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Load–deformation prediction for eccentrically loaded bolt groups by a kinematic hardening approach
Journal of Constructional Steel Research 65 (2009) 436–442
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Journal of Constructional Steel Research journal homepage: www.elsevier.com/locate/jcsr
Load–deformation prediction for eccentrically loaded bolt groups by a kinematic hardening approach W.H. Siu, R.K.L. Su ∗ Department of Civil Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, China
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Article history: Received 21 August 2007 Accepted 8 March 2008 Keywords: Non-linear Bolt groups Partial interaction Kinematic hardening Slip
a b s t r a c t The use of steel plates anchored on the sides of beams is effective in strengthening existing beams when they have deteriorated. The behaviour of the side plated beam is highly affected by the level of partial interaction between the components, which is provided by the use of bolts acting in groups. To determine the structural response of composite beams, the interaction through bolts or the load deformation response of bolt groups should be modelled. This study aims to develop a procedure to predict the nonlinear load–deformation response of bolt groups subjected to combined in-plane moment and shear. The in-plane behaviour of bolts is first established by adopting the analogy from material plasticity. Then the incremental load–deformation responses of bolt groups are derived. Numerical examples are carried out and the results are verified by the available experimental results. The theory developed herein is able to model the load–deformation response of bolt groups and can potentially be applied to non-linear analyses of bolted side plated structures. © 2008 Elsevier Ltd. All rights reserved.
1. Introduction Composite structures are widely used in building and bridge constructions as they utilize the advantage of using concrete in compression and steel in tension at the same time. One of the relatively new forms of composite structures is produced by anchoring steel plates on the sides of reinforced concrete beams using sets of bolt groups [1–5]. It is shown that, by using the steel plates, the RC beam gains additional strength and stiffness while maintaining a satisfactory level of ductility, and has been used in retrofitting reinforced concrete beams like floor beams [2] and coupling beams [5]. The behaviour of side plated beams with bolt anchors is highly affected by the longitudinal and transverse partial interaction between concrete and steel components. In a standard composite beam, the partial interaction is dominant in the longitudinal direction. But, for a side plated beam, the components were displaced relative to each other in the longitudinal and transverse directions, as both the axial and shear forces are transferred through the shear action of bolts. It has been pointed out that the transverse partial interaction results in a reduction in curvature in the steel component at the critical section, leading to a reduction in strength and stiffness when compared with the case of full
∗ Corresponding author. Fax: +852 2559 5337. E-mail address: [email protected] (R.K.L. Su). 0143-974X/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.jcsr.2008.03.002
interaction [2–4]. To closely predict the behaviour of the composite beam, the behaviour of the partially interacting bolt groups should be considered in a structural analysis for this type of composite structures. In a bolted side plated beam, the connecting bolt groups are not only subjected to translational deformation, but also rotational deformation as the deflection profiles of the steel and concrete components are different. This implies that the bolt groups take up both in-plane shear and moment. While previous research on anchor bolts and bolt groups was mainly focused on their behaviours under pure shear [16–18], research on bolt groups in steelwork subjected to combined shear and moment was carried out and was aimed at determining the strength [6,7,9–11] and stiffness [13] of connections. For the strength analyses, the deformations of individual bolts were related to the rigid body movement of the bolt group, as defined by the centre of rotation. Then, by assuming the load deformation relationship of bolts as elastic, plastic, elasto-plastic or non-linear, the coordinates of the centre of rotation were solved by considering equilibrium, and the ultimate strength was subsequently found. On the other hand, studies were also carried out to describe the rotational stiffness of shear-type bolted moment-connections in cold form steel portal frames [13]. Formulas relating the geometry and the stiffness of bolt groups were developed and the importance of considering the semi-rigidity of bolt groups was pointed out. The purpose of this study is to derive a new numerical algorithm for predicting both the strength and deformation of bolt groups under general in-plane loads. A new model describing the nonlinear in-plane behaviour of bolts when being deformed along
W.H. Siu, R.K.L. Su / Journal of Constructional Steel Research 65 (2009) 436–442
Fig. 1. Idealized tri-linear load–deformation relationship of bolts.
any non-linear path is established based on an analogy from the material plasticity of bolts. Then, the governing equations for the load–deformation relationship of bolt groups under combined in-plane moment and shear are derived and expressed in an incremental form. The mathematical model is implemented via an original computer program to simulate the response of bolt groups and the accuracy of the proposed model is validated against the available experimental data in the literature. The method developed is able to be incorporated into a non-linear program to study the effect of partial interaction on reinforced concrete beams strengthened by bolted side plates. 2. Non-linear load deformation relationship of bolt groups In this section, a kinematic hardening model for simulating a bolt subjected to a multi-directional shear and deformed along a non-linear path is established. Based on that, incremental relationships between the responses of bolt groups and the applied in-plane moment and shear are derived by considering the force equilibrium in the x and y directions as well as the moment equilibrium of bolt groups. By applying finite incremental loads to the bolt groups, the complete load–deformation responses can then be solved by summing up the incremental responses. 2.1. Assumptions The following assumptions are made within the framework of the present formulation. (1) Bolts are connected by a plate that acts as if it is rigid. (2) The media to where the bolts connect are frictionless. (3) The load–deformation relationship of bolts, including the effect of bolt hole elongation, is assumed to be tri-linear (see Fig. 1) and kinematically hardened. (4) The bolts are separated widely such that the interference between them can be neglected. (5) Failure of individual bolts, and hence the bolt group, is governed by internal bolt forces. The first assumption idealizes the bolts as being perfectly fitted in the connection component. Hence, only rigid body motion is permissible for the bolt group. As a result, the relative positions of all the bolts remain unchanged after deformation. For ordinary bolt connections, clearance holes are often used to facilitate bolt installation. Consequently, bolt slip occurs when the load is applied. Such influence is found to be small [12] when compared with the overall response of the bolt group and is often neglected in eccentrically-loaded bolt group analyses [7,9–11]. The second assumption limits the usage of the theory developed for bolt groups comprising ordinary bolts where the precompressive force, and hence the friction between media, is
437
negligible. For high strength friction grip bolt connections, the theory developed herein is not applicable as the applied shear is primarily resisted by the friction between the connecting media. The slip would be insignificant due to the high stiffness provided by the frictional resistance. In this case, the connections can be considered to be rigid. The third statement assumes bolts have a kinematic hardening behaviour, which is widely accepted by engineers to describe the non-linear behaviour of metals and is commonly used in modelling the cyclic behaviour of steelworks. As bolts are, in general, made of metals, the assumption is likely to be valid in the following study. According to the first and fourth assumptions, the force exerted on each bolt can be readily obtained from the deformation of the bolt. After calculating the polar moment of inertia and the stiffness of the bolt group, the relationship between the rigid body movement of the bolt group and the eccentric shear can be established. The fifth assumption is different from the normal assumption that bolt failure is governed by the slip capacity. A limiting strength is chosen as the failure criterion in the present study, because the bolts actually slip in curved paths when subjected to eccentric shear, and the deformation capacity of bolts within a bolt group will be reduced [7]. Hence, it is difficult to define a specific slip capacity for the bolts, and the use of strength as the failure criterion appears to be more appropriate. 2.2. Numerical model for bolts under multidirectional shear Prior to solving the non-linear response of bolt groups, the behaviour of bolts under multidirectional shear has to be defined since the deformation path of bolts is not straight [7]. In this study, the bolt behaviour, which assumes individual bolts are kinematically hardened when the applied load exceeds their yield force, is postulated based on multilayer plasticity as developed by Mroz [14]. As an analogy to multilayer plasticity, the behaviour of bolts is controlled by two surfaces on the force plane, namely the yield surface and the bounding surface. The yield surface represents the occurrence of the plastic stage, while the bounding surface governs the stage of yielding, which will be explained in the following section. The two surfaces are both circular and centred at the origin of the force plane, as illustrated in Fig. 2(a), since bolts under shear generally behave invariantly with respect to in-plane directions. The radii of the yield and bounding surfaces are defined as Pyα and Pyβ , respectively, from the idealized tri-linear load–deformation curve of bolts, as shown in Fig. 1. According to Mroz’s translation rule [14], the yield surface never overlaps with the bounding surface, and the rule implies that the yield surface always nests within the bounding surface. When the initial loading is applied to the bolt, the force point P = (Px , Py )T lies within the yield surface. In this case, the load–deformation response is elastic and can be expressed as
1P = ke 1u,
(1)
where ke is the elastic stiffness of the bolt, and 1P = (1Px , 1Py )T and 1u = (1ux , 1uy )T are the incremental load and deformation, respectively, of the bolt. As further load is applied, the force point gradually shifts away from the origin and will touch the yield surface, and the bolt yields. In general, the stiffness of the bolt in the directions normal and tangential to the yield surface at the force point can be written as 1Pn k 0 1un = p , (2) 1Pt 0 ke 1ut where the subscripts n and t denote the normal and tangential directions, respectively, and kp is the corresponding plastic stiffness in the idealized load–deformation curve, depending on the current yielding stage. The post-yield behaviour of bolts can be divided into two stages as described as follows.
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438
(a) Elastic stage.
(b) First stage yielding.
(c) Second stage yielding.
Fig. 2. Translation rule of yield surfaces of bolts.
2.2.1. First stage yielding First stage yielding refers to the state when the force point of a bolt (denoted by P) is on the yield surface (active surface), as illustrated in Fig. 2(b). The corresponding plastic stiffness equals kpα , as shown in Fig. 1. The unitary normal vector on the yield ˆ where surface at point P is denoted by n,
ˆ = n
P−α kP − αk
(3)
and α is the centre of the yield surface. When an infinitesimal force 1P is exerted on the bolt, the centre of the yield surface ¯ a point on the will translate along the direction joining P and P, bounding surface (target surface), such that the normal to the ˆ by an amount 1α to the new centre bounding surface at P¯ equals n, α¯ . Denoting the translation direction as m, we have m = P¯ − P
(4)
ˆ 1α = α¯ − α = λm
(5)
ˆ is the translation unitary direction and is given in Eq. (6), where m ˆ = m
m
kmk
.
(6)
As the final force P + 1P lies on the new yield surface, we have
¯ . Pyα = kP + 1P − αk
(7)
From Eqs. (4) to (7), the updated location of the centre of the yield surface is readily solved with the given P, 1P and α . 2.2.2. Second stage yielding Second stage yielding refers to the state when the yield surface translates outward from the origin and touches the bounding surface, as shown in Fig. 2(c). In this stage, the stiffness of the bolt in the direction normal to the yield surface is further reduced from the first stage plastic stiffness, kpα , to kpβ (see Fig. 1), and P lies on both surfaces. This implies that the two surfaces touch each other at P. Presented mathematically, we have P¯ = P.
(8)
The common normal to the surfaces at P, is calculated as
ˆ = n
P−α P−β = , kP − αk kP − βk
(9)
where β is the centre of the bounding surface. If an infinitesimal force 1P is applied, the centre of bounding surface translates from β to β¯ along the common normal. In mathematical terms,
1β = β¯ − β = λnˆ .
(10)
Fig. 3. Rigid body movement of bolt group.
When the final force P + 1P is on the new bounding surface, we have
(11) Pyβ = P + 1P − β¯ . From Eqs. (9) to (11), the updated location of the centre of the bounding surface β¯ is readily solved. The yield surface translates accordingly so that the two surfaces remain touching at the force point, P + 1P. Using Eq. (9), we have P + 1P − α¯ P + 1P − β¯
. =
kP + 1P − αk ¯
P + 1P − β¯
(12)
Substituting Eqs. (7) and (11) into (12) and simplifying, we have α¯ = β¯ + Pyβ − Pyα P + 1P − β¯ . (13) From Eq. (13), the new location of the yield surface can be solved once the new location of the bounding surface and the force increment are known. 2.3. Incremental load–deformation relationship of bolt groups Consider a general bolt group, as shown in Fig. 3, subjected to an external load (Fx , Fy , M). The external load is divided into equal, infinitesimal load increments (1Fx , 1Fy , 1M) and is applied to the bolt group sequentially. In a particular loading step j, the elastic and plastic stiffness of the ith bolt are ke and kp respectively. The lateral movements of the bolt group at the origin of the coordinate system are denoted by (1x, 1y, 1θ), as illustrated in Fig. 3. By classical bolt group theory, the relationship between the deformation of the ith bolt (1ux , 1uy )i and the rigid body movement of the bolt group (1x, 1y, 1θ) can be expressed as 1x 1ux 1 0 −yi 1y . = (14) 1uy i 0 1 xi
1θ
W.H. Siu, R.K.L. Su / Journal of Constructional Steel Research 65 (2009) 436–442
439
Fig. 4. Post yield load–deformation relationship of bolts.
The load–deformation response of the ith bolt can be represented by Eq. (2) as,
1Pn k = p 1Pt i 0
0 ke
i
1un . 1ut i
(15)
Applying a coordinate transformation to the global x and y directions, and using Eq. (14), the following relationship can be obtained 1x 1Px c −s kp 0 c s 1 0 −yi 1y . = 1Py i s c i 0 ke i −s c i 0 1 xi
1θ
i
(16) Here, c and s corresponds to cos φ and sin φ respectively, with the ˆ i ). value φ as indicated in Fig. 4 and equal to arg(n Expanding Eq. (16) and substituting k = ke − kp , where k represents the stiffness degradation, results in
1Px 1Py
Fig. 5. Schematic diagram of the computer program BOGAN.
i
" k − c2 k = e −csk
−csk ke − s2 k
# 1x −(ke − c2 k)yi − (csk)xi 1y . (ke − s2 k)xi + (csk)yi
1θ
(17)
i
Considering the global force equilibrium of the applied loads and the bolt forces, we have X X 1Fx = 1Pxi , 1Fy = 1Pyi and i
1M =
i
X
(18)
Substituting Eq. (17) into Eq. (18) and rearranging, we have
1x 1y 1θ
Kcc −Kcs
= −Rccy − Rcsx
−Kcs Kss Rssx + Rcsy
dj =
j X
(1x, 1y, 1θ)n .
(20)
n=1
3. Implementation of non-linear bolt group theory
(1Pyi xi − 1Pxi yi ).
i
response of the general bolt group dj after the jth loading step is readily solved by summing up incremental responses, represented mathematically by
−1
−Rccy − Rcsx Rssx + Rcsy Iccy + Issx + 2Ics
1F x 1Fy , 1M
(19)
P P P 2 where Kcc = c2 k, Kss = i k, Rssx = i ke − i ke − sP i ci sP i k, Kcs = P Pi 2 2 si k)xi , Rccy = i (ke − ci k)yi , Rcsx = i ci si xi k, Rcsy = i ci si yi k, i (ke − P P P Iccy = i (ke − ci2 k)y2i , Issx = i (ke − s2i k)x2i , and Ics = i ci si xi yi k. Eq. (19) represents the incremental load–deformation relationship of the bolt group under an in-plane shear force in any direction and eccentricity at a single load step. In particular, when k equals zero (i.e. all bolts are elastic), all non-diagonal terms become zero and it represents the elastic case where the behaviour of bolt groups can be simulated by independent springs. The deformation
Following the aforementioned non-linear bolt group theory, a computer program, BOGAN, written in FORTRAN, was developed to carry out numerical simulations of bolt groups under in-plane shear. The incremental load step approach is used in the program; a schematic diagram showing the major steps of BOGAN is presented in Fig. 5, and the details are presented herein. Step 1: The parameters for the analysis are defined. These include: (1) The properties of the bolts (e.g. the stiffnesses of the bolts and the two yield points of the bolts); (2) The geometries of the bolt group (i.e. the number and coordinates of bolts with respect to an origin defined by the user); (3) Details of the simulation (e.g. the total applied load and the number of load steps). Step 2: To begin with, an incremental load is applied to the bolt group. Step 3: The yield conditions of the bolts are checked and the corresponding value kpi is set in the stiffness matrix in Eq. (2). Using Eq. (19) and the value φij , the global stiffness matrix is assembled.
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440
Table 1 Specimen details in experiment by Crawford and Kulak Specimen number
Bolt group arrangement (row × column)
Row separation (mm)
Column separation (mm)
Eccentricity (mm)
B1 B2 B3 B4 B5 B6 B7 B8
5×1 5×1 5×1 6×1 6×1 4×2 4×2 5×2
63.5 76.2 76.2 76.2 76.2 76.2 76.2 63.5
– – – – – 63.5 63.5 63.5
203.2 254.0 304.8 330.2 381.0 304.8 381.0 381.0
Step 4: The rigid body movement (1x, 1y, 1θ)j of the jth loading step is solved using Eq. (19). Then the incremental deformations and forces on individual bolts are calculated by Eqs. (14) and (16). Step 5: The yield conditions of the bolts are checked by considering the updated bolt force vector. For the ith yielded bolts at the 1st stage yielding, the unitary translational direction ˆ ij is computed using Eq. (4). Using of the yield surface m ˆ ij , the updated locations of the centre of the computed m yield surface α¯ ij are solved using Eqs. (5) and (7). For the ith yielded bolts at the 2nd stage yielding and using the value ˆ ij computed at the previous load step, the centres of the n yield and bounding surfaces are solved using Eqs. (10), (11) and (13). Step 6: The centres of the yield surface and bounding surface α i(j+1) and β i(j+1) are set as α¯ ij and β¯ ij , respectively. The normal to the yield surface at the updated force point of the ˆ i(j+1) is solved using Eq. (3) and the values of φi(j+1) bolts n are also computed. These values are used to assemble the stiffness matrices in the j + 1th loading step. Step 7: The cumulative bolt forces (Px , Py )i and deformations (ux , uy )i are computed and updated. The cumulative deformations are then used to update the instantaneous locations of the bolts. Step 8: The bolt forces are checked against the ultimate resistance of the bolts. If any of the bolt forces exceed the limiting resistance, the program will be terminated. Otherwise, the program proceeds to the j + 1th loading step (Step 2) and further loads are applied to the bolt group.
Fig. 6. Experimental setup in Kulak and Crawford’s study.
4. Numerical simulation by computer program BOGAN In this section, numerical simulations carried out by the program BOGAN are discussed. The experimental study of nonlinear deformations of bolt groups, performed by Crawford and Kulak [8], was modelled by BOGAN to verify the applicability of the theory to the practical problem. 4.1. Experimental investigation In Crawford and Kulak’s experiments [8], eight bolt groups were tested by a setup in which two bolt groups were used to connect the supports to the two ends of a deep beam with the load applied at its mid-span, as shown in Fig. 6. The eccentric load was applied to the bolt groups by the reaction at the supports. The load-rotation relationship and the ultimate strength of the bolt groups were recorded in the experiments. The parameters varied in the test included the dimensions of the bolt groups, the separation between bolts, and the eccentricities of the applied loads. The detailed arrangements of the specimens, which consist of 1–2 columns with 4–6 bolts in each column, are given in Table 1. More details concerning the experiment, including the test setup and the shear test of the bolts, can be found in reference [8].
Fig. 7. Load–deformation relationship of bolts adopted in the simulations.
4.2. Simulation by BOGAN To simulate the load–deformation response of the bolt groups, the properties of the bolt groups, including the coordinates of the bolts and the load–deformation relationship of the bolts, have to be defined. In this analysis, the load–deformation relationship of the bolts is based on the shear test results obtained by Crawford and Kulak, in which a compression jig is used. It had been revealed that using a compression jig in the shear test results in a 10% increase in shear strength with little effect on initial stiffness [15], and the failure behaviour of bolts was better reflected by a tension jig experiment [12]; hence, the load–deformation relationship of bolts used for simulations was adjusted accordingly and idealized as a tri-linear function, as shown in Fig. 7. It should be noted that most of the bolts behave with gradual stiffness drops as the deformation increases; a clear yield point and bounding point are difficult to define on the load–deformation curve. Hence, a notional yield point and bounding point that could fit
W.H. Siu, R.K.L. Su / Journal of Constructional Steel Research 65 (2009) 436–442
(a) Specimen B1.
441
(b) Specimen B4.
(c) Specimen B7. Fig. 8. Comparison of moment rotation curves. Table 2 Mechanical properties of bolt used in the simulations k (kN/mm)
Table 3 Summary of the numerical and experimental results
P (kN)
Ultimate Load (kN)
ke
kp α
kp β
Pyα
Pyβ
Pu
526.7
66.3
5.7
133.8
260.0
297.0
the actual load–deformation curve were used for computational convenience. A proportional load defined by the corresponding eccentricities was applied to the bolt groups until failure, and the geometries of the bolt groups and the mechanical properties of bolts are summarized in Tables 1 and 2 respectively. The loadcarrying capacity, deformability, and complete load–deformation relationship of the bolts are compared with the experimental results in the following section. 4.3. Comparison of the numerical and experimental results Due to the symmetric arrangements of bolt groups about the yaxis, only the bolt group on the left hand side of the experimental setup (see Fig. 6) was modelled with the applied load acting along the y axis, which is also the axis of symmetry. Moreover, the applied load P was also reduced by half in the numerical simulation. The calculated load–rotation curves and the measured results [8] of specimen B1, B4 and B7 are plotted in Fig. 8(a)–(c). In general, the measured and calculated results agreed well with each other. However, the simulated elastic stiffness is consistently lower than the experimental results. The difference may be due to
B1 B2 B3 B4 B5 B6 B7 B8
Stiffness (Secant stiffness at 40% ultimate loading) (kN/0.01rad)
Experimental
Numerical
Exp a
Exp b
Numerical
504.0 515.2 425.6 562.2 495.0 591.4 474.9 595.8
530.9 508.5 430.1 575.7 501.8 566.7 497.3 633.9
1408.5 1791.5 1476.5 1923.5 3594.2 1294.5 1168.6 2248.0
2304.9 1743.3 1476.5 2169.8 3594.2 1503.1 1284.5 2236.8
937.9 1076.2 921.5 1477.2 1263.8 1112.1 898.1 1238.6
ignoring the frictional force between the connecting media in the numerical simulation. When preload was applied on bolts in the experimental study, the external loading would be first resisted by the friction between the connecting media [12], the magnitude of which depends on factors like the preload in the bolts and the conditions of the fraying surfaces, and results in a higher stiffness for the bolt connection. A detailed comparison of the calculated and measured elastic stiffness of the bolt groups is summarized in Table 3. When the external loading applied on the bolt group increases and approaches its ultimate state, the numerical results agree closely with the experimental results, as illustrated in Fig. 8(a)–(c) and listed in Table 3. It may be due to the fact that at high shear load level, relatively larger shear deformations occur in bolts, releasing the axial elongation, and hence the bolt preload [19–21]. So the
442
W.H. Siu, R.K.L. Su / Journal of Constructional Steel Research 65 (2009) 436–442
responses of bolt groups indicates that the theory developed can be readily applied to non-linear analyses of bolted side plated beams with consideration of the effect of the partial interactions of the bolts. Acknowledgements The work described in this paper has been fully supported by The University of Hong Kong through the Small Project Funding 2005-06 and by the Research Grants Council of Hong Kong SAR (Project Nos. HKU7129/03E and HKU7168/06E). References
Fig. 9. Load rotation response of B7 simulated using different load increments.
friction between the fraying surfaces gradually decreases and has no significant effect on the shear strength of bolts. From the consistent numerical and experimental results of the individual bolt groups, it can be concluded that the proposed numerical technique is generally accurate and reliable for determining the load–deformation relationship of bolt groups. However, when the particular focus is on the initial stiffness of the bolt groups, the effect of the frictional behaviour between the fraying surfaces should be considered. The load–deformation relationship of B7 is simulated using different load increment sizes ranging from 1% to 7% of the loadcarrying capacity of the bolt group and is plotted in Fig. 9. It is observed that variations in load increment size have little effect on the predicted ultimate strength and the shape of the load–deformation curve. 5. Conclusions This paper presents a new solution algorithm to calculate the non-linear load–deformation response of bolt groups subjected to combined in-plane moment and shear. The bolts are assumed to be kinematically hardened when yielded. By postulating the post-yield behaviour of bolts using Mroz’s translation rules for the yield surface, the stiffness of bolts in the x and y directions was derived. Further application of the classical theory of bolt groups, the complete non-linear response of bolt groups, and the behaviour of individual bolts were evaluated accordingly. The computer program BOGAN, which simulates the non-linear responses of bolt groups under combined in-plane moment and shear, has been implemented to evaluate the effectiveness of the theories in real applications. Its reliability and accuracy have been verified by comparing the numerical results with those from an experiment in the literature. The success of using BOGAN to simulate the
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Journal of Constructional Steel Research journal homepage: www.elsevier.com/locate/jcsr
Load–deformation prediction for eccentrically loaded bolt groups by a kinematic hardening approach W.H. Siu, R.K.L. Su ∗ Department of Civil Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, China
article
info
Article history: Received 21 August 2007 Accepted 8 March 2008 Keywords: Non-linear Bolt groups Partial interaction Kinematic hardening Slip
a b s t r a c t The use of steel plates anchored on the sides of beams is effective in strengthening existing beams when they have deteriorated. The behaviour of the side plated beam is highly affected by the level of partial interaction between the components, which is provided by the use of bolts acting in groups. To determine the structural response of composite beams, the interaction through bolts or the load deformation response of bolt groups should be modelled. This study aims to develop a procedure to predict the nonlinear load–deformation response of bolt groups subjected to combined in-plane moment and shear. The in-plane behaviour of bolts is first established by adopting the analogy from material plasticity. Then the incremental load–deformation responses of bolt groups are derived. Numerical examples are carried out and the results are verified by the available experimental results. The theory developed herein is able to model the load–deformation response of bolt groups and can potentially be applied to non-linear analyses of bolted side plated structures. © 2008 Elsevier Ltd. All rights reserved.
1. Introduction Composite structures are widely used in building and bridge constructions as they utilize the advantage of using concrete in compression and steel in tension at the same time. One of the relatively new forms of composite structures is produced by anchoring steel plates on the sides of reinforced concrete beams using sets of bolt groups [1–5]. It is shown that, by using the steel plates, the RC beam gains additional strength and stiffness while maintaining a satisfactory level of ductility, and has been used in retrofitting reinforced concrete beams like floor beams [2] and coupling beams [5]. The behaviour of side plated beams with bolt anchors is highly affected by the longitudinal and transverse partial interaction between concrete and steel components. In a standard composite beam, the partial interaction is dominant in the longitudinal direction. But, for a side plated beam, the components were displaced relative to each other in the longitudinal and transverse directions, as both the axial and shear forces are transferred through the shear action of bolts. It has been pointed out that the transverse partial interaction results in a reduction in curvature in the steel component at the critical section, leading to a reduction in strength and stiffness when compared with the case of full
∗ Corresponding author. Fax: +852 2559 5337. E-mail address: [email protected] (R.K.L. Su). 0143-974X/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.jcsr.2008.03.002
interaction [2–4]. To closely predict the behaviour of the composite beam, the behaviour of the partially interacting bolt groups should be considered in a structural analysis for this type of composite structures. In a bolted side plated beam, the connecting bolt groups are not only subjected to translational deformation, but also rotational deformation as the deflection profiles of the steel and concrete components are different. This implies that the bolt groups take up both in-plane shear and moment. While previous research on anchor bolts and bolt groups was mainly focused on their behaviours under pure shear [16–18], research on bolt groups in steelwork subjected to combined shear and moment was carried out and was aimed at determining the strength [6,7,9–11] and stiffness [13] of connections. For the strength analyses, the deformations of individual bolts were related to the rigid body movement of the bolt group, as defined by the centre of rotation. Then, by assuming the load deformation relationship of bolts as elastic, plastic, elasto-plastic or non-linear, the coordinates of the centre of rotation were solved by considering equilibrium, and the ultimate strength was subsequently found. On the other hand, studies were also carried out to describe the rotational stiffness of shear-type bolted moment-connections in cold form steel portal frames [13]. Formulas relating the geometry and the stiffness of bolt groups were developed and the importance of considering the semi-rigidity of bolt groups was pointed out. The purpose of this study is to derive a new numerical algorithm for predicting both the strength and deformation of bolt groups under general in-plane loads. A new model describing the nonlinear in-plane behaviour of bolts when being deformed along
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Fig. 1. Idealized tri-linear load–deformation relationship of bolts.
any non-linear path is established based on an analogy from the material plasticity of bolts. Then, the governing equations for the load–deformation relationship of bolt groups under combined in-plane moment and shear are derived and expressed in an incremental form. The mathematical model is implemented via an original computer program to simulate the response of bolt groups and the accuracy of the proposed model is validated against the available experimental data in the literature. The method developed is able to be incorporated into a non-linear program to study the effect of partial interaction on reinforced concrete beams strengthened by bolted side plates. 2. Non-linear load deformation relationship of bolt groups In this section, a kinematic hardening model for simulating a bolt subjected to a multi-directional shear and deformed along a non-linear path is established. Based on that, incremental relationships between the responses of bolt groups and the applied in-plane moment and shear are derived by considering the force equilibrium in the x and y directions as well as the moment equilibrium of bolt groups. By applying finite incremental loads to the bolt groups, the complete load–deformation responses can then be solved by summing up the incremental responses. 2.1. Assumptions The following assumptions are made within the framework of the present formulation. (1) Bolts are connected by a plate that acts as if it is rigid. (2) The media to where the bolts connect are frictionless. (3) The load–deformation relationship of bolts, including the effect of bolt hole elongation, is assumed to be tri-linear (see Fig. 1) and kinematically hardened. (4) The bolts are separated widely such that the interference between them can be neglected. (5) Failure of individual bolts, and hence the bolt group, is governed by internal bolt forces. The first assumption idealizes the bolts as being perfectly fitted in the connection component. Hence, only rigid body motion is permissible for the bolt group. As a result, the relative positions of all the bolts remain unchanged after deformation. For ordinary bolt connections, clearance holes are often used to facilitate bolt installation. Consequently, bolt slip occurs when the load is applied. Such influence is found to be small [12] when compared with the overall response of the bolt group and is often neglected in eccentrically-loaded bolt group analyses [7,9–11]. The second assumption limits the usage of the theory developed for bolt groups comprising ordinary bolts where the precompressive force, and hence the friction between media, is
437
negligible. For high strength friction grip bolt connections, the theory developed herein is not applicable as the applied shear is primarily resisted by the friction between the connecting media. The slip would be insignificant due to the high stiffness provided by the frictional resistance. In this case, the connections can be considered to be rigid. The third statement assumes bolts have a kinematic hardening behaviour, which is widely accepted by engineers to describe the non-linear behaviour of metals and is commonly used in modelling the cyclic behaviour of steelworks. As bolts are, in general, made of metals, the assumption is likely to be valid in the following study. According to the first and fourth assumptions, the force exerted on each bolt can be readily obtained from the deformation of the bolt. After calculating the polar moment of inertia and the stiffness of the bolt group, the relationship between the rigid body movement of the bolt group and the eccentric shear can be established. The fifth assumption is different from the normal assumption that bolt failure is governed by the slip capacity. A limiting strength is chosen as the failure criterion in the present study, because the bolts actually slip in curved paths when subjected to eccentric shear, and the deformation capacity of bolts within a bolt group will be reduced [7]. Hence, it is difficult to define a specific slip capacity for the bolts, and the use of strength as the failure criterion appears to be more appropriate. 2.2. Numerical model for bolts under multidirectional shear Prior to solving the non-linear response of bolt groups, the behaviour of bolts under multidirectional shear has to be defined since the deformation path of bolts is not straight [7]. In this study, the bolt behaviour, which assumes individual bolts are kinematically hardened when the applied load exceeds their yield force, is postulated based on multilayer plasticity as developed by Mroz [14]. As an analogy to multilayer plasticity, the behaviour of bolts is controlled by two surfaces on the force plane, namely the yield surface and the bounding surface. The yield surface represents the occurrence of the plastic stage, while the bounding surface governs the stage of yielding, which will be explained in the following section. The two surfaces are both circular and centred at the origin of the force plane, as illustrated in Fig. 2(a), since bolts under shear generally behave invariantly with respect to in-plane directions. The radii of the yield and bounding surfaces are defined as Pyα and Pyβ , respectively, from the idealized tri-linear load–deformation curve of bolts, as shown in Fig. 1. According to Mroz’s translation rule [14], the yield surface never overlaps with the bounding surface, and the rule implies that the yield surface always nests within the bounding surface. When the initial loading is applied to the bolt, the force point P = (Px , Py )T lies within the yield surface. In this case, the load–deformation response is elastic and can be expressed as
1P = ke 1u,
(1)
where ke is the elastic stiffness of the bolt, and 1P = (1Px , 1Py )T and 1u = (1ux , 1uy )T are the incremental load and deformation, respectively, of the bolt. As further load is applied, the force point gradually shifts away from the origin and will touch the yield surface, and the bolt yields. In general, the stiffness of the bolt in the directions normal and tangential to the yield surface at the force point can be written as 1Pn k 0 1un = p , (2) 1Pt 0 ke 1ut where the subscripts n and t denote the normal and tangential directions, respectively, and kp is the corresponding plastic stiffness in the idealized load–deformation curve, depending on the current yielding stage. The post-yield behaviour of bolts can be divided into two stages as described as follows.
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(a) Elastic stage.
(b) First stage yielding.
(c) Second stage yielding.
Fig. 2. Translation rule of yield surfaces of bolts.
2.2.1. First stage yielding First stage yielding refers to the state when the force point of a bolt (denoted by P) is on the yield surface (active surface), as illustrated in Fig. 2(b). The corresponding plastic stiffness equals kpα , as shown in Fig. 1. The unitary normal vector on the yield ˆ where surface at point P is denoted by n,
ˆ = n
P−α kP − αk
(3)
and α is the centre of the yield surface. When an infinitesimal force 1P is exerted on the bolt, the centre of the yield surface ¯ a point on the will translate along the direction joining P and P, bounding surface (target surface), such that the normal to the ˆ by an amount 1α to the new centre bounding surface at P¯ equals n, α¯ . Denoting the translation direction as m, we have m = P¯ − P
(4)
ˆ 1α = α¯ − α = λm
(5)
ˆ is the translation unitary direction and is given in Eq. (6), where m ˆ = m
m
kmk
.
(6)
As the final force P + 1P lies on the new yield surface, we have
¯ . Pyα = kP + 1P − αk
(7)
From Eqs. (4) to (7), the updated location of the centre of the yield surface is readily solved with the given P, 1P and α . 2.2.2. Second stage yielding Second stage yielding refers to the state when the yield surface translates outward from the origin and touches the bounding surface, as shown in Fig. 2(c). In this stage, the stiffness of the bolt in the direction normal to the yield surface is further reduced from the first stage plastic stiffness, kpα , to kpβ (see Fig. 1), and P lies on both surfaces. This implies that the two surfaces touch each other at P. Presented mathematically, we have P¯ = P.
(8)
The common normal to the surfaces at P, is calculated as
ˆ = n
P−α P−β = , kP − αk kP − βk
(9)
where β is the centre of the bounding surface. If an infinitesimal force 1P is applied, the centre of bounding surface translates from β to β¯ along the common normal. In mathematical terms,
1β = β¯ − β = λnˆ .
(10)
Fig. 3. Rigid body movement of bolt group.
When the final force P + 1P is on the new bounding surface, we have
(11) Pyβ = P + 1P − β¯ . From Eqs. (9) to (11), the updated location of the centre of the bounding surface β¯ is readily solved. The yield surface translates accordingly so that the two surfaces remain touching at the force point, P + 1P. Using Eq. (9), we have P + 1P − α¯ P + 1P − β¯
. =
kP + 1P − αk ¯
P + 1P − β¯
(12)
Substituting Eqs. (7) and (11) into (12) and simplifying, we have α¯ = β¯ + Pyβ − Pyα P + 1P − β¯ . (13) From Eq. (13), the new location of the yield surface can be solved once the new location of the bounding surface and the force increment are known. 2.3. Incremental load–deformation relationship of bolt groups Consider a general bolt group, as shown in Fig. 3, subjected to an external load (Fx , Fy , M). The external load is divided into equal, infinitesimal load increments (1Fx , 1Fy , 1M) and is applied to the bolt group sequentially. In a particular loading step j, the elastic and plastic stiffness of the ith bolt are ke and kp respectively. The lateral movements of the bolt group at the origin of the coordinate system are denoted by (1x, 1y, 1θ), as illustrated in Fig. 3. By classical bolt group theory, the relationship between the deformation of the ith bolt (1ux , 1uy )i and the rigid body movement of the bolt group (1x, 1y, 1θ) can be expressed as 1x 1ux 1 0 −yi 1y . = (14) 1uy i 0 1 xi
1θ
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439
Fig. 4. Post yield load–deformation relationship of bolts.
The load–deformation response of the ith bolt can be represented by Eq. (2) as,
1Pn k = p 1Pt i 0
0 ke
i
1un . 1ut i
(15)
Applying a coordinate transformation to the global x and y directions, and using Eq. (14), the following relationship can be obtained 1x 1Px c −s kp 0 c s 1 0 −yi 1y . = 1Py i s c i 0 ke i −s c i 0 1 xi
1θ
i
(16) Here, c and s corresponds to cos φ and sin φ respectively, with the ˆ i ). value φ as indicated in Fig. 4 and equal to arg(n Expanding Eq. (16) and substituting k = ke − kp , where k represents the stiffness degradation, results in
1Px 1Py
Fig. 5. Schematic diagram of the computer program BOGAN.
i
" k − c2 k = e −csk
−csk ke − s2 k
# 1x −(ke − c2 k)yi − (csk)xi 1y . (ke − s2 k)xi + (csk)yi
1θ
(17)
i
Considering the global force equilibrium of the applied loads and the bolt forces, we have X X 1Fx = 1Pxi , 1Fy = 1Pyi and i
1M =
i
X
(18)
Substituting Eq. (17) into Eq. (18) and rearranging, we have
1x 1y 1θ
Kcc −Kcs
= −Rccy − Rcsx
−Kcs Kss Rssx + Rcsy
dj =
j X
(1x, 1y, 1θ)n .
(20)
n=1
3. Implementation of non-linear bolt group theory
(1Pyi xi − 1Pxi yi ).
i
response of the general bolt group dj after the jth loading step is readily solved by summing up incremental responses, represented mathematically by
−1
−Rccy − Rcsx Rssx + Rcsy Iccy + Issx + 2Ics
1F x 1Fy , 1M
(19)
P P P 2 where Kcc = c2 k, Kss = i k, Rssx = i ke − i ke − sP i ci sP i k, Kcs = P Pi 2 2 si k)xi , Rccy = i (ke − ci k)yi , Rcsx = i ci si xi k, Rcsy = i ci si yi k, i (ke − P P P Iccy = i (ke − ci2 k)y2i , Issx = i (ke − s2i k)x2i , and Ics = i ci si xi yi k. Eq. (19) represents the incremental load–deformation relationship of the bolt group under an in-plane shear force in any direction and eccentricity at a single load step. In particular, when k equals zero (i.e. all bolts are elastic), all non-diagonal terms become zero and it represents the elastic case where the behaviour of bolt groups can be simulated by independent springs. The deformation
Following the aforementioned non-linear bolt group theory, a computer program, BOGAN, written in FORTRAN, was developed to carry out numerical simulations of bolt groups under in-plane shear. The incremental load step approach is used in the program; a schematic diagram showing the major steps of BOGAN is presented in Fig. 5, and the details are presented herein. Step 1: The parameters for the analysis are defined. These include: (1) The properties of the bolts (e.g. the stiffnesses of the bolts and the two yield points of the bolts); (2) The geometries of the bolt group (i.e. the number and coordinates of bolts with respect to an origin defined by the user); (3) Details of the simulation (e.g. the total applied load and the number of load steps). Step 2: To begin with, an incremental load is applied to the bolt group. Step 3: The yield conditions of the bolts are checked and the corresponding value kpi is set in the stiffness matrix in Eq. (2). Using Eq. (19) and the value φij , the global stiffness matrix is assembled.
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Table 1 Specimen details in experiment by Crawford and Kulak Specimen number
Bolt group arrangement (row × column)
Row separation (mm)
Column separation (mm)
Eccentricity (mm)
B1 B2 B3 B4 B5 B6 B7 B8
5×1 5×1 5×1 6×1 6×1 4×2 4×2 5×2
63.5 76.2 76.2 76.2 76.2 76.2 76.2 63.5
– – – – – 63.5 63.5 63.5
203.2 254.0 304.8 330.2 381.0 304.8 381.0 381.0
Step 4: The rigid body movement (1x, 1y, 1θ)j of the jth loading step is solved using Eq. (19). Then the incremental deformations and forces on individual bolts are calculated by Eqs. (14) and (16). Step 5: The yield conditions of the bolts are checked by considering the updated bolt force vector. For the ith yielded bolts at the 1st stage yielding, the unitary translational direction ˆ ij is computed using Eq. (4). Using of the yield surface m ˆ ij , the updated locations of the centre of the computed m yield surface α¯ ij are solved using Eqs. (5) and (7). For the ith yielded bolts at the 2nd stage yielding and using the value ˆ ij computed at the previous load step, the centres of the n yield and bounding surfaces are solved using Eqs. (10), (11) and (13). Step 6: The centres of the yield surface and bounding surface α i(j+1) and β i(j+1) are set as α¯ ij and β¯ ij , respectively. The normal to the yield surface at the updated force point of the ˆ i(j+1) is solved using Eq. (3) and the values of φi(j+1) bolts n are also computed. These values are used to assemble the stiffness matrices in the j + 1th loading step. Step 7: The cumulative bolt forces (Px , Py )i and deformations (ux , uy )i are computed and updated. The cumulative deformations are then used to update the instantaneous locations of the bolts. Step 8: The bolt forces are checked against the ultimate resistance of the bolts. If any of the bolt forces exceed the limiting resistance, the program will be terminated. Otherwise, the program proceeds to the j + 1th loading step (Step 2) and further loads are applied to the bolt group.
Fig. 6. Experimental setup in Kulak and Crawford’s study.
4. Numerical simulation by computer program BOGAN In this section, numerical simulations carried out by the program BOGAN are discussed. The experimental study of nonlinear deformations of bolt groups, performed by Crawford and Kulak [8], was modelled by BOGAN to verify the applicability of the theory to the practical problem. 4.1. Experimental investigation In Crawford and Kulak’s experiments [8], eight bolt groups were tested by a setup in which two bolt groups were used to connect the supports to the two ends of a deep beam with the load applied at its mid-span, as shown in Fig. 6. The eccentric load was applied to the bolt groups by the reaction at the supports. The load-rotation relationship and the ultimate strength of the bolt groups were recorded in the experiments. The parameters varied in the test included the dimensions of the bolt groups, the separation between bolts, and the eccentricities of the applied loads. The detailed arrangements of the specimens, which consist of 1–2 columns with 4–6 bolts in each column, are given in Table 1. More details concerning the experiment, including the test setup and the shear test of the bolts, can be found in reference [8].
Fig. 7. Load–deformation relationship of bolts adopted in the simulations.
4.2. Simulation by BOGAN To simulate the load–deformation response of the bolt groups, the properties of the bolt groups, including the coordinates of the bolts and the load–deformation relationship of the bolts, have to be defined. In this analysis, the load–deformation relationship of the bolts is based on the shear test results obtained by Crawford and Kulak, in which a compression jig is used. It had been revealed that using a compression jig in the shear test results in a 10% increase in shear strength with little effect on initial stiffness [15], and the failure behaviour of bolts was better reflected by a tension jig experiment [12]; hence, the load–deformation relationship of bolts used for simulations was adjusted accordingly and idealized as a tri-linear function, as shown in Fig. 7. It should be noted that most of the bolts behave with gradual stiffness drops as the deformation increases; a clear yield point and bounding point are difficult to define on the load–deformation curve. Hence, a notional yield point and bounding point that could fit
W.H. Siu, R.K.L. Su / Journal of Constructional Steel Research 65 (2009) 436–442
(a) Specimen B1.
441
(b) Specimen B4.
(c) Specimen B7. Fig. 8. Comparison of moment rotation curves. Table 2 Mechanical properties of bolt used in the simulations k (kN/mm)
Table 3 Summary of the numerical and experimental results
P (kN)
Ultimate Load (kN)
ke
kp α
kp β
Pyα
Pyβ
Pu
526.7
66.3
5.7
133.8
260.0
297.0
the actual load–deformation curve were used for computational convenience. A proportional load defined by the corresponding eccentricities was applied to the bolt groups until failure, and the geometries of the bolt groups and the mechanical properties of bolts are summarized in Tables 1 and 2 respectively. The loadcarrying capacity, deformability, and complete load–deformation relationship of the bolts are compared with the experimental results in the following section. 4.3. Comparison of the numerical and experimental results Due to the symmetric arrangements of bolt groups about the yaxis, only the bolt group on the left hand side of the experimental setup (see Fig. 6) was modelled with the applied load acting along the y axis, which is also the axis of symmetry. Moreover, the applied load P was also reduced by half in the numerical simulation. The calculated load–rotation curves and the measured results [8] of specimen B1, B4 and B7 are plotted in Fig. 8(a)–(c). In general, the measured and calculated results agreed well with each other. However, the simulated elastic stiffness is consistently lower than the experimental results. The difference may be due to
B1 B2 B3 B4 B5 B6 B7 B8
Stiffness (Secant stiffness at 40% ultimate loading) (kN/0.01rad)
Experimental
Numerical
Exp a
Exp b
Numerical
504.0 515.2 425.6 562.2 495.0 591.4 474.9 595.8
530.9 508.5 430.1 575.7 501.8 566.7 497.3 633.9
1408.5 1791.5 1476.5 1923.5 3594.2 1294.5 1168.6 2248.0
2304.9 1743.3 1476.5 2169.8 3594.2 1503.1 1284.5 2236.8
937.9 1076.2 921.5 1477.2 1263.8 1112.1 898.1 1238.6
ignoring the frictional force between the connecting media in the numerical simulation. When preload was applied on bolts in the experimental study, the external loading would be first resisted by the friction between the connecting media [12], the magnitude of which depends on factors like the preload in the bolts and the conditions of the fraying surfaces, and results in a higher stiffness for the bolt connection. A detailed comparison of the calculated and measured elastic stiffness of the bolt groups is summarized in Table 3. When the external loading applied on the bolt group increases and approaches its ultimate state, the numerical results agree closely with the experimental results, as illustrated in Fig. 8(a)–(c) and listed in Table 3. It may be due to the fact that at high shear load level, relatively larger shear deformations occur in bolts, releasing the axial elongation, and hence the bolt preload [19–21]. So the
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responses of bolt groups indicates that the theory developed can be readily applied to non-linear analyses of bolted side plated beams with consideration of the effect of the partial interactions of the bolts. Acknowledgements The work described in this paper has been fully supported by The University of Hong Kong through the Small Project Funding 2005-06 and by the Research Grants Council of Hong Kong SAR (Project Nos. HKU7129/03E and HKU7168/06E). References
Fig. 9. Load rotation response of B7 simulated using different load increments.
friction between the fraying surfaces gradually decreases and has no significant effect on the shear strength of bolts. From the consistent numerical and experimental results of the individual bolt groups, it can be concluded that the proposed numerical technique is generally accurate and reliable for determining the load–deformation relationship of bolt groups. However, when the particular focus is on the initial stiffness of the bolt groups, the effect of the frictional behaviour between the fraying surfaces should be considered. The load–deformation relationship of B7 is simulated using different load increment sizes ranging from 1% to 7% of the loadcarrying capacity of the bolt group and is plotted in Fig. 9. It is observed that variations in load increment size have little effect on the predicted ultimate strength and the shape of the load–deformation curve. 5. Conclusions This paper presents a new solution algorithm to calculate the non-linear load–deformation response of bolt groups subjected to combined in-plane moment and shear. The bolts are assumed to be kinematically hardened when yielded. By postulating the post-yield behaviour of bolts using Mroz’s translation rules for the yield surface, the stiffness of bolts in the x and y directions was derived. Further application of the classical theory of bolt groups, the complete non-linear response of bolt groups, and the behaviour of individual bolts were evaluated accordingly. The computer program BOGAN, which simulates the non-linear responses of bolt groups under combined in-plane moment and shear, has been implemented to evaluate the effectiveness of the theories in real applications. Its reliability and accuracy have been verified by comparing the numerical results with those from an experiment in the literature. The success of using BOGAN to simulate the
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