Moment-rotation behaviour of top-seat angle bolted connections

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beam, bolt size and length of angle were considered. The results ...... [67] J.H. Bickford, S. Nassar, Handbook of Bolts and Bolted Joints, Marcel Dekker, Inc,.
Journal of Constructional Steel Research 136 (2017) 149–161

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Moment-rotation behaviour of top-seat angle bolted connections produced from austenitic stainless steel

MARK

Mohammad Jobaer Hasana, Mahmud Ashrafa,⁎, Brian Uya,b a b

School of Engineering and Information Technology, The University of New South Wales, Canberra, ACT 2610, Australia School of Civil Engineering, The University of Sydney, NSW 2006, Australia

A R T I C L E I N F O

A B S T R A C T

Keywords: Analytical model Austenitic stainless steel, bolted connection Finite Element (FE) modelling Moment-rotation (M − ϕ) behaviour Semi-rigid connection Top-seat angle

Use of stainless steel alloys for structural applications could provide a more efficient balance between design life expenses in addition to in-service performance. To achieve safe and economic design rules, an appropriate understanding of the mechanical response of structural components, as well as connection systems produced from stainless steel, is required. Over the last two decades, significant research has been carried out on stainless steel members, but the moment-rotation (M − ϕ) behaviour of stainless steel connections is yet to be thoroughly investigated. Top-seat angle connections are widely used in traditional bare steel constructions due to their ability to transfer both the vertical reaction as well as the end moment, and their behaviour is classified as partially restrained or semi-rigid in nature. The current study investigated the M − ϕ behaviour of austenitic stainless steel top-seat angle bolted connections. The Finite Element (FE) modelling technique using ABAQUS was used to develop appropriately validated numerical models based on the experimental evidence available from the relevant literature. Significant strain hardening exhibited by the austenitic grades was considered in FE modelling. The effects of a number of key geometric parameters such as angle thickness, gauge distance, depth of beam, bolt size and length of angle were considered. The results obtained from the parametric study were used to develop an analytical model based on the power model approach suggested by Richard–Abbott (1975); and the performance of the developed model was also verified. The proposed simple connection model will enable predicting the M − ϕ behaviour of stainless steel top-seat angle bolted connections using easily obtainable geometric and material properties.

1. Introduction In the conventional analysis and design, connections in bare steel frames are often considered either as pinned or fixed. In the ideal state, pinned connections assume that no moment will be transferred between the beam and the column. Since rotation is a great concern for such joints, the beam and the column that are connected together by a pin must conform to serviceability limit states. At the other extreme, a fully rigid (FR) condition indicates that no relative rotation will develop between the adjoining members, an assumption which in many cases, overestimates the end moments of the beam. These idealized models are popular due to their simplistic formulations in the analysis and design of steel frameworks. However, evidence from experimental investigations during the last few decades clearly indicates that all beam-tocolumn connections used in current practice exhibit some flexibility, and fall between two extreme cases of fully rigid and ideally pinned: such connections are referred to as semi-rigid connections [1,2]. An over prediction of joint rigidity could result in underestimating the ⁎

lateral sway, storey drift, and the probability of failure, whilst underestimating the connection capacity could lead to erroneous design forces for beams and columns [3]. Unrealistic predictions of the structural response and reliability of steel frames could be significantly affected due to inaccurate prediction of the connection behaviour [4]. From the point of view of the structural assembly, fully rigid connections are conceived as the connections where the beam is directly welded onto the flange of the column. By comparison, connections that have very little connection stiffness, for instance, single and double web-angles and header plate connections, are typical pinned connections. Beam-to-column connections, which are fastened through bolts or rivets using top-seat angles in web plates and/or angles, show nonlinear behaviour and lie somewhere between the fully rigid and perfectly pinned conditions; these connections are generally categorised as semi-rigid or partially restrained connections. Rigid connections completely welded connections, which force the hinges to be formed in the connected beam and make it susceptible to brittle failure due to low cycle fatigue as was observed in the Northridge [5]

Corresponding author. E-mail addresses: [email protected] (M.J. Hasan), [email protected] (M. Ashraf), [email protected] (B. Uy).

http://dx.doi.org/10.1016/j.jcsr.2017.05.014 Received 1 October 2016; Received in revised form 14 May 2017; Accepted 19 May 2017 0143-974X/ © 2017 Elsevier Ltd. All rights reserved.

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Fig. 1. Test setup of specimens A1 and A2 by Azizinamini [20].

achieving a reliable and efficient design for building frames.

and Kobe [6] earthquakes. Semi-rigid connections can dissipate hysteretic energy through deformation of connection components and have been recommended by many researchers [7–12]. Stainless steel sections, in many cases, can be a promising alternative to ordinary carbon steel beam-column structures in consort with suitable fasteners. In recent times, stainless steel has been gaining popularity in structural applications due to their corrosion resistance, better performance at elevated temperature, ease of maintenance and pleasing appearance. Austenitic grades provide a large strain-hardening domain up to 50% elongation at fracture, which could potentially make these alloys suitable for seismic resistant framed structures [13–16]. Relatively high initial cost could be justified considering the life cycle cost of a structure for its low maintenance requirements and excellent durability [17–19]. Analysis and design of bare stainless steel building frames require a clear understanding of the M − ϕ behaviour of beam-to-column connections. A considerable numbers of studies have been done on ordinary carbon steel to predict the M − ϕ behaviour of top-seat angle bolted connections. Azizinamini [20] reported experimental investigations under monotonic and cyclic loadings, which were extensively used in devising analytical models for carbon steel connections [21–24]. Numerical modelling techniques, especially the finite element method (FEM) [25–30], were widely used to investigate the behaviour of top-seat angle bolted connections. A number of analytical models including linear [31], bi-linear [32], piecewise multilinear [33], polynomial [34] and exponential [35] are available in the literature; all of these proposed models, however, require too many variables and constants, making the practical application of them difficult. A threeparameter power model proposed by Kishi and Chen [21] is a simple and useful model. Chen and Kishi [22] also developed a very useful data bank for semi-rigid connections based on experimental evidence and developed a computer program to predict the M − ϕ behaviour of carbon steel connections. A number of researchers have recently investigated the behaviour of stainless steel bolted connections. Kim and Kuwamura [36] investigated bolted connections between cold-formed stainless steel sheets. Bouchair et al. [37] conducted experimental and numerical investigations of cover plate connections for austenitic grade stainless steel. Kim et al. [38,39] performed parametric studies on various geometric features to observe the effect of curling (out-of-plane deformation) on bolted connections. Salih et al. [40,41] reported the importance of the various parameters of austenitic and ferritic stainless steel, then studying net section and bearing failure modes of connections experimentally and numerically. Parametric studies on stainless steel angle and gusset plate bolted connections were also performed numerically by Salih et al. [42]. Cai and Young [43,44] investigated the structural behaviour of cold-formed stainless steel bolted connections at room temperature as well as under fire. However, no research has been conducted on typical beam-to-column connections produced from stainless steel alloys. The current study numerically investigated the M − ϕ behaviour of austenitic top-seat angle connections and paves the way for reliable analytical modelling of the observed behaviour. Appropriate understanding of this basic connection behaviour is a prerequisite for

2. FE modelling of top-seat angle bolted connections Commercial FE software ABAQUS/CAE [55] was used in the current study to simulate the M − ϕ behaviour of top-seat angle bolted connections. In the first step, FE models were developed to validate the adopted modelling technique by using experimental results available on similar types of connections produced from ordinary carbon steel. Once verified, the modelling technique was used to carry out parametric studies for austenitic stainless steel connections. The following sections present details of the FE modelling approach that was adopted in the current study. 2.1. Model geometry Fig. 1 shows the experimental setup reported by Azizinamini [20], where test members consisted of a pair of beam sections attached to a centrally placed stub column. To simplify the analysis, a 3D FE model representing a quarter of the experimental setup for the top-seat angle bolted connection was developed with appropriate boundary conditions. Fig. 2 shows a typical top-seat angle bolted connection in which two angles are used to connect the top and bottom flanges of the beam to the supporting column by using the required number of bolts based on appropriate design principles. Bolt hole diameters that considered were 1.6 mm (1/16 in.) larger than the nominal diameter of the corresponding bolt in accordance with test data. 2.2. Material properties Material properties were taken from Azizinamini [20] and Harper [45] as verification models, whilst in the subsequent parametric analysis material properties for beams, columns and angles were representative of austenitic grades (EN1.4435/AISI316L) as reported by Real et al. [46]. For bolt elements, material properties suggested by BS EN ISO 3506-1:2009 [47] (grade A4 property class 80) were adopted. Modulus of Elasticity and Poison's ratio were considered as 200GPa and 0.3, respectively. Table 1 shows the key material properties used in the current study for the parametric analysis. 2.3. Material modelling A number of material models have been proposed during the last two decades to simulate the nonlinear stress–strain behaviour exhibited by different grades of stainless steel. EN 1993-1-4 [48] contains some useful extracts to model material nonlinearity. Most of the models are based on the Ramberg–Osgood [49] formulation proposed for aluminium, as shown in Eq. (1), where E0 is Young's modulus, σ0.2 is the 0.2% proof stress (conventionally considered as the yield stress), and n is the strain hardening exponent, usually calculated using Eq. (2).

ε=

150

⎛ σ ⎞n σ + 0.002 ⎜ ⎟ ⎝ σ0.2 ⎠ E0

(1)

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Fig. 2. A typical top-seat angle bolted connection.

the experimental curve and the analytical models; Eqs. (7) to (10) show the suggested modifications.

Table 1 Material properties considered in the current study. σ0.2 (MPa)

Connection components

Beam, column, angle Bolt

n=

322 600

σu (MPa)

593 800

σ0.01 (MPa)

228 –

σ0.05 (MPa)

282 –

Strain hardening exponent n m 10.62 –

ln

2.23 –

ln

( )

(2)

The Ramberg-Osgood model showed close agreement for up to 0.2% stress, but significantly over predicted stresses for larger strains. This discrepancy led to modifications proposed by Mirambell-Real [50] and Rasmussen [51], who proposed two-stage expressions to accurately model the nonlinear stress–strain behaviour exhibited by stainless steel. Eq. (3) shows the modification proposed by Rasmussen [48] for stresses beyond σ0.2. The additional strain-hardening exponent m can be determined using Eq. (4), whilst the ultimate strain εu and stress σu can be estimated using Eqs. (5) and (6), and three basic RambergOsgood parameters (E, σ0.2 and n). This proposal is included in EN 1993-1-4, Annex C [48] for modelling the material response of stainless steel.

ε=

⎛ σ − σ0.2 ⎞m σ − σ0.2 + εu ⎜ ⎟ + ε0.2 for σ > σ0.2 ⎝ σu − σ0.2 ⎠ E0.2

m = 1 + 3.5 εu = 1 −

σ0.2 σu

σ0.2 σu

⎧ 0.2 + 185 σ 0.2 , for austenitic and duplex E0 ⎪ σ0.2 = ⎨ 0.2 + 185 σ 0.2 E0 σu ⎪ ⎩ 1 − 0.0375(n − 5) , for all stainless steel alloys

( ) σ 0.2 σ 0.05

(7)

σ m = 1 + 2.8 0.2 σu

(8)

⎧ 0.2 + 185 σ 0.2 , for austenitic, duplex and lean duplex ⎪ σ0.2 E0 =⎨ σ σu ⎪ 0.46 + 145 E0.2 , for ferritic grades ⎩ 0

(9)

⎧1 − ⎪ εu = ⎨ ⎪ ⎩

ln (20) σ 0.2 σ 0.01

ln (4)

n=

σ 0.2 , σu

for austenitic, duplex and lean duplex

(

0.6 1 −

σ 0.2 σu

),

for ferritic grades

(10)

To exploit the benefits of strain hardening evident through a continuous design curve, the Continuous Strength Method (CSM) is a newly proposed strain-based design technique for nonlinear metallic materials [53]. Recently, Afshan and Gardner [54] proposed CSM guidelines which adopt a simple elastic, linear hardening material model, to produce accurate and consistent predictions at the crosssection level for stocky sections. In this model, the yield stress point is defined as (σy, εy), where σy is taken as the material 0.2% proof stress and εy is the corresponding elastic strain εy = σy/E, where E is the material's Young's modulus. The strain hardening slope (Esh), as expressed in Eq. (11), is the slope of a line between the 0.2% proof stress point (σy, εy) and a specified maximum point (εmax, σmax) with εmax taken as 0.16εu, where εu is the ultimate tensile strain, and σmax is taken as the ultimate tensile stress σu. The ultimate strain (εu) corresponds to the material ultimate tensile stress (σu) as given by Eq. (12). A schematic diagram of the suggested bi-linear material model is shown in Fig. 3.

(3) (4)

Esh =

(5)

σu − σy 0.16εu − εy

εu = 1 −

(11)

σy σu

(12)

Both the modified nonlinear model proposed by Rasmussen and the bi-linear strain hardening material model were initially adopted in the current study; Fig. 4 shows the results obtained for five different topseat angles bolted connections. A consistent discrepancy was observed

(6)

Arrayago et al. [52] recently modified strain-hardening exponents through a least square adjustment which minimizes the error between 151

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Nut

Bolt Shank

Defined section for bolt pretension Head Fig. 5. Various parts of bolt used in FEM.

(39 kips) for the 22.3 mm (7/8 in) diameter A325 bolts. Fig. 3. Elastic, linear hardening material model adopted in FEA.

2.5. Boundary conditions and applied loads

in the nonlinear part of the M − ϕ curve for all connection models, although it was not very significant. A bi-linear hardening model produced relatively higher strain hardening stiffness leading to marginally higher connection moments: for all the considered connections, an increase in moment resistance of only 4% was observed at a 20mrad rotation. The bi-linear material model was eventually adopted for the parametric study due to its simplicity in FE simulation.

All web plane nodes were restrained against out-of-plane displacement to prevent the structure as well as to restrain the beam against lateral-torsional buckling. Beam sections used in connection models were compact, which eliminated the possibility of instabilities due to local buckling. Von Misses stress distributions obtained from FE models showed that the beam sections remained elastic under the considered loading conditions except for the vicinity of fastener holes of the connected beam flanges. Hence, local buckling failure modes were ignored in current FE investigation. Developed connection models were analysed in two loading steps. In the first step, bolt pretension was applied to a pre-defined section of bolt shank. In the current study, bolt pretension force was applied 70% of minimum tensile strength as suggested by RCSC [57]. In the second step, displacements were gradually increased up to 50 mm at a distance of 1500 mm from column surface as shown in Fig. 6. The value of the beam end displacement yielded a rotation close to 0.03 rad. Connection moment M was determined by multiplying the reaction force (P) at the end of the beam with the instantaneous centre of rotation, which is located at the heel of angle leg adjacent to the compression beam flange as illustrated in Fig. 7. Relative rotation of the connection ϕr calculated from the results of FE analysis was estimated by using the Eq. (13):

2.4. Contact modelling and bolt preload ABAQUS offers a “master-slave” type algorithm for general contact formulation which identifies the surfaces that are in contact or interpenetrate or slip and executes constraints on the nodes of the slave surface such that they do not penetrate the master surface [55]. Hence, contact modelling plays a significant role in beam-to-column bolted angle connections. To observe their individual effects on connection behaviour, bolts were divided into shank, head and nut elements as shown in Fig. 5. The contact areas are the bolt, shank-tobolt holes and bolt head and/or nut-to-components. During analysis, the finite sliding option was adopted between the contact surfaces such as vertical legs of angles and column flange, horizontal legs of angles and beam flanges, and bolts and surrounding bolt hole elements. In addition, “slip critical - Class A” faying surfaces were considered for high strength bolts, and hence Coulomb's friction coefficient of the contacting surfaces was taken as 0.3 to define the tangential behaviour of contact surfaces as allowed by AISC [56] and Research Council on Structural Connections (RCSC) [57]. Since Azizinamini [20] only mentioned tightening the bolts by an air wrench using the turn-ofthe-nut method, common design values were used for modelling the pretension as suggested by AISC and RCSC. For instance, 173.5 kN

ϕr =

δt − δb d

(13)

where δt and δb are the horizontal displacements at the top and bottom edges of beam flanges, respectively, and d is the depth of connected beam.

Fig. 4. Observed variations in M − ϕ response for bi-linear and nonlinear material models.

152

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Plane of Y symmetry and Pin support

Plane of X symmetry

Direction for applied displacement

Z

Y

X

Fig. 6. Boundary conditions used for top-seat angle connections in FE models.

Fig. 8. Element selection criteria for top-seat angle connection.

of freedom are able to eliminate artificial stiffening due to the Poisson's effect in bending. C3D8I elements were observed to perform particularly well both in the elastic and the inelastic regimes as shown in Fig. 8. C3D8I elements were hence deemed suitable for representing the behaviour of bending dominated top-seat angle bolted connections. C3D8I elements have also been recommended by other researchers [60–63] for simulating structures primarily dominated by bending and contact phenomena. Mesh sensitivity analysis was also carried out for C3D8I elements and the obtained results for the FE model replicating test A1 [20] are shown in Fig. 9. Some variations are noticeable due to changes in mesh size with coarse mesh showing relatively higher stiffness when compared to the stiffness of relatively finer meshes that were considered in the study. Moderately fine mesh comprising 14,585 C3D8I elements, which was adopted in the current study, showed similar performance in second order elements (C3D20 and C3D20R) with significantly less computational time and produced a close replication of the experimental results.

Fig. 7. Deformation of top-seat angle connection at the ultimate condition.

2.6. Element selection ABAQUS offers a large variety of 3D hexahedron (brick), shell, contact and beam elements endowed with different features depending on the application. Kukreti et al. [58] and Gebbeken et al. [59] observed that the hexahedron (brick) elements are more suitable to modelling the continuum behaviour of bolted connections rather than standard shell elements. In the ABAQUS library, different continuum or solid elements are available including eight-node linear (first-order) elements (C3D8, C3D8H, C3D8I, C3D8R) and twenty-node quadratic (second-order) elements (C3D20, C3D20H, C3D20R). In the current study, careful thought was given to choosing appropriate elements for simulating the behaviour of connections. In the current FE modelling of a top-seat angle bolted connection, the performance of each of the aforementioned elements was compared against Azizinamini's test result A1 [20]. In Fig. 8, it can be observed that the C3D8R elements with reduced integration (1 G point) underestimate connection resistance as they are not suited for contact zone modelling due to their inherent rank deficiency, and the elements can deform without any resistance to load. By comparison, C3D8 elements with full integration (8 G points) are accurate in terms of constitutive law integration, but the shear-locking phenomenon is commonly associated with this type of element when simulating bending-dominated structures. C3D8I elements with full integration (8 G points) and incompatible modes have 13 additional degrees of freedom, and the primary effect of these degrees of freedom is to eliminate the parasitic shear stresses that are observed in regular displacement elements whilst analysing bending dominated problems [55]. In addition, these degrees

Fig. 9. Mesh sensitivity analysis of C3D8I element.

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Table 2 Geometric properties of basic elements used in FE model verifications. Reference

Specimen designation

Azizinamini et al. [20] Harper et al. [45]

Bolt dia.

A1 A2 TEST3

(mm)

AISC Sections Column

Beam

22.3 22.3 22.3

W12 × 96 W12 × 96 W8 × 24

W14 × 38 W14 × 38 W8 × 21

3. Verification of the FE modelling technique for top-seat angle carbon steel connections

1. In the FE analysis, due to the absence of actual nonlinear behaviour of connection materials, an approximate isotropic hardening model was incorporated; 2. As the actual material properties were not available, typical material properties for A36 steel was considered for the beams, columns and angles, whilst A325 grade was used for bolts; 3. The friction coefficient and the level of pre-tensioning also have a

A1 A2 TEST3

Angle

Bolt

365 277 297

636 636 636

550 530 517

830 830 830

206

0.3

(mm)

(mm)

L6 × 4 × 3/8 L6 × 4 × 1/2 L6 × 3.5 × 3/8

203.2 203.2 152.4

63.5 63.5 50.8

139.7 139.7 88.9

4. Parametric studies Fig. 2 shows a typical top-seat angle connection and key parameters that are reported to affect its structural response. It is usually assumed that this type of connection can only transfer a vertical reaction through the seat angle, whilst the top angle provides lateral stability, thus neglecting the contribution of supporting gravity loads. However, according to experimental results reported in [20,45,64,65], this type of connection rotates at the critical section of the seat angle, and that the top angle provides resistance to bending forces at the end of the beam. Analytical predictions reported by Kishi and Chen [21], which were based on experimental evidence obtained for traditional steel topseat connections, showed that the geometrical parameters, such as gauge distance (g), bolt diameter (D), depth of beam (d), angle

Table 3 Material properties for various components used in the verifications of FE models. Poisson's ratio

(mm)

The deformation pattern, for specimen A1 tested by Azizinamini, is shown in Fig. 11 for an in-depth comparison between FE simulation and test reports. The connection was rotated around the centre of the horizontal leg of the seat angle, and the maximum horizontal displacement was observed at the heel of the top angle. However, from the Von Mises stress distribution for the top angle, it was observed that the vicinity of the bolt hole in the vertical leg of the top angle was severely deformed, and higher stresses developed near the bolt hole and the fillets of the top angle that were subjected to tensile force. Beams and columns mostly remained elastic, except for areas adjacent to the bolt hole of beam and column flanges. Stress distributions of each angle revealed that the toe of the horizontal leg of top angle yielded first, and this eventually propagated to the toe of the vertical leg of the top angle. The vertical leg of the seat angle, which was subjected to compression, experienced significantly less stress, although the horizontal leg of seat angle experienced higher stress at the ultimate condition. Similar yielding phenomena were reported by Azizinamini through experimental investigations [20] as well as by Kishi-Chen through analytical investigations [21] using the power model. These observations provide further evidence that the adopted FE modelling technique can be considered reliable for simulating the M − ϕ behaviour of top-seat angle bolted connections.

Fig. 10 compares the M − ϕ behaviour obtained using the recently developed numerical models with those reported in the literature. The overall performance of the developed FE models shows good agreement in the elastic range, although the tested connection for Azizinamini A2 showed higher stiffness than those obtained using FE simulation. A power model proposed by Kishi and Chen [21] as well as FE models developed by Ahmed et al. [25,30] were considered to evaluate the observed minor discrepancies; the comparisons clearly showed that the adopted numerical technique can produce a reliable approximation of the M − ϕ behaviour in top-seat angle bolted connections. For A1 and TEST3 specimens, the results obtained from FE models showed very good agreement in the elastic range, although minor deviations were observed between the test and the FEM results at larger rotations between 10–30mrad. The following issues may have caused the observed discrepancies:

Modulus of Elasticity (GPa)

Angle

3.3. Observed deformations and stress distributions in connection

3.2. Moment-rotation behaviour

Ultimate strength (MPa) Angle Bolt

Bolt spacing

Although some minor deviation was observed for the A2 specimen in the linear range, the overall performance of the FE models may be considered acceptable as both the Kishi-Chen power model and Ahmed et al. FE model were in close agreement with the current FE model predictions. Since the moment-rotation curves gradually progressed with no specific ultimate resistance, moment capacities for the connections were compared at 0.02 rad as suggested by AISC (360–10). FEM underestimated the test result by 4.7% for the A1 specimen and overestimated the response of A2 and TEST3 specimens by 4.4% and 4.3%, respectively. This clearly shows that the developed FEM technique is capable of simulating the M − ϕ behaviour of top-seat angle connections with an acceptable level of accuracy.

The modelling technique adopted in the current study was verified using test results reported on top-seat angle bolted connections produced from ordinary carbon steel. To evaluate the accuracy of the FEM approach, numerically obtained results were compared with available test results as well as with the power model data reported in [2,20,45]. Details of key model parameters considered in verifying the FE models are tabulated in Tables 2 and 3. Since no coupon test evidence was reported in the research reports for specimens A1, A2 [20] and TEST3 [45], a bilinear stress–strain relation with isotropic hardening characteristics representing the plastic behaviour of all connection members was considered. A strain hardening constant was determined considering the ultimate strains as 10% and 20% for bolts, and for the other connection members respectively as recommended in [30].

Yield stress (MPa)

Gage

significant effect on the connection response, and this was observed especially in the nonlinear range of the A2 specimen.

3.1. Details of the connections used for verification

Specimen designation

Length

154

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(a) Azizinamini test A1

(b) Azizinamini test A2

(c) Harper TEST3 Fig. 10. FE model verifications with experimental top-seat connection test results.

thickness (t), length of angle (L) as well as material strength had a significant influence on connection stiffness, strength and ductility. Bolt pretension or the clamping force was also reported to have a significant influence on connection behaviour [61], especially in the nonlinear range, and hence was examined in the current parametric analysis. A total of 324 combinations of FE models were considered in the current study as shown in Table 4. AISC steel sections W12 × 96 was used as a column, W14 × 38 and W8 × 21 were used as beam and L6 × 4 with variable thickness angle was considered to model top-seat angles.

Table 4 Variations of geometric parameters considered in FEA. Gauge distance g (mm)

Bolt dia. D (mm)

Beam depth d (mm)

Thickness of the angle t (mm)

Length of the angle L (mm)

Bolt pretension (% of minimum tensile strength)

63.5 50.8 69.9

22.2 19.1 15.9

358.2 210.3 –

9.5 7.9 12.7

203.2 228.6 –

70 80 90

tensile strength following relevant guidelines. AISC [56] and RCSC [57] recommends bolt preload be 70% of the minimum tensile strength, where the tensile strength is equal to 75% of the bolt ultimate strength. Eurocode 3 [66] suggest 70% of bolt ultimate strength as a bolt pretension, which is significantly higher than those suggested by AISC

5. Influence of bolt pretension on top-seat angle connection In the current study, bolt pretension was applied to the predefined surface for bolt shank in the initial load step. In the parametric analysis, bolt pretension was varied by 70%, 80% and 90% of the minimum

Fig. 11. Deformation of (a) connection, (b) stress distribution in top angle FEA, at the 30 mrad rotation for A1 specimen and (c) Azizinamini's test top angle from experiment.

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Fig. 12. Observed variations in M − ϕ curves due to changes in bolt pretension.

capacity of stainless steel connections. The current study adopted Eq. (14) to explicitly recognise the effects of strain hardening using Kp.

and RCSC. Fig. 12 shows an obvious positive trend in connection stiffness and strength with increasing bolt pretension. As bolt pretension was increased, higher clamping forces between the faying surfaces transferring through friction contributed to the increased connection resistance when the load was gradually increased. In the current study, for top-seat angle austenitic bolted connections, an average increase of 6.3% of moment resistance was observed when bolt pretension was increased from 70% to 80% of the minimum tensile strength, whilst this increase in moment resistance nearly doubled (11.6%) when the applied bolt pretension was made 90% of tensile strength. In practice, a constant bolt pretension should be adopted by applying tightening torque to the nut, and for stainless steel fasteners, it is recommended [67] that pretension should not exceed 90% of yield strength. To be on the conservative side, the AISC recommended bolt pretension i.e. 70% of the minimum tensile strength was used in the parametric analysis.

M=

(Ki − Kp ) ϕr ⎡ ⎢⎣1 +

Ki − Kp M0

1

n⎤ n

+ Kp ϕr

× ϕr ⎥⎦

Where reference plastic rotation,

(14)

ϕ0 =

M0 Ki − Kp

(15)

In Eq. (14), M is the connection moment at any instance, M0 is the reference moment, Ki is the initial connection stiffness, Kp is the hardening stiffness of the connection, ϕr is the relative rotation, and n is the shape parameter. 6.2. Boundary conditions of the moment-rotation (M − ϕ) curve

6. Analytical prediction

The general moment-rotation curve as illustrated in Fig. 13 for a top-seat angle connection should satisfy the following boundary conditions:

6.1. Moment-rotation model functions The four parameter power model, originally proposed by Richard and Abbott [68] for elastic–plastic stress-strain relationship, was considered in the current study to formulate the M − ϕ behaviour of austenitic top-seat angle bolted connections. This model requires only four parameters; its formulation is simple and straightforward, and it incorporates the strain-hardening behaviour which is significant for stainless steel alloys. Neglecting this beneficial effect of strain hardening will produce inaccurate predictions by underestimating the

a) The M − ϕ curve should pass through the origin; M(ϕ = 0) = 0. b) The M − ϕ curve should pass through the ultimate point; M(ϕ = ϕu) = Mu. c) The slope of the M − ϕ curve at the origin is equal to the initial = Ki . stiffness: dM dϕ (ϕ =0)

d) As the rotation becomes large (ϕ → ϕu), the M − ϕ curve tends to a straight line, represented by M = M0 + Kpϕ, where M0 is defined as the reference moment or the intercept constant moment, and Kp is the strain hardening stiffness of the M − ϕ curve in the plastic zone. e) The reference plastic rotation ϕ0 is the rotation at the intersecting points between the tangents passing through the initial portion of moment-rotation curve and the line ( M0 + Kpϕ ). 6.3. Determination of moment-rotation key functions Moment-rotation functions were derived in terms of easily obtainable geometric parameters of a top-seat angle connection. The key connection parameters such as initial stiffness (Ki), hardening stiffness (Kp), reference moment (M0) and shape factor (n) were initially expressed in terms of the geometric parameters lumped as shown in Eq. (16a). Such forms of the equation were chosen for its simplicity in statistical calibration as well as to observe the influence of individual geometric parameters on strength, stiffness and shape on the M − ϕ curve.

Fig. 13. A typical M − ϕ curve showing key parameters for the power model.

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logn = d1 logp1 + d2 logp2 + ………+d m logpm

(17d)

Multiple linear regression analyses were performed based on numerically generated results to obtain appropriate values for the coefficients aj, bj, cj and dj. 6.4. Formulation of key parameters of M − ϕ functions Key connection parameters Ki, Kp, M0 and n were initially extracted from FE modelling data for 108 top-seat angles bolted stainless steel connections. Multiple linear regression analyses were conducted to obtain a global optimal solution for the exponents. Geometric input parameters were sequentially organised with an appropriate FE ID for each of the models. Objective functions, which depend on the geometric inputs, were created in the same spreadsheet cell and exponents were enacted upon the decision functions of the objective cell. The necessary constraints for decision variables were also generated. Several iterations considering a specified tolerance of 0.001% was adopted to trigger a move from one iteration cycle to the next iteration for a convergence study. A generalized reduce gradient (GRG) nonlinear data solving tool was adopted to minimize the deviation between FE data and statistically predicted data. Summation of the square of deviation from FE key parameters was calculated and subsequently used to achieve an optimal solution for a general expression for key connection parameters Ki, Kp, M0 and n. Once the coefficients of the geometric parameters were obtained, the sensitivity of each coefficient was tested through examining the physical response in connection behaviour. For example, negative values for exponents associated with gauge distance (g) indicated an inverse relation of g with the connection strength and stiffness; this was in line with the experimental evidence. After a comprehensive statistical analysis, Eq. (18a) was proposed for the key connection parameters Ki, Kp, M0 and n, where all the geometric parameters should be expressed in ‘mm’. Nut width Do depends on bolt diameter (D), and should be calculated according to the AISC specification for bolts [69]. For instance, if 15.9, 19.1 and 22.3 mm diameter bolts are used, the values of Do should be taken as 27.0, 31.8 and 36.5 mm, respectively. Once the geometric configuration of a topseat connection is known, the key connection parameters could be determined easily by using Eq. (18a) as shown below.

(a) Initial and strain hardening stiffness

(b) Reference moment and shape factor Fig. 14. Comparison of connection key parameters prediction. m

∏ pjaj

Ki =

(16a)

J =1

m

Kp =

∏ pjbj

(16b)

J =1

Ki = 3.656 × 10−5 × L1.449 × t1.343 × g−1.192 × D1.033 × d1.804 (kN − m rad ) o (18a)

m

M0 =

∏ pjcj

Kp = 9.572 ×

(16c)

J =1

10−4

×

L0.591

×

t 0.597

×

g−0.590

×

D1.354 o

×

d1.067 (kN

− m rad ) (18b)

m

n=

∏ pjdj

Mo = 8.690 × 10−4 × L0.257 × t1.568 × g−0.937 × D1.237 × d0.936 (kN − m ) o

(16d)

J =1

(18c)

where, pj is the jth size parameter, aj, bj, cj and dj are the dimensionless exponents that indicate the effect of the jth geometric parameter, and m is the number of connection geometric parameters considered. Taking logarithms on both sides of Eq. (16a), the corresponding function parameters can be expressed as shown in Eq. (17a)

logKi = a1 logp1 + a2 logp2 + ………+am logpm

(17a)

logKp = b1 logp1 + b 2 logp2 + ………+bm logpm

(17b)

logM0 = c1 logp1 + c2 logp2 + ………+cm logpm

(17c)

n = 0.535 ×

L0.092

×

t−0.082

×

g0.196

×

D0.079 o

×

d−0.013

(Dimensionless ) (18d)

7. Performance of the suggested formulations to predict key parameters Ki, Kp, M0 and n Initial stiffness Ki plays a significant role by providing rigidity of a connection in the elastic region of M − ϕ curves; higher initial stiffness was observed for larger beams and angles, but the initial stiffness

Table 5 Performance of the proposed models in predicting key parameters Ki, Kp, M0 and n. Total number of validations

Statistical parameters

108

Average COV

Ki, FE Ki, Analytical

1.038 0.097

157

Kp, FE Kp, Analytical

0.990 0.154

M0, FE M0, Analytical

1.001 0.067

n, FE nAnalytical

1.001 0.060

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(a) Variation in gauge distance, g

(b) Variations in angle thickness, t

(c) Variation in bolt diameter, D

(d) Variations in depth of beam, d

(e) Variation in length of angle, L Fig. 15. Comparison of M − ϕ curves for various geometric dimensions.

decreased as the gauge distance was increased. Other key connection parameters Mo and Kp control the moment-rotation behaviour in the nonlinear range, where the connection gradually reaches its ultimate

capacity due to successive yielding in angles and bolts, leading to the formation of several plastic hinges. Hence, connections with thicker angles, larger bolts size along with considerably deeper beams pro158

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Table 6 Performance of the proposed models in predicting key parameters Ki, Kp, M0 and n. ID

Geometric parameters

Ratio of FE/analytical key parameters predictions Ki Kp Mo n

M20, FE (kN-m)

M20, Analytical (kN-m)

Discrepancies in moment at 20 mrad (%)

67

L = 228.6, t = 7.9, g = 63.5, D = 19.1, d = 203.2 L = 228.6, t = 9.5, g = 69.9, D = 19.1, d = 358.2 L = 228.6, t = 9.5, g = 63.5, D = 15.9, d = 358.2 L = 203.2, t = 12.7, g = 50.8, D = 15.9, d = 203.2 L = 228.6, t = 12.7, g = 50.8, D = 15.9, d = 203.2 L = 228.6, t = 9.5, g = 69.9, D = 15.9, d = 358.2 L = 228.6, t = 9.5, g = 69.9, D = 15.9, d = 203.2 L = 228.6, t = 12.7, g = 69.9, D = 15.9, d = 203.2

0.99

0.68

0.98

0.81

21.839

23.426

7.27

0.90

1.31

1.03

1.16

49.670

50.762

2.20

0.97

0.91

1.14

0.96

47.046

42.661

− 9.32

1.20

1.19

0.88

1.02

40.313

43.279

7.36

0.98

1.26

0.86

1.01

41.346

45.104

9.09

0.93

0.93

1.13

0.96

43.180

39.248

− 9.11

1.00

1.00

1.13

0.93

25.386

22.931

− 9.67

1.03

0.86

1.08

1.20

17.911

17.673

− 1.33

199 235 268 286 307 316 319

nonlinear post-elastic regime is difficult to predict with uncertainties involving strain hardening and eventual formation of plastic hinges. It is worth noting that the prediction of Mo and n showed the fewer discrepancies and less scatteredness between FE and the analytical study, as shown in Table 5.

duced significantly higher moment capacities with considerable strain hardening stiffness in the inelastic range. Elastic and inelastic stages of M − ϕ curves were linked through the selection of a suitable shape parameter n. The prediction of key parameters using the proposed analytical formulations as given in Eq. (18a), clearly, reveals the contribution of each geometric parameter in the overall M − ϕ behaviour of a top-seat angle connection. The proposed analytical model also supports the observed experimental mechanism of top-seat angle connections [20] [45] in addition to those reported in other analytical investigations [21–24,70]. The key parameters Ki, Kp, Mo and n predicted by the analytical model and those obtained from FEA are compared in Fig. 14 and the results are summarised in Table 5. It was observed that the predictions for Ki are relatively more accurate than those for Kp as the

8. Prediction of complete M − ϕ curves for top-seat austenitic stainless steel connections 8.1. Prediction of M − ϕ curves for the variation of geometric parameters The performance of the proposed analytical model in predicting the complete M − ϕ behaviour of top-set angle connections produced from austenitic stainless steel was verified against results obtained through

Fig. 16. Comparison of M − ϕ curves when analytical key parameters predict worst values.

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Fig. 16. (continued)

9. Conclusions

FE simulation. Effects of the considered key parameters were evaluated separately using both the analytical model and the FEM technique. Fig. 15 shows the complete range of M − ϕ curves for a number of connection configurations; in each case, one of the parameters was changed whilst keeping others constant. Fig. 15(a) shows an inverse relationship between gauge distance g with connection strength and stiffness, whilst Fig. 15(b, c and d) show that thickness of the angle t, bolt diameter D and depth of the beam d influence connection moment capacity in a proportionate way. However, length of the angles L did not show any significant effect for the considered cases. All M − ϕ curves shown in Fig. 15 clearly illustrate that the proposed analytical model provided good agreement with FE predictions for top-seat angle connections.

Appropriate knowledge on the moment-rotation behaviour of connections is one of the prerequisites for reliable analysis and the design of bare stainless steel structures. The beneficial effects of stainless steel members can only be exploited if the connection is compatible in transferring resulting actions. A top-seat angle connection is one of the most common types used in construction, and the current study proposes an analytical model to predict complete M − ϕ curves for such connections produced from austenitic stainless steel. The proposed model was based on numerical results on carbon steel connections, as there is no experimental evidence currently available on such stainless steel connections. The effects of extensive strain hardening properties of stainless steel were appropriately taken into consideration in the FE models and the subsequent development of the four-parameter analytical model. Equations were proposed to calculate the key connection parameters such as initial connection stiffness Ki, plastic connection stiffness Kp reference moment M0 and shape parameter n. The performance of the proposed formulations was compared against those obtained numerically and showed reasonable agreements with the FE data. The proposed technique requires easily obtainable geometric and material information to generate a complete M− ϕ curve. This simple approach has been extensively verified using numerical techniques, and very good agreement has been observed in predicting complete moment-rotation behaviour of top seat angle connections. This approach should pave the way to develop further analytical models for other connection types in order to recognise the semi-rigid behaviour of stainless steel connections.

8.2. Sensitivity of M − ϕ curve to individual parameter predictions The performance of the analytical prediction of M − ϕ curves was critically investigated when predictions obtained by the Eq. (18a) for individual parameters provided significant discrepancies from the corresponding FE results. A number of connections were identified to evaluate the overall accuracy of the proposed analytical model. Table 6 presents eight different connection types where analytical key parameters showed significant variation (from 13 to 32%) to those obtained from FE predictions. However, in Fig. 16, it is observed that M − ϕ curves show insignificant difference at the initial stage with some noticeable discrepancies observed at post yielding state. It should, however, be noted that the average error in predicting the moment at the 20 mrad (M20) was less than 10%. Given the complexity of connection type and material nonlinearity, this discrepancy is considered acceptable for structural applications. Once more experimental evidence become available, the suggested technique could be easily recalibrated to achieve more accurate predictions.

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