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Partial interaction stresses in continuous composite beams under serviceability loads
Journal of Constructional Steel Research 60 (2004) 1525–1543 www.elsevier.com/locate/jcsr
Partial interaction stresses in continuous composite beams under serviceability loads Rudolf Seracino , Chow T. Lee, Tze C. Lim, Jwo Y. Lim School of Civil and Environmental Engineering, University of Adelaide, Adelaide, SA 5005, Australia Received 21 May 2003; received in revised form 19 December 2003; accepted 9 January 2004
Abstract The number of fatigue assessments of composite bridges is growing rapidly worldwide due to increasing allowable load limits and because many of the bridges are reaching the end of their anticipated design life. Special attention is usually given to predicting the residual endurance or strength of the shear connection because it cannot be visually inspected. However, the increased stresses in the steel and concrete components due to partial interaction must also be considered in a fatigue assessment. Based on linear elastic partial interaction theory, this paper develops a simplified procedure to predict the partial interaction curvature in continuous composite beams. When used in conjunction with focal points, the partial interaction flexural stresses in the steel and concrete components can be determined and used to more accurately predict the residual strength or endurance of the composite section. This research extends the tiered assessment approach previously published for simply supported beams so that it is now applicable to composite beams with any number of spans, span lengths, shear connection distribution and cross-section. The technique is validated using a finite element program developed to model the behaviour of composite structures and the procedure is demonstrated in an illustrative assessment. # 2004 Elsevier Ltd. All rights reserved. Keywords: Assessment; Continuous composite beams; Curvature; Fatigue; Flexural Stresses; Partial interaction; Slip
Corresponding author. Tel.: +61-8-83203-5451; fax: +61-8-8303-4359. E-mail address: [email protected] (R. Seracino).
0143-974X/$ - see front matter # 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jcsr.2004.01.002
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Nomenclature + 1 2 3 A C c dc E Ext fi fp I i Int K k L Ls l M MF n ni P p pi S s V V x y y a b e /
positive (sagging) moment (or curvature) when used as a superscript negative (hogging) moment (or curvature) when used as a superscript left shear span when used as a subscript middle shear span when used as a subscript right shear span when used as a subscript cross-sectional area integration constant concrete when used as a subscript distance between centroids of the steel beam and concrete deck modulus of elasticity external support when used as a subscript full interaction focal point second moment of area shear span containing the design point when used as a subscript internal support when used as a subscript integration constant stiffness of a shear connector total length of a continuous beam length of a span length of a shear span bending moment; mid-span when used as a subscript curvature magnification factor modular ratio; transformed composite section when used as a subscript no interaction concentrated load longitudinal spacing of the shear connectors partial interaction support when used as a subscript slip (relative displacement between the steel and concrete component at the interface); steel when used as a subscript support reaction total vertical shear force at a design point distance of design point from left support distance of a focal point from the steel-concrete interface distance of a centroid from the steel-concrete interface geometric and material parameter geometric and material parameter strain curvature
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1. Introduction Results of theoretical analyses and experimental investigation published by Newmark et al. in 1951 [1] demonstrated that the strains in the steel and concrete components of composite beams are greater than that predicted by standard full interaction analysis. The larger strains are a result of the relative displacement, or slip, between the steel and concrete components at the interface because the shear connectors must deform to resist the longitudinal shear forces. Although most composite beams tested from the 1960s to the early 1980s were simply supported, some tests were carried out on two and three-span continuous composite beams [2–7]. The purpose of most of the tests was to investigate ultimate failure modes and validate the use of the plastic design technique for composite structures. Some of these early tests were coupled with elementary computer analyses, but it was not until the 1990s that advanced non-linear computer techniques were being used to analyse composite structures [8–10]. Much of the current research on continuous composite beams is on developing simplified techniques for predicting the increased deflections allowing for partial interaction [11–13]. One of the reasons for this trend is that the introduction of high strength steels and high performance concrete has allowed longer spans, particularly in bridges, such that the deflection serviceability limit state often governs design. Most research is related to improving the design of new structures however, as many composite bridges are approaching the end of their anticipated design life, coupled with increasing allowable load limits, the assessment of existing structures is rapidly becoming more important worldwide. Research has recently been published [14] presenting a practical procedure to predict the increased strains in the steel and concrete components of simply supported composite beams allowing for partial interaction. Related research included quantifying the reduction in the stress range resisted by the shear connection due to partial interaction in simply supported beams [15], which has recently been extended to continuous beams [16]. As a result of this research, a tiered approach to assess the remaining strength or endurance of existing simply supported composite steel-concrete beams was developed [17]. After careful site inspection to establish the condition of the bridge, the tiered approach can be used to determine whether load testing of the structure is required. If it cannot be demonstrated that the remaining endurance or strength of the structure is adequate, then a load test could be used to quantify the beneficial effect of other factors contributing to the reserve strength, such as the effect of railings or barriers if cast integrally with the concrete deck. Alternatively, advanced non-linear computer analyses can be made as described in the papers referred to in this introduction. Unfortunately, the current tiered assessment approach does not include the analysis of the partial interaction flexural stresses in continuous structures. The research presented in this paper completes the extension of the tiered assessment approach to continuous composite steel-concrete beams of any number of spans with varying span lengths, connector distribution and cross-section. Although partial interaction theory was developed over fifty years ago it has not been considered in the assessment of composite structures until recently. The complexity of the
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equations required to model partial interaction behaviour, even for simply supported beams, is one of the reasons why partial interaction theory has not been used in the past [18]. As is demonstrated in this paper, the complexity of the equations increase with the number of spans however, simplified models are proposed that enhances the practical applicability of this approach. A mathematical model of the partial interaction curvature for symmetrical twospan continuous beams subjected to the traversal of a single concentrated load is presented. As this is a linear elastic analysis, the principal of superposition can be used to predict the response for any combination of applied (axle) loads. A simplified model to predict the partial interaction curvature applicable to any continuous beam is then developed so that when used in conjunction with the focal points [14], the partial interaction strain distribution is determined. The theoretically derived models are validated at all stages with a finite element program using linkage elements to model the shear connection. The program has been used in previous research on composite beams [14,15,19] and is similar in structure to other computer based analysis techniques referred to in this paper. Finally, application of the technique in an assessment is demonstrated. 2. Linear elastic partial interaction theory The widely used linear elastic partial interaction theory originally developed by Newmark et al. [1] has been adopted. For a given total vertical shear V at a design point x measured from the left support of a composite beam, the following expression can be developed for slip [18] s ¼ K1 sinhðaxÞ þ K2 coshðaxÞ þ bV
ð1Þ
where K1 and K2 are integration constants, and the parameters a and b are given by [18] a2 ¼ b¼
l pEs Io A0
dc pA0 k
ð2Þ ð3Þ
where 1 Io ¼ dc2 þ A0 Ao Ic Io ¼ Is þ n 1 n 1 ¼ þ Ao Ac As
ð4Þ ð5Þ ð6Þ
and dc is the distance between the centroid of the steel beam and the centroid of the concrete deck as shown in Fig. 1, k is the stiffness of a shear connector, p is the longitudinal spacing of the shear connectors, I is the second moment of area, A is
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Fig. 1. Partial interaction strain distribution.
the cross sectional area, and n is the modular ratio given by Es/Ec where E is the modulus of elasticity, and the subscripts s and c refer to the steel and concrete components, respectively. 3. Partial interaction curvature in symmetrical two-span continuous composite beams The partial interaction curvature /pi is determined using a magnification factor MF [14] applied to the full interaction curvature /fi for the same moment at a section and is defined as MF ¼
/pi /fi
ð7Þ
The curvature along the beam is obtained by integrating the following equation [18] with respect to x d/ ðks=pÞdc V ¼ dx E s Io
ð8Þ
For a symmetrical two-span continuous composite beam subjected to a concentrated load P, the distribution of s in Eq. (1) is defined by three equations, one for each shear span. As shown in Fig. 2, l is the length of a shear span and subscripts 1, 2 and 3 refer to the left, middle and right shear spans, respectively, L is the total length of the continuous beam, Ls is the length of a span and V is the support reaction. The three equations required to model the distribution of slip along the length of a symmetrical two-span continuous composite beam for a load
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Fig. 2. Symmetrical two-span continuous beam.
P located at a distance l1 from the left support are given by 1 aL s1 ¼ b Pcoshðal1 Þ Psinhðal1 ÞcothðaLÞ V2 sech coshðaxÞ bV1 2 2 s2 ¼ b P sinh ðal1 Þ sinh ðaxÞ bcoshðaxÞ 1 aL bðV1 PÞ Psinh ðal1 ÞcothðaLÞ þ V2 sech 2 2
ð9Þ
ð10Þ
tanhðaLÞsinh ðaxÞ þ coshðaxÞ aL b Psinh ðal1 Þ 1 coth ðaLÞtanh aL 2 tanhðaLÞ þ tanh 2 1 aL aL V2 sech ð11Þ tanh bðV1 P þ V2 Þ 2 2 2
s3 ¼
where s1, s2 and s3 give the slip distribution within shear spans 1, 2 and 3, respectively and P, V1, V2 and V3 are all taken as positive. Because the continuous beam is symmetrical, the results of this model can be mirrored about the internal support to obtain the slip distribution when the load is located on the right span. The derivation of Eqs. (9)–(11) can be found elsewhere [16]. Substituting Eqs. (9)–(11) into Eq. (8) and integrating gives the following three equations defining the curvature distribution dc2 A0 P 1 V2 aL coshðal1 Þ sinhðal1 ÞcothðaLÞ sech sinhðaxÞ V1 x /1 ¼ 2 P 2 Es Io a V1 x þ þ C1 ð12Þ Es Io
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where the integration constant C 1 ¼ 0, for the boundary condition /1 ¼ 0 at x ¼ 0 dc2 A0 P PsinhðaxÞ sinhðal1 ÞcoshðaxÞ /2 ¼ sinhðal1 ÞcothðaLÞ a Es Io a 1 V2 aL ðV1 PÞx þ sech þ C2 ðV1 PÞx þ 2 P 2 Es Io
ð13Þ
where the integration constant C2 is given by the following for the boundary condition /1 ¼ /2 at x ¼ l 1 C2 ¼
Pl1 1 dc2 A0 Es Io
ð14Þ
and finally, 8 > > dc2 A0 < P ½tanhðaLÞcoshðaxÞ þ sinhðaxÞ
/3 ¼ aL a Es Io > > : tanh ðaLÞ þ tanh 2 aL 1 V2 aL aL sinhðal1 Þ 1 cothðaLÞtanh sech tanh 2 2 P 2 2 ) V3 x V3 x þ þ C3 ð15Þ Es Io where the integration constant C3 is given by the following for the boundary condition /3 ¼ 0 at x ¼ L C3 ¼
V3 L 2 0 d A 1 Es Io c
ð16Þ
The variation of curvature along the length of a symmetrical two-span 50.4 m beam when a 320 kN load is at the mid-span of the left span (l 1 ¼ 12:6 m) is shown in Fig. 3. The cross-sectional geometry of the composite beam and shear connection stiffness used in this example is such that a ¼ 0:183 103 mm1 , dc ¼ 1410 mm and ð1=A0 Þ ¼ 2:8 106 mm2 . As expected for this analysis, there is excellent agreement between the theoretical distribution (Eqs. (12), (13) and (15)) and that of the finite element computer simulation. When a load moves across the left span, the peak positive (sagging) moment, or curvature, for a design point on the left span (0 x < 0:5L) occurs when the load is at the design point such that l 1 ¼ x. Under this load condition, the right span is subjected to negative (hogging) moment, or curvature. When the design point is located on the right span (0:5L x L), the location of the load l1 on the left span that results in the peak negative moment at x can be
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Fig. 3. Partial interaction curvature.
determined by differentiating Eq. (15) with respect to l1 and equating to zero giving 8 > > dc2 A0 < P½tanh ða LÞcosh ðaxÞ þ sinhðaxÞ
0¼ aL E s Io > > : a tanh ðaLÞ þ tanh 2 aL 1 V2 0 aL aL coshðal1 Þ 1 cot hðaLÞtanh sech tanh 2 2 P 2 2 ) V3 0 x V3 0 L 2 0 V3 0 x þ þ dc A 1 ð17Þ Es Io Es Io Because l1 cannot be expressed explicitly, an iterative solution technique in a spreadsheet analysis could be used. As the peak positive curvature is determined by substituting l 1 ¼ x into Eq. (12) and the peak negative curvature is determined by substituting l1 from Eq. (17) into Eq. (15), the magnification factors for these two cases can be quantified. The full interaction curvatures (/1)fi and (/2)fi for 0 x < 0:5L are given by ð/1 Þfi ¼ ð/2 Þfi ¼
xV1 1 dc2 A0 Es Io
ð18Þ
where the reactions V1 and V2 are determined from equilibrium and can be expressed as ðV 1 Þi ¼ P
2:5Pðl1 Þi 2Pðl1 Þ3i þ L L3
ð19Þ
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and ðV2 Þi ¼
3Pðl1 Þi 4Pðl1 Þ3i L L3
ð20Þ
where the subscript i refers to the shear span the design point is located in. As the peak positive curvature occurs when l 1 ¼ x for 0 x < 0:5L, V1 in Eq. (18) is given by Eq. (19) in the following form V1 ¼ P
2:5Px 2Px3 þ 3 L L
ð21Þ
The full interaction curvature (/3)fi for 0:5L x L is given by ð/3 Þfi ¼
V3 1 dc2 A0 ðx LÞ E s Io
ð22Þ
The location of the load to give the peak full interaction negative curvature for 0:5 L x < L can be determined by differentiating Eq. (22) with respect to l1 and equating to zero to give dð/3 Þfi dl1
¼
V3 0 1 dc2 A0 ðx LÞ ¼ 0 Es Io
ð23Þ
where l1 is obtained by solving V 3 0 ¼ 0. From vertical equilibrium, V 3 ¼ V 1 þ V 2 P and by differentiating with respect to l1 0
V3 0 ¼ ðV1 þ V2 PÞ ¼ V1 0 þ V2 0 ¼ 0
ð24Þ
The derivatives of V1 and V2 are given by dðV1 Þi 2:5P 6Pðl1 Þ2i þ ¼ ¼ ðV 1 Þi 0 L dðl1 Þi L3
ð25Þ
dðV2 Þi 3P 12Pðl1 Þ2i ¼ ¼ ðV2 Þi 0 L dðl1 Þi L3
ð26Þ
and
pffiffiffiffiffi so that finally, by substituting Eqs. (25) and (26) into Eq. (24) gives l1 ¼ L= 12. The magnification factors must be derived for both positive and negative curvatures because each design point in the two-span beam is subjected to both hogging and sagging moments as a load traverses the beam. The magnification factor for the positive moment region is determined by dividing Eq. (12) with Eq. (18) giving dc2 A0 P V2 aL þ coshðal1 Þ sinhðal1 ÞcothðaLÞ sech MF ¼ 2 0 2 2P 1 dc A xV1 a sinhðaxÞ þ 1 ð27Þ
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The magnification factor for the negative pffiffiffiffiffi moment region is determined by dividing Eq. (15) with Eq. (22) where l1 ¼ L= 12 to give 8 > > < ½tanhðaLÞcoshðaxÞ þ sinhðaxÞ
10dc2 A0 MF ¼ aL ðx LÞ 1 dc2 A0 > > : a tanhðaLÞ þ tanh 2 aL sinhðal1 Þ 1 cothðaLÞtanh 2 ) V2 aL aL 10V3 þ sech ð28Þ tanh 2 2 2P P Figs. 4 and 5 show the distribution of MF + and MF respectively, along with the results of the computer simulation for the same two-span 50.4 m beam described in Fig. 3. It can be observed in Fig. 4 that MF+ near the mid-spans, where the positive moments are largest, the theoretical model and computer simulation are in good agreement. The results diverge near the supports with the theoretical model predicting conservative increases of /pi. However, as the positive moment near the supports is relatively small, the apparent overly conservative discrepancy is not an issue. Conversely, the negative moment is maximum at the internal support and it can be seen in Fig. 5 that the theoretical distribution of MF is in good agreement with the computer simulation. Figs. 4 and 5 demonstrate that the partial interaction curvature can be substantially larger than the full interaction curvature because MF is always greater than unity. This highlights the importance of considering the partial interaction stresses in an assessment of continuous composite beams.
Fig. 4. MF+ distribution.
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Fig. 5. MF distribution.
Unfortunately, an iterative solution procedure is required to solve for l1 limiting the application of this approach in practice. Hence, the following section derives a simplified model that is able to predict the distributions with reasonable accuracy. 4. Simplified mathematical model Simplified models for the MF distributions are developed by defining the factor at key design points along a two-span beam. The distribution between these points is modelled with straight-line segments. 4.1. Simplified MF+ model At the mid-span where the increase in the positive curvature is most critical, the peak positive full interaction curvature occurs when the load is acting at the midspan. Therefore, x ¼ ðl 1 Þ1 ¼ L=4 and by substituting Eq. (19) into Eq. (18) the following full interaction curvature is obtained at the mid-span LP ð29Þ ð/1 Þfi ¼ 1 dc2 A0 10Es Io Similarly, the peak positive partial interaction curvature at the same design point is determined by taking x ¼ ðl 1 Þ1 ¼ L=4 and substituting Eqs. (19) and (20) into Eq. (12) giving 3 2 aL 0:34sinh 2 0 6 LP d A P aL aL 4 7 6e 4 sinh 7 ð/1 Þpi ¼ 1 dc2 A0 þ c ð30Þ 4 5 aL 10Es Io E s Io a 4 cosh 2
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For typical values of aL, the term in the square brackets of Eq. (30) is approximately 0.5 and hence, Eq. (30) can be simplified to ð/1 Þpi ¼
d 2 A0 P LP 1 dc2 A0 þ c 10Es Io 2Es Io a
ð31Þ
By dividing Eq. (31) with Eq. (29), the positive curvature magnification factor at mid-span is simply given by 5 dc2 A0 5 dc2 A0 þ ¼1þ MFM ¼ 1 þ ð32Þ aL 1 dc2 A0 2aLS 1 dc2 A0 which is expressed in terms of span length Ls to cater for continuous beams with varying span lengths and number of spans as demonstrated later. For simplicity, the following magnification factor originally derived for simply supported beams [14] can also be used to predict the distribution at the supports of continuous beams with reasonable accuracy MFSþ ¼
1 1 dc2 A0
ð33Þ
The simplified model for the positive curvature magnification factor distribution is validated by superimposing the predictions of Eqs. (32) and (33) on the results of þ the two-span 50.4 m beam shown in Fig. 4. At the mid-spans, MFM ¼ 2:3 using Ls ¼ 25:2 m which is in very good agreement with the more rigorous analysis methods. At the supports, MFSþ ¼ 3:4 which is also in good agreement. As shown in Fig. 4, the factors defined by Eqs. (32) and (33) are connected by straight-line segments to model the distribution elsewhere along the beam. 4.2. Simplified MF model The location of the load to give the peak negative full interaction curvature at the internal support is obtained by taking x ¼ L=2 and substituting Eqs. (19) and (20) into Eq. (22), differentiating with respect to (l1)3 and equating to zero to give pffiffiffiffiffi pffiffiffiffiffi ðl1 Þ3 ¼ L= 12. Therefore, by substituting x ¼ L=2 and ðl1 Þ3 ¼ L= 12 into Eq. (22) gives the following at the internal support ð/3 Þfi ¼
0:048PL 2 0 dc A 1 E s Io
ð34Þ
Similarly, the location of the load to give the peak negative partial interaction curvature at the internal support, is determined by taking x ¼ L=2, differentiating Eq. (13) with respect to (l1)2 and equating to zero such that 3Pdc2 A0 P 2 0 3P aL dc A 1 þ ðl1 Þ22 2 1 dc2 A0 dc2 A0 Pe 2 cosh aðl1 Þ2 þ 4 L 2aL 2 0 6Pdc A þ ¼0 ð35Þ aL3
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Because (l1)2 cannot be solved for explicitly, an iterative solution technique would again be required. However, as cosh aðl1 Þ2 a2 ðl1 Þ22 , Eq. (35) can be rewritten in the following explicit form sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 1:5dc2 A0 0:25 dc2 A0 1 aL ðl1 Þ2 L aL dc2 A0 e 2 ðaLÞ3 þ3 1 dc2 A0 aL þ 6dc2 A0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3dc2 A0 dc2 A0 1 aLS ð36Þ ¼ LS 4dc2 A0 eaLS ðaLS Þ3 þ3 1 dc2 A0 aLS þ 3dc2 A0 At the design point x ¼ L=2, substituting Eqs. (19), (20) and (36) into Eq. (13) gives the following expression for the peak negative partial interaction curvature at the internal support 2 0 dc A 1 P 5ðl1 Þ2 ðl1 Þ32 2 ð/2 Þpi ¼ ðl1 Þ2 þ Es Io 4 L 2 0 1:04 dc A P 3ðl1 Þ2 2ðl1 Þ32 aL þ þ sinhðaðl1 Þ2 Þe 2 ð37Þ aEs Io 2L L3 By dividing Eq. (37) with Eq. (34), the negative curvature magnification factor at the internal support is given by 5:2ðl1 Þ2 20:8ðl1 Þ32 L L3 0 20:8 dc2 A 2ðl Þ3 3ðl Þ aL sinhðaðl1 Þ2 Þe 2 1 2 þ 13 2 þ 2 0 2L L aL dc A 1 2:6ðl1 Þ2 2:6ðl1 Þ32 ¼ 3 LS 2 L0 S 10:4 dc A 3ðl1 Þ2 ðl1 Þ32 aL S sinhðaðl1 Þ2 Þe þ þ 4LS 4L3S aLS dc2 A0 1
¼ MFInt
ð38Þ
where Eqs. (36) and (38) are also expressed in terms of Ls for convenience. To simplify the calculation, the hyperbolic sine term in the square brackets of Eq. (38) can be conservatively ignored. As the negative moment approaches zero towards the external supports, the magnification factor of the negative curvature at external supports can simply be taken as ¼ 1:0 MFExt
ð39Þ
The simplified model for the negative curvature magnification factor distribution is validated by superimposing the predictions of Eqs. (38) and (39) on the results of the two-span 50.4 m beam shown in Fig. 5. In this example ðl 1 Þ2 ¼ 15:7 m and ¼ 2:75 using Ls ¼ 25:2 m and it can be seen that the agreement with the MFInt other analysis methods is again very good.
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The simplification of the symmetrical two-span continuous composite beam model has resulted in a much more convenient analysis tool. The application of the simplified model for various realistic continuous beam conditions is discussed in the following section. 4.3. Application of the simplified MF model Although the simplified model was developed for symmetrical two-span beams, it can safely be used to analyse the partial interaction response of continuous beams with varying span lengths, number of spans, connector distribution and cross-section. For varying connector distributions or cross-section, a conservative approach is to use the geometric and material properties in the vicinity of the magnification factor that give the maximum MF. For structures with varying span þ lengths Ls, MFM is calculated using the length of the span in question and MFInt is based on the combination of span length and parameters of the adjacent span that gives the maximum value. The remaining factors, MFSþ and MFExt , are independent of Ls. Figs. 6 and 7 illustrate the distribution of MF + and MF respectively for a three-span beam. The points defining the simplified distribution are calculated as before using Ls ¼ 16:8 m. The only difference not already discussed is the distribution of MF for internal spans. The minimum value of MF in an internal span is defined by the intersection of the distributions for overlapping two-span models as illustrated in Fig. 7. The maximum MF at a design point within an internal span defines the distribution as shown by the thick dashed line. In an assessment using a fatigue vehicle with several axle loads, the principal of superposition can be used as this is a linear elastic analysis. But even though this is
Fig. 6. MF+ distribution for a three-span beam.
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Fig. 7. MF distribution for a three-span continuous beam.
an elastic analysis, the non-linear shear load-slip response of the connectors can be allowed for by using an appropriate secant stiffness and cracking or long term effects of the concrete can be allowed for by defining an appropriate flexural rigidity [10]. As the stress range is required in a fatigue assessment, the partial interaction strain distribution at a design point can be determined from the maximum curvature found using the simplified model described in this paper coupled with the focal points shown in Fig. 1. The focal points define two points on the strain distribution that remain fixed for a given moment at a section regardless of the stiffness of the shear connection and are given by the intersection points of the full interaction and no interaction bounds as shown in Fig. 1. Because the focal points have been derived elsewhere [14], only the equations defining the points are reproduced for completeness of this paper. The focal point in the concrete component is given by ðEI Þ yc P n yn þ EI ðyc Þfp ¼ ðEI Þn P 1 EI
ð40Þ
and
ðec Þfp ¼
i M h ðyc Þfp yn ðEI Þn
ð41Þ
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and similarly, the focal point in the steel component is given by ðEI Þ ys P n yn EI ðys Þfp ¼ ðEI Þ 1P n EI
ð42Þ
and ðes Þfp ¼
i M h ðys Þfp yn ðEI Þn
ð43Þ
where (yc)fp and (ys)fp are the distances of the focal points measured from the steelconcrete interface, (ec)fp and (es)fp are the corresponding strains at the focal points, yn , yc and ys are the location of the centroids of the transformed composite section, the concrete component and the steel component, respectively, from the steelconcrete interface, (EI)n is the flexural rigidity of the fully composite transformed section, REI is the flexural rigidity of the section assuming no interaction, that is (Ec Ic þ Es Is ), and M is the applied bending moment at the design point. The bending moment distribution determined from a standard structural analysis procedure assuming full interaction can be used in this analysis [11]. Eqs. (40) and (42) define the location where the full interaction and no interaction strain distributions intersect with respect to the steel-concrete interface, and Eqs. (41) and (43) are simply the strains at these points using the full interaction curvature, as illustrated in Fig. 1.
5. Illustrative fatigue assessment Suppose a two-span 50.4 m long composite beam (Ls ¼ 25:2 m) has reached the end of its anticipated design life. Hence, an assessment is undertaken to determine whether the remaining strength or endurance is sufficient to extend the life of the structure or remedial measures are necessary. The cross-section is such that ys ¼ 1290 mm and yc ¼ 125 mm so that dc ¼ 1415 mm, yn ¼ 474 mm (in the steel component), ð1=A’Þ ¼ 2:8 106 mm2 , ðEIÞn ¼ 2:64 1016 Nmm2 and REI ¼ 7:6 1015 Nmm2 . A uniform distribution of shear connectors was used such that a ¼ 0:409 103 mm1 and the standard fatigue loading is such that the maximum positive moment at the mid-span is 1651 kNm. From Eq. (42), ðys Þfp ¼ 1620 mm measured from the steel-concrete interface. The full interaction curvature is 6:25 108 mm1 so that the full interaction strains at the top and bottom of the 1750 mm deep steel section are 30 le and 80 le respectively where compressive strains are negative. Using Eq. (43) the strain þ at the focal point in the steel component is ðes Þfp ¼ 72 le. From Eq. (32), MFM ¼ 7 1 1:61 and therefore the partial interaction curvature is 1:01 10 mm . Finally, using (ys)fp and (es)fp as the reference point on the partial interaction strain
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distribution, the partial interaction strains at the top and bottom of the steel section are 92 le and 85 le respectively. Therefore, the maximum tensile stress at the bottom of the steel section has increased by 6%, which does not appear to be significant. However, taking the exponent of the fatigue equation for steel sections as 3, the endurance is reduced by a factor of ð1:06Þ3 ¼ 0:83 compared with that predicted by a standard full interaction analysis. The magnitude of the stress difference is larger near the steel-concrete interface because the focal points are located towards the top and bottom of the composite section. Consequently, it can be seen in this example that the maximum compressive stress in the top flange increases three fold. Even with this significant increase in compressive stress, buckling of the top flange should not be a concern as it is restrained by the shear connectors and the concrete. To complete the assessment at the mid-span, this procedure must be repeated for the maximum negative moment at the design point. In negative moment regions, the focal points are located the same distance from the steel–concrete interface so, the situation is similar to that described in the positive moment region. The obvious difference is that the sense of the stress changes from tension to compression and vice versa. Hence, the compression stresses increase in the bottom flange in the negative moment region, which may be unrestrained and therefore buckling may be a concern. Furthermore, the tensile stresses in the steel at the interface in negative moment regions are increased more than in the positive moment region, which will have a significant impact on the residual strength or endurance. A benefit of partial interaction observed in negative moment regions is that the tensile strains in the concrete reduce, limiting the expected extent of concrete cracking. As in any assessment, it is important to perform a thorough site investigation to visually determine the extent of any fatigue damage. Information on the extent of concrete cracking is important to this analysis as it enables the use of an appropriate flexural rigidity for the concrete component. If the partial interaction assessment demonstrates that the remaining endurance or strength of the composite section is not sufficient to safely extend the fatigue life of the structure, a load test could be performed. The load test will allow for the other factors that contribute to the strength and stiffness of the structure that cannot be considered easily in such a theoretical approach. Furthermore, a realistic representation of the actual vehicle weights and frequencies throughout the life of the bridge should be used to improve the accuracy of the assessment. If the structure requires strengthening or stiffening to extend its fatigue life, the assessment approach outlined in this section can also be used to determine the effectiveness of remedial measures.
6. Conclusions A simple mathematical model based on linear elastic partial interaction theory has been developed to more accurately predict the stresses in the steel and concrete components of continuous composite beams under serviceability loads. When used with recently published research on the assessment of the shear connection in
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continuous composite beams, the tiered fatigue assessment approach previously restricted to simply supported beams, is completed. Even though this is a linear elastic analysis, the non-linear shear load-slip behaviour of stud shear connectors can be modelled using an appropriate secant stiffness, and long term effects and cracking of the concrete can be included by using an appropriate flexural rigidity. The approach presented is applicable to continuous beams with any number of spans, span lengths, connector distribution and cross-section. Furthermore, when used with an appropriate fatigue damage equation it can also be used to quantify the effectiveness of remedial measures. Following a thorough site investigation, it is suggested to use this procedure to decide whether load testing is required. Therefore load tests, which are disruptive, costly and time consuming, need only be considered in the later stages of an assessment if it shown theoretically, allowing for partial interaction, that the residual strength or endurance is inadequate. Load tests will establish the extent of unintentional strengthening and stiffening by other factors not easily considered theoretically but contribute to the residual strength or endurance of composite bridges.
References [1] Newmark NM, Siess CP, Viest IM. Tests and analysis of composite beams with incomplete interaction. Proceedings Society for Experimental Stress Analysis 1951;9(1):75–92. [2] Slutter RG, Driscoll GC. Flexural strength of steel-concrete composite beams. Journal of the Structural Division, ASCE 1965;91(ST2):71–99. [3] Yam LCP, Chapman JC. The inelastic behaviour of continuous composite beams of steel and concrete. Proceedings of the Institution of Civil Engineers 1972;2(53):487–501. [4] Plum DR, Horne MR. The analysis of continuous composite beams with partial interaction. Proceedings of the Institution of Civil Engineers 1975;2(59):625–43. [5] Hope-Gill MC, Johnson RP. Tests on three three-span continuous composite beams. Proceedings of the Institution of Civil Engineers 1976;2(61):367–81. [6] Hamada S, Longworth J. Ultimate strength of continuous composite beams. Journal of the Structural Division, ASCE 1976;102(ST7):1463–78. [7] Ansourian P. Experiments on continuous composite beams. Proceedings of the Institution of Civil Engineers 1981;2(71):25–51. [8] Oven VA, Burgess IW, Plank RJ, Abdul Wali AA. An analytical model for the analysis of composite beams with partial interaction. Computers and Structures 1997;62(3):493–504. [9] Fabbrocino G, Manfredi G, Cosenza E. Analysis of continuous composite beams including partial interaction and bond. Journal of Structural Engineering, ASCE 2000;126(11):1288–94. [10] Faella C, Martinelli E, Nigro E. Steel and concrete composite beams with flexible shear connection: ‘‘exact’’ analytical expression of the stiffness matrix and applications. Computers and Structures 2002;80:1001–9. [11] Jasim NA, Atalla A. Deflections of partially composite continuous beams: A simple approach. Journal of Constructional Steel Research 1999;49:291–301. [12] Faella C, Martinelli E, Nigro E. Shear connection non-linearity and deflections of steel-concrete composite beams: A simplified method. Journal of Structural Engineering, ASCE 2003;192(1):12–20. [13] Nie J, Cai CS. Steel-concrete composite beams considering shear slip effects. Journal of Structural Engineering, ASCE 2003;129(4):495–506. [14] Seracino R, Oehlers DJ, Yeo MF. Partial-interaction flexural stresses in composite steel and concrete bridge beams. Engineering Structures 2001;23:1186–93.
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[15] Seracino R, Oehlers DJ, Yeo MF. Partial-interaction fatigue assessment of stud shear connectors in composite bridge beams. Structural Engineering and Mechanics 2002;13(4):455–64. [16] Seracino R, Lee CT, Tan Z. Partial interaction shear flow forces in continuous composite steel-concrete beams. Journal of Structural Engineering, ASCE. (submitted for publication, April 2003) [17] Oehlers DJ, Seracino R. A tiered approach to the fatigue assessment of composite steel and concrete bridge beams. Proceedings of the Institution of Civil Engineers, Structures and Buildings 2002;152(3):249–57. [18] Johnson RP. Composite structures of steel and concrete vol. 1. Oxford: Blackwell Scientific Publications Ltd; 1994. [19] Oehlers DJ, Seracino R, Yeo MF. Effect of friction on shear connection in composite bridge beams. Journal of Bridge Engineering, ASCE 2000;5(2):91–8.
Partial interaction stresses in continuous composite beams under serviceability loads Rudolf Seracino , Chow T. Lee, Tze C. Lim, Jwo Y. Lim School of Civil and Environmental Engineering, University of Adelaide, Adelaide, SA 5005, Australia Received 21 May 2003; received in revised form 19 December 2003; accepted 9 January 2004
Abstract The number of fatigue assessments of composite bridges is growing rapidly worldwide due to increasing allowable load limits and because many of the bridges are reaching the end of their anticipated design life. Special attention is usually given to predicting the residual endurance or strength of the shear connection because it cannot be visually inspected. However, the increased stresses in the steel and concrete components due to partial interaction must also be considered in a fatigue assessment. Based on linear elastic partial interaction theory, this paper develops a simplified procedure to predict the partial interaction curvature in continuous composite beams. When used in conjunction with focal points, the partial interaction flexural stresses in the steel and concrete components can be determined and used to more accurately predict the residual strength or endurance of the composite section. This research extends the tiered assessment approach previously published for simply supported beams so that it is now applicable to composite beams with any number of spans, span lengths, shear connection distribution and cross-section. The technique is validated using a finite element program developed to model the behaviour of composite structures and the procedure is demonstrated in an illustrative assessment. # 2004 Elsevier Ltd. All rights reserved. Keywords: Assessment; Continuous composite beams; Curvature; Fatigue; Flexural Stresses; Partial interaction; Slip
Corresponding author. Tel.: +61-8-83203-5451; fax: +61-8-8303-4359. E-mail address: [email protected] (R. Seracino).
0143-974X/$ - see front matter # 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jcsr.2004.01.002
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Nomenclature + 1 2 3 A C c dc E Ext fi fp I i Int K k L Ls l M MF n ni P p pi S s V V x y y a b e /
positive (sagging) moment (or curvature) when used as a superscript negative (hogging) moment (or curvature) when used as a superscript left shear span when used as a subscript middle shear span when used as a subscript right shear span when used as a subscript cross-sectional area integration constant concrete when used as a subscript distance between centroids of the steel beam and concrete deck modulus of elasticity external support when used as a subscript full interaction focal point second moment of area shear span containing the design point when used as a subscript internal support when used as a subscript integration constant stiffness of a shear connector total length of a continuous beam length of a span length of a shear span bending moment; mid-span when used as a subscript curvature magnification factor modular ratio; transformed composite section when used as a subscript no interaction concentrated load longitudinal spacing of the shear connectors partial interaction support when used as a subscript slip (relative displacement between the steel and concrete component at the interface); steel when used as a subscript support reaction total vertical shear force at a design point distance of design point from left support distance of a focal point from the steel-concrete interface distance of a centroid from the steel-concrete interface geometric and material parameter geometric and material parameter strain curvature
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1. Introduction Results of theoretical analyses and experimental investigation published by Newmark et al. in 1951 [1] demonstrated that the strains in the steel and concrete components of composite beams are greater than that predicted by standard full interaction analysis. The larger strains are a result of the relative displacement, or slip, between the steel and concrete components at the interface because the shear connectors must deform to resist the longitudinal shear forces. Although most composite beams tested from the 1960s to the early 1980s were simply supported, some tests were carried out on two and three-span continuous composite beams [2–7]. The purpose of most of the tests was to investigate ultimate failure modes and validate the use of the plastic design technique for composite structures. Some of these early tests were coupled with elementary computer analyses, but it was not until the 1990s that advanced non-linear computer techniques were being used to analyse composite structures [8–10]. Much of the current research on continuous composite beams is on developing simplified techniques for predicting the increased deflections allowing for partial interaction [11–13]. One of the reasons for this trend is that the introduction of high strength steels and high performance concrete has allowed longer spans, particularly in bridges, such that the deflection serviceability limit state often governs design. Most research is related to improving the design of new structures however, as many composite bridges are approaching the end of their anticipated design life, coupled with increasing allowable load limits, the assessment of existing structures is rapidly becoming more important worldwide. Research has recently been published [14] presenting a practical procedure to predict the increased strains in the steel and concrete components of simply supported composite beams allowing for partial interaction. Related research included quantifying the reduction in the stress range resisted by the shear connection due to partial interaction in simply supported beams [15], which has recently been extended to continuous beams [16]. As a result of this research, a tiered approach to assess the remaining strength or endurance of existing simply supported composite steel-concrete beams was developed [17]. After careful site inspection to establish the condition of the bridge, the tiered approach can be used to determine whether load testing of the structure is required. If it cannot be demonstrated that the remaining endurance or strength of the structure is adequate, then a load test could be used to quantify the beneficial effect of other factors contributing to the reserve strength, such as the effect of railings or barriers if cast integrally with the concrete deck. Alternatively, advanced non-linear computer analyses can be made as described in the papers referred to in this introduction. Unfortunately, the current tiered assessment approach does not include the analysis of the partial interaction flexural stresses in continuous structures. The research presented in this paper completes the extension of the tiered assessment approach to continuous composite steel-concrete beams of any number of spans with varying span lengths, connector distribution and cross-section. Although partial interaction theory was developed over fifty years ago it has not been considered in the assessment of composite structures until recently. The complexity of the
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equations required to model partial interaction behaviour, even for simply supported beams, is one of the reasons why partial interaction theory has not been used in the past [18]. As is demonstrated in this paper, the complexity of the equations increase with the number of spans however, simplified models are proposed that enhances the practical applicability of this approach. A mathematical model of the partial interaction curvature for symmetrical twospan continuous beams subjected to the traversal of a single concentrated load is presented. As this is a linear elastic analysis, the principal of superposition can be used to predict the response for any combination of applied (axle) loads. A simplified model to predict the partial interaction curvature applicable to any continuous beam is then developed so that when used in conjunction with the focal points [14], the partial interaction strain distribution is determined. The theoretically derived models are validated at all stages with a finite element program using linkage elements to model the shear connection. The program has been used in previous research on composite beams [14,15,19] and is similar in structure to other computer based analysis techniques referred to in this paper. Finally, application of the technique in an assessment is demonstrated. 2. Linear elastic partial interaction theory The widely used linear elastic partial interaction theory originally developed by Newmark et al. [1] has been adopted. For a given total vertical shear V at a design point x measured from the left support of a composite beam, the following expression can be developed for slip [18] s ¼ K1 sinhðaxÞ þ K2 coshðaxÞ þ bV
ð1Þ
where K1 and K2 are integration constants, and the parameters a and b are given by [18] a2 ¼ b¼
l pEs Io A0
dc pA0 k
ð2Þ ð3Þ
where 1 Io ¼ dc2 þ A0 Ao Ic Io ¼ Is þ n 1 n 1 ¼ þ Ao Ac As
ð4Þ ð5Þ ð6Þ
and dc is the distance between the centroid of the steel beam and the centroid of the concrete deck as shown in Fig. 1, k is the stiffness of a shear connector, p is the longitudinal spacing of the shear connectors, I is the second moment of area, A is
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Fig. 1. Partial interaction strain distribution.
the cross sectional area, and n is the modular ratio given by Es/Ec where E is the modulus of elasticity, and the subscripts s and c refer to the steel and concrete components, respectively. 3. Partial interaction curvature in symmetrical two-span continuous composite beams The partial interaction curvature /pi is determined using a magnification factor MF [14] applied to the full interaction curvature /fi for the same moment at a section and is defined as MF ¼
/pi /fi
ð7Þ
The curvature along the beam is obtained by integrating the following equation [18] with respect to x d/ ðks=pÞdc V ¼ dx E s Io
ð8Þ
For a symmetrical two-span continuous composite beam subjected to a concentrated load P, the distribution of s in Eq. (1) is defined by three equations, one for each shear span. As shown in Fig. 2, l is the length of a shear span and subscripts 1, 2 and 3 refer to the left, middle and right shear spans, respectively, L is the total length of the continuous beam, Ls is the length of a span and V is the support reaction. The three equations required to model the distribution of slip along the length of a symmetrical two-span continuous composite beam for a load
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Fig. 2. Symmetrical two-span continuous beam.
P located at a distance l1 from the left support are given by 1 aL s1 ¼ b Pcoshðal1 Þ Psinhðal1 ÞcothðaLÞ V2 sech coshðaxÞ bV1 2 2 s2 ¼ b P sinh ðal1 Þ sinh ðaxÞ bcoshðaxÞ 1 aL bðV1 PÞ Psinh ðal1 ÞcothðaLÞ þ V2 sech 2 2
ð9Þ
ð10Þ
tanhðaLÞsinh ðaxÞ þ coshðaxÞ aL b Psinh ðal1 Þ 1 coth ðaLÞtanh aL 2 tanhðaLÞ þ tanh 2 1 aL aL V2 sech ð11Þ tanh bðV1 P þ V2 Þ 2 2 2
s3 ¼
where s1, s2 and s3 give the slip distribution within shear spans 1, 2 and 3, respectively and P, V1, V2 and V3 are all taken as positive. Because the continuous beam is symmetrical, the results of this model can be mirrored about the internal support to obtain the slip distribution when the load is located on the right span. The derivation of Eqs. (9)–(11) can be found elsewhere [16]. Substituting Eqs. (9)–(11) into Eq. (8) and integrating gives the following three equations defining the curvature distribution dc2 A0 P 1 V2 aL coshðal1 Þ sinhðal1 ÞcothðaLÞ sech sinhðaxÞ V1 x /1 ¼ 2 P 2 Es Io a V1 x þ þ C1 ð12Þ Es Io
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where the integration constant C 1 ¼ 0, for the boundary condition /1 ¼ 0 at x ¼ 0 dc2 A0 P PsinhðaxÞ sinhðal1 ÞcoshðaxÞ /2 ¼ sinhðal1 ÞcothðaLÞ a Es Io a 1 V2 aL ðV1 PÞx þ sech þ C2 ðV1 PÞx þ 2 P 2 Es Io
ð13Þ
where the integration constant C2 is given by the following for the boundary condition /1 ¼ /2 at x ¼ l 1 C2 ¼
Pl1 1 dc2 A0 Es Io
ð14Þ
and finally, 8 > > dc2 A0 < P ½tanhðaLÞcoshðaxÞ þ sinhðaxÞ
/3 ¼ aL a Es Io > > : tanh ðaLÞ þ tanh 2 aL 1 V2 aL aL sinhðal1 Þ 1 cothðaLÞtanh sech tanh 2 2 P 2 2 ) V3 x V3 x þ þ C3 ð15Þ Es Io where the integration constant C3 is given by the following for the boundary condition /3 ¼ 0 at x ¼ L C3 ¼
V3 L 2 0 d A 1 Es Io c
ð16Þ
The variation of curvature along the length of a symmetrical two-span 50.4 m beam when a 320 kN load is at the mid-span of the left span (l 1 ¼ 12:6 m) is shown in Fig. 3. The cross-sectional geometry of the composite beam and shear connection stiffness used in this example is such that a ¼ 0:183 103 mm1 , dc ¼ 1410 mm and ð1=A0 Þ ¼ 2:8 106 mm2 . As expected for this analysis, there is excellent agreement between the theoretical distribution (Eqs. (12), (13) and (15)) and that of the finite element computer simulation. When a load moves across the left span, the peak positive (sagging) moment, or curvature, for a design point on the left span (0 x < 0:5L) occurs when the load is at the design point such that l 1 ¼ x. Under this load condition, the right span is subjected to negative (hogging) moment, or curvature. When the design point is located on the right span (0:5L x L), the location of the load l1 on the left span that results in the peak negative moment at x can be
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Fig. 3. Partial interaction curvature.
determined by differentiating Eq. (15) with respect to l1 and equating to zero giving 8 > > dc2 A0 < P½tanh ða LÞcosh ðaxÞ þ sinhðaxÞ
0¼ aL E s Io > > : a tanh ðaLÞ þ tanh 2 aL 1 V2 0 aL aL coshðal1 Þ 1 cot hðaLÞtanh sech tanh 2 2 P 2 2 ) V3 0 x V3 0 L 2 0 V3 0 x þ þ dc A 1 ð17Þ Es Io Es Io Because l1 cannot be expressed explicitly, an iterative solution technique in a spreadsheet analysis could be used. As the peak positive curvature is determined by substituting l 1 ¼ x into Eq. (12) and the peak negative curvature is determined by substituting l1 from Eq. (17) into Eq. (15), the magnification factors for these two cases can be quantified. The full interaction curvatures (/1)fi and (/2)fi for 0 x < 0:5L are given by ð/1 Þfi ¼ ð/2 Þfi ¼
xV1 1 dc2 A0 Es Io
ð18Þ
where the reactions V1 and V2 are determined from equilibrium and can be expressed as ðV 1 Þi ¼ P
2:5Pðl1 Þi 2Pðl1 Þ3i þ L L3
ð19Þ
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and ðV2 Þi ¼
3Pðl1 Þi 4Pðl1 Þ3i L L3
ð20Þ
where the subscript i refers to the shear span the design point is located in. As the peak positive curvature occurs when l 1 ¼ x for 0 x < 0:5L, V1 in Eq. (18) is given by Eq. (19) in the following form V1 ¼ P
2:5Px 2Px3 þ 3 L L
ð21Þ
The full interaction curvature (/3)fi for 0:5L x L is given by ð/3 Þfi ¼
V3 1 dc2 A0 ðx LÞ E s Io
ð22Þ
The location of the load to give the peak full interaction negative curvature for 0:5 L x < L can be determined by differentiating Eq. (22) with respect to l1 and equating to zero to give dð/3 Þfi dl1
¼
V3 0 1 dc2 A0 ðx LÞ ¼ 0 Es Io
ð23Þ
where l1 is obtained by solving V 3 0 ¼ 0. From vertical equilibrium, V 3 ¼ V 1 þ V 2 P and by differentiating with respect to l1 0
V3 0 ¼ ðV1 þ V2 PÞ ¼ V1 0 þ V2 0 ¼ 0
ð24Þ
The derivatives of V1 and V2 are given by dðV1 Þi 2:5P 6Pðl1 Þ2i þ ¼ ¼ ðV 1 Þi 0 L dðl1 Þi L3
ð25Þ
dðV2 Þi 3P 12Pðl1 Þ2i ¼ ¼ ðV2 Þi 0 L dðl1 Þi L3
ð26Þ
and
pffiffiffiffiffi so that finally, by substituting Eqs. (25) and (26) into Eq. (24) gives l1 ¼ L= 12. The magnification factors must be derived for both positive and negative curvatures because each design point in the two-span beam is subjected to both hogging and sagging moments as a load traverses the beam. The magnification factor for the positive moment region is determined by dividing Eq. (12) with Eq. (18) giving dc2 A0 P V2 aL þ coshðal1 Þ sinhðal1 ÞcothðaLÞ sech MF ¼ 2 0 2 2P 1 dc A xV1 a sinhðaxÞ þ 1 ð27Þ
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The magnification factor for the negative pffiffiffiffiffi moment region is determined by dividing Eq. (15) with Eq. (22) where l1 ¼ L= 12 to give 8 > > < ½tanhðaLÞcoshðaxÞ þ sinhðaxÞ
10dc2 A0 MF ¼ aL ðx LÞ 1 dc2 A0 > > : a tanhðaLÞ þ tanh 2 aL sinhðal1 Þ 1 cothðaLÞtanh 2 ) V2 aL aL 10V3 þ sech ð28Þ tanh 2 2 2P P Figs. 4 and 5 show the distribution of MF + and MF respectively, along with the results of the computer simulation for the same two-span 50.4 m beam described in Fig. 3. It can be observed in Fig. 4 that MF+ near the mid-spans, where the positive moments are largest, the theoretical model and computer simulation are in good agreement. The results diverge near the supports with the theoretical model predicting conservative increases of /pi. However, as the positive moment near the supports is relatively small, the apparent overly conservative discrepancy is not an issue. Conversely, the negative moment is maximum at the internal support and it can be seen in Fig. 5 that the theoretical distribution of MF is in good agreement with the computer simulation. Figs. 4 and 5 demonstrate that the partial interaction curvature can be substantially larger than the full interaction curvature because MF is always greater than unity. This highlights the importance of considering the partial interaction stresses in an assessment of continuous composite beams.
Fig. 4. MF+ distribution.
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Fig. 5. MF distribution.
Unfortunately, an iterative solution procedure is required to solve for l1 limiting the application of this approach in practice. Hence, the following section derives a simplified model that is able to predict the distributions with reasonable accuracy. 4. Simplified mathematical model Simplified models for the MF distributions are developed by defining the factor at key design points along a two-span beam. The distribution between these points is modelled with straight-line segments. 4.1. Simplified MF+ model At the mid-span where the increase in the positive curvature is most critical, the peak positive full interaction curvature occurs when the load is acting at the midspan. Therefore, x ¼ ðl 1 Þ1 ¼ L=4 and by substituting Eq. (19) into Eq. (18) the following full interaction curvature is obtained at the mid-span LP ð29Þ ð/1 Þfi ¼ 1 dc2 A0 10Es Io Similarly, the peak positive partial interaction curvature at the same design point is determined by taking x ¼ ðl 1 Þ1 ¼ L=4 and substituting Eqs. (19) and (20) into Eq. (12) giving 3 2 aL 0:34sinh 2 0 6 LP d A P aL aL 4 7 6e 4 sinh 7 ð/1 Þpi ¼ 1 dc2 A0 þ c ð30Þ 4 5 aL 10Es Io E s Io a 4 cosh 2
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For typical values of aL, the term in the square brackets of Eq. (30) is approximately 0.5 and hence, Eq. (30) can be simplified to ð/1 Þpi ¼
d 2 A0 P LP 1 dc2 A0 þ c 10Es Io 2Es Io a
ð31Þ
By dividing Eq. (31) with Eq. (29), the positive curvature magnification factor at mid-span is simply given by 5 dc2 A0 5 dc2 A0 þ ¼1þ MFM ¼ 1 þ ð32Þ aL 1 dc2 A0 2aLS 1 dc2 A0 which is expressed in terms of span length Ls to cater for continuous beams with varying span lengths and number of spans as demonstrated later. For simplicity, the following magnification factor originally derived for simply supported beams [14] can also be used to predict the distribution at the supports of continuous beams with reasonable accuracy MFSþ ¼
1 1 dc2 A0
ð33Þ
The simplified model for the positive curvature magnification factor distribution is validated by superimposing the predictions of Eqs. (32) and (33) on the results of þ the two-span 50.4 m beam shown in Fig. 4. At the mid-spans, MFM ¼ 2:3 using Ls ¼ 25:2 m which is in very good agreement with the more rigorous analysis methods. At the supports, MFSþ ¼ 3:4 which is also in good agreement. As shown in Fig. 4, the factors defined by Eqs. (32) and (33) are connected by straight-line segments to model the distribution elsewhere along the beam. 4.2. Simplified MF model The location of the load to give the peak negative full interaction curvature at the internal support is obtained by taking x ¼ L=2 and substituting Eqs. (19) and (20) into Eq. (22), differentiating with respect to (l1)3 and equating to zero to give pffiffiffiffiffi pffiffiffiffiffi ðl1 Þ3 ¼ L= 12. Therefore, by substituting x ¼ L=2 and ðl1 Þ3 ¼ L= 12 into Eq. (22) gives the following at the internal support ð/3 Þfi ¼
0:048PL 2 0 dc A 1 E s Io
ð34Þ
Similarly, the location of the load to give the peak negative partial interaction curvature at the internal support, is determined by taking x ¼ L=2, differentiating Eq. (13) with respect to (l1)2 and equating to zero such that 3Pdc2 A0 P 2 0 3P aL dc A 1 þ ðl1 Þ22 2 1 dc2 A0 dc2 A0 Pe 2 cosh aðl1 Þ2 þ 4 L 2aL 2 0 6Pdc A þ ¼0 ð35Þ aL3
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Because (l1)2 cannot be solved for explicitly, an iterative solution technique would again be required. However, as cosh aðl1 Þ2 a2 ðl1 Þ22 , Eq. (35) can be rewritten in the following explicit form sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 1:5dc2 A0 0:25 dc2 A0 1 aL ðl1 Þ2 L aL dc2 A0 e 2 ðaLÞ3 þ3 1 dc2 A0 aL þ 6dc2 A0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3dc2 A0 dc2 A0 1 aLS ð36Þ ¼ LS 4dc2 A0 eaLS ðaLS Þ3 þ3 1 dc2 A0 aLS þ 3dc2 A0 At the design point x ¼ L=2, substituting Eqs. (19), (20) and (36) into Eq. (13) gives the following expression for the peak negative partial interaction curvature at the internal support 2 0 dc A 1 P 5ðl1 Þ2 ðl1 Þ32 2 ð/2 Þpi ¼ ðl1 Þ2 þ Es Io 4 L 2 0 1:04 dc A P 3ðl1 Þ2 2ðl1 Þ32 aL þ þ sinhðaðl1 Þ2 Þe 2 ð37Þ aEs Io 2L L3 By dividing Eq. (37) with Eq. (34), the negative curvature magnification factor at the internal support is given by 5:2ðl1 Þ2 20:8ðl1 Þ32 L L3 0 20:8 dc2 A 2ðl Þ3 3ðl Þ aL sinhðaðl1 Þ2 Þe 2 1 2 þ 13 2 þ 2 0 2L L aL dc A 1 2:6ðl1 Þ2 2:6ðl1 Þ32 ¼ 3 LS 2 L0 S 10:4 dc A 3ðl1 Þ2 ðl1 Þ32 aL S sinhðaðl1 Þ2 Þe þ þ 4LS 4L3S aLS dc2 A0 1
¼ MFInt
ð38Þ
where Eqs. (36) and (38) are also expressed in terms of Ls for convenience. To simplify the calculation, the hyperbolic sine term in the square brackets of Eq. (38) can be conservatively ignored. As the negative moment approaches zero towards the external supports, the magnification factor of the negative curvature at external supports can simply be taken as ¼ 1:0 MFExt
ð39Þ
The simplified model for the negative curvature magnification factor distribution is validated by superimposing the predictions of Eqs. (38) and (39) on the results of the two-span 50.4 m beam shown in Fig. 5. In this example ðl 1 Þ2 ¼ 15:7 m and ¼ 2:75 using Ls ¼ 25:2 m and it can be seen that the agreement with the MFInt other analysis methods is again very good.
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The simplification of the symmetrical two-span continuous composite beam model has resulted in a much more convenient analysis tool. The application of the simplified model for various realistic continuous beam conditions is discussed in the following section. 4.3. Application of the simplified MF model Although the simplified model was developed for symmetrical two-span beams, it can safely be used to analyse the partial interaction response of continuous beams with varying span lengths, number of spans, connector distribution and cross-section. For varying connector distributions or cross-section, a conservative approach is to use the geometric and material properties in the vicinity of the magnification factor that give the maximum MF. For structures with varying span þ lengths Ls, MFM is calculated using the length of the span in question and MFInt is based on the combination of span length and parameters of the adjacent span that gives the maximum value. The remaining factors, MFSþ and MFExt , are independent of Ls. Figs. 6 and 7 illustrate the distribution of MF + and MF respectively for a three-span beam. The points defining the simplified distribution are calculated as before using Ls ¼ 16:8 m. The only difference not already discussed is the distribution of MF for internal spans. The minimum value of MF in an internal span is defined by the intersection of the distributions for overlapping two-span models as illustrated in Fig. 7. The maximum MF at a design point within an internal span defines the distribution as shown by the thick dashed line. In an assessment using a fatigue vehicle with several axle loads, the principal of superposition can be used as this is a linear elastic analysis. But even though this is
Fig. 6. MF+ distribution for a three-span beam.
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Fig. 7. MF distribution for a three-span continuous beam.
an elastic analysis, the non-linear shear load-slip response of the connectors can be allowed for by using an appropriate secant stiffness and cracking or long term effects of the concrete can be allowed for by defining an appropriate flexural rigidity [10]. As the stress range is required in a fatigue assessment, the partial interaction strain distribution at a design point can be determined from the maximum curvature found using the simplified model described in this paper coupled with the focal points shown in Fig. 1. The focal points define two points on the strain distribution that remain fixed for a given moment at a section regardless of the stiffness of the shear connection and are given by the intersection points of the full interaction and no interaction bounds as shown in Fig. 1. Because the focal points have been derived elsewhere [14], only the equations defining the points are reproduced for completeness of this paper. The focal point in the concrete component is given by ðEI Þ yc P n yn þ EI ðyc Þfp ¼ ðEI Þn P 1 EI
ð40Þ
and
ðec Þfp ¼
i M h ðyc Þfp yn ðEI Þn
ð41Þ
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and similarly, the focal point in the steel component is given by ðEI Þ ys P n yn EI ðys Þfp ¼ ðEI Þ 1P n EI
ð42Þ
and ðes Þfp ¼
i M h ðys Þfp yn ðEI Þn
ð43Þ
where (yc)fp and (ys)fp are the distances of the focal points measured from the steelconcrete interface, (ec)fp and (es)fp are the corresponding strains at the focal points, yn , yc and ys are the location of the centroids of the transformed composite section, the concrete component and the steel component, respectively, from the steelconcrete interface, (EI)n is the flexural rigidity of the fully composite transformed section, REI is the flexural rigidity of the section assuming no interaction, that is (Ec Ic þ Es Is ), and M is the applied bending moment at the design point. The bending moment distribution determined from a standard structural analysis procedure assuming full interaction can be used in this analysis [11]. Eqs. (40) and (42) define the location where the full interaction and no interaction strain distributions intersect with respect to the steel-concrete interface, and Eqs. (41) and (43) are simply the strains at these points using the full interaction curvature, as illustrated in Fig. 1.
5. Illustrative fatigue assessment Suppose a two-span 50.4 m long composite beam (Ls ¼ 25:2 m) has reached the end of its anticipated design life. Hence, an assessment is undertaken to determine whether the remaining strength or endurance is sufficient to extend the life of the structure or remedial measures are necessary. The cross-section is such that ys ¼ 1290 mm and yc ¼ 125 mm so that dc ¼ 1415 mm, yn ¼ 474 mm (in the steel component), ð1=A’Þ ¼ 2:8 106 mm2 , ðEIÞn ¼ 2:64 1016 Nmm2 and REI ¼ 7:6 1015 Nmm2 . A uniform distribution of shear connectors was used such that a ¼ 0:409 103 mm1 and the standard fatigue loading is such that the maximum positive moment at the mid-span is 1651 kNm. From Eq. (42), ðys Þfp ¼ 1620 mm measured from the steel-concrete interface. The full interaction curvature is 6:25 108 mm1 so that the full interaction strains at the top and bottom of the 1750 mm deep steel section are 30 le and 80 le respectively where compressive strains are negative. Using Eq. (43) the strain þ at the focal point in the steel component is ðes Þfp ¼ 72 le. From Eq. (32), MFM ¼ 7 1 1:61 and therefore the partial interaction curvature is 1:01 10 mm . Finally, using (ys)fp and (es)fp as the reference point on the partial interaction strain
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distribution, the partial interaction strains at the top and bottom of the steel section are 92 le and 85 le respectively. Therefore, the maximum tensile stress at the bottom of the steel section has increased by 6%, which does not appear to be significant. However, taking the exponent of the fatigue equation for steel sections as 3, the endurance is reduced by a factor of ð1:06Þ3 ¼ 0:83 compared with that predicted by a standard full interaction analysis. The magnitude of the stress difference is larger near the steel-concrete interface because the focal points are located towards the top and bottom of the composite section. Consequently, it can be seen in this example that the maximum compressive stress in the top flange increases three fold. Even with this significant increase in compressive stress, buckling of the top flange should not be a concern as it is restrained by the shear connectors and the concrete. To complete the assessment at the mid-span, this procedure must be repeated for the maximum negative moment at the design point. In negative moment regions, the focal points are located the same distance from the steel–concrete interface so, the situation is similar to that described in the positive moment region. The obvious difference is that the sense of the stress changes from tension to compression and vice versa. Hence, the compression stresses increase in the bottom flange in the negative moment region, which may be unrestrained and therefore buckling may be a concern. Furthermore, the tensile stresses in the steel at the interface in negative moment regions are increased more than in the positive moment region, which will have a significant impact on the residual strength or endurance. A benefit of partial interaction observed in negative moment regions is that the tensile strains in the concrete reduce, limiting the expected extent of concrete cracking. As in any assessment, it is important to perform a thorough site investigation to visually determine the extent of any fatigue damage. Information on the extent of concrete cracking is important to this analysis as it enables the use of an appropriate flexural rigidity for the concrete component. If the partial interaction assessment demonstrates that the remaining endurance or strength of the composite section is not sufficient to safely extend the fatigue life of the structure, a load test could be performed. The load test will allow for the other factors that contribute to the strength and stiffness of the structure that cannot be considered easily in such a theoretical approach. Furthermore, a realistic representation of the actual vehicle weights and frequencies throughout the life of the bridge should be used to improve the accuracy of the assessment. If the structure requires strengthening or stiffening to extend its fatigue life, the assessment approach outlined in this section can also be used to determine the effectiveness of remedial measures.
6. Conclusions A simple mathematical model based on linear elastic partial interaction theory has been developed to more accurately predict the stresses in the steel and concrete components of continuous composite beams under serviceability loads. When used with recently published research on the assessment of the shear connection in
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continuous composite beams, the tiered fatigue assessment approach previously restricted to simply supported beams, is completed. Even though this is a linear elastic analysis, the non-linear shear load-slip behaviour of stud shear connectors can be modelled using an appropriate secant stiffness, and long term effects and cracking of the concrete can be included by using an appropriate flexural rigidity. The approach presented is applicable to continuous beams with any number of spans, span lengths, connector distribution and cross-section. Furthermore, when used with an appropriate fatigue damage equation it can also be used to quantify the effectiveness of remedial measures. Following a thorough site investigation, it is suggested to use this procedure to decide whether load testing is required. Therefore load tests, which are disruptive, costly and time consuming, need only be considered in the later stages of an assessment if it shown theoretically, allowing for partial interaction, that the residual strength or endurance is inadequate. Load tests will establish the extent of unintentional strengthening and stiffening by other factors not easily considered theoretically but contribute to the residual strength or endurance of composite bridges.
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[15] Seracino R, Oehlers DJ, Yeo MF. Partial-interaction fatigue assessment of stud shear connectors in composite bridge beams. Structural Engineering and Mechanics 2002;13(4):455–64. [16] Seracino R, Lee CT, Tan Z. Partial interaction shear flow forces in continuous composite steel-concrete beams. Journal of Structural Engineering, ASCE. (submitted for publication, April 2003) [17] Oehlers DJ, Seracino R. A tiered approach to the fatigue assessment of composite steel and concrete bridge beams. Proceedings of the Institution of Civil Engineers, Structures and Buildings 2002;152(3):249–57. [18] Johnson RP. Composite structures of steel and concrete vol. 1. Oxford: Blackwell Scientific Publications Ltd; 1994. [19] Oehlers DJ, Seracino R, Yeo MF. Effect of friction on shear connection in composite bridge beams. Journal of Bridge Engineering, ASCE 2000;5(2):91–8.