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T-stub behavior under out-of-plane bending. II: Parametric study and analytical characterization
Engineering Structures xxx (2015) xxx–xxx
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T-stub behavior under out-of-plane bending. II: Parametric study and analytical characterization Beatriz Gil a,⇑, Frans Bijlaard b, Eduardo Bayo a a b
Department of Construction, Facilities and Structures, School of Architecture, University of Navarra, 31080 Pamplona, Spain Structural and Building Engineering, Department of Design and Construction, Faculty of Civil Engineering, Delft University of Technology, 2600GA Delft, The Netherlands
a r t i c l e
i n f o
Article history: Available online xxxx Keywords: Semi-rigid connections Components method Minor axis bending T-stub Bolted connection
a b s t r a c t Beam to column connections subjected to loads in the beam minor axis direction are usually considered either as pinned or rigid for both resistance and stiffness checks. This simplification is also assumed at the time of considering the stability of the column and the lateral buckling of the beams. However, as proven in the companion paper by means of laboratory tests and numerical simulations, these types of connections are actually semi-rigid and partial strength. The consideration of their real semi-rigid behavior can lead to a more accurate global analysis and, consequently, more optimized structures. Their behavior relies mainly on the characterization of the components acting on the T-stub under out-of-plane bending. In this paper a parametric analysis is performed to define the components involved. Analytical expressions are specified to describe each one of the components and the assembly process following a proposed mechanical model. The stiffness and strength that are obtained compare very satisfactorily with the experimental results of the T-stubs described in the companion paper. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Since the appearance of the Eurocode 3, part 1–8 [1], steel connections cannot only be classified as either pinned or rigid. The general case implies that the connection is semi-rigid and partial strength, and the process of classification implies the calculation of the connection stiffness and strength. The semi-rigidity condition of plane connections in the major axis has been widely investigated and documented since the seventies [2–6]. It is considered in modern codes including EC3, and in this case by means of the component method [1,3]. This method, despite the complexity that it entails, has been accepted and is widely used by structural designers in everyday practice. However, the three-dimensional behavior of semi-rigid steel joints [7–10] as well as the behavior of the minor axis [11,12] has not been fully investigated. As a consequence, they are not currently contemplated in modern codes, and their behavior is not included in 3D structural analysis and design software. In order to extrapolate the component method to the three-dimensional case, additional components need to be identified and studied. Until this is done, the connections of beams attached to the minor axis of the column or beams under out-of-plane bending ⇑ Corresponding author. E-mail address: [email protected] (B. Gil).
will have to be considered either as pinned or rigid. These approximations may have a strong influence not only when the beams are loaded in their minor axis but also as boundary conditions for the stability checks of both the columns (buckling in and out of plane) and beams (lateral buckling). In the companion paper [13], it is shown by means of experimental tests of T-stubs under out-of-plane bending and through finite element simulations that these kinds of joints actually have a semi-rigid behavior, thus, their stiffness and resistance should be included to perform accurate and reliable structural analyses. In this paper we intend to characterize these semi-rigid properties within the context of the component method. An intrinsic part of this process is the characterization of the T-Stub components that appear under out-of-plane bending. This is carried out in this paper as part of the characterization of the whole joint, as previously done for 2D joints [14–20]. The characterization of the new components under out-ofplane bending would require a large number of real scale tests resulting from the variation of all the necessary parameters involved in each component. In order to avoid this expensive process, a parametric analysis is performed by means of finite elements models which are previously calibrated with the experimental results of five T-stub tests. Both the experimental results and the finite element models are described in the companion paper [13]. The present paper describes the results of the
http://dx.doi.org/10.1016/j.engstruct.2015.03.039 0141-0296/Ó 2015 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Gil B et al. T-stub behavior under out-of-plane bending. II: Parametric study and analytical characterization. Eng Struct (2015), http://dx.doi.org/10.1016/j.engstruct.2015.03.039
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B. Gil et al. / Engineering Structures xxx (2015) xxx–xxx
Nomenclature a aw bb bc bep C C0 dc dl
E ec e eep el
F bf ;c;Rd F cf ;b;Rd F cf ;t;Rd F cw;b;Rd Fts,Rd fy fyd g hb hc hep hl Icw J kbf ;c
distance from the beam web to the edge of the beam flange weld throat thickness beam flange width column flange width end plate width coefficient used to calculate the stiffness coefficient of the column flange in bending at the compression zone coefficient that affects the effective length of the column web in bending distance between the upper and lower root radii of the column web distance from the centroid of the loading area for the column flange in bending at the compression zone to the axis of the column Young modulus of the steel material distance from the bolt axis to the lateral edge of the column distance from the bolt axis to the free edge of the column flange distance from the bolt axis to the lateral edge of the end plate distance from the centroid of the loading area of the column flange in bending at the compression zone to the outer edge resistance of the beam flange in compression resistance of the column flange under bending in the compression zone resistance of the column flange in torsion resistance of the column web in bending resistance of the stiffeners yield stress of the steel material design value of the yield stress of steel material distance between the two bolts in a row, measured horizontally beam height column height end plate height height of the loading area for the column flange in bending in the compression zone moment of inertia of the effective section of the column web torsional modulus of the column flange stiffness coefficient for the beam flange in compression
parametric study, carried out by means of the finite element program Abaqus. These results, together with the tests, have led to the identification, subdivision and analytical characterization of the stiffness and strength of each of the components involved. The assembly of all the components into a mechanical model provides the global stiffness and strength of the T-stub, which are compared with those of the tests showing a good agreement. Once the behavior of the T-stub is defined, that of the whole joint follows readily.
2. Parametric study by means of finite element models An experimental program was carried out in order to assess the real behavior of the T-stub under out-of-plane bending. Finite element models with the same characteristics as those of the tests were developed and calibrated with the experimental results. Both, the experimental program and the finite element modeling
kcf ;b kcf ;t kcw;b kts K rot;cf ;t Lc leff,bf,c leff ;cf ;b leff ;cw;b leff ;ep;b leff,nc mc mep ml
Mj,Rd Mp M pl;cw;Rd p pl Sj,ini T tep tfb tfc twb twc T pl;Rd W T;pl z
cM0 kp
q rcr sy
stiffness coefficient of the column flange under bending in the compression zone stiffness coefficient for the column flange in torsion stiffness coefficient for the column web in bending stiffness coefficient for the stiffeners rotational stiffness of the column flange in torsion total length of the column effective length for the beam flange in compression effective length of the column flange in bending at the compression zone effective length for the column web in bending effective length of the end plate in bending at the compression zone effective length corresponding to a non-circular pattern for a bolt row distance from the bolt axis to a line placed to 0.8 times the root radius from the beam web distance from the bolt axis to a line placed to 0.8 times the weld from the beam web distance from the centroid of the loading area of the column flange in bending for the compression zone to the inner edge plastic moment resistance of the T-stub plastic moment plastic resistant moment of the effective section of the column web distance between bolt rows measured vertically width of the loading area for the column flange in bending in the compression zone initial stiffness of the T-stub torques end plate thickness beam flange thickness column flange thickness beam web thickness column web thickness plastic resistance of a rectangular section in torsion plastic torsional modulus lever arm of the joint coefficient = 1.05 plate slenderness reduction factor for plate buckling elastic critical plate buckling stress tangential stress
are described in the companion paper [13]. They showed a good agreement, demonstrating that the finite element models provided a useful tool to develop a parametric study and appraise the design formulae, avoiding the need for additional and costly experiments. The finite element models have the same characteristics that were explained in Section 3 of the companion paper, that is, the type of the elements are solid 3D elements with 8 nodes and reduced integration (element C3D8R) that avoids shear locking. The type of interactions between surfaces, boundary conditions, and mesh are described in detail in the companion paper. In regard to the material properties, the elastic–plastic nominal properties of the steel are used in the parametric study. The geometric characteristics of the proposed models are summarised in Table 1. Fifty models, divided in two series: S1 and S2, have been analyzed. In both series, the column section varies from HEB 160 to HEB 240, meanwhile the half beam of the T-stub is kept unchanged and equal to half of an IPE300. The thickness of the end plate is 10 mm in series S1 and 20 mm in series S2. The length of
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B. Gil et al. / Engineering Structures xxx (2015) xxx–xxx Table 1 Models for the parametric study. Parametric study (FEM) Series
Model
Column
Beam (Half beam)
End plate (Extended) (mm)
Column length (mm)
Steel
Bolts
S1
S1-1
HEB 240
IPE 300
10
S275
£ 16 (10.9)
S1-2
HEB 220
IPE 300
10
S275
£ 16 (10.9)
S1-3
HEB 200
IPE 300
10
S275
£ 16 (10.9)
S1-4
HEB 180
IPE 300
10
S275
£ 16 (10.9)
S1-5
HEB 160
IPE 300
10
1200 1800 2400 3000 4000 1200 1800 2400 3000 4000 1200 1800 2400 3000 4000 1200 1800 2400 3000 4000 1200 1800 2400 3000 4000
S275
£ 16 (10.9)
S2-1
HEB 240
IPE 300
20
S275
£ 16 (10.9)
S2-2
HEB 220
IPE 300
20
S275
£ 16 (10.9)
S2-3
HEB 200
IPE 300
20
S275
£ 16 (10.9)
S2-4
HEB 180
IPE 300
20
S275
£ 16 (10.9)
S2-5
HEB 160
IPE 300
20
S275
£ 16 (10.9)
S2
1200 1800 2400 3000 4000 1200 1800 2400 3000 4000 1200 1800 2400 3000 4000 1200 1800 2400 3000 4000 1200 1800 2400 3000 4000
the column varies from 1200 mm to 4000 mm, whereas the length of the beam remains the same in all the models. The following results are obtained, through the parametric study:
The rest of the curves are shown below in this paper, wherever they are needed to explain a specific component of the connection.
– Moment–rotation curves of the connections (Fig. 2). – Von Misses stress in the column web close to the root radius along the column length (Fig. 1a). – Contact pressure in the column flange in its perpendicular direction along the column flange width (Fig. 1b). – Contact pressure near de edge of the column flange along the end plate length in the plastic stage (Fig. 1c).
Following the same criteria as that proposed in Eurocode 3 [1], the experimental program and the finite element models help to identify the main components that have to be either modified or defined in order to describe the out of plane bending behavior of the T-stub. The minor axis moment is equivalent to a pair of two forces, one of them pulls out from the joint on one side, and the other one pushes in the joint on the opposite side. Accordingly, the following components are activated: Components in the compression side:
Fig. 2(a) and (b) shows the moment–rotation curves for the models for only one set of column lengths (3000 mm). As expected, the higher the column the less stiff the joint is, and similarly, the bigger the column section, the stiffer the joint is. The models for S2 are stiffer than S1 because the end plate is thicker.
3. Components for the T-stub in out of plane bending
– End plate in bending. – Column flange in bending. – Beam flange in compression.
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B. Gil et al. / Engineering Structures xxx (2015) xxx–xxx
– Column web in bending. – Column flange in torsion. The necessary parameters that define the T-stub components are shown in Fig. 3. The components corresponding to the tension zone (end plate in bending, column flange in bending and bolts in tension) are the same as those already defined in EC3-1.8, Tables 6.4, 6.5 and 6.6, taking into account that the effective length of the end plate is different for each one of the two bolts in the tension side. The description of the rest of the components, which will need to be characterized, is presented below. 3.1. Beam flange in compression and buckling This component corresponds to the beam flange in compression and buckling (Fig. 4) and can be characterized in an analogous way to the existing EC3 component for the column web in transverse compression: EC3-1.8, 6.2.6.2 for resistance, and EC3-1.8 Table 6.11 for stiffness. Consequently, the strength and stiffness corresponding to this component can be defined in the following form:
Fig. 1. Lines where the stresses are measured.
(a)
Moment -Rotation S1
18
F bf ;c;Rd ¼ qat fb f y =cM0
ð1Þ
0:7at fb Leff ;bf ;c
ð2Þ
kbf ;c ¼
16 wb where a ¼ bb t and Leff,bf,c can be calculated, assuming a dispersion 2 of a = 45° (see Fig. 2), equal to a. Then, Eq. (2) could be simplified as kbf ;c ¼ 0:7tfb . The reduction factor for plate buckling q, can be obtained following EC3-1-5, clause 4.4 Table 4.2 and Annex A.1 [21], thus:
Moment (kN m)
14 12 10 8 6
2 0
q ¼ ðkp 0:2Þ=k2p for outstand compression elements
S1-1 L300 S1-2 L300 S1-3 L300 S1-4 L300 S1-5 L300
4
0
20
40
60
80
100
120
140
160
180
kp is the plate slenderness, kp ¼ where 200
Rotation (mrd)
(b)
Moment -Rotation S2
18
Moment (kN m)
14 12
8 6
S2-1 L300 S2-2 L300 S2-3 L300 S2-4 L300 S2-5 L300
4 2
rcr is the elastic
ð3Þ
0
20
40
60
80
100
120
140
160
180
3.2. Column flange in bending
200
Rotation (mrd) Fig. 2. Moment–rotation curves of one set of lengths for the parametric study.
Components in the tension side: – – – –
and
The Eqs. (1) and (2) will also be valid for the strength and stiffness of a connection with transverse stiffeners placed between the column flanges.
10
0
fy
rcr
critical plate buckling stress (see Eq. (A.1) in EC3-1.5 Annex A.1(2) and Table 4.1 and Table 4.2 in EC3-1.5). For the case of the beam flange in compression, the plate slenderness can be simplified using the following expression:
sffiffiffiffiffiffiffiffiffi 2 kp ¼ 1:4 a f y Et2fb
16
qffiffiffiffiffi
Bolts in tension. End plate in bending. Column flange in bending. Beam flange in tension. Other components:
In order to characterize this component we need to distinguish between the part that is in compression and the part that is in tension. The component for an unstiffened column flange in bending in the tension zone has already been established in EC3, Part 1–8 Clause 6.2.6.4 for the strength and Clause 6.3.2 for the stiffness. These formulae deal with the T-stub behavior corresponding to each bolt row under tension. A different situation arises in the case of a compression zone, for which the corresponding plastic mechanism has been identified by means of the test results and the numerical parametric study. Accordingly, a plastic loading area is established considering a dispersion angle of 45° through the end plate, measured from the welds of the beam flange. This results in a width of pffiffiffi pl ¼ tfb þ 2aw 2 þ 2t ep as shown in Fig. 5.
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B. Gil et al. / Engineering Structures xxx (2015) xxx–xxx
bc g
ec
ec
hc
t fc
t ep
t fc
1 2
beam
e MSd
t fb
bb
p
bc
z
t wc
h ep
bb
hb /2
eep mep t wb
column
bep
Fig. 3. Dimensions of the T-Stub for out of plane characterization.
hc
t fc
tfc 1 2
t wc α
beam
M
a
leff,bf,c
column
Fig. 4. Beam flange in compression.
tfc tep
0,5(bc-twc)
M
M
dl
pl
pl Loading area
hl
Fig. 5. Column flange in bending for the compression zone.
Contact pressure CPRESS - Column flange - S1 (IPE 300/HEB160-240/EP10mm)
Contact pressure CPRESS - Column flange - S2 (IPE 300/HEB160-240/EP20mm)
16
S1-1 HEB240 S1-2 HEB 220 S1-3 HEB 200 S1-4 HEB 180 S1-5 HEB 160
8
Contact pressure
12
Contact Pressure
10
S2-1 HEB 240 S2-2 HEB 220 S2-3 HEB 200 S2-4 HEB 180 S2-5 HEB 160
8
4
6
4
2 0 -12
-8
-4
0
4
Posion (cm)
(a) plate thickness of 10 mm
8
12
0 -12
-8
-4
0
Posion (cm)
4
8
12
(b) plate thickness of 20 mm
Fig. 6. Contact pressure in column flange: (a) plate thickness of 10 mm, and (b) plate thickness of 20 mm.
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The height hl of this loading area can be obtained from the results of the parametric study that are illustrated in the graphs of Fig. 6. These show the contact pressure, within the plastic range, measured in the middle of the column flange for a T-Stub formed by half of an IPE300 and columns whose sizes range from HEB160 to HEB240, and for end plate thicknesses of 10 mm (a) and 20 mm (b), respectively. These graphs reveal that, for a thin end plate, the loaded part of the column flange starts approximately at a distance of 0.5twc + c and spreads until the edge of the end plate with almost a constant pressure (see finite element stress contour plots in Fig. 6). For a thick end plate, the loading area starts at a distance of around 0.5twc and extends until the edge of the end plate. Thus, it may be considered with sufficient accuracy that the resultant load is applied at a distance dl ¼ 0:5bep 0:5hl measured from the axis of the column web. Then, the lever arm ‘‘z’’ for the joint (see Fig. 3) becomes: z ¼ 0:5g þ dl . In the aim of determining the effective length that governs the resistance of this isolated component, a finite element analysis has been carried out based on the models shown in Fig. 7. These models consisted of a steel plate with the measures of half a column flange (80 13 mm for a HEB160) with lengths varying between 200 mm and 1200 mm (same as the column in the tests). This plate is clamped at both ends as well as in one longitudinal side, thus simulating that the column flange is clamped to the web. A uniform pressure is applied in a portion of surface in the middle of the plate. The resulting stiffness and maximum load applied is the same for each different studied length from 1200 to 400 mm. For the models with smaller lengths (300 mm and 200 mm) both, the stiffness and the maximum load increase with respect to the previous cases. A second model is considered which is the result of clamping the longitudinal side only and leaving the edges of the plates free (see Fig. 7). The results obtained are very similar to those of the previous case. After performing these simulations, it is found that for the case of the test T-stub 01 (column HEB160) the effective length of the column flange in bending (in the compression zone) is around 360 mm, and for the case of T-stub 03 (column HEM160) the effective length is around 400 mm. Also, through the finite element simulations and test results, the yielding lines of this component are identified. These lines are similar to those considered in the EC3-1.8, table 6.4 for noncircular patterns of a group of two bolts in an unstiffened column flange, for the tension zone (Fig. 8). The effective length in this case, for every bolt, is leff ;nc ¼ ð2m þ 0:625e þ 0:5pÞ where p is the distance between bolt rows, measured vertically, e is the distance from the bolt axis to the free edge of the column flange, and m ¼ 0:5bc e 0:5t wc 0:8r c . However, this effective length is a simplification that corresponds to the failure mode 1 mentioned in EC3-1.8 Table 6.2, that is, when four plastic hinges are formed. In our case only two plastic hinges are formed in the column flange and then the total effective length for the group of two bolts becomes leff ;nc ¼ ð5:5m þ 4e þ pÞ, as proposed by Zoetemeijer [2]. Since the yield lines produced in the compression zone are similar to those for the bolts, (see Fig. 8), the same expression can be used for the column flange in bending in the compression zone by Clamped side
0 (20 th ng e L
0,5(bc-twc) M p
pl Loading area
e m
ml el hl
Tension zone
Compression zone
Fig. 8. Yielding lines of the column flange in bending for the tension and compression zone.
replacing p by pl (as the width of the loading area), m by ml ¼ ð0:5bep 0:5twc 0:8rc Þ=2 and e by el ¼ 0:5bc ml ð0:5twc þ 0:8rc Þ. Using these data the effective length can be readily obtained. In the case of the T-stub01 the length turns out to be 355 mm, which fits very well with the isolated finite element models of the column flange under bending in the compression zone. After describing the effective lengths and other parameters, the failure mechanism is studied, as mentioned above, using a cantilever model (Fig. 9) with a length of 0.5bc and a load F applied at a distance dl. The failure is produced when a plastic hinge is formed. Using the principle of virtual work, VW ext ¼ VW int ! Fdl h ¼ M p h which leads to the following expression for the resistance of the column flange in bending: 2
F cf ;b;Rd ¼
M p leff ;cf ;b t fc f yd ¼ dl 4dl
ð4Þ
where leff ;cf ;b ¼ ð5:5ml þ 4el þ pl Þ. With regard to the stiffness, elastic theory is used but, in order to avoid complex calculations for structural designers in everyday practice, a simpler model of a cantilever with a section of tfcx leff C and a load applied to a distance of dl from the clamped side has been considered. The purpose of the coefficient C is to reduce the effective length used in the resistance calculation and adjust it to the characterization of the stiffness. By means of a parametric study the value of C is estimated to be 0.9. Consequently, the stiffness coefficient becomes:
kcf ;b ¼
3I 3 dl
¼
3Cleff ;cf ;b t3fc 3 12dl
¼
Cleff ;cf ;b t 3fc
ð5Þ
3
4dl
If the column has transverse stiffeners the compression is transmitted directly from the beam flange to the stiffener avoiding
Mp
F
Clamped side
t
) mm 00 2 o1
o 0t (20 h t ng Le
m 00 12
Fig. 7. Contact pressure in the middle line of the column flange.
υ
m)
δ
dl 0,5b c Fig. 9. Failure mechanism for the column flange in bending.
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B. Gil et al. / Engineering Structures xxx (2015) xxx–xxx
bending of the column flange in the compression zone, hence, the stiffness and strength becomes that of the transverse stiffener:
F ts;Rd ¼ t ts 0:5ðbc t wc Þf y =cM0
ð6Þ
The plastic resistance of a rectangular section in torsion is T pl;Rd ¼ W T;pl sy , where the plastic torsional modulus is W T;pl ¼ t 2fc ð3bc t fc Þ
sy;max ¼ t ts 0:5ðbc t wc Þ kts ¼ ðhc 2tfc Þ
is:
T pl;Rd ¼ W T;pl
3.3. End plate in bending The strength and stiffness of the end plate in bending in the compression zone is directly obtained from that of the column flange, by replacing the thickness of the column flange tfc by the thickness of the end plate tep. Consequently: 2 M p leff ;cf ;b tep f yd ¼ dl 4dl
ð8Þ
with leff ;ep;b ¼ leff ;cf ;b ¼ ð5:5ml þ 4el þ pl Þ
kep;b ¼
3Iep 3
dl
¼
3Cleff ;ep;b t 3ep 3 12dl
f pyffiffi. 3
For a clamped column T pl;Rd ¼ Fz . 2 Therefore, the maximum torsional moment that can be applied
ð7Þ
where tts is the thickness of the transverse stiffener.
F ep;b;Rd ¼
(see Ref. [22]) and the maximum tangential stress is
6
¼
ð9Þ
with C = 0.9. If the column has stiffeners, the stiffness coefficient of this component is not taken into account because its combination with the one of the column flange in bending is replaced by the characteristics (strength and stiffness) of the transverse stiffener (Eqs. (6) and (7)). 3.4. Column flange in torsion The rotation produced in the column side comes mainly from a combination of the rotation of the column web in bending and both column flanges in torsion. With the aim of characterizing the torsion in the column flange, the flange is modeled as an isolated rectangular beam of dimensions bct and clamped at both edges. The torque is applied as shown in Fig. 10.
ð10Þ
And consequently, the maximum force applied in the connection is:
F cf ;t;Rd ¼
2W T;pl f y t 2fc ð3bc t fc Þf y pffiffiffi ¼ pffiffiffi zcM0 3 3 3 z cM0
ð11Þ
Regarding the stiffness, the torsional rotation for a clamped colTLc . Therefore the stiffness coefficient is: umn is h ¼ 4GJ
kcf ;t ¼
Cleff ;ep;b t 3ep 3 4dl
fy F z pffiffiffi ¼ cf ;t;Rd 2 cM0 3
4 0:38 J J ¼ 1:52 2 L c z2 Lc z
ð12Þ
where Lc is the total length of the column, z is the lever arm and J is the torsional modulus of the column flange. J ¼ 13 bc t 3fc (for thin walled rectangular sections). 3.5. Column web in bending After considering the column flange in torsion, the next component to consider is the column web in bending, which happens to be the other main source of deformation and which usually is the first one to yield near the root radius. Due to the two forces acting on both sides of the column flange (see Fig. 11), a bending moment appears in the column web. This moment is higher at the side of the joint and becomes gradually smaller at the opposite side (Fig. 11). A plastic hinge is formed where the moment is the highest and the section is smallest, that is, where the root radius meets the column web. Accordingly, the strength of the column web in bending becomes:
F cw;b;Rd ¼
M pl;cw;Rd leff ;cw;b t2wc f yd ¼ z 4z
ð13Þ
where M pl;cw;Rd is the plastic resistant moment of the effective section of the column web. z is the lever arm. g is the distance between bolts, measured horizontally.
T/2
T Lc
hc eep F MSd
dc
T/2
g
1 2
beam
z
eep F column Mp
F
z
F
K cf,t
K cf,t
M
dc Fig. 10. Application of torsional moments in the column flange.
Fig. 11. Column web in bending.
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S2_L180
S1_L180
30
30
S1-1 HEB 240 S1-2 HEB 220 S1-3 HEB 200 S1-4 HEB 180 S1-5 HEB 160
25
25
15 10
-150
-100
-50
0
20
Stress (kN/cm2)
Stress (kN/cm2)
20
5
-200
0
50
100
150
S2-1 HEB 240 S2-2 HEB 220 S2-3 HEB 200 S2-4 HEB 180 S2-5 HEB 160
200
-200
15 10 5
-150
-100
0
-50
0
50
30
30
S1-1 HEB 240 S1-3 HEB 200
25
25
S1-4 HEB 180 S1-5 HEB 160
20
Stress (kN/cm2)
Stress (kN/cm2)
20 15 10 5
-100
-50
0
0
50
100
150
200
-200
15 10 5
-150
-100
0
-50
0
50
30
30
S1-1 HEB 240 S1-3 HEB 200
25
25
S1-4 HEB 180 S1-5 HEB 160
20
Stress (kN/cm2)
Stress (kN/cm2)
20 15 10 5
-50
0
0
50
100
150
200
-200
15 10 5
-150
-100
0
-50
0
50
30
30
S1-1 HEB 240 S1-3 HEB 200
25
25
S1-4 HEB 180 S1-5 HEB 160
20
Stress (kN/cm2)
Stress (kN/cm2)
20 15 10 5
-50
0
0
200
S2-1 HEB 240 S2-2 HEB 220 S2-3 HEB 200 S2-4 HEB 180 S2-5 HEB 160
S1-2 HEB 220
-100
150
S2_L400
S1_L400
-150
100
posion X (cm)
posion X (cm)
-200
200
S2-1 HEB 240 S2-2 HEB 220 S2-3 HEB 200 S2-4 HEB 180 S2-5 HEB 160
S1-2 HEB 220
-100
150
S2_L300
S1_L300
-150
100
posion X (cm)
posion X (cm)
-200
200
S2-1 HEB 240 S2-2 HEB 220 S2-3 HEB 200 S2-4 HEB 180 S2-5 HEB 160
S1-2 HEB 220
-150
150
S2_L240
S1_L240
-200
100
posion X (cm)
posion X (cm)
50
posion X (cm)
100
150
200
-200
15 10 5
-150
-100
-50
0
0
50
100
150
200
Posion X (cm)
Fig. 12. Von Misses Stress in the column web along its length.
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B. Gil et al. / Engineering Structures xxx (2015) xxx–xxx
cf,b,t
bf,t ep,b,t
cw,b M cf,b,c
bf,c
ep,b,c Fig. 14. Mechanical model for a T-stub in out-of-plane bending.
components involved in the T-Stub under out-of-plane bending are assembled as series of springs according to the mechanical model illustrated in Fig. 14. The parameters appearing in this figure are: Fig. 13. T-Stub in out-of-plane bending. Effective length of the column web.
leff ;cw;b is the effective length for the column web in bending which is obtained by means of finite element parametric study as explained in the next paragraphs and in Figs. 12 and 13. The graphs of Fig. 12 show the Von Misses stress levels along the column web length obtained from the parametric study. As mentioned before the series S1 has a T-Stub formed with half an IPE300 with a 10 mm end-plate. The series S2 has a T-Stub formed with half an IPE300 with a 20 mm end-plate. The section of the column varies in both series from HEB160 to HEB240 and different lengths are studied as well. It may be observed that the yielded length in the column web is around 700–800 mm in all the cases, even with different column sections, column lengths or end plate thickness. The effective length for the column web in bending, leff ;cw;b , fits quite accurately when it is equal to twice the effective length for the column flange in compression, and not becoming larger than the total column length Lc.
leff ;cw;b ¼ 2leff ;cf ;b 6 Lc
ð14Þ
Regarding the stiffness, a simple model is considered as it is shown in Fig. 11. The column web is modeled as a beam with a rotational spring in each side equal to the rotational stiffness of the column flange in torsion with a stiffness coefficient kcf ;t which is defined in Eq. (12). Then, the stiffness coefficient for the column web in bending can be calculated by means of standard structural analysis as:
kcw;b
! ð4Icw þ kcf ;t dc z2 Þ 1 4I2cw ¼ 2 þ z dc ð4Icw þ kcf ;t dc z2 Þ dc
ð15Þ
where Icw is the inertia of the effective section of the column web C0 l
t3
wc Icw ¼ eff ;cw;b . 12 The coefficient C 0 , that affects the effective length, takes the value of 0.5 and is obtained from the parametric study taking into account the elastic response (for stiffness purposes) rather than the plastic response used for resistance calculation. If the column has transverse stiffeners, there is no rotation between the column flange and the column web, and the stiffness coefficient of this component becomes infinite.
4. Assembly of the components Finally, following the principles of the component method [1,3] and after characterizing the properties (stiffness and strength), the
cw,b – spring corresponding to the column web in bending, that includes the column flange in torsion, as explained in Section 3.5 and Eq. (15) cf,b,t – spring corresponding to the column flange in bending for the tension zone, ep,b,t – spring corresponding to the end plate in bending for the tension zone, bf,t – spring corresponding to the beam flange in tension, cf,b,c – spring corresponding to the column flange in bending for the compression zone, ep,b,c – spring corresponding to the end plate in bending for the compression zone, bf,c – spring corresponding to the beam flange in compression.
5. Comparison between the test and the analytical model Table 2 shows a comparison of the values obtained for initial stiffness and strength in the tests and by means of the proposed analytical model implementing the components described above and Table 3 shows a comparison of the values obtained by means of finite element models and by means of the proposed analytical model. The experimental values are subtracting the rotation due to torsion in the column to properly compare with those of the analytical model. The plastic moment is considered as 1.5 times the value of the elastic moment because that is the value of the shape coefficient for a rectangular section, which in the T-stub is bb ⁄ tfb. Both tests, FEM and analytical models show very good agreement. More specifically the stiffness predicted by the analytical model is slightly lower than that resulting from the tests and FEM. On the other hand, the predicted strength is somehow greater than that of the tests and FEM.
Table 2 Comparison of the initial stiffness and strength between the experimental and analytical model. Joint
Characteristics
T-Stub01
HEB160/IPE300 EP10 mm/Bolted
T-Stub03
HEM160/IPE300 EP10 mm/Bolted
Initial stiffness Sj,ini (kNm)
Plastic moment Mj,Rd (kNm)
Test
Analytical
Test
Analytical
119
117.9
8.4
8.6
546
525.1
13.5
14.2
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B. Gil et al. / Engineering Structures xxx (2015) xxx–xxx
Table 3 Comparison of the initial stiffness and strength between the finite element model and analytical model. Joint
Characteristics
S1-3 H = 240 cm S1-4 H = 180 cm C-1 H = 300 cm C-2 H = 300 cm C-3 H = 300 cm
HEB200/IPE300 EP10 mm/Bolts 16 mm HEB180/IPE300 EP10 mm/Bolts 16 mm HEB200/IPE200 EP16 mm/Bolts 16 mm HEB200/IPE270 EP18 mm/Bolts 20 mm HEB260/IPE400 EP20 mm/Bolts 24 mm
Initial stiffness Sj,ini (kNm)
Plastic moment Mj,Rd (kNm)
FEM
Analytical
FEM
Analytical
131.1
123.8
12.8
12.7
117.5
119.2
10.5
10.6
98.5
89.4
7.8
7.9
134.0
116.1
12.9
12.5
197.8
173.8
18.5
21.3
6. Conclusions This paper deals with the mechanical model and analytical characterization of the bolted T-stub of a semi-rigid steel connection subjected to out-of-plane bending. To reach its complete characterization, an experimental program and finite element models were developed and presented in the companion paper [13]. Then, a parametric study is carried out where different parameters such as end plate thickness, column length and column section are varied. With these tools, the components that are involved in the whole T-Stub are identified and independently studied. The test and finite element models reveal that the sources of deformation for joints are, in the compression side: the end plate in bending, the column flange in bending and the beam flange in compression; in the tension side: the bolts in tension, the end plate in bending, the column flange in bending and the beam flange in tension. The other components involved are the column web in bending and the column flange in torsion. In fact, the latest two are the main sources of deformation in unstiffened joint and are components that are not contemplated in Eurocode 3. These components are analyzed and characterized based on the component method. New effective lengths and analytical expressions have been established for the new components, and also for those already described in Eurocode 3 that needed to be adapted for joints in out-of-plane bending. Finally, the components involved in the T-stub are assembled in a proposed mechanical model and compared with the experimental results and finite elemen models showing that the proposed model predict the behavior of the T-Stub with good accuracy. Further research must be done about T-stubs under torsion to end the complete characterization of 3D steel joints. Acknowledgements The authors gratefully acknowledge the financial support provided by the Spanish Ministerio de Ciencia e Innovación under the contract number BIA2010-20839-C02-02. References [1] CEN. Eurocode 3: design of steel structures. Part 1–8: Design of joints (en 1993-1-8:2005). CEN; 2005.
[2] Zoetemeijer P. A design method for the tension side of statically loaded, bolted beam-to-colum connections. HERON 1974;20(1). [3] Faella C, Piluso V, Rizzano G. Structural steel semirigid connections: theory, design and software. CRC Press LLC; 1999. [4] Jaspart JP. Etude de la semi-rigidité des noeuds poutre-colonne et son influence sur la résistance et la stabilité des ossatures en acier. Belgique: Université de Liège; 1991. [5] Davison JB, Kirby PA, Nethercot DA. Rotational stiffness characteristics of steel beam to column connections. J Constr Steel Res 1987;8:17–54. [6] Stark J, Bijlaard FSK. Design rules for beam-to-column connections in europe. J Constr Steel Res 1988;10:415–62. [7] Nguyen P-C, Seung-Eock Kim. An advanced analysis method for threedimensional steel frames with semi-rigid connections. Finite Elem Anal Des 2014;80:23–32. [8] Loureiro A, Moreno A, Gutiérrez R, Reinosa JM. Experimental and numerical analysis of three-dimensional semi-rigid steel joints under non-proportional loading. Eng Struct 2012;38:68–77. [9] Gil B, Goñi R, Bayo E. Experimental and numerical validation of a new design for three-dimensional semi-rigid composite joint under general loads. Eng Struct 2013;48:55–69. [10] Cabrero JM, Bayo E. The semi-rigid behaviour of three-dimensional steel beam-to-column joints subjected to proportional loading. Part II: Theoretical model and validation. J Constr Steel Res 2007;63(9):1254–67. [11] de Lima LRO, de Andrade SAL, Vellasco PCG, da Silva LS. Experimental and mechanical model for predicting the behaviour of minor axis beam-to-column semi-rigid joints. Int J Mech Sci 2002;44:1047–65. [12] de Lima LRO, de F, Freire JL, Vellasco PCG, de Andrade SAL, da Silva JGS. Structural assessment of minor axis steel joints using photoelasticity and finite elements. J Constr Steel Res 2009;65:466–78. [13] Gil B, Goñi R. T-stub behaviour under out-of-plane bending. I: experimental research and finite element modelling. Eng Struct 2014; submitted for publication. [14] Girão Coelho A, Simoes da Silva L, Bijlaard FSK. Finite-element modeling of the nonlinear behavior of bolted T-Stub connections. J Struct Eng 2006;132(6): 918–28. [15] Lee J, Goldsworthy HM, Gad EF. Blind bolted t-stub connections to unfilled hollow section columns in low rise structures. J Constr Steel Res 2010;66:981–92. [16] Loureiro A, Gutiérrez R, Reinosa JM, Moreno A. Axial stiffness prediction of non-preloaded T-Stubs: AN analytical frame approach. J Constr Steel Res 2010;66:1516–22. [17] Al-Khatab Z, Bouchaïr A. Analysis of a bolted t-stub strengthened by backingplates with regard to eurocode 3. J Constr Steel Res 2013;63:1603–15. [18] Loureiro A, López M, Gutiérrez R, Reinosa JM. A new analytical formulation for the e-stub strength calculation in three dimensional steel joints with additional plates welded to the weak axis. Eng Struct 2013;56:2263–72. [19] Wan Hu J, Leon RT, Park T. Mechanical models for the analysis of bolted T-Stub connections under cyclic loads. J Constr Steel Res 2012;78:45–57. [20] Girão Coelho A, Bijlaard FSK, Gresnigt N, Simoes da Silva L. Experimental assessment of the behaviour of bolted T-Stub connections made up of welded plates. J Constr Steel Res 2004:60. [21] CEN. Eurocode 3: design of steel structures. Part 1. 5: Plated structural elements (en 1993-1-5:2006); 2006. [22] Case J, Chilver L, Ross CTF. Strength of materials and structures. ButterworthHeinemann; 1999.
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Contents lists available at ScienceDirect
Engineering Structures journal homepage: www.elsevier.com/locate/engstruct
T-stub behavior under out-of-plane bending. II: Parametric study and analytical characterization Beatriz Gil a,⇑, Frans Bijlaard b, Eduardo Bayo a a b
Department of Construction, Facilities and Structures, School of Architecture, University of Navarra, 31080 Pamplona, Spain Structural and Building Engineering, Department of Design and Construction, Faculty of Civil Engineering, Delft University of Technology, 2600GA Delft, The Netherlands
a r t i c l e
i n f o
Article history: Available online xxxx Keywords: Semi-rigid connections Components method Minor axis bending T-stub Bolted connection
a b s t r a c t Beam to column connections subjected to loads in the beam minor axis direction are usually considered either as pinned or rigid for both resistance and stiffness checks. This simplification is also assumed at the time of considering the stability of the column and the lateral buckling of the beams. However, as proven in the companion paper by means of laboratory tests and numerical simulations, these types of connections are actually semi-rigid and partial strength. The consideration of their real semi-rigid behavior can lead to a more accurate global analysis and, consequently, more optimized structures. Their behavior relies mainly on the characterization of the components acting on the T-stub under out-of-plane bending. In this paper a parametric analysis is performed to define the components involved. Analytical expressions are specified to describe each one of the components and the assembly process following a proposed mechanical model. The stiffness and strength that are obtained compare very satisfactorily with the experimental results of the T-stubs described in the companion paper. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Since the appearance of the Eurocode 3, part 1–8 [1], steel connections cannot only be classified as either pinned or rigid. The general case implies that the connection is semi-rigid and partial strength, and the process of classification implies the calculation of the connection stiffness and strength. The semi-rigidity condition of plane connections in the major axis has been widely investigated and documented since the seventies [2–6]. It is considered in modern codes including EC3, and in this case by means of the component method [1,3]. This method, despite the complexity that it entails, has been accepted and is widely used by structural designers in everyday practice. However, the three-dimensional behavior of semi-rigid steel joints [7–10] as well as the behavior of the minor axis [11,12] has not been fully investigated. As a consequence, they are not currently contemplated in modern codes, and their behavior is not included in 3D structural analysis and design software. In order to extrapolate the component method to the three-dimensional case, additional components need to be identified and studied. Until this is done, the connections of beams attached to the minor axis of the column or beams under out-of-plane bending ⇑ Corresponding author. E-mail address: [email protected] (B. Gil).
will have to be considered either as pinned or rigid. These approximations may have a strong influence not only when the beams are loaded in their minor axis but also as boundary conditions for the stability checks of both the columns (buckling in and out of plane) and beams (lateral buckling). In the companion paper [13], it is shown by means of experimental tests of T-stubs under out-of-plane bending and through finite element simulations that these kinds of joints actually have a semi-rigid behavior, thus, their stiffness and resistance should be included to perform accurate and reliable structural analyses. In this paper we intend to characterize these semi-rigid properties within the context of the component method. An intrinsic part of this process is the characterization of the T-Stub components that appear under out-of-plane bending. This is carried out in this paper as part of the characterization of the whole joint, as previously done for 2D joints [14–20]. The characterization of the new components under out-ofplane bending would require a large number of real scale tests resulting from the variation of all the necessary parameters involved in each component. In order to avoid this expensive process, a parametric analysis is performed by means of finite elements models which are previously calibrated with the experimental results of five T-stub tests. Both the experimental results and the finite element models are described in the companion paper [13]. The present paper describes the results of the
http://dx.doi.org/10.1016/j.engstruct.2015.03.039 0141-0296/Ó 2015 Elsevier Ltd. All rights reserved.
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B. Gil et al. / Engineering Structures xxx (2015) xxx–xxx
Nomenclature a aw bb bc bep C C0 dc dl
E ec e eep el
F bf ;c;Rd F cf ;b;Rd F cf ;t;Rd F cw;b;Rd Fts,Rd fy fyd g hb hc hep hl Icw J kbf ;c
distance from the beam web to the edge of the beam flange weld throat thickness beam flange width column flange width end plate width coefficient used to calculate the stiffness coefficient of the column flange in bending at the compression zone coefficient that affects the effective length of the column web in bending distance between the upper and lower root radii of the column web distance from the centroid of the loading area for the column flange in bending at the compression zone to the axis of the column Young modulus of the steel material distance from the bolt axis to the lateral edge of the column distance from the bolt axis to the free edge of the column flange distance from the bolt axis to the lateral edge of the end plate distance from the centroid of the loading area of the column flange in bending at the compression zone to the outer edge resistance of the beam flange in compression resistance of the column flange under bending in the compression zone resistance of the column flange in torsion resistance of the column web in bending resistance of the stiffeners yield stress of the steel material design value of the yield stress of steel material distance between the two bolts in a row, measured horizontally beam height column height end plate height height of the loading area for the column flange in bending in the compression zone moment of inertia of the effective section of the column web torsional modulus of the column flange stiffness coefficient for the beam flange in compression
parametric study, carried out by means of the finite element program Abaqus. These results, together with the tests, have led to the identification, subdivision and analytical characterization of the stiffness and strength of each of the components involved. The assembly of all the components into a mechanical model provides the global stiffness and strength of the T-stub, which are compared with those of the tests showing a good agreement. Once the behavior of the T-stub is defined, that of the whole joint follows readily.
2. Parametric study by means of finite element models An experimental program was carried out in order to assess the real behavior of the T-stub under out-of-plane bending. Finite element models with the same characteristics as those of the tests were developed and calibrated with the experimental results. Both, the experimental program and the finite element modeling
kcf ;b kcf ;t kcw;b kts K rot;cf ;t Lc leff,bf,c leff ;cf ;b leff ;cw;b leff ;ep;b leff,nc mc mep ml
Mj,Rd Mp M pl;cw;Rd p pl Sj,ini T tep tfb tfc twb twc T pl;Rd W T;pl z
cM0 kp
q rcr sy
stiffness coefficient of the column flange under bending in the compression zone stiffness coefficient for the column flange in torsion stiffness coefficient for the column web in bending stiffness coefficient for the stiffeners rotational stiffness of the column flange in torsion total length of the column effective length for the beam flange in compression effective length of the column flange in bending at the compression zone effective length for the column web in bending effective length of the end plate in bending at the compression zone effective length corresponding to a non-circular pattern for a bolt row distance from the bolt axis to a line placed to 0.8 times the root radius from the beam web distance from the bolt axis to a line placed to 0.8 times the weld from the beam web distance from the centroid of the loading area of the column flange in bending for the compression zone to the inner edge plastic moment resistance of the T-stub plastic moment plastic resistant moment of the effective section of the column web distance between bolt rows measured vertically width of the loading area for the column flange in bending in the compression zone initial stiffness of the T-stub torques end plate thickness beam flange thickness column flange thickness beam web thickness column web thickness plastic resistance of a rectangular section in torsion plastic torsional modulus lever arm of the joint coefficient = 1.05 plate slenderness reduction factor for plate buckling elastic critical plate buckling stress tangential stress
are described in the companion paper [13]. They showed a good agreement, demonstrating that the finite element models provided a useful tool to develop a parametric study and appraise the design formulae, avoiding the need for additional and costly experiments. The finite element models have the same characteristics that were explained in Section 3 of the companion paper, that is, the type of the elements are solid 3D elements with 8 nodes and reduced integration (element C3D8R) that avoids shear locking. The type of interactions between surfaces, boundary conditions, and mesh are described in detail in the companion paper. In regard to the material properties, the elastic–plastic nominal properties of the steel are used in the parametric study. The geometric characteristics of the proposed models are summarised in Table 1. Fifty models, divided in two series: S1 and S2, have been analyzed. In both series, the column section varies from HEB 160 to HEB 240, meanwhile the half beam of the T-stub is kept unchanged and equal to half of an IPE300. The thickness of the end plate is 10 mm in series S1 and 20 mm in series S2. The length of
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B. Gil et al. / Engineering Structures xxx (2015) xxx–xxx Table 1 Models for the parametric study. Parametric study (FEM) Series
Model
Column
Beam (Half beam)
End plate (Extended) (mm)
Column length (mm)
Steel
Bolts
S1
S1-1
HEB 240
IPE 300
10
S275
£ 16 (10.9)
S1-2
HEB 220
IPE 300
10
S275
£ 16 (10.9)
S1-3
HEB 200
IPE 300
10
S275
£ 16 (10.9)
S1-4
HEB 180
IPE 300
10
S275
£ 16 (10.9)
S1-5
HEB 160
IPE 300
10
1200 1800 2400 3000 4000 1200 1800 2400 3000 4000 1200 1800 2400 3000 4000 1200 1800 2400 3000 4000 1200 1800 2400 3000 4000
S275
£ 16 (10.9)
S2-1
HEB 240
IPE 300
20
S275
£ 16 (10.9)
S2-2
HEB 220
IPE 300
20
S275
£ 16 (10.9)
S2-3
HEB 200
IPE 300
20
S275
£ 16 (10.9)
S2-4
HEB 180
IPE 300
20
S275
£ 16 (10.9)
S2-5
HEB 160
IPE 300
20
S275
£ 16 (10.9)
S2
1200 1800 2400 3000 4000 1200 1800 2400 3000 4000 1200 1800 2400 3000 4000 1200 1800 2400 3000 4000 1200 1800 2400 3000 4000
the column varies from 1200 mm to 4000 mm, whereas the length of the beam remains the same in all the models. The following results are obtained, through the parametric study:
The rest of the curves are shown below in this paper, wherever they are needed to explain a specific component of the connection.
– Moment–rotation curves of the connections (Fig. 2). – Von Misses stress in the column web close to the root radius along the column length (Fig. 1a). – Contact pressure in the column flange in its perpendicular direction along the column flange width (Fig. 1b). – Contact pressure near de edge of the column flange along the end plate length in the plastic stage (Fig. 1c).
Following the same criteria as that proposed in Eurocode 3 [1], the experimental program and the finite element models help to identify the main components that have to be either modified or defined in order to describe the out of plane bending behavior of the T-stub. The minor axis moment is equivalent to a pair of two forces, one of them pulls out from the joint on one side, and the other one pushes in the joint on the opposite side. Accordingly, the following components are activated: Components in the compression side:
Fig. 2(a) and (b) shows the moment–rotation curves for the models for only one set of column lengths (3000 mm). As expected, the higher the column the less stiff the joint is, and similarly, the bigger the column section, the stiffer the joint is. The models for S2 are stiffer than S1 because the end plate is thicker.
3. Components for the T-stub in out of plane bending
– End plate in bending. – Column flange in bending. – Beam flange in compression.
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B. Gil et al. / Engineering Structures xxx (2015) xxx–xxx
– Column web in bending. – Column flange in torsion. The necessary parameters that define the T-stub components are shown in Fig. 3. The components corresponding to the tension zone (end plate in bending, column flange in bending and bolts in tension) are the same as those already defined in EC3-1.8, Tables 6.4, 6.5 and 6.6, taking into account that the effective length of the end plate is different for each one of the two bolts in the tension side. The description of the rest of the components, which will need to be characterized, is presented below. 3.1. Beam flange in compression and buckling This component corresponds to the beam flange in compression and buckling (Fig. 4) and can be characterized in an analogous way to the existing EC3 component for the column web in transverse compression: EC3-1.8, 6.2.6.2 for resistance, and EC3-1.8 Table 6.11 for stiffness. Consequently, the strength and stiffness corresponding to this component can be defined in the following form:
Fig. 1. Lines where the stresses are measured.
(a)
Moment -Rotation S1
18
F bf ;c;Rd ¼ qat fb f y =cM0
ð1Þ
0:7at fb Leff ;bf ;c
ð2Þ
kbf ;c ¼
16 wb where a ¼ bb t and Leff,bf,c can be calculated, assuming a dispersion 2 of a = 45° (see Fig. 2), equal to a. Then, Eq. (2) could be simplified as kbf ;c ¼ 0:7tfb . The reduction factor for plate buckling q, can be obtained following EC3-1-5, clause 4.4 Table 4.2 and Annex A.1 [21], thus:
Moment (kN m)
14 12 10 8 6
2 0
q ¼ ðkp 0:2Þ=k2p for outstand compression elements
S1-1 L300 S1-2 L300 S1-3 L300 S1-4 L300 S1-5 L300
4
0
20
40
60
80
100
120
140
160
180
kp is the plate slenderness, kp ¼ where 200
Rotation (mrd)
(b)
Moment -Rotation S2
18
Moment (kN m)
14 12
8 6
S2-1 L300 S2-2 L300 S2-3 L300 S2-4 L300 S2-5 L300
4 2
rcr is the elastic
ð3Þ
0
20
40
60
80
100
120
140
160
180
3.2. Column flange in bending
200
Rotation (mrd) Fig. 2. Moment–rotation curves of one set of lengths for the parametric study.
Components in the tension side: – – – –
and
The Eqs. (1) and (2) will also be valid for the strength and stiffness of a connection with transverse stiffeners placed between the column flanges.
10
0
fy
rcr
critical plate buckling stress (see Eq. (A.1) in EC3-1.5 Annex A.1(2) and Table 4.1 and Table 4.2 in EC3-1.5). For the case of the beam flange in compression, the plate slenderness can be simplified using the following expression:
sffiffiffiffiffiffiffiffiffi 2 kp ¼ 1:4 a f y Et2fb
16
qffiffiffiffiffi
Bolts in tension. End plate in bending. Column flange in bending. Beam flange in tension. Other components:
In order to characterize this component we need to distinguish between the part that is in compression and the part that is in tension. The component for an unstiffened column flange in bending in the tension zone has already been established in EC3, Part 1–8 Clause 6.2.6.4 for the strength and Clause 6.3.2 for the stiffness. These formulae deal with the T-stub behavior corresponding to each bolt row under tension. A different situation arises in the case of a compression zone, for which the corresponding plastic mechanism has been identified by means of the test results and the numerical parametric study. Accordingly, a plastic loading area is established considering a dispersion angle of 45° through the end plate, measured from the welds of the beam flange. This results in a width of pffiffiffi pl ¼ tfb þ 2aw 2 þ 2t ep as shown in Fig. 5.
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B. Gil et al. / Engineering Structures xxx (2015) xxx–xxx
bc g
ec
ec
hc
t fc
t ep
t fc
1 2
beam
e MSd
t fb
bb
p
bc
z
t wc
h ep
bb
hb /2
eep mep t wb
column
bep
Fig. 3. Dimensions of the T-Stub for out of plane characterization.
hc
t fc
tfc 1 2
t wc α
beam
M
a
leff,bf,c
column
Fig. 4. Beam flange in compression.
tfc tep
0,5(bc-twc)
M
M
dl
pl
pl Loading area
hl
Fig. 5. Column flange in bending for the compression zone.
Contact pressure CPRESS - Column flange - S1 (IPE 300/HEB160-240/EP10mm)
Contact pressure CPRESS - Column flange - S2 (IPE 300/HEB160-240/EP20mm)
16
S1-1 HEB240 S1-2 HEB 220 S1-3 HEB 200 S1-4 HEB 180 S1-5 HEB 160
8
Contact pressure
12
Contact Pressure
10
S2-1 HEB 240 S2-2 HEB 220 S2-3 HEB 200 S2-4 HEB 180 S2-5 HEB 160
8
4
6
4
2 0 -12
-8
-4
0
4
Posion (cm)
(a) plate thickness of 10 mm
8
12
0 -12
-8
-4
0
Posion (cm)
4
8
12
(b) plate thickness of 20 mm
Fig. 6. Contact pressure in column flange: (a) plate thickness of 10 mm, and (b) plate thickness of 20 mm.
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B. Gil et al. / Engineering Structures xxx (2015) xxx–xxx
The height hl of this loading area can be obtained from the results of the parametric study that are illustrated in the graphs of Fig. 6. These show the contact pressure, within the plastic range, measured in the middle of the column flange for a T-Stub formed by half of an IPE300 and columns whose sizes range from HEB160 to HEB240, and for end plate thicknesses of 10 mm (a) and 20 mm (b), respectively. These graphs reveal that, for a thin end plate, the loaded part of the column flange starts approximately at a distance of 0.5twc + c and spreads until the edge of the end plate with almost a constant pressure (see finite element stress contour plots in Fig. 6). For a thick end plate, the loading area starts at a distance of around 0.5twc and extends until the edge of the end plate. Thus, it may be considered with sufficient accuracy that the resultant load is applied at a distance dl ¼ 0:5bep 0:5hl measured from the axis of the column web. Then, the lever arm ‘‘z’’ for the joint (see Fig. 3) becomes: z ¼ 0:5g þ dl . In the aim of determining the effective length that governs the resistance of this isolated component, a finite element analysis has been carried out based on the models shown in Fig. 7. These models consisted of a steel plate with the measures of half a column flange (80 13 mm for a HEB160) with lengths varying between 200 mm and 1200 mm (same as the column in the tests). This plate is clamped at both ends as well as in one longitudinal side, thus simulating that the column flange is clamped to the web. A uniform pressure is applied in a portion of surface in the middle of the plate. The resulting stiffness and maximum load applied is the same for each different studied length from 1200 to 400 mm. For the models with smaller lengths (300 mm and 200 mm) both, the stiffness and the maximum load increase with respect to the previous cases. A second model is considered which is the result of clamping the longitudinal side only and leaving the edges of the plates free (see Fig. 7). The results obtained are very similar to those of the previous case. After performing these simulations, it is found that for the case of the test T-stub 01 (column HEB160) the effective length of the column flange in bending (in the compression zone) is around 360 mm, and for the case of T-stub 03 (column HEM160) the effective length is around 400 mm. Also, through the finite element simulations and test results, the yielding lines of this component are identified. These lines are similar to those considered in the EC3-1.8, table 6.4 for noncircular patterns of a group of two bolts in an unstiffened column flange, for the tension zone (Fig. 8). The effective length in this case, for every bolt, is leff ;nc ¼ ð2m þ 0:625e þ 0:5pÞ where p is the distance between bolt rows, measured vertically, e is the distance from the bolt axis to the free edge of the column flange, and m ¼ 0:5bc e 0:5t wc 0:8r c . However, this effective length is a simplification that corresponds to the failure mode 1 mentioned in EC3-1.8 Table 6.2, that is, when four plastic hinges are formed. In our case only two plastic hinges are formed in the column flange and then the total effective length for the group of two bolts becomes leff ;nc ¼ ð5:5m þ 4e þ pÞ, as proposed by Zoetemeijer [2]. Since the yield lines produced in the compression zone are similar to those for the bolts, (see Fig. 8), the same expression can be used for the column flange in bending in the compression zone by Clamped side
0 (20 th ng e L
0,5(bc-twc) M p
pl Loading area
e m
ml el hl
Tension zone
Compression zone
Fig. 8. Yielding lines of the column flange in bending for the tension and compression zone.
replacing p by pl (as the width of the loading area), m by ml ¼ ð0:5bep 0:5twc 0:8rc Þ=2 and e by el ¼ 0:5bc ml ð0:5twc þ 0:8rc Þ. Using these data the effective length can be readily obtained. In the case of the T-stub01 the length turns out to be 355 mm, which fits very well with the isolated finite element models of the column flange under bending in the compression zone. After describing the effective lengths and other parameters, the failure mechanism is studied, as mentioned above, using a cantilever model (Fig. 9) with a length of 0.5bc and a load F applied at a distance dl. The failure is produced when a plastic hinge is formed. Using the principle of virtual work, VW ext ¼ VW int ! Fdl h ¼ M p h which leads to the following expression for the resistance of the column flange in bending: 2
F cf ;b;Rd ¼
M p leff ;cf ;b t fc f yd ¼ dl 4dl
ð4Þ
where leff ;cf ;b ¼ ð5:5ml þ 4el þ pl Þ. With regard to the stiffness, elastic theory is used but, in order to avoid complex calculations for structural designers in everyday practice, a simpler model of a cantilever with a section of tfcx leff C and a load applied to a distance of dl from the clamped side has been considered. The purpose of the coefficient C is to reduce the effective length used in the resistance calculation and adjust it to the characterization of the stiffness. By means of a parametric study the value of C is estimated to be 0.9. Consequently, the stiffness coefficient becomes:
kcf ;b ¼
3I 3 dl
¼
3Cleff ;cf ;b t3fc 3 12dl
¼
Cleff ;cf ;b t 3fc
ð5Þ
3
4dl
If the column has transverse stiffeners the compression is transmitted directly from the beam flange to the stiffener avoiding
Mp
F
Clamped side
t
) mm 00 2 o1
o 0t (20 h t ng Le
m 00 12
Fig. 7. Contact pressure in the middle line of the column flange.
υ
m)
δ
dl 0,5b c Fig. 9. Failure mechanism for the column flange in bending.
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B. Gil et al. / Engineering Structures xxx (2015) xxx–xxx
bending of the column flange in the compression zone, hence, the stiffness and strength becomes that of the transverse stiffener:
F ts;Rd ¼ t ts 0:5ðbc t wc Þf y =cM0
ð6Þ
The plastic resistance of a rectangular section in torsion is T pl;Rd ¼ W T;pl sy , where the plastic torsional modulus is W T;pl ¼ t 2fc ð3bc t fc Þ
sy;max ¼ t ts 0:5ðbc t wc Þ kts ¼ ðhc 2tfc Þ
is:
T pl;Rd ¼ W T;pl
3.3. End plate in bending The strength and stiffness of the end plate in bending in the compression zone is directly obtained from that of the column flange, by replacing the thickness of the column flange tfc by the thickness of the end plate tep. Consequently: 2 M p leff ;cf ;b tep f yd ¼ dl 4dl
ð8Þ
with leff ;ep;b ¼ leff ;cf ;b ¼ ð5:5ml þ 4el þ pl Þ
kep;b ¼
3Iep 3
dl
¼
3Cleff ;ep;b t 3ep 3 12dl
f pyffiffi. 3
For a clamped column T pl;Rd ¼ Fz . 2 Therefore, the maximum torsional moment that can be applied
ð7Þ
where tts is the thickness of the transverse stiffener.
F ep;b;Rd ¼
(see Ref. [22]) and the maximum tangential stress is
6
¼
ð9Þ
with C = 0.9. If the column has stiffeners, the stiffness coefficient of this component is not taken into account because its combination with the one of the column flange in bending is replaced by the characteristics (strength and stiffness) of the transverse stiffener (Eqs. (6) and (7)). 3.4. Column flange in torsion The rotation produced in the column side comes mainly from a combination of the rotation of the column web in bending and both column flanges in torsion. With the aim of characterizing the torsion in the column flange, the flange is modeled as an isolated rectangular beam of dimensions bct and clamped at both edges. The torque is applied as shown in Fig. 10.
ð10Þ
And consequently, the maximum force applied in the connection is:
F cf ;t;Rd ¼
2W T;pl f y t 2fc ð3bc t fc Þf y pffiffiffi ¼ pffiffiffi zcM0 3 3 3 z cM0
ð11Þ
Regarding the stiffness, the torsional rotation for a clamped colTLc . Therefore the stiffness coefficient is: umn is h ¼ 4GJ
kcf ;t ¼
Cleff ;ep;b t 3ep 3 4dl
fy F z pffiffiffi ¼ cf ;t;Rd 2 cM0 3
4 0:38 J J ¼ 1:52 2 L c z2 Lc z
ð12Þ
where Lc is the total length of the column, z is the lever arm and J is the torsional modulus of the column flange. J ¼ 13 bc t 3fc (for thin walled rectangular sections). 3.5. Column web in bending After considering the column flange in torsion, the next component to consider is the column web in bending, which happens to be the other main source of deformation and which usually is the first one to yield near the root radius. Due to the two forces acting on both sides of the column flange (see Fig. 11), a bending moment appears in the column web. This moment is higher at the side of the joint and becomes gradually smaller at the opposite side (Fig. 11). A plastic hinge is formed where the moment is the highest and the section is smallest, that is, where the root radius meets the column web. Accordingly, the strength of the column web in bending becomes:
F cw;b;Rd ¼
M pl;cw;Rd leff ;cw;b t2wc f yd ¼ z 4z
ð13Þ
where M pl;cw;Rd is the plastic resistant moment of the effective section of the column web. z is the lever arm. g is the distance between bolts, measured horizontally.
T/2
T Lc
hc eep F MSd
dc
T/2
g
1 2
beam
z
eep F column Mp
F
z
F
K cf,t
K cf,t
M
dc Fig. 10. Application of torsional moments in the column flange.
Fig. 11. Column web in bending.
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S2_L180
S1_L180
30
30
S1-1 HEB 240 S1-2 HEB 220 S1-3 HEB 200 S1-4 HEB 180 S1-5 HEB 160
25
25
15 10
-150
-100
-50
0
20
Stress (kN/cm2)
Stress (kN/cm2)
20
5
-200
0
50
100
150
S2-1 HEB 240 S2-2 HEB 220 S2-3 HEB 200 S2-4 HEB 180 S2-5 HEB 160
200
-200
15 10 5
-150
-100
0
-50
0
50
30
30
S1-1 HEB 240 S1-3 HEB 200
25
25
S1-4 HEB 180 S1-5 HEB 160
20
Stress (kN/cm2)
Stress (kN/cm2)
20 15 10 5
-100
-50
0
0
50
100
150
200
-200
15 10 5
-150
-100
0
-50
0
50
30
30
S1-1 HEB 240 S1-3 HEB 200
25
25
S1-4 HEB 180 S1-5 HEB 160
20
Stress (kN/cm2)
Stress (kN/cm2)
20 15 10 5
-50
0
0
50
100
150
200
-200
15 10 5
-150
-100
0
-50
0
50
30
30
S1-1 HEB 240 S1-3 HEB 200
25
25
S1-4 HEB 180 S1-5 HEB 160
20
Stress (kN/cm2)
Stress (kN/cm2)
20 15 10 5
-50
0
0
200
S2-1 HEB 240 S2-2 HEB 220 S2-3 HEB 200 S2-4 HEB 180 S2-5 HEB 160
S1-2 HEB 220
-100
150
S2_L400
S1_L400
-150
100
posion X (cm)
posion X (cm)
-200
200
S2-1 HEB 240 S2-2 HEB 220 S2-3 HEB 200 S2-4 HEB 180 S2-5 HEB 160
S1-2 HEB 220
-100
150
S2_L300
S1_L300
-150
100
posion X (cm)
posion X (cm)
-200
200
S2-1 HEB 240 S2-2 HEB 220 S2-3 HEB 200 S2-4 HEB 180 S2-5 HEB 160
S1-2 HEB 220
-150
150
S2_L240
S1_L240
-200
100
posion X (cm)
posion X (cm)
50
posion X (cm)
100
150
200
-200
15 10 5
-150
-100
-50
0
0
50
100
150
200
Posion X (cm)
Fig. 12. Von Misses Stress in the column web along its length.
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cf,b,t
bf,t ep,b,t
cw,b M cf,b,c
bf,c
ep,b,c Fig. 14. Mechanical model for a T-stub in out-of-plane bending.
components involved in the T-Stub under out-of-plane bending are assembled as series of springs according to the mechanical model illustrated in Fig. 14. The parameters appearing in this figure are: Fig. 13. T-Stub in out-of-plane bending. Effective length of the column web.
leff ;cw;b is the effective length for the column web in bending which is obtained by means of finite element parametric study as explained in the next paragraphs and in Figs. 12 and 13. The graphs of Fig. 12 show the Von Misses stress levels along the column web length obtained from the parametric study. As mentioned before the series S1 has a T-Stub formed with half an IPE300 with a 10 mm end-plate. The series S2 has a T-Stub formed with half an IPE300 with a 20 mm end-plate. The section of the column varies in both series from HEB160 to HEB240 and different lengths are studied as well. It may be observed that the yielded length in the column web is around 700–800 mm in all the cases, even with different column sections, column lengths or end plate thickness. The effective length for the column web in bending, leff ;cw;b , fits quite accurately when it is equal to twice the effective length for the column flange in compression, and not becoming larger than the total column length Lc.
leff ;cw;b ¼ 2leff ;cf ;b 6 Lc
ð14Þ
Regarding the stiffness, a simple model is considered as it is shown in Fig. 11. The column web is modeled as a beam with a rotational spring in each side equal to the rotational stiffness of the column flange in torsion with a stiffness coefficient kcf ;t which is defined in Eq. (12). Then, the stiffness coefficient for the column web in bending can be calculated by means of standard structural analysis as:
kcw;b
! ð4Icw þ kcf ;t dc z2 Þ 1 4I2cw ¼ 2 þ z dc ð4Icw þ kcf ;t dc z2 Þ dc
ð15Þ
where Icw is the inertia of the effective section of the column web C0 l
t3
wc Icw ¼ eff ;cw;b . 12 The coefficient C 0 , that affects the effective length, takes the value of 0.5 and is obtained from the parametric study taking into account the elastic response (for stiffness purposes) rather than the plastic response used for resistance calculation. If the column has transverse stiffeners, there is no rotation between the column flange and the column web, and the stiffness coefficient of this component becomes infinite.
4. Assembly of the components Finally, following the principles of the component method [1,3] and after characterizing the properties (stiffness and strength), the
cw,b – spring corresponding to the column web in bending, that includes the column flange in torsion, as explained in Section 3.5 and Eq. (15) cf,b,t – spring corresponding to the column flange in bending for the tension zone, ep,b,t – spring corresponding to the end plate in bending for the tension zone, bf,t – spring corresponding to the beam flange in tension, cf,b,c – spring corresponding to the column flange in bending for the compression zone, ep,b,c – spring corresponding to the end plate in bending for the compression zone, bf,c – spring corresponding to the beam flange in compression.
5. Comparison between the test and the analytical model Table 2 shows a comparison of the values obtained for initial stiffness and strength in the tests and by means of the proposed analytical model implementing the components described above and Table 3 shows a comparison of the values obtained by means of finite element models and by means of the proposed analytical model. The experimental values are subtracting the rotation due to torsion in the column to properly compare with those of the analytical model. The plastic moment is considered as 1.5 times the value of the elastic moment because that is the value of the shape coefficient for a rectangular section, which in the T-stub is bb ⁄ tfb. Both tests, FEM and analytical models show very good agreement. More specifically the stiffness predicted by the analytical model is slightly lower than that resulting from the tests and FEM. On the other hand, the predicted strength is somehow greater than that of the tests and FEM.
Table 2 Comparison of the initial stiffness and strength between the experimental and analytical model. Joint
Characteristics
T-Stub01
HEB160/IPE300 EP10 mm/Bolted
T-Stub03
HEM160/IPE300 EP10 mm/Bolted
Initial stiffness Sj,ini (kNm)
Plastic moment Mj,Rd (kNm)
Test
Analytical
Test
Analytical
119
117.9
8.4
8.6
546
525.1
13.5
14.2
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B. Gil et al. / Engineering Structures xxx (2015) xxx–xxx
Table 3 Comparison of the initial stiffness and strength between the finite element model and analytical model. Joint
Characteristics
S1-3 H = 240 cm S1-4 H = 180 cm C-1 H = 300 cm C-2 H = 300 cm C-3 H = 300 cm
HEB200/IPE300 EP10 mm/Bolts 16 mm HEB180/IPE300 EP10 mm/Bolts 16 mm HEB200/IPE200 EP16 mm/Bolts 16 mm HEB200/IPE270 EP18 mm/Bolts 20 mm HEB260/IPE400 EP20 mm/Bolts 24 mm
Initial stiffness Sj,ini (kNm)
Plastic moment Mj,Rd (kNm)
FEM
Analytical
FEM
Analytical
131.1
123.8
12.8
12.7
117.5
119.2
10.5
10.6
98.5
89.4
7.8
7.9
134.0
116.1
12.9
12.5
197.8
173.8
18.5
21.3
6. Conclusions This paper deals with the mechanical model and analytical characterization of the bolted T-stub of a semi-rigid steel connection subjected to out-of-plane bending. To reach its complete characterization, an experimental program and finite element models were developed and presented in the companion paper [13]. Then, a parametric study is carried out where different parameters such as end plate thickness, column length and column section are varied. With these tools, the components that are involved in the whole T-Stub are identified and independently studied. The test and finite element models reveal that the sources of deformation for joints are, in the compression side: the end plate in bending, the column flange in bending and the beam flange in compression; in the tension side: the bolts in tension, the end plate in bending, the column flange in bending and the beam flange in tension. The other components involved are the column web in bending and the column flange in torsion. In fact, the latest two are the main sources of deformation in unstiffened joint and are components that are not contemplated in Eurocode 3. These components are analyzed and characterized based on the component method. New effective lengths and analytical expressions have been established for the new components, and also for those already described in Eurocode 3 that needed to be adapted for joints in out-of-plane bending. Finally, the components involved in the T-stub are assembled in a proposed mechanical model and compared with the experimental results and finite elemen models showing that the proposed model predict the behavior of the T-Stub with good accuracy. Further research must be done about T-stubs under torsion to end the complete characterization of 3D steel joints. Acknowledgements The authors gratefully acknowledge the financial support provided by the Spanish Ministerio de Ciencia e Innovación under the contract number BIA2010-20839-C02-02. References [1] CEN. Eurocode 3: design of steel structures. Part 1–8: Design of joints (en 1993-1-8:2005). CEN; 2005.
[2] Zoetemeijer P. A design method for the tension side of statically loaded, bolted beam-to-colum connections. HERON 1974;20(1). [3] Faella C, Piluso V, Rizzano G. Structural steel semirigid connections: theory, design and software. CRC Press LLC; 1999. [4] Jaspart JP. Etude de la semi-rigidité des noeuds poutre-colonne et son influence sur la résistance et la stabilité des ossatures en acier. Belgique: Université de Liège; 1991. [5] Davison JB, Kirby PA, Nethercot DA. Rotational stiffness characteristics of steel beam to column connections. J Constr Steel Res 1987;8:17–54. [6] Stark J, Bijlaard FSK. Design rules for beam-to-column connections in europe. J Constr Steel Res 1988;10:415–62. [7] Nguyen P-C, Seung-Eock Kim. An advanced analysis method for threedimensional steel frames with semi-rigid connections. Finite Elem Anal Des 2014;80:23–32. [8] Loureiro A, Moreno A, Gutiérrez R, Reinosa JM. Experimental and numerical analysis of three-dimensional semi-rigid steel joints under non-proportional loading. Eng Struct 2012;38:68–77. [9] Gil B, Goñi R, Bayo E. Experimental and numerical validation of a new design for three-dimensional semi-rigid composite joint under general loads. Eng Struct 2013;48:55–69. [10] Cabrero JM, Bayo E. The semi-rigid behaviour of three-dimensional steel beam-to-column joints subjected to proportional loading. Part II: Theoretical model and validation. J Constr Steel Res 2007;63(9):1254–67. [11] de Lima LRO, de Andrade SAL, Vellasco PCG, da Silva LS. Experimental and mechanical model for predicting the behaviour of minor axis beam-to-column semi-rigid joints. Int J Mech Sci 2002;44:1047–65. [12] de Lima LRO, de F, Freire JL, Vellasco PCG, de Andrade SAL, da Silva JGS. Structural assessment of minor axis steel joints using photoelasticity and finite elements. J Constr Steel Res 2009;65:466–78. [13] Gil B, Goñi R. T-stub behaviour under out-of-plane bending. I: experimental research and finite element modelling. Eng Struct 2014; submitted for publication. [14] Girão Coelho A, Simoes da Silva L, Bijlaard FSK. Finite-element modeling of the nonlinear behavior of bolted T-Stub connections. J Struct Eng 2006;132(6): 918–28. [15] Lee J, Goldsworthy HM, Gad EF. Blind bolted t-stub connections to unfilled hollow section columns in low rise structures. J Constr Steel Res 2010;66:981–92. [16] Loureiro A, Gutiérrez R, Reinosa JM, Moreno A. Axial stiffness prediction of non-preloaded T-Stubs: AN analytical frame approach. J Constr Steel Res 2010;66:1516–22. [17] Al-Khatab Z, Bouchaïr A. Analysis of a bolted t-stub strengthened by backingplates with regard to eurocode 3. J Constr Steel Res 2013;63:1603–15. [18] Loureiro A, López M, Gutiérrez R, Reinosa JM. A new analytical formulation for the e-stub strength calculation in three dimensional steel joints with additional plates welded to the weak axis. Eng Struct 2013;56:2263–72. [19] Wan Hu J, Leon RT, Park T. Mechanical models for the analysis of bolted T-Stub connections under cyclic loads. J Constr Steel Res 2012;78:45–57. [20] Girão Coelho A, Bijlaard FSK, Gresnigt N, Simoes da Silva L. Experimental assessment of the behaviour of bolted T-Stub connections made up of welded plates. J Constr Steel Res 2004:60. [21] CEN. Eurocode 3: design of steel structures. Part 1. 5: Plated structural elements (en 1993-1-5:2006); 2006. [22] Case J, Chilver L, Ross CTF. Strength of materials and structures. ButterworthHeinemann; 1999.
Please cite this article in press as: Gil B et al. T-stub behavior under out-of-plane bending. II: Parametric study and analytical characterization. Eng Struct (2015), http://dx.doi.org/10.1016/j.engstruct.2015.03.039