shear connection are given in various design codes such as BS5950 (1), AS2327 (2),. Eurocode 4 (3), and the Hong Kong Steel Code, CoPSteel (4). Design ...
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PERFORMANCE-BASED DESIGN AND ANALYSIS OF COMPOSITE BEAMS IN BUILDING STRUCTURES K F Chung1 and A J Wang2 1
Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, Hong Kong SAR, China 2
Hyder Consulting Limited, Hong Kong SAR, China
ABSTRACT In order to enable effective design and construction of composite beams in building structures, advanced three dimensional non-linear finite element models are established to assist designers to examine and understand the deformation characteristics of composite beams during the entire loading history. These models are provided to facilitate the performancebased design and analysis of long span composite beams with practical constructional features. Details of the advanced three dimensional finite element models of a composite beam as well as a composite joint are presented together with careful calibration against test data. Moreover, the effects of the deformation characteristics of both shear connectors and tensile reinforcement are also examined and presented. The proposed numerical analysis and design models are demonstrated to be effective for detailed analyses and design of composite beams and joints with practical geometrical dimensions and arrangements. Designers are strongly encouraged to employ these models in their practical work to exploit the full advantages offered by composite construction. KEYWORDS Composite beams, finite element models, integrated analysis and design.
COMPOSITE BEAMS IN BUILDING STRUCTURES Composite beams are strong and stiff flexural members with long spanning capacities. The structural form of a composite beam is essentially a thin wide concrete flange connected with a steel section where the concrete flange is in compression while the steel section is largely in tension. Shear connectors, usually headed studs, are welded to the top flange of the steel
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section and embedded in the concrete flange. Depending on the number of shear connectors provided along the interface between the steel section and the concrete flange which is either a solid concrete slab or a composite slab, the composite beam may operate in either full shear connection or partial shear connection, and hence, exhibit a wide range of deformation characteristics according to the flexibility of the shear connectors. Prescriptive Design of Composite Beams In many structural design codes, plastic design principles are adopted in designing composite beams, and their moment resistances under sagging and hogging moments are determined according to plastic stress blocks. In general, flexibility of shear connectors is often ignored in strength assessment while stringent requirements on the slippage ductility of shear connectors are imposed in order to justify uniform distribution of shear resistances along shear spans at ultimate limit state. In continuous composite beams, the amount of moment re-distribution is specified in a prescriptive manner according to the geometrical dimensions of the composite cross-sections as well as the provision of tensile reinforcement. Moreover, while both full and partial shear connection may be adopted in composite beams under sagging moments, full shear connection is usually required in composite beams under hogging moments. In general, this is readily achieved through the provision of few shear connectors in transferring the tensile resistances of steel reinforcement over the hogging moment regions. At present, design methods for composite beams using plastic stress blocks with full or partial shear connection are given in various design codes such as BS5950 (1), AS2327 (2), Eurocode 4 (3), and the Hong Kong Steel Code, CoPSteel (4). Design handbooks for composite beams with either solid concrete slabs or composite slabs with profiled steel decking may also be found in the literature (5-9). Practical Issues in Composite Beam Design In general, it is often necessary for designers to refer to specialist design guides in designing composite beams and floor systems with practical constructional features:
composite beams with asymmetric rolled or fabricated I sections, composite beams with fabricated I sections with tapered webs, composite beams with web openings for full integration with building services, composite beams with partial continuity offered by the connections with columns, and long span composite beams against floor vibration in service.
Although there are many design methods available in the literature, their use in practice are fairly limited. Many methods require extensive design efforts together with a deep learning curve to achieve high structural efficiency. Some of them are product specific, and hence, their applicability is rather limited. Moreover, little information on associated failure criteria is provided. In general, it is highly desirable to develop performance-based analysis and design tools for practical design of composite beams. This allows designers to understand the structural behaviour of the composite beams as well as to monitor their ‘stress and strain’ condition during the entire loading history. Moreover, different failure criteria in designing composite
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beams and joints with specific mechanical properties, geometrical dimensions, and member configurations as well as constructional features may be adopted as required in various projects.
OBJECTIVES AND SCOPE OF WORK In this paper, a research and development project is reported which aims to develop advanced numerical analysis and design models for practical design of long span composite beams in building structures. Details of advanced finite element models of composite beams and joints are presented and established to assist designers to examine and understand the deformation characteristics of composite beams and joints during the entire loading history. Moreover, the effects of the deformation characteristics of both shear connectors and tensile reinforcement are also examined and presented. The project aims to develop these models to facilitate the performance-based design and analysis of long span composite beams with practical constructional features.
FINITE ELEMENT MODELLING In order to simulate numerically the structural behaviour of composite beams and joints with practical member configurations and loading conditions, three dimensional finite element models are established using the general purpose finite element package ABAQUS (Version 6.4, 2004) (10). In general, the steel sections are modelled with shell element S8, and the concrete flange and the profiled steel decking are modelled with solid element C3D8. Material Models It is important to have a suitable material model for each material in the composite beams and joints. For steel under uni-axial loading condition, a bi-linear stress-strain curve, as shown in Figure 1a), is adopted as the material model. Moreover, failure of the steel is assumed to follow the von Mises failure criteria which failure surface is also shown in Figure 1a). For concrete under uni-axial loading condition, a non-linear stress-strain curve as shown in Figure 1b) is adopted in the material model. The compressive strength of concrete is taken to be equal to its cylinder strength while its tensile strength is taken as only 10 % of its compressive value. The limiting compressive strain of concrete against crushing is taken to be 0.35 %. As shown in Figure 1b), failure of the concrete is assumed to follow the DruckerPrager failure criteria (10,11) when subjected to tri-axial loading. In order to simulate controlled concrete cracking in the presence of longitudinal reinforcement and profiled steel decking, the concept of ‘smeared layers’ is introduced in which smeared reinforced concrete layers and smeared composite decking layers are adopted. The mechanical properties of the smeared layers, namely, the equivalent compressive strength, the equivalent tensile strength and the equivalent Young’s modulus, are evaluated (10,11) according to the respective areas and the respective material curves of concrete, steel reinforcement and profiled steel decking as shown in Figure 2. It should be noted that the adoption of smeared layers and the corresponding modified material curves is very effective in suppressing numerical divergence during solution iterations in the finite element analyses.
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For simplicity, transverse reinforcement and profiled steel decking in the transverse direction are ignored in the material models. σ Esh = 0.005Eo py
σ1
1 Eo 1
ε
σ2 -py
σ3 Tri-axial loading
Uni-axial loading
a) Steel
σ
( / c' ) p 1 ( / c ' ) c
pc
E 5.5 f cu
0.3 pc
0.001
σ1
ε
1
' c 0.0035
0.025
c' 2.4 10 4
p
f cu
σ2
3
c 1.55 32.4 σ3
pc = 0.8 fcu Uni-axial loading
Tri-axial loading
b) Concrete
Figure 1: Material models Material and Geometrical Non-linearities With both material and geometrical non-linearities incorporated into the finite element models, large deformation in any severely yielded regions of the steel sections can be modelled accurately. In addition, the first eigen-mode of the finite element model is adopted as the initial geometrical imperfection, and the magnitude of the maximum initial imperfection is taken as 25% of the web thicknesses of the steel beams. The presence of the
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Smeared reinforced concrete layer for concrete and reinforcement
Smeared reinforced concrete layer for concrete and reinforcement
Layers of concrete
Layers of concrete
Layers of steel
Layers of steel
i. Cross-section between troughs
ii. Cross-section at troughs
a) Composite beams under sagging moments
Layers of steel
Smeared composite decking layer for concrete and profiled steel decking
Layers of concrete Smeared reinforced concrete layer for concrete and reinforcement
C σs(εi) σc(εi) and σt(εi) As Ac Es(εi) Ec(εi)
is the strength of steel reinforcement or profiled steel decking at a strain of i; are the compressive and the tensile strengths of concrete at a strain of i; is the area of steel reinforcement or profiled steel decking in the smeared layer; is the area of concrete in the smeared layer; is the Young’s modulus of steel reinforcement or profiled steel decking at a strain of i; and is the Young’s modulus of concrete at a strain of i .
Equivalent compressive strength of the smeared layer: ( ) A s c ( i ) A c eq, c (i ) s i As Ac Equivalent tensile strength of the smeared layer: ( ) A s t ( i ) A c eq, t ( i ) s i As Ac Equivalent Young’s modulus of the smeared layer: E ( ) A s E c ( i ) A c E eq ( i ) s i As Ac
Figure 2: Smeared layers of reinforced concrete and composite decking
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initial geometrical imperfection in the finite element model will facilitate solution iterations during non-linear analyses. Furthermore, it should be noted that in order to avoid local inclusion between the finite elements of the concrete flanges and the steel sections during non-linear analyses, axial spring elements with extremely high compressive stiffness but zero tensile stiffness are provided along the interfaces between the concrete flanges and the steel sections. Shear Connectors Every shear connector is modelled with one horizontal spring, one transverse spring and one vertical spring in order to simulate both the longitudinal and the transverse shear forces as well as the pull-out force of the shear connector. The load-slippage curves of the horizontal and the transverse springs are obtained from the normalized load-slippage curve proposed by Ollgaard et al (12) as follows: Fh
=
Ps (1 – e -S )
(1)
where Fh Ps α
is the longitudinal shear force developed in the shear connector at a slippage of S (mm); is the shear resistance of the shear connectors; is a non-dimensional parameter with its value between 0.5 and 1.5; and is a parameter with a unit of mm-1; its value is typically between 0.5 and 2.5.
In general, the typical load-slippage curve of headed shear connectors reported by Lawson (13) may be represented by Equation (1) with α = 1.2 and = 2.0.
NUMERICAL INVESTIGATION ON COMPOSITE BEAMS AND JOINTS In the present study, the structural behaviour of a continuous composite beam and a semirigid composite joint are examined, and effects of the deformation characteristics of flexible shear connectors and tensile reinforcement are thoroughly studied. Among dozens of continuous composite beams and semi-rigid composite joints tested and reported in the literature, the following are adopted in the present study:
Beam CTB4 reported by Ansourian (14), a continuous composite beam exhibiting significant moment re-distribution.
Joint B5 reported by Brown & Anderson (15), a symmetrically load semi-rigid composite joint with beam end-plate connections.
Finite Element Study on Composite Beam Figure 3 illustrates the overall test arrangement of the continuous composite beam, Beam CTB4, together with the three dimensional finite element model. The predicted loaddeflection curve of Beam CTB4 is plotted in Figure 4 together with the measured data for
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IPB200 HE A 200
800 At = 804 mm2
P/2 2250
2250
100
2250
Ab = 767 mm2
3 5 7
HE 3 × 28 @ 320 c/cA 200
190
2250
P/2
3 5 7
C
200 10 6.5
Beam CTB4
Beam CTB4. py = 236.0 N/mm2, pc = 27.2 N/mm2.
Section C-C
Details of test specimen P/2
Δ
P/2
Δ
3 × 28 @ 320 c/c
Finite element mesh Shell elements: C3D8
three shear connectors
three shear connectors
Shell elements: S8 Contact spring elements not shown for clarity
Shear connector: 19 mm headed shear connector with as-welded height equal to 75 mm.
Initial imperfection
Figure 3: Beam CTB4
500
500 PJulA07
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400 600
400
Stage 1 Stage 2 495.3 (Δs)max = 0 mm 450.8 (Δs)max = 0.24 mm
300
300
200
Apllied load, P (kN)
Total load, P (kN) Total load, P (kN)
Total load, P (kN)
500 455.0 431.2
400
430.0 (Δs)max = 0.45 mm
405.6
200 300
200
100
100
E-R Test
100
0
P-R-72P-R-72
E-100 E-100
P-100-72 P-100-72
Test Test
P-50-72 P-50-72
N-72 N-72
0
0
0
E-R
0
0
5
510
10
20 10
15 3015
2040 20
Deflection at mid-span (mm) Vertical deflection at mid-span, Δ (mm) Vertical deflection at mid-span, Δ (mm)
Figure 4: Load deflection curves of Beam CTB4 direct comparison. It is shown that there is good agreement between the predicted and the measured data. As shown in Figure 5, it is found that failure of a continuous composite beam often involves a two-stage mechanism as follows:
Stage 1 Failure at internal support under hogging moment; and Stage 2 Failure at mid-span under sagging moment. Stage 1 Failure at internal support under hogging moment
von Mises stress (N/mm2)
Stage 2 Failure at mid-span under sagging moment. Figure 5: Failure mode of Beam CTB4
1.0 py 0.8 py 0.6 py 0.4 py 0.2 py 0
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Hence, the continuous composite beam is considered to be failed when plastic hinges are formed at both the internal support and near the mid-span, i.e. when both the hogging and the sagging hogging moment capacities of the composite beam are fully mobilized. A plastic hinge is regarded to be fully developed at a critical cross-section when its maximum strain reaches the limiting value, max , which is defined as follows: max
= 6
fy Es
(2)
where fy Es
is the yield strength of the steel; and is the Young’s modulus of the steel.
Hence, it is shown that the proposed three-dimensional finite element model is able to provide accurate prediction to the structural behaviour of continuous composite beams in both linear and nonlinear deformation stages. Moment re-distribution in a continuous composite beam It is interesting to examine the moment re-distribution behaviour in continuous composite beams, and the development of moment re-distribution in Beam CTB4 is illustrated in Figure 6. It should be noted that the first plastic hinge is formed at the internal support under a total load of 455.0 kN while the hogging and the sagging moments at the critical cross-sections are 159.7 and 165.3 kNm respectively. Upon further increase of the applied load, the second plastic hinge is formed near the mid-span under a total load of 495.3 kN while the hogging and the sagging moments at the critical cross-sections are 159.7 and 195.5 kNm respectively. Owing to moment re-distribution in the continuous composite beam, the total load carrying capacity is increased from 455.0 kN at Stage 1 failure to 495.3 kN at Stage 2 failure, i.e. an increase of 8.9%. The degree of moment re-distribution at the internal support is found to be 19%. Refer to Chung & Wang (16) for further details on the numerical analysis and design models of continuous composite beams. It should be noted that plastic local buckling is also successfully captured in the compressive flange of the steel sections near internal supports well after plastic hinges have been fully developed. In general, the occurrence of such plastic local buckling is only important to composite beams with wide flanges in steel sections in which the hogging moment capacities of the composite beams may decrease significantly after the onset of plastic local buckling. Flexibility of shear connectors In order to examine the effects of flexible shear connectors to the structural behaviour of continuous composite beams, shear connectors with different load-slippage characteristics as shown in Figure 3 are incorporated into the finite element model. The corresponding predicted load-deflection curves are also plotted in the same graph in Figure 4 for direct comparison. It is shown that there is a significant variation in the load carrying capacities among all these beams, and the maximum difference among the load carrying capacities is found to be 15%. Hence, it is demonstrated that the load-slippage characteristics of shear connectors are
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300
Stage 1 Stage 2
Moment, M (kNm)
250
FEM: Hogging moment FEM: Sagging moment Mhog,e = 197.2
200
Msag1 = 165.3
Msag2 = 195.5
Mhog1 = 159.7 Mhog2 = 159.7
Hogging moment from elastic analysis
150
100
Stage 1
Stage 2
50
455.0
0 0
200
495.3
400
600
800
Total load, P (kN)
Elastic analysis Nonlinear analysis P/2
197.2 kNm 159.7 kNm
37.5 kNm
176.8 kNm 195.5 kNm
b) Applied moment at failure The degree of hogging moment redistributed from internal support, mr, is given by: mr
= (Mhog, e - Mhog2) / Mhog,e
where Mhog, e Mhog2
mr
is the applied moment at the internal support at failure according to elastic analysis, and is the applied moment at the internal support at failure according to nonlinear analysis. = (197.2 – 159.7) / 197.2 = 19 %
Figure 6: Moment redistribution in Beam CTB4
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important in assessing the load carrying capacities of continuous composite beams as the internal force distribution depends not only on the flexural rigidities of the composite beams but also on the flexibility of shear connectors. On the contrary, the flexibility of shear connectors is considered not important in predicting the load carrying capacities of simply supported composite beams, although it will affect their deflections. Finite Element Study on Composite Joint Figure 7 illustrates the overall test arrangement of a semi-rigid composite joint, Joint BA5. The corresponding three dimensional finite element model is presented in Figure 8 together with the details of the beam end-plate connections (17). Figure 9 presents the deformed mesh of the finite element model, Joint BA5, at failure. It should be noted that in physical tests, composite connections often fail owing to the rupture of tensile reinforcement. This is readily predicted in the finite element models. In addition, severe yielding and stress concentration are found in the following locations:
the upper portion of the end-plate of the steel beam under the pull-out action of the bolt forces; and
part of the flange to web junction of the steel column under direct bearing of the lower portion of the end-plate of the steel beam.
The predicted load-deflection curve of the composite joint is plotted in Figure 10 together with the measured data for direct comparison. It is shown that there is good agreement between the predicted and the measured data. Hence, it is shown that the proposed three-dimensional finite element model is able to provide accurate prediction to the structural behaviour of semi-rigid composite joints with beam end-plate connections in both linear and nonlinear deformation stages. Moment capacities of composite joints According to experimental investigations reported in the literature, most of the tests on composite joints are terminated due to excessive deformation in the connections or rupture of tensile reinforcement. Hence, in order to establish the moment capacities of composite joints, the moment capacities are defined to be the applied moments at which the strain in the tensile reinforcement reaches a limiting value, εt , at 5%. Rupture of the tensile reinforcement, and hence, failure of the composite joint is likely to happen beyond that value. Development of internal forces It is interesting to examine the tensile forces in the bolts and the tensile reinforcement as well as the compressive (bearing) forces near the bottom flanges of the steel beams during the entire loading history; the development of various internal forces in the composite joint is presented in Figure 11. It is found that the forces in the tensile reinforcement are mobilized in the early loading stage, as shown in Figure 11, because of the relatively large deformation in the tensile reinforcement, when compared with the deformation in the bolts. This leads to early yielding of the tensile reinforcement, and the tensile forces in the bolts are subsequently developed at
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P
120
1100 At = 804 mm2 P
208.7 13.2
235 typ
UC 254×254×73
528.3
9.6
1410
45 45
80 90 328
Bolts M20 Grade 8.8 End plate: 578 ×250 ×15 250
Connection BA5：py = 355 N/mm2, pc = 38 N/mm2.
Fs (kN) Shear force, Ph (kN)
100
Load-slippage curve of shear connector
Connections details
80
Measured yield strength (N/mm2)
60
Fs = F(1 - e
40
)
Steel beam Flange Web 351
Test FEM
20
-2 ΔS 0.8
385
Steel column Flange Web 285
331
0 0
1
2
Slippage, s (mm) Δh (mm)
3
Figure 7: Joint BA5
End- plate
Reinforcement
Measured cylinder strength of concrete (N/mm2)
305
504
38.4
578
80
25
1410
UB 533 × 210 × 82
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Smeared layer for reinforced concrete (Solid elements: C3D8) Concrete flange (Solid elements: C3D8)
P
Steel decking (Shell elements: S8)
Steel beam (Shell elements: S8)
Steel column (Shell elements: S8)
Steel beam
End-plate
Flange of column
Web of column
Note: Spring contact elements are not shown for clearity.
Figure 8: Finite element model of Joint BA5
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a) Perspective view von Mises stress (N/mm2) 1.0 py 0.8 py 0.6 py 0.4 py 0.2 py 0
b) Side view
von Mises stress (N/mm2) 1.0 py 0.8 py 0.6 py 0.4 py 0.2 py 0
Figure 9: Typical failure mode of Joint BA5
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600
Moment, M (kNm)
500
400
300
200
100
0 0
10
20
Rotation, θ (×10-3 rad)
30
40
Figure 10: Moment-rotation curves of Joint BA5
1000
Fc = 817.1 kN Compressive contact force, Fc Tensile force in reinforcement, Fr Tensile force in bolts, Ft
Internal force, F (kN)
800
Tensile force in reinforcement, Fr
600
Tensile force in bolts, Ft
Fr = 410.5 kN 400
Ft = 406.6 kN
200
Compressive contact force, Fc M3D = 440.1 kNm
0 0
100
200
300
400
500
600
Moment, M (kNm)
Figure 11: Development of internal forces in Joint BA5
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large deformation stage of the composite joint. Thus, tensile reinforcement with sufficient ductility should be provided in order to fully mobilize the moment capacities of the composite joints.
PARAMETRIC STUDIES After careful verification of the finite element models for both composite beams and joints, it is possible to examine the effects of deformation characteristics of both shear connectors and tensile reinforcement on the structural behaviour of composite beams and joints through systematic parametric studies. A total of 120 non-linear finite element analyses on composite joints, continuous composite beams and semi-continuous composite beams with shear connectors and tensile reinforcement having different deformation characteristic were conducted (17). Owing to the limited space in this paper, only some of the key findings of the parametric studies are presented. In general, three different shear connectors, namely, Shear connectors A, B and C, with different slippage limits are considered in the parametric studies, and their deformation characteristics are plotted in Figure 12a). Moreover, two different tensile reinforcement, namely, tensile reinforcement N and H, with different deformation limits are considered, and their deformation characteristics are plotted in Figure 12b). Both the slippage limits of the shear connectors and the deformation limits of the tensile reinforcement are considered to range within the corresponding limits in practice. Composite Joints with Different Shear Connectors and Tensile Reinforcement Figure 13a) illustrates the overall arrangement of a composite joint with beam end-plate connections; details of the connections are also presented. It should be noted that in the composite joints, tensile reinforcement H is adopted while three different shear connectors are used for comparison. The moment-rotation curves of the composite joints with different shear connectors and tensile reinforcement are plotted in Figure 13b) for comparison. It is shown that:
The composite joint with Shear connector A exhibits very ductile deformation along the entire loading history owing to the ductile slippage characteristics of Shear connector A as well as the ductile deformation characteristics of tensile reinforcement H.
In the composite joint with Shear connector B, reduction in the moment capacity of the composite joint at large deformation is found. This may well be explained by the reduced shear resistance of Shear connector B at any slippage larger than 5 mm.
Despite the ductile deformation characteristics of tensile reinforcement H, the composite joint with Shear connector C is shown to have severe reduction in its moment capacity at large deformation owing to the rupture of shear connectors at a slippage larger than 7 mm. As no composite action is possible, only the moment capacity of the steel beam is readily mobilized.
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100
Lawson (13) Ollgaard (12)
FP (kN) h h(kN)
80 72
Shear connector A
60
Shear connector B
40 36
20 Shear connector C
0 0
2
4
5
6 ΔSh(mm) (mm)
7
8
10
12
a) Assumed load-slippage curves of shear connectors
500 500
450 400 400
Stress, σ (N/mm2)
Tensile reinforcement N 300 300
Tensile reinforcement H
205 kN/mm2 1
200 200
100 100
110
50
00 00
2 20 10
40 4 20
60 6 30
80 8 40
10 100 50
12 120
60
-2
Strain, ε (× 10 )
b) Assumed stress-strain curves of tensile reinforcement
Figure 12: Material models of shear connectors and tensile reinforcement
Similarly, the structural behaviour of the composite joints with different tensile reinforcement is also studied; Shear connector A is used in both cases. The moment-rotation curves of the composite joints are also plotted in Figure 13b), and it is shown that:
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P
P
15
UC 356 × 368 ×153
210 typ
15
UB 610 × 305 × 179
1500
1500
92 92
12
307.0 23.6
365.1 62
617.5
14.1
641.5
62
130
70
2500 18Y6@175, At = 508 mm2
368.4 Notes: Concrete cylinder strength, pc = 24 N/mm2. Yield strength of steel, py = 355 N/mm2 .
a) Details of composite joint Failure, t = 5%
800
Moment, M (kNm)
1000
Tensile reinforcement H Shear connector A Shear connector B Shear connector C
575.4
600
502.4 400
Shear connector A Tensile reinforcement H Tensile reinforcement N
800
Moment, M (kNm)
1000
452.4
575.4
600
400
200
200
0
0 0
20
40
60
0
Rotation, θ (×10-3 rad)
20
40
60
Rotation, θ (×10-3 rad)
b) Moment-rotation curves
Figure 13: Parametric study on a composite joint
The composite joint with tensile reinforcement H exhibits very ductile deformation along the entire loading history owing to the ductile deformation characteristics of tensile reinforcement H as well as the ductile slippage characteristics of Shear connector A.
In the composite joint with tensile reinforcement N, despite the ductile slippage characteristics of Shear connector A, the composite joint is shown to have severe reduction in the moment capacity of the composite joint at large deformation owing to
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the rupture of tensile reinforcement. Hence, only the moment capacity of the steel beam is readily mobilized. Consequently, it is shown that the proposed models are able to predict the detailed structural behaviour of composite joints based on the deformation characteristics of both the shear connectors and the tensile reinforcement. Composite Beams with Different Shear Connectors and Tensile Reinforcement The overall arrangement of the internal span of a semi-continuous composite with beam endplate connections is illustrated in Figure 14; details of the connections are also presented. It should be noted that in the composite beam, tensile reinforcement H is adopted while three different shear connectors are used for comparison. The predicted moment-rotation curves of the composite beams with different shear connectors and tensile reinforcement are plotted in Figure 14 for comparison. In general, the structural behaviour of the composite beams is found to be very similar to those of the composite joints, i.e. both the slippage characteristics of shear connectors and the deformation characteristics of tensile reinforcement have significant effects on the structural behaviour of composite beams. Moreover, the maximum values of slippage in the shear connectors in various cases are summarized in Figure 14 for easy comparison. It is shown that
For composite beams with tensile reinforcement H, the maximum values of slippage on Shear connectors A, B and C are found to be 12.6, 15.2 and 19.1 mm respectively. Hence, in the presence of non-ductile shear connectors, larger slippage is often needed to develop the full failure mechanism in the composite beams while lower load carrying capacities of composite beams is normally obtained.
In the composite beam with Shear connector A and tensile reinforcement N, the composite beam is shown to have small reduction in its load carrying capacity at large deformation owing to partial yielding of the tensile reinforcement. It should be noted that there is a steady load transfer from the tensile reinforcement to the shear connectors at large deformation. Moreover, partial composite action is developed in the composite beam at failure.
As demonstrated in the parametric studies, the proposed models are able to predict the detailed structural behaviour of composite beams based on the deformation characteristics of both the shear connectors and the tensile reinforcement.
CONCLUSIONS This paper presents the development of three dimensional finite element models which are advanced numerical analysis and design models for composite beams under practical member configurations and loading conditions. These models are provided to facilitate the performance-based design and analysis of long span composite beams with practical constructional features.
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UC 356 × 368 ×153
UC 356 × 368 ×153
W
UB 610HEA × 305200 × 179 47 @ 210 c/c 10000 Internal span
2500 18Y6@175, At = 508 mm2 130
70
617.5
307.0 23.6 14.1 Notes: Concrete cylinder strength, pc = 24 N/mm2. Yield strength of steel, py = 355 N/mm2.
a) Details of composite beam
6000
6000
First crack Stage 1 Stage 2
4742.2 4104.1
4000
3765.1
3489.2 3000
2000
1765.4 Tensile reinforcement H Shear connector A Shear connector B Shear connector C
1000
4742.2
5000
Applied load, W (kN)
Applied load, W (kN)
5000
4504.2 4000
3489.2 3000
2000
1765.4 Shear connector A Tensile reinforcement H Tensile reinforcement N
1000
0
0 0
50
100
150
200
250
300
0
50
Beam Shear connector A Shear connector B Shear connector C
100
150
200
250
Deflection at mid-span, Δ (mm)
Deflection at mid-span, Δ (mm)
Maximum slippage of shear connectors at failure, Smax (mm) Stage 1 Stage 2 4.9 12.6 4.9 15.2 4.9 19.1
Beam Tensile reinforcement H Tensile reinforcement N
Maximum slippage of shear connectors at failure, Smax (mm) Stage 1 Stage 2 4.9
12.6
4.9
12.2
b) Load-deflection curves
Figure 14: Parametric study on a semi-continuous composite beam
300
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It is shown that: 1.
Prescriptive design methods are often presented in a simplistic formulation as they operate with certain implicit assumptions on both material and structural behaviour. Hence, they are conservative, but easy to use. The proposed performance-based analysis and design models allow design to operate beyond what are currently permitted by those prescriptive design methods. While the analysis and design models are more rigorous in their prediction capabilities, their effective use certainly requires a thorough understanding on the material models as well as the structural behaviour of the structures.
2.
Through extensive calibration against a wide range of test data, the proposed models are able to provide detailed information on the structural behaviour of composite joints, continuous composite beams and semi-continuous composite beams. Based on the material models of the steel and the concrete as well as the deformation characteristics of the shear connectors and the tensile reinforcement, the loaddeflection curves of the structures can be obtained along the entire loading history.
Designers are strongly encouraged to employ the models in their practical work to exploit the full advantages offered by composite construction.
ACKNOWLEDGEMENTS The project leading to the publication of this paper is supported by the Research Committee of the Hong Kong Polytechnic University (Project No. G-W039).
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