold-formed steel strips, the hotrolled steel plates, the bolt and also the washers, in orderto capture yielding propagation throughout the materialthickness. Such elements are especially suitable for theplasticity type problem since they allow discontinuousstrain fields in simulating shear bands. Furthermore, thenormal stresses acting on the cold-formed steel strips dueto the clamping forces in bolt shanks and also the tan-gential stresses due to frictional forces between contactinterfaces may also be incorporated. Contact interfacesbetween the cold-formed steel strips and the bolt, thewasher and the hot rolled steel plates are modeled bycontact elements CONTAC49 so that intuitive assump-tions on the position and the size of contact area arenot required.

In order to simplify the model, only half of the lapshear test specimen is modeled as shown in Fig. 6a aftertaking symmetry along the longitudinal axis of the testspecimen. The hot rolled steel plate is assumed to berigid and fixed in space; it is modeled by a singleelement with all its degrees of freedom restrained. Thebolt is assumed to be threadless and forms an integralcomponent with the washer. The bolt–washer componentis assumed to be linear elastic throughout the analysis.Furthermore, the root of the bolt–washer component isfixed in space by constraining the associated nodes. Forease of meshing, an artificial small hole of 1 mm diam-eter is provided through the centreline of the bolt–washercomponent. Furthermore, the finite element mesh isrefined locally in the vicinity of the bolt hole, as shownin Fig. 6b, for improved resolution of stresses and defor-

Fig. 6. Finite element mesh for CFS-HRS connection (specimenA11A).

mations; local buckling in the strips may also be incor-porated. The aspect ratios of those elements near the bolthole are kept small to reduce the shear locking effect inthe elements. This is particularly important for elementson the contact surfaces as they have to conform with theshape of the bolt shank when simulating the bolt–holeinteraction. Table 3 summarizes the types and the totalnumber of elements used in the models.

The ‘true’ stress–strain curves [26] established fromcoupon tests are adopted for the cold-formed steel stripsin order to establish both yielding and strength degra-dation during deformation. Non-linear material, geo-metrical and contact analysis is then carried out to pre-dict the load–extension curves of the bolted connections.The finite element modeling may provide detailed infor-mation on the yield zones of the cold-formed steel strips,the stress distribution in the strips undergoing end bear-ing failure, and also the resistance contributions of thebearing and the frictional actions, respectively. Further-more, the effect of clamping forces in bolt shanks on theload–extension curves of the bolted connections will alsobe examined.


Table 3Details of finite element meshesa

Model Elements A11A/A21A A11B/A21B

No. of nodes – 1878 1917No. of solid elements SOLID45 933 CFS/264 B+ W/1 HRS 933 CFS/288 B+ W/1 HRSNo. of contact elements CONTAC49 981 1083

a CFS denotes cold-formed steel strips, HRS denotes hot rolled steel plates, B+ W denotes bolt and washers.

6.2. Stress–strain curves

Based on the results of the coupon tests, a bi-linearstress–strain curve is first incorporated into the finiteelement model with the yield strength taken as the meas-ured yield strength,py, of the cold-formed steel strips.The finite element model with such a material curve isreferred to as FEA-py. However, it is found that themaximum resistance of the bolted connections is typi-cally 15% higher than the results in G550 test specimens.

Consequently, an engineered stress–strain curve withreduced strength at large strain after yielding, i.e.strength degradation, is proposed which is incorporatedinto the finite element models in subsequent analyses.While the ultimate tensile strength and the correspondingstrain are based on coupon (tensile) tests, the samecurves are assumed to be valid to the same materialunder compression. In order to incorporate the effect ofstrength degradation at large compressive strain, a nega-tive slope is added to the curve after the ultimate com-pressive strength is reached. The value of the negativeslope in the strength degradation portion is selected astwice that of the post-yielding tangent modulus of thesteel. This was established after a trial-and-error processin matching the predicted load–extension curves to themeasured ones. The compressive strength was thenreduced linearly to zero at a strain around 50%; the finiteelement model with such a material curve is referred toas FEA-pr.

A third finite element model was also used with a bi-linear stress–strain curve. The yield strength is based onthe measured tensile strength,Us, of the materialobtained in coupon tests. The finite element model withsuch a material curve is referred to as FEA-Us.

Table 4 summarizes the material properties of the

Table 4Properties of components of bolted connections in finite element analyses

Components Elastic modulus Yield strength Tensile strength Elongation limit Formulation of elements(kN/mm2) (N/mm2) (N/mm2)

CFS stripsG550 (t = 1.56 mm) 216.5 620.7 638.4 1.25% Material & geometrical non-linearG300 (t = 1.46 mm) 204.2 320.5 453.6 20.0% Material & geometrical non-linear

Bolt–washer 205.0 – – – Elastic linearHot rolled steel plates – – – – Rigid

cold-formed steel strips, the bolt–washer component,and also the hot rolled steel plate used in the finiteelement models. The proposed stress–strain curves arepresented in Fig. 1b and 1c for G550 and G300 stripsteel, respectively; the proposed stress–strain curves areassumed to be rate-independent. It should be noted thatthe proposed stress–strain curves are conservative whencompared with a similar approach reported in a finiteelement modeling of local fracture [27].

Due to large local deformation in the cold-formedsteel strips around the bolt holes in direct contact withbolt shanks, plasticity is considered by incorporating thevon Mises yield criterion, the Prandtl–Reuss flow ruletogether with isotropic (work) hardening rule. The cri-terion determines the stress level at the onset of yieldingwhile the flow rule relates the stress increments to strainincrements during plastic deformation. This allows theyield surface to change in size with progressive yieldingin the cold-formed steel strips in the vicinity of boltholes.

6.3. Contact stiffness, frictional coefficient andclamping force in the bolt

In order to model the contact condition between thecold-formed steel strips and the bolt–washer component,the generation of contact elements according to the pen-alty technique recommended by Cook [28] wasemployed in the finite element models. Each contactelement is defined by a node (contact node) on one sur-face and four other nodes on another surface (targetsurface), and thus, each contact element is tetrahedral inshape. Compatibility between the interfaces is securedby preventing each contact node from over-penetratingthe associated target surfaces through the development


of contact forces. These forces act on contact nodes andhave components both normal and tangential to the asso-ciated target surface in general. While normal forces aredeveloped to fulfil the compatibility requirement, thecompatibility control is governed by a contact stiffnesswhich should be large enough to maintain accuracy butsmall enough to safeguard convergence.

In the present investigation, all the nodes associatedwith the cold-formed steel strips around bolt holes areassigned as contact nodes while facets of the shank areselected as target surfaces. Contact elements were gener-ated between each contact node and all the target faces.In a similar manner, contact conditions for the cold-for-med steel strips to the washer and also to the hot rolledsteel plate are established. It should be noted that themaximum number of contact nodes to each target surfacewas controlled by initial separations between the compo-nents.

The value of the contact stiffness is assigned to be2000 N/mm after a trial-and-error process in comparingthe measured and the predicted load–extension curves ofa number of test specimens. Tangential forces are alsodeveloped as a result of friction and the elastic Coulombfriction coefficient,m, is taken to be 0.2 for all contactinterfaces. Both the values of the contact stiffness andthe friction coefficient are assumed to remain constantthroughout the analysis.

Bolt clamping is also incorporated by considering abolt shank whose length is less than the thickness of thecold-formed steel strip typically by 5%, and thus,initially the washer penetrates into the cold-formed steelstrips in the geometry of the finite element model. Dur-ing the first iteration of the analysis, the washer and thecold-formed steel strips will push against each other byinducing tensile stresses in the bolt shank while com-pressive stresses in areas beneath the washer are estab-lished. Equilibrium should be achieved in order to pro-duce the clamping forces in the bolt shank due to boltinstallation before the finite element model is loaded inshear. Experience shows that successful production ofthe clamping forces in bolt shanks will eliminate numeri-cal divergence in subsequent iterations, enabling thesmooth computation of the entire load–extension curves.

6.4. Solution procedure

The finite element model incorporates material, geo-metrical and contact non-linearity, and therefore, non-linear analysis is required. In the present investigation,the solution procedure requires the full load to be appliedin a series of small increments so that the solutions mayfollow the load–extension curves closely. A value of 5%is recommended as the maximum plastic strainincrement after each incremental load. In order to simu-late the lap shear test results, a series of displacementincrements is applied to the end of the cold-formed steel

strips up to a total extension of 3 mm. The size of eachincrement is automatically adjusted by the programwhich limits the increment for plastic strain to be lessthan 5%. On average, the displacement increment isfound to be 0.25 mm, corresponding to an increment of3.5% plastic strain. With this strain level, the individualelements may exhibit significant changes in shape andorientation so that their stiffnesses and nodal displace-ments will affect each other considerably.

Consequently, this is a highly non-linear problem andat each sub-step, the solution is obtained through a num-ber of equilibrium iterations. This is accomplished by thefull Newton–Raphson procedure in which all the nodaldisplacements, the out-of-balance forces, and the tangentstiffness matrix of the structure are updated after eachequilibrium iteration. A force-based convergence cri-terion is adopted which requires the square root of thesum of squares (SRSS) of the load imbalance to be lessthan 1% of the SRSS of the applied loads in an equilib-rium iteration. Attempts with other solution schemessuch as the modified Newton–Raphson procedure arefound to be unsuccessful due to numerical divergence.

7. Results of finite element analyses

Three finite element models with different stress–strain curves, namely, FEA-py, FEA-Us and FEA-pr, areemployed to examine the load–extension curves of lapshear test specimens with cold-formed steel strips of dif-ferent grades and thicknesses, and washers of differentsizes. Typical deformed shape of the bolted connectionis presented in Fig. 6c and the results of the finiteelement analyses are presented below.

7.1. Load–extension curves

The load–extension curves of a number of bolted con-nections are successfully predicted and they are plottedin the same graphs of the relevant test data, i.e. in Figs.4 and 5, for direct comparison. It is shown that both themeasured and the predicted load–extension curves fol-low each other very closely in terms of both initial andfinal slopes, and also maximum load carrying capacities,after allowing for variation in material properties andgeometrical dimensions of the test specimens.

7.2. Load carrying capacities

In order to avoid excessive deformation in connec-tions, the maximum resistance of the bolted connectionsis assumed to be equal to the applied load at 3 mm exten-sion. The resistances of the finite element models at spe-cific extensions are presented in Table 5 together withthe average load carrying capacities of the test specimensfor direct comparison.


Table 5Results of lap shear tests and finite element analyses

Test Lap Finite element analysessheartests

PFEA C orPLST/PFEA

PLST py Us pr py Us pr

Connection resistance of bolted connections at 1 mm extension(kN)A11A 20.14 20.00 21.04 20.20 1.01 0.96 1.00A11B 17.48 20.78 21.00 20.18 0.84 0.83 0.87A21A 14.83 12.36 12.80 12.80 1.20 1.16 1.16A21B 14.77 12.46 12.88 12.88 1.18 1.15 1.15



A model factor,C, is established to measure the effec-tiveness of the finite element modeling which is definedas follows:

C 5Resistance measured from lap shear test

Resistance predicted by finite element model

A unity model factor implies that the finite elementmodeling is accurate in predicting the load carryingcapacities of bolted connections. For model factorslarger than unity, the finite element modeling is con-servative. It is shown in Table 5 that the model factorsof all three finite element models at various extensionsvary within a narrow range between 0.83 and 1.20, indi-cating the general effectiveness of the finite elementmodels. The findings of the finite element analyses arepresented below.


It is shown that the finite element models with bi-linear stress–strain curves, i.e. FEA-Us and FEA-py,always give a resistance at 3 mm extension about 15%higher than that of FEA-pr, i.e. with strength degra-dation. The model factors of both FEA-py and FEA-Us

are around 0.87 while those of FEA-pr are very close tounity. This confirms the suitability of the proposedstress–strain curves with strength degradation for cold-

formed steel strips with high yield strength and low duc-tility. The finite element model FEA-pr is used in thesubsequent analysis for G550 test specimens.


It is shown that the finite element models FEA-Us andFEA-pr tend to give a resistance at 3 mm extension about10% higher than that of FEA-py. The model factors ofboth FEA-Us and FEA-pr are around 0.89 while that ofFEA-py is around 0.96. It should be noted that asobserved from lap shear tests, strip curling occurs at anextension of about 2 mm, and before the peak loads arereached. However, in the finite element models, curlingof the ends of the cold-formed steel strips occurs at anextension larger than 4 mm. This implies that the finiteelement models slightly overestimate the load carryingcapacities at 3 mm as strip curling has not yet been cap-tured. However, strip curling is unlikely to occur in typi-cal connection arrangements in practice, and accuratemodeling of strip curling is, thus, considered not to becritical.

It is interesting to note that the model factors of finiteelement models FEA-py, FEA-Us and FEA-pr at 2 mmextension are 1.10, 0.99 and 0.99, respectively. It isapparent that finite element model FEA-py is very con-servative due to low yield strength value. In contrast, thefinite element model FEA-Us gives better results due tohigher yield strength. However, the variation in localdeformation and diversified strain levels within thebolted connections are not considered at all in the model.As the finite element model FEA-pr has allowed forstrength degradation at large strain and achieved a modelfactor of almost unity, it is thus suggested that the pro-posed stress–strain curve with strength degradation isalso suitable for cold-formed steel strips with low yieldstrength and high ductility.

It is shown that for cold-formed steel strips with highstrength and low ductility, the resistance of the boltedconnections is typically 15% lower than those steels withbi-linear stress–strain curves, i.e. without strength degra-dation. As most of the existing design expressions aredeveloped and calibrated against test data of cold-formedsteel strips without strength degradation, the same designrules should be applied cautiously when calculating thebearing resistance of high strength cold-formed steelstrips with low ductility.

7.5. Stress distribution

The von Mises stress distribution around bolt holes ofG550 test specimens A11A in the cold-formed steelstrips at three different extensions are presented in Fig.7. It is shown that for those elements around the bolthole in direct contact with the bolt shank, the stressesare typically several times the applied stress, revealing


Fig. 7. Stress distribution of bolted connection G550 specimen A11Awith extension of: (a) 0.50 mm; (b) 1.50 mm; (c) 3.00 mm.

high stress concentration due to direct end bearing. Asthe extension increases, yielding occurs in theseelements and the yield zone increases in size rapidly asshown in Fig. 7. It should be noted that at 3 mm exten-sion, the stresses of these elements are only one-third ofthe yield strength of the cold-formed steel strips. Fig. 8illustrates the large strain variations in the vicinity ofthese elements. A 40% strain is predicted as themaximum direct strainex, confirming large plastic flowof materials around the bolt hole in direct contact withthe bolt shank, as observed in lap shear tests. The strainsof those elements of interest may also be obtainedthrough the proposed stress–strain curves based on thestress distribution in Fig. 7.

Consequently, the finite element model is consideredeffective in predicting the bearing failure of the cold-formed steel strips in the bolted connections. It is inter-esting to plot the von Mises stress distribution along across-section containing the elements of interest at vari-ous extensions. Fig. 9 illustrates the extent of materialyielding in the vicinity of bolt hole across the width ofthe test specimen A11A at various extensions.

The von Mises stress distributions around a bolt holeof the G300 test specimen A21A at three different exten-sions are also presented in Fig. 10. It is shown that at 3mm extension, yielding is only confined to a small area

Fig. 8. Strain components of bolted connection G550 specimenA11A at 3 mm extension.

Fig. 9. Von Mises stress distribution of bolted connection G550specimen A11A.

of material when compared with the G550 test speci-mens. Therefore, strength degradation is less significantfor those steel with high ductility.

7.6. Yield zones

The von Mises stress distribution of the finite elementmodels FEA-py and FEA-pr for test specimen A11A at3 mm extension are shown in Fig. 11. It should be notedthat while the yield zones in both models are of similarsizes, strength degradation in finite element model FEA-pr is apparent due to large local strains in the elementsin direct contact with the bolt shank. The presence ofstrength degradation leads to 15% reduction in the load


Fig. 10. Stress distribution of bolted connection G300 specimenA21A.

Fig. 11. Stress distribution of bolted connection G550 specimenA11A at 3 mm extension.

carrying capacity of the connection compared with thefinite element model FEA-py.

It should be noted that for connections with bolts atclose spacing, interaction between the yield zones of

individual bolts may occur, and the load carryingcapacities of the bolted connections may be reduced.Furthermore, the effectiveness of the yield zones mayalso be affected by the confinement provided by thewashers, depending on their sizes and stiffnesses(thicknesses).

7.7. Resistance contribution of bearing and frictionalactions

In addition to the end bearing resistance of the cold-formed steel strips in direct contact with the bolt shank,frictional resistance may be developed along the contactinterfaces with the washers. In general, the frictionalforces are dependent on the clamping forces in the bolts,the frictional coefficient between the contact interfaces,and also the sizes of the washers.

Table 6 summarizes the results of finite element analy-ses for test specimens A11A and A21A with differentvalues of frictional coefficient,m, at the contact inter-faces of the bolted connections at various extensions. Itis shown that the frictional resistance is well developedin the early stage of deformation, such as 0.5 mm exten-sion, and remains reasonably constant thereafter. Thebearing resistance is only a fraction of the frictionalresistance initially but increases rapidly at subsequentextensions.

It should be noted that with a frictional coefficient of0.2, the resistance contribution of frictional actionaccounts for 18% and 22% of the load carryingcapacities for G550 and G300 test specimens, respect-ively. This is broadly consistent with the findings of thelap shear tests.

In the case whenm is increased to 0.4, the frictionalforces will increase typically by 70% as shown in Table6 while the bearing resistance of the cold-formed steelstrips remains reasonably constant. The load carryingcapacities of the connections will increase by 15%.Consequently, the load carrying capacity is not sensitiveto the value of the friction coefficient.

It should be noted that the clamping forces in the boltsincrease significantly during deformation as shown inTable 7. For test specimen A11A at 3 mm extension,the clamping force in the bolt is increased by over 80%when compared with the initial value. However, as theyield strength of the bolts is always much higher thanthose of the steel strips, the tensile capacities of the boltsare rarely critical.

It is interesting to note that an increase of clampingforces in bolt shanks does not produce an increase in thefrictional resistance, as shown in Table 6. This may beexplained by the fact that the effective frictional coef-ficient between materials undergoing progressive yield-ing is actually diminishing, rather than remaining con-stant, as assumed in the finite element analyses.


Table 6Resistance contributions of both bearing and friction actions in bolted connectionsa

Test Extension Friction coefficient= 0.0 Friction coefficient= 0.2 Friction coefficient= 0.4

(mm) P Pbf Pbf/Pm P Pff Pff /Pm Pbf Pbf/Pm P Pff Pff /Pm Pbf Pbf/Pm

(kN) (kN) (kN) (kN) (kN) (kN) (kN) (kN)

A11A 0.50 9.46 9.42 0.37 12.88 4.88 0.17 8.00 0.28 17.62 9.60 0.30 8.02 0.251.00 15.78 15.70 0.62 20.20 5.00 0.17 15.20 0.53 23.76 9.52 0.30 14.24 0.451.50 19.92 19.84 0.79 24.26 5.16 0.18 19.10 0.66 27.58 9.52 0.30 18.06 0.572.00 22.70 22.62 0.90 25.82 5.24 0.18 20.58 0.71 30.30 9.56 0.30 20.74 0.653.00 25.20 25.16 1.00 28.94 5.04 0.17 23.90 0.83 31.82 8.76 0.28 23.06 0.72

A21A 0.50 4.26 4.22 0.22 9.74 4.48 0.20 5.26 0.24 13.34 8.80 0.37 4.54 0.191.00 10.66 10.62 0.56 12.80 4.48 0.20 8.32 0.38 16.30 8.60 0.36 7.70 0.321.50 12.34 12.26 0.64 16.86 4.72 0.21 12.14 0.55 19.74 8.52 0.35 11.22 0.472.00 15.98 15.94 0.83 18.46 4.40 0.20 14.06 0.64 21.30 8.40 0.35 12.90 0.543.00 19.12 19.08 1.00 22.04 4.36 0.20 17.68 0.80 24.10 7.40 0.31 16.70 0.69

a P, Resistance of connection predicted in FEM;Pff , resistance contribution of friction action;Pbf, resistance contribution of bearing action;Pm,maximum resistance of connection at 3 mm extension.

Table 7Clamping force developed in bolt shank at various extensionsa

Test Extension Friction coefficient= 0.2

P PBT PBT/Pm

mm (kN) (kN)

A11A 0.00 0.00 10.90 0.380.50 12.88 12.36 0.431.00 20.20 13.38 0.461.50 24.26 14.70 0.512.00 25.82 15.42 0.533.00 28.94 18.00 0.62

A21A 0.00 0.00 9.84 0.450.50 9.74 11.34 0.511.00 12.80 11.72 0.531.50 16.86 12.70 0.582.00 18.46 12.80 0.583.00 22.04 13.16 0.60

a P, Resistance of connection predicted in FEM;PBT, clampingforce developed in bolt shank;Pm, maximum resistance of connectionat 3 mm extension.

8. Recommended procedure of finite elementmodeling

In order to effectively model a bolted connectionbetween cold-formed steel strips and hot rolled steelplate, the following data are recommended.

1. The tensile stress–strain curve may be a bi-linearstress–strain curve based on results from coupon tests.The compressive and tensile stress–strain curves aresimilar, but strength degradation should be incorpor-ated with a negative slope of twice the post-yieldingtangent modulus after the maximum compressivestrength has been reached.

2. The contact stiffness of the contact interfaces may be

assigned with a value of 2000 N/mm, provided thatthe elastic modulus of the cold-formed steel is around205 kN/mm2.

3. The frictional coefficient may be assigned a value of0.2, corresponding to a smooth contact surface. Theclamping force is typically in the range 5–10 kN.

9. Conclusions

A finite element modeling on bolted connectionsbetween cold-formed steel strips and hot rolled steelplates was carried out with non-linear material, geo-metrical and contact analysis. Based on the results ofcoupon tests, an engineered stress–strain curve withstrength degradation for cold-formed steel under bothtension and compression is proposed and incorporatedinto the model. Twelve lap shear tests with two steelgrades, one bolt diameter and two washer sizes were car-ried out to calibrate the finite element models. It is foundthat for extensions up to 3 mm, the load–extensioncurves of the bolted connections compare well with testdata in terms of both the initial and the final slopes, andalso the maximum load carrying capacities.

With the help of finite element modeling, the patternsof yielding and strength degradation, and the strain dis-tribution around the connections are established in detail.Typical strain levels in the cold-formed steel strips inthe vicinity of bolt holes are found to be 40%. Therefore,it is important to incorporate reduced strength at largestrains for accurate prediction of the load carryingcapacities of bolted connections. Furthermore, it is alsofound that the frictional resistance contributes typically20% of the load carrying capacities of the bolted connec-tions. This depends on the clamping forces in bolts, the


frictional coefficient between contact interfaces, and alsothe sizes of washers.

Consequently, the finite element model is an effectivetool to investigate the structural performance of boltedconnections between cold-formed steel strips and hotrolled steel plates under static shear loading. The failuremode under the present investigation is the bearing fail-ure of cold-formed steel strips around the bolt holeswhile other modes of failure may be readily investigated.A recommended finite element modeling procedure isalso presented.

It is shown that for cold-formed steel strips with highstrength and low ductility, the resistance of the boltedconnections is typically 15% lower than those steels withbi-linear stress–strain curves, i.e. without strength degra-dation. As most of the existing design expressions aredeveloped and calibrated against test data of cold-formedsteel strips without strength degradation, it is rec-ommended that these design rules are applied in calculat-ing the bearing resistance of high strength cold-formedsteel strips with low ductility.

Acknowledgements

The work described in this paper was supported by agrant for the project “Moment connections among cold-formed steel members in buildings” from the ResearchGrants Council of the Hong Kong Special Administrat-ive Region (Project No. PolyU5031/98E) and a grant forthe project “Mechanical enhancement of connectionsbetween cold-formed steel members” of the Hong KongPolytechnic University Research Committee (ProjectNo. G-S565).

Appendix A. True stress and true strain

The engineering stress–strain curve obtained fromconventional coupon tests does not give a true indicationof the deformation characteristics of a material at largestrain as it is based entirely on the original dimensionsof the specimens whilst the dimensions of the materialchange continuously during the test. Consequently, boththe ‘true’ strain,e, and the ‘true’ stress,s, should beused for any strain level greater than 0.01 and may beevaluated as follows [28]:

e 5 ln(1 1 e), wheree 5DLL

, s 5PAo

(1 1 e)

whereDL is the change in length,L is the original length,Ao is the original area, andP is the applied force.

References

[1] European Convention for Constructional Steelwork, Europeanrecommendations for steel construction—the design and testingof connections in steel sheeting and sections. Publ No 21, ECCSCommittee TC7, Working Group TWG 7.2, 1983.

[2] European Convention for Constructional Steelwork, Recommen-dations for steel construction—Mechanical fasteners for use insteel sheeting and sections. Publ No 35, ECCS Committee TC7,Working Group TWG 7.2, 1983.

[3] BS5950, Structural use of steelwork in buildings: Part 5. Code ofpractice for the design of cold-formed sections. London: BritishStandards Institution, 1998.

[4] Cold-formed steel structure code AS/NZ 4600: 1996, StandardsAustralia Standards New Zealand, Sydney, 1996.

[5] Load and resistance factor design specification for cold-formedsteel structural members. LRFD Cold-formed Steel Design Man-ual, Part 1. Washington DC: American Iron and Steel Institute,1996.

[6] Eurocode 3: Design of steel structures: Part 1.3: General rules—Supplementary rules for cold-formed thin gauge members andsheeting. ENV 1993-1-3, European Committee for Standardis-ation.

[7] Rhodes J. Design of cold-formed steel members. Barking, UK:Elsevier, 1991.

[8] Yu WW. Cold-formed steel design. 2nd ed. John Wiley andSons, 1991.

[9] Hancock GJ. Design of cold-formed steel structures. 3rd ed. Aus-tralian Institute of Steel Construction, 1998.

[10] Chung KF. Building design using cold-formed steel sections:Worked examples to BS 5950: Part 5: 1987. Steel ConstructionInstitute, 1993.

[11] Brunig M. Numerical analysis and modeling of large deformationand necking behavior of tensile specimens. Finite Elements AnalDesign 1998;28:303–19.

[12] Nagtegaal JC, De Jong JE. Some computational aspects of elas-tic-plastic large strain analysis. Int J Numer Methods Eng1981;17:15–41.

[13] Mitchell GP, Owen DRJ. Numerical solution for elastic–plasticproblems. Eng Comput 1988;5:274–84.

[14] Needleman A. On finite element formulations for large elastic–plastic deformations. Comput Struct 1985;20:247–57.

[15] Lee BC, Kwak BM. A computational method for elasto-plasticcontact problems. Comput Struct 1984;35(5):757–65.

[16] Tsai P, Chen WH. Finite element analysis of elasto-plastic con-tact problems with friction. AIAA J 1986;24:344–6.

[17] Zienkiewics OC, Liu YC, Huang GC. Error estimation and adap-tivity in flow formulation for forming problems. Int J NumerMethods Eng 1988;25:23–42.

[18] Petersen SB, Gouveia BPPA, Rodrigues JMC, Martins PAF. Ametal-forming approach to automatic generation of graded initialquadrilateral finite element meshes. Eng Comput1998;15(5):577–87.

[19] Krishnamurthy N, Graddy D. Correlation between 2 and 3 dimen-sional finite element analysis of steel bolted end plate connec-tions. Comput Struct 1976;6:381–9.

[20] Gebbeken N, Rothert H, Binder B. On the numerical analysis ofend plate connections. J Construct Steel Res 1994;30:177–96.

[21] Bursi S, Leonelli L. A finite element model for the rotationalbehaviour of end plate steel connections. In: Proceedings ofSSRC Annual Technical Session, Chicago, 1994:163–75.

[22] Chung KF, Ip KH. Finite element modelling of cold-formed steelbolted connections. In: Proceedings of the Second European Con-ference on Steel Structures, Praha, May 1999, 1999:503–6.

[23] Ip KH, Chung KF. Failure modes of bolted cold-formed steelconnections under static shear loading. Proceeding of the Second


International Conference on Advances in Steel Structures, HongKong, December 1999, in press.

[24] Chung KF, Ip KH. Finite element modelling of double boltedconnections between cold-formed steel strips under static shearloading. Proceedings of the Second International Conference onAdvances in Steel Structures, Hong Kong, December 1999, inpress.

[25] User Manual of ANSYS for Revision 5.0—Procedures VolumeI. Swanson Analysis Systems, Inc., USA, 1994.

[26] Bathe KJ. Finite element procedures. Prentice-Hall: EnglewoodCliffs (NJ), 1996.

[27] Fan L, Rondal J, Cescotto S. Finite element modeling of singlelap screw connections in steel sheeting under static shear. Thin-Walled Struct 1997;27(2):165–85.

[28] Cook RD, Malkus DS, Plesha ME. Concepts and applications offinite element analysis. 3rd ed. John Wiley and Sons, 1989.

Finite Element Modeling of Bolted Connections Between Coldformed

Documents

Transcript of Finite Element Modeling of Bolted Connections Between Coldformed