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Simplified elasto-plastic analysis of composite beams and cellular beams to Eurocode 4
Journal of Constructional Steel Research 67 (2011) 1426–1434
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Journal of Constructional Steel Research
Simplified elasto-plastic analysis of composite beams and cellular beams to Eurocode 4 R. Mark Lawson a,b,⁎, A.H. Anthony Saverirajan c a b c
Dept Civil Engineering, University of Surrey, Guildford GU 2 7XH, UK Steel Construction Institute, UK TEP Consultants Pte Ltd, Singapore
a r t i c l e
i n f o
Article history: Received 20 October 2010 Accepted 16 March 2011 Available online 24 May 2011 Keywords: Composite Beams Elasto-plastic Eurocodes Strain Bending resistance
a b s t r a c t The elasto-plastic analysis of composite beams is important when considering the increase in bending resistance of the beam and the end slip between the steel and concrete at higher strains. This paper provides a simplified method of elasto-plastic analysis by considering equilibrium of the composite cross-section as a function of its strain profile. A parabolic–rectangular stress block for concrete is used in this model with a declining concrete strength at strains exceeding 0.0035. The bending resistance of the composite beam is expressed as a function of the bottom flange strain, and is compared to fully plastic design to EN 1994-1-1: Eurocode 4 and the AISC LRFD Code. The effect of various parameters on the development of the plastic bending resistance of composite beams is investigated, such as asymmetry of the section, the steel strength, the influence of propped or un-propped construction, strain hardening in the steel and reducing concrete strength at high strains, interface slip, and the effect of openings in the web of the beams. It was found that a moment of 95% of the plastic bending resistance of a composite beam (0.95Mpℓ) is reached at a flange strain of 2 to 4 × yield strain for propped beams and 5 to 10 × yield strain for un-propped beams. When strain hardening in the steel is included in the analysis, bottom flange strains at a moment of 0.95Mpℓ are reduced by up to 30% relative to the case without strain hardening. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction The plastic design of composite beams is based on the development of idealised ‘plastic’ or rectangular stress blocks in the steel section and in the concrete slab. The transition from elastic to fully plastic behaviour occurs by the gradual development of plasticity in the composite cross-section as the steel strains increase above the yield strain, εy for the particular grade of steel. Theoretically, it is only possible to reach the full plastic resistance of the composite crosssection Mpℓ at infinitely high strains in the bottom flange. The principles of plastic design may be invalidated for composite sections of highly asymmetric shape or for composite cellular beams with large openings when the plastic neutral is low in the section depth because: • the strains in the concrete may exceed the limiting strain of 0.0035 before the plastic stress blocks are fully developed, which leads to a reducing bending resistance at high strains. • the influence of un-propped construction causes pre-existing strains in the steel section, which means that the plastic resistance of the composite beam is developed at higher strains than for propped beams. ⁎ Corresponding author at: Dept Civil Engineering, University of Surrey, Guildford GU 2 7XH, UK. Tel.: + 44 1483 686617. E-mail address: [email protected] (R.M. Lawson). 0143-974X/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jcsr.2011.03.016
• slip deformations at the interface between the concrete and steel may increase, leading to higher strains in the beam at a given bending moment. • these effects may be accentuated for asymmetric cellular beams with large web openings, where neutral axis depth may be lower in the section than in an unperforated beam. For practical design purposes when using the precise stress–strain models for steel and concrete, the plastic resistance of the composite beam, Mpℓ may be assumed to be developed at a steel strain corresponding to a bending resistance of 0.95Mpℓ. This is reached at a bottom flange strain of up to 10εy. This equivalent plastic moment of 0.95Mpℓ was also used by Aribert [1] in the calibration of the partial shear connection rules of EN 1994-1-1: Eurocode 4 [2]. However, in his calibration study, Aribert considered only propped beams with a span: depth ratio in the range of 30 to 33, which is higher than the typical ratio of 20 to 25 for un-propped beams. The beam proportions have an important effect on end slip. Also in the Eurocode 4 calibration study, Johnson and Molenstra [3] used different models for concrete strength and strain hardening in the steel, and also the beam performance was evaluated at Mpℓ rather than 0.95Mpℓ. In a summary by Banfi [4] of the calibration studies in Eurocode 4, the difference in end slip between un-propped and propped beams was identified, which was not considered in the development of the Eurocode 4 shear connection rules. Banfi also showed that end slip
R.M. Lawson, A.H.A. Saverirajan / Journal of Constructional Steel Research 67 (2011) 1426–1434
varies approximately as the cube of the moment ratio M/Mpℓ acting on the beam. The various assumptions in this paper and in previous analyses are summarised in Table 1. Therefore this paper presents an opportunity to examine the sensitivity of various geometric and material parameters on the development of the plastic resistance of composite beams. The following analysis considers the development of elasto-plastic behaviour of composite beams based on an exact equilibrium analysis. It allows for strain hardening in the steel, the declining strength of the concrete slab at high strains, and interface slip between the steel and the concrete. In un-propped construction, a pre-existing strain is applied to the steel section, which modifies the development of the plastic resistance, although the bending resistances of the propped and un-propped beams tend to the same plastic resistance. This type of behaviour would be extremely complex to model using Finite Element Analysis. The concrete properties are taken from the idealised stress–strain curve in BS EN 1992-1-1: Eurocode 2 [5]. A further unloading branch in the stress–strain relationship is considered in which it is assumed that the strength of concrete declines linearly from its design strength at a strain of 0.0035 to zero at a strain of 0.025. 2. Elasto-plastic analysis — propped beams In the elasto-plastic analysis of a propped composite beam with a solid slab, the bottom flange strain is given by nεy (where n ≥ 1, and defines the multiple of the elastic yield strain, εy). Various cases of neutral axis depth, ye (measured from the top of the slab) may be considered in the calculation of the elasto-plastic properties of the composite section. Ignoring the effects of slip, the strain εc at the top of the concrete slab is established from a linear strain distribution, as function of the bottom flange stain, nεy and neutral axis position, ye, as follows: εc = nεy
ye : hs + hc −ye
NA Fwb
hs
<εy
Fc Fft Fwt
hc Aft
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ye-hc
ye
(hs +hc -ye ) /n
tw
hs+hc-ye
εy
Fwp,b Ffb
nεy
Afb
Cross-section
εc
Stress blocks
Strain profile
Fig. 1. Elasto-plastic stress blocks in a composite section.
axis position, ye. This bending resistance may be compared to the idealised plastic bending resistance of the composite section, Mpℓ , which is developed at an infinitely high value of n (or in practice for n N 10). 3. Forces and moments in steel section Consider firstly the case where the neutral axis lies in the steel web, but the top flange is elastic (i.e. top flange strain is less than εy), as shown in Fig. 1. The forces in the steel section are established from the plastic and elastic stress blocks, and the moments of each stress block around the neutral axis position, are determined as follows: Tension force in bottom flange (plastic): Ffb = Afb fy Moment in bottom flange: Mfb = Afb fy hs + hc −ye −0:5tfb Tension force in bottom (plastic) part of web:
The forces in the steel elements are established from the particular strain profile, as illustrated in Fig. 1. For all cases, the extent of the plastic zone in the web is defined geometrically by the parameter, n. The compression force in the concrete Fc is calculated depending on the strains at the top and bottom of the slab — see later. Equilibrium is established at a value of ye at which the tension and compression forces in the composite section are equal. This is an iterative process. In all cases, ye is less than the elastic neutral axis depth measured from the top of the slab, which can be used as a starting point in the iteration. The elasto-plastic bending resistance of the composite beam is obtained by taking moments of each stress block around the neutral
ðn−1Þ −tfb Fwb;p = tw fy ðhs + hc −ye Þ n Moment in bottom (plastic) part of web: ðn−1Þ ðn + 1Þ −tfb −tfb Mwb = 0:5tw fy ðhs + hc −ye Þ ðhs + hc −ye Þ n n
Tension force in bottom (elastic) part of web: Fwb = 0:5tw fy ðhs + hc ye Þ = n
Table 1 Summary of various elasto-plastic analyses of composite beams. Properties used in models
Johnson and Molenstra [3]
Aribert [1]
This paper
Concrete
• Empirical polynomial stress–strain curve • Declining strength after εcu = 0.0035
• Parabolic–plastic stress–strain curve to EC2 • No declining strength after εcu = 0.0035
• Partial factor = 1.5 • Strain hardening immediately after yield • Strain hardening modulus of E/70 to E/200 • Partial factor = 1.0
• • • •
• Parabolic–plastic stress–strain curve to EC2 • Declining strength after εcu = 0.0035 to zero at 2.5% strain • Partial factor = 1.5 • Plastic deformation from εy to 6εy • Strain hardening after a strain of 6εy • Strain hardening modulus of 2.7 kN/mm2
Steel
Bending resistance Beam proportions studied
• Plastic bending resistance, Mpℓ • Existing composite beam tests
• • • • •
Partial factor = 1.0 Plastic deformation from εy to 10 εy Strain hardening after 10 εy Strain hardening modulus obtained from fy at 10 εy and fu at 25 εy Partial factor = 1.0 0.95 Mpℓ considered in parametric study Beams with span: depth ratio of 30 to 33 Propped beams Flange ratios of 1:1 and 1:3
• • • • •
Partial factor = 1.0 0.95 Mpℓ considered in parametric study Beams with span: depth ratio of 25 Propped and unpropped beams Flange ratios of 1:1 and 1:3
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Concrete strength
Moment in bottom (elastic) part of web: 2
0.85 fcyl/γc
2
Mwb = 0:33tw fy ðhs + hc −ye Þ =n
Assumed linear strength reduction from εc>0.0035
Compression force in top flange (elastic):
Fft = Aft fy n
0.002
2 ye −hc −0:5tft
0.0035
0.025
ðhs + hc −ye Þ
Fwt = 0:5tw fy n
2 ye −hc −tft ðhs + hc −ye Þ
Moment in the top (elastic) part of web:
Mwt = 0:33tw fy n
3 ye −hc −tft ðhs + hc −ye Þ
Equilibrium is satisfied for a value ye when: Ffb + Fwb;p + Fwb = Fft + Fwt + Fc For highly asymmetric sections, plasticity can occur in the top part of the web at high strains, as presented in Fig. 2. The top flange strain exceeds εy, when; hs n t : + hc + ð1 + nÞ ft ð1 + nÞ
There are two factors applied to the concrete strength to obtain its design compression resistance. The partial safety factor for the material, γc, is 1.5 for design at the ultimate limit state. In addition, the compression strength of concrete in a uniform stress field is taken as a factor of 0.85 times the concrete's cylinder strength. This factor also takes account of the difference between the area under the parabolic stress–strain curve and an equivalent rectangular stress block. In the design of concrete beams in bending to Eurocode 2 [5], this factor is taken as 1.0. However, in the design of composite beams to Eurocode 4, the 0.85 factor is retained. In this paper, the analyses are based on a concrete strength of 0.85 factor times the cylinder strength divided by a partial factor of 1.5, and the bending resistance at a given strain is compared to a notional bending resistance of 0.95Mpℓ, where Mpℓ is the design plastic bending resistance of the composite beam using the same material strengths. The sensitivity to the 0.85 factor is examined when considering the more precise concrete stress–strain relationship in Fig. 3. Consider firstly the case where the neutral axis lies in the steel section i.e. ye N hc. For a concrete strain of εc ≤ 0.002, the force in the concrete is given by the integral of the parabolic stress–strain curve, as follows:
y
Fc = 0:85fcyℓ = γc ⋅Be ∫y e−h e
Similar equations may be derived for the forces in the steel section and the moments of each stress block around the neutral axis position. 4. Forces and moment in the concrete slab The forces and moments in the concrete slab are calculated from the idealised stress–strain curve for concrete to EN 1992-1-1:Eurocode 2 [5], which is illustrated in Fig. 3. The first part of the stress–strain curve for concrete is parabolic up to a plateau, which corresponds to its design strength. Eurocode 2 gives no information on the performance of concrete after a limiting strain of 0.0035 when the concrete slab is assumed to have reached its maximum compression resistance. This was the case in the previous calibration studies [1,3].
Fc
hc Aft NA hs
Concrete strain
Fig. 3. Parabolic–linear stress–strain curve for concrete with declining strength at high strains.
Compression force in top (elastic) part of web:
ye ≥
Declining strength at high strain
ðhs + hc −ye Þ
Moment in top flange (elastic):
Mft = Aft fy n
Constant strength plateau
Parabolic stress-strain curve
ye −hc −0:5tft
Fwb
Cross-section
ye
1 εc y −hc 2 1 ye −hc 3 εc − e : × 1− + 3 0:002 3 ye ye 0:002
The bending resistance due to the parabolic concrete stress block is given by:
e
Stress blocks
c
2
= 0:566fcyℓ = γc ⋅Be ⋅ye × 1−0:375
2 2 εc 2y εc y − y⋅dy ye ye 0:002 0:002 ε
y
Mc = 0:85fcyℓ = γc ⋅Be ∫y e−h
c
0:002
εc y −hc 3 εc ye −hc 4 − e : + 0:375 0:002 ye 0:002 ye
For the case where the neutral axis lies in the concrete i.e. ye b hc, and εc ≤ 0.002, the force and moment in the concrete reduces to:
hs +hc -ye nεy
Afb
0:002
εc εy Fft Fwp,t Fwt = Fwb(hs +hc -ye)/n
εy
Fwp,b Ffb
2 2 2y εc y − dy ye ye 0:002
c
= 0:85fcyℓ = γc ⋅Be ⋅ye
(hs +hc -ye)/n
tw
εc c 0:002 ε
Fc = 0:85fcyℓ = γc ⋅Be ⋅ye
ε 1 εc c 1− 3 0:002 0:002
Strain profile 2
Fig. 2. Stress blocks with plasticity in the top and bottom flanges.
Mc = 0:566fcyℓ = γc ⋅Be ⋅ye
ε h ε i c c 1−0:375 : 0:002 0:002
R.M. Lawson, A.H.A. Saverirajan / Journal of Constructional Steel Research 67 (2011) 1426–1434
For εc N 0.002, but εc ≤ 0.0035, the concrete force is given by the integral over the complete parabolic stress–strain curve including the rectangular stress block, as follows; "
0:002 εc ye −hc 2 Fc = 0:85fcyℓ = γc ⋅Be ⋅ye 1−0:33 − εc 0:002 ye 2 3 # εc ye −hc + 0:33 : 0:002 ye The bending resistance of the concrete slab over the parabolic– rectangular stress–strain curve is given by: " Mc =
2 0:425fcyℓ=γc ⋅Be ⋅ye
+ 0:5
εc 0:002
2
0:002 2 εc ye −hc 3 −1:33 1−0:17 εc 0:002 ye
ye −hc ye
4 # :
Aft hs
5. Behaviour of concrete at high strains At high steel strains, concrete strains may exceed 0.0035 and theoretically the analysis should stop as the concrete has reached its limiting strain. However, by using the concrete stress–strain relationship in Fig. 3, it is possible to consider the effect of the declining concrete strength on the behaviour of the composite beam. For εc N 0.0035 but ≤ 0.025, the same formulae are used as for the case of a parabolic–rectangular stress block, except that the force and moment in the concrete slab are reduced according to the declining concrete strength. For the case where ye N hc, the force in the concrete, Fc is calculated using the same formula as for εc N 0.002, but is given by subtracting the last term in the following equation: Fc;ðεc N 0:0035Þ = Fc;ðεc N 0:002Þ −0:425fcyℓ = γc ⋅Be ⋅ye
ðεc −0:0035Þ2 : 0:0215εc
The bending resistance of the concrete slab, Mc, is calculated using the same formula as for εc N 0.002, but is given by subtracting the last term in the following equation: N 0:002Þ 2
−0:283fcyℓ = γc ⋅Be ⋅ye
2
ðεc −0:0035Þ 0:0215εc
0:0035 1+ 2εc
where Fc,(εc N 0.002) and Mc,(εc N 0.002) are the force and moment given by the previous equations for 0.002 b εc b 0.0035, but using the actual value of εc N 0.0035. A further case arises when the strain at the top of the slab is so high that the strain in the bottom of the concrete slab exceeds 0.002, which occurs when; ye N
hc : ð1−0:002 = εc Þ
ys
+
εt
yeff
=
tw
hs +hc -ye
εs
A fb
Cross-section
εc ye
Strains acting on steel beam
nεy-εs
Strains acting on composite section
nεy
Combined strains
Fig. 4. Strains in an un-propped beam combining the non-composite and composite strains.
In this case, the above equations are simplified as follows: Fc = 0:85fcyℓ = γc ⋅Be ⋅hc −0:425fcyℓ = γc ⋅Be ⋅ye
ðεc −0:0035Þ2 0:0215εc
2
−0:283fcyℓ = γc ⋅Be ⋅ye
ðεc −0:0035Þ2 0:0035 1+ : 2εc 0:0215εc
6. Influence of un-propped construction
1 0:002 2 2 Mc = 0:425fcyℓ = γc ⋅Be ⋅ye 1− : 6 εc
= Mc;ðεc
εc
Mc = 0:85fcyℓ = γc ⋅Be ⋅hc ⋅ðye −0:5hc Þ
1 0:002 Fc = 0:85fcyℓ = γc ⋅Be ⋅ye 1− 3 εc
N 0:0035Þ
εs ys (hs - ys)
hc
When the neutral axis lies in the slab i.e. ye b hc, and εc N 0.002 but εc ≤ 0.0035, the force and moment in the concrete slab reduce to:
Mc;ðεc
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In un-propped construction, the stresses due to the self-weight of the slab and beam act on the steel section. This has the effect of reducing the stress in the bottom flange that is available for composite action under the loads applied after construction. Conversely in unpropped construction, the slip at the ends of the beam is reduced for a given flange strain. The build-up of strains in an un-propped beam is illustrated in Fig. 4. Assume a pre-existing strain of DLS εY in the bottom flange, where DLS is the dead load strain factor due to the self weight of the beam and slab. DLS is in the range of 0.2 to 0.4 for a typical composite beam, equivalent to a stress = DLS fy in the bottom flange after construction. When the bottom flange strain is εb = nεy, the strain due to composite action of the un-propped beam is given by (n − DLS)εy.The top flange strain, εt, (compression positive), due to composite action combined with the initial stresses on the steel beam, is now given by: εt =
ye −hc ys ⋅ðn−DLSÞεy + DLS⋅εy h + hc −ye h−ys
where ys is the neutral axis depth of the steel section measured from the top of the steel beam — see Fig. 4. The concrete forces and moments are obtained as previously, but are calculated using a reduced concrete strain of: εc =
ye ⋅ðn−DLSÞεy : h + hc −ye
Also, for the case of a composite slab using profiled decking of height, hp, the same formulae for concrete force and moment are used provided hc is defined as the depth of solid concrete above the decking. In the previous formulae for the forces and moments in the steel section, and in formulae for εc and εt above, the total depth of the slab (hc + hp) replaces hc. 7. Slip at the interface between the steel and concrete For cases of partial shear connection, slip occurs between the steel and the concrete. This discontinuity in the strain profile is given by an interface strain εslip, as illustrated in Fig. 5. The integral of the slip strain, εslip over the half span of the beam defines the end slip. For a
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Fc
hc Aft NA
εc Fft Fwp,t Fwt = Fwb (hs +hc -ye) /n
Fwb
hs
hc
hc ye
ht
(hs +hc -ye) /n
tw
hs +hc -ye
εy
Fwp,b Ffb
n εy
Afb
Cross-section
εslip (ye/(ye- hc)) εslip
Stress blocks
hs
A ft
NA
εc
Fc Fft Fwt
ho
hc ht ye -hc -h t h s +hc -hb -ye
Fwp,b Ffb
hb
hb
Cross-section Fig. 5. Stress blocks for an interface slip, εslip between the steel and concrete.
sinusoidal deflected beam shape, the end slip is given approximately by εslipL/π, where L is the beam span. Often in cases of partial shear connection, and particularly in unpropped beams, the upper part of the steel section develops plastic blocks at a bottom flange strain that is less than for the case of full shear connection, as illustrated in Fig. 5.
ye
h s +hc -ye >εy
nεy
A fb
Strain profile
< εy
Stress blocks
Strain profile
Fig. 7. Elasto-plastic stress profile in a composite cellular beam (neutral axis lies within the opening depth).
opening depth is illustrated in Fig. 7. In this figure, the strain profile is such that the bottom Tee is fully plastic, but the top Tee is still elastic. Other design cases may be considered in which the top flange is plastic, as considered earlier for unperforated beams.
8. Strain hardening in steel 10. Results of elasto-plastic analyses Strain hardening in steel may be considered at high strains according to two alternative models. In the Annex to EN 1993-1-5 [6], the strain hardening modulus is taken as Es/100 = 2 kN/mm2. In the model by Byfield et al. [7], the steel strength is taken as its yield strength for strains of εy to 6εy and then increases according to a strain hardening modulus of 2.7 kN/mm2. The EN 1993-1-5 model leads to higher strengths in the normal range of plastic strains. In this study, the Byfield model will be used, and in this model shown in Fig. 6, the steel stress in the bottom flange increases by approximately 1.8% at a strain of 10εy. Although a relatively small effect, strain hardening can be important when the composite bending resistance at constant yield strength asymptotes to 0.95Mpℓ. The results for the two strain hardening models are considered later. 9. Application of elasto-plastic analysis to cellular beams Cellular beams are rolled or fabricated sections with regular circular openings for services distribution. They are designed as composite beams of 12 to 20 m span, and are most efficient as secondary beams with openings typically of 60 to 70% of the depth of the beam. Often cellular beams are highly asymmetric in shape, which are efficient when designed compositely. The partial shear connection rules in EN 1994-1-1: Eurocode 4 clause 6.6.1.2 were developed for I sections with solid webs, which are designed for their plastic bending resistance. For cellular beams, the bending resistance at an opening is the critical design case rather than for the unperforated section. The previous stress block approach may be modified for composite cellular beams by ignoring the contribution of the web within the opening depth. The case where the Neutral axis (NA) lies in the
The following parametric studies were carried out using the elasto-plastic analysis models presented above.
10.1. Bending resistance as a function of bottom flange strain The increase of bending resistance of a composite beam with bottom flange strain is illustrated in Fig. 8 for the case of an un-propped and propped symmetric beam of 800 mm depth. The flange thickness is 18 mm and the web thickness is 10 mm. The line corresponding to 0.95Mpℓ is the bending resistance that is sensibly achieved in plastic design when considering codified steel and concrete stress–stain relationships. It is apparent that the propped beam reaches 0.95Mpℓ at a strain of 3εy, whereas the same un-propped beam reaches this moment at a strain of 7εy. For an asymmetric beam with flange area ratio of 3:1 (bottom flange thickness of 36 mm and top flange thickness of 12 mm), Fig. 9 shows that the propped beam reaches 0.95Mpℓ at less than 2εy, whereas the same un-propped beam reaches this moment at 3εy. However, the rapid decline in bending resistance is apparent at higher strains due to the loss in strength of the concrete. The corresponding steel and concrete strains for a 600 mm deep symmetric composite beam are presented in Table 2 for moment ratios of 0.8 to 0.95Mpℓ. In comparison, the elastic bending resistance of the composite section is approximately 0.75Mpℓ. It is apparent that strains reduce rapidly with reducing moment ratio, M/Mpℓ, particularly in un-propped construction.
Stress fu BS EN 1993-1-5 fsh
Esh = 2.1 kN/mm2
Esh = 2.7 kN/mm2
fy Byfield et al
Es = 210 kN/mm2
εy
6εy
~22εy
Strain,ε
Fig. 6. Strain hardening models for steel (EN 1993-1-5 annex and Byfield).
Fig. 8. Influence of propped and un-propped construction on the bending resistance of 800 mm deep symmetric composite beam.
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Table 3 Influence of beam depth on steel and concrete strains for symmetric un-propped composite beams depending on analysis method. Method
Fig. 9. Influence of propped and un-propped construction on the bending resistance of 800 mm deep asymmetric composite beam.
10.2. Influence of steel strain hardening and the concrete strength factor The influence of strain hardening of the steel and of the 0.85 factor on concrete strength on the development of the plastic bending resistance of a composite beam are presented in Table 3 for the case of un-propped symmetric beams of 400 to 800 mm depth. This shows that the steel strain at a moment of 0.95Mpℓ is insensitive to the section depth but that the concrete strain is higher for shallower sections. Concrete strains can be well above 0.0035 before a moment of 0.95Mpℓ is reached. When strain hardening using the Byfield [7] model is included in the analysis, the maximum steel and concrete strains both reduce by 20 to 30% at a moment of 0.95Mpℓ, which shows the importance of strain hardening on the development of the plastic bending resistance of the composite beam. The effect of the two strain-hardening relationships for steel is presented in Fig. 10 for un-propped symmetric beams. At a moment of 0.95Mpℓ, the bottom flange strain is 5.5εy using the strain hardening model of BS EN 1993-1-5 [6] and 7.5εy using the Byfield [7] strain hardening model. This analysis shows that the strain at a moment close to Mpℓ is sensitive to the precise form of strain hardening considered in the model. The concrete strength factor of 0.85 in the previous formulae is taken directly from Eurocode 4 and takes account of the difference between the parabolic–rectangular stress strain curve for concrete and the equivalent plastic stress block. It may be argued that the 0.85 factor can be set to 1.0 when using the precise stress–strain curve for concrete rather than an idealised rectangular stress block. In Table 3, it is shown that the effect of the 0.85 factor on concrete cylinder strength is significant as it influences both the neutral axis position and also the contribution of the concrete slab to the bending resistance. When taking this factor as 1.0 rather than 0.85, the bending resistance of the composite beam is increased by up to 5%, as shown in Fig. 10, and at a moment of 0.95Mpℓ, the strains in the steel and the concrete are reduced by over 40%.
No strain hardening in steel and with 0.85 factor on concrete strength With strain hardening in steel and with 0.85 factor on concrete strength No strain hardening in steel. Factor on concrete strength set to 1.0
Max. strains
Beam depth 400 mm
600 mm
800 mm
Steel strain, nεy Concrete strain, εc Steel strain, nεy Concrete strain, εc Steel strain, nεy Concrete strain, εc
9.0εy =0.0156
9.0εy =0.0156
8.0εy =0.0138
0.0065
0.0050
0.0041
7.5εy =0.0130
7.1εy =0.0122
6.8εy =0.0118
0.0053
0.0039
0.0031
5.6εy =0.0097
6.1εy =0.0105
6.0εy =0.0104
0.0031
0.0026
0.0026
Data for S355 Steel and C30 concrete. Slab depth = 130 mm with 60 mm deck profile depth. Bending moment = 0.95 Mpℓ.
In Table 4, the effect of using S275 steel for a 600 mm deep asymmetric section is examined, based on the same parameters of strain hardening and concrete strength factor (Fig. 11). It is shown for S275 steel that the steel and concrete strains at a moment of 0.95Mpℓ are only slightly lower than for S355 steel, which indicates that the effect of steel grade on the strains close to the plastic resistance of the beam is not so significant. 10.3. Influence of declining concrete strength The effect of including the declining concrete strength at high strain is illustrated in Fig. 12 for 800 mm deep symmetric un-propped beam. It is apparent that the bending resistance begins to fall at a bottom flange strain of about 4εy, whereas without this declining strength, the bending resistance would otherwise increase gradually. It is important therefore that the effect of declining concrete strength is included for deeper or highly asymmetric beams, as the composite beam may ‘unload’ due to concrete crushing. 10.4. Influence of slip between the beam and the slab In this model, the interface slip between the steel beam and concrete slab is represented by a slip factor, k, multiplied by the concrete strain, εc at the top of the slab, such that the interface slip strain, εslip = k εc. The end slip, δs, may be obtained by integration of εslip along the span, and is given for a uniform loading case by the approximate formula: L δs = k⋅ ⋅εc π
Table 2 Influence of moment ratio on steel and concrete strains for a 600 mm deep symmetric composite section in S355 steel. Method
Maximum Moment ratio M/Mpl strains 0.95 0.9
Un-propped Steel beams strain, nεy Concrete strain, εc Propped Steel beams strain, nεy Concrete strain, εc
0.85
0.8
7.1εy =0.012 3.5εy =0.006 2.2εy =0.004 1.7εy =0.003 0.0039
0.0020
0.0014
0.0011
2.8εy =0.005 1.9εy =0.003 1.2εy =0.002 εy =0.0017 0.0022
0.0016
Data for S355 Steel and C30 concrete. Slab depth = 130 mm with 60 mm deck profile depth. Dead load strain factor = 0.3 in unpropped construction.
0.0011
0.0008
Fig. 10. Comparison of strain hardening relationships on the bending resistance of a 600 mm deep symmetric un-propped composite beam.
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Table 4 Influence of steel grade and section asymmetry for 600 mm deep un-propped composite beams. Method
No strain hardening in steel and with 0.85 factor on concrete strength With strain hardening in steel and with 0.85 factor on concrete strength No strain hardening in steel. Factor on concrete strength set to 1.0
Max. strains
Symmetric beams S275 steel
S355 steel
Asymmetric beam S355 steel
Steel strain, nεy Concrete strain, εc Steel strain, nεy Concrete strain,εc Steel strain, nεy Concrete strain,εc
11εy =0.0147
9εy =0.0156
4.0εy =0.0069
0.0052
0.0050
0.0048
8.1εy =0.0109
7.1εy =0.0122
4.0εy =0.0069
0.0030
0.0039
0.0048
8.6εy =0.0115
6.1εy =0.0105
3.6εy =0.0062
0.0025
0.0026
0.0023
Asymmetric section has 3:1 ratio of flange areas. Slab depth = 130 mm with 60 mm deck depth. Bending moment = 0.95 Mpℓ.
where L is the beam span, which may be taken as approximately 25 × beam depth, hs, for a uniformly loaded composite beam. The analyses were repeated for symmetric and asymmetric beams (with 1:1 and 3:1 flange asymmetry) and slip factors of k = 0, 0.3 and 0.5, as shown in Fig. 13. It is apparent that the influence of slip is relatively small at high steel strains as the bending resistance is dominated by the development plasticity in the steel section. For a uniformly loaded 600 mm deep symmetric beam of 15 m span, the end slip δs is 3.5 mm using the above formula with a slip factor of 0.3, and 6.5 mm for a slip factor of 0.5. In the Eurocode 4 calibration studies, a limiting slip of 6 mm was considered when developing the partial shear connection rules. It should be noted that Eurocode 4 would require a minimum degree of shear connection of 70% for a 15 m span beam of symmetric shape. 10.5. Longitudinal force in the slab as function of moment ratio At the plastic bending resistance of the composite beam, the longitudinal shear force transferred between the steel and the concrete is equal to the smaller of the compression resistance of the slab or the tensile resistance of the steel beam. Using the model based on the steel and concrete stress–strain relationships, the longitudinal forces in the slab reduce with moment ratio, as shown in Fig. 14 for a symmetric unpropped beam. Three slip factors are considered, and the highest longitudinal forces occur for the case of no slip. It may be assumed that the longitudinal forces in the slab reduce approximately linearly with moment ratio, M/Mpℓ. Fig. 13 shows that this linear assumption is conservative for lower moment ratios. As noted earlier, the design moment in the calibration studies is taken as 0.95Mpℓ,
Fig. 11. Influence of concrete strength factor on the bending resistance of a 600 mm deep symmetric composite beam.
Fig. 12. Influence of declining concrete strength on the bending resistance of an 800 mm deep composite beam.
at which point the force in the concrete reaches its design resistance (Table 5). It is proposed that the total longitudinal shear force Fc may be taken as a function of moment ratio as follows; 1:5 Fc = Fc ;pℓ = M =Mpℓ where Fc,pℓ is the compression force in the concrete slab at the plastic bending resistance of the composite beam. This reduction in longitudinal shear force with moment ratio is very important when considering the shear connector forces and requirements for transverse reinforcement in composite beams designed for serviceability limits of deflection or vibration, where moment ratios are generally below 0.8.
11. Results of elasto-plastic analyses for cellular beams The variation of bending resistance of cellular composite beams with bottom flange strain is considered for beams of 800 mm depth and with regular openings of 500 mm diameter. The results are presented in Fig. 15 for symmetric propped and un-propped cellular beams, and in Fig. 16 for asymmetric cellular beams with a bottom: top flange asymmetry ratio of 3:1. For a propped cellular beam, a bending resistance of 0.95Mpℓ is reached at a bottom flange strain of 2 to 3εy. For an un-propped cellular beam, this moment is reached at a bottom flange strain of 8 to 10εy. The corresponding concrete and steel strain are presented in Table 6 for both symmetric and asymmetric cellular beams. For asymmetric cellular beams, the concrete strain, εc does not exceed 0.0035, unlike the same composite section without openings.
Fig. 13. Influence of interface slip factor for a 600 mm deep symmetric un-propped composite beam.
R.M. Lawson, A.H.A. Saverirajan / Journal of Constructional Steel Research 67 (2011) 1426–1434
Fig. 14. Longitudinal force in the slab as a function of moment ratio and slip factor for a 600 mm deep symmetric composite beams.
12. Comparison of eurocode 4 with the American (AISC) code The AISC Limit State (LRFD) Code [8] is similar in general principles of composite beam design to Eurocode 4, but differs in terms of how the partial safety factors for materials are applied. The concrete strength is defined as 0.85fc, but partial safety factors for both concrete and steel are taken as unity to calculate the plastic bending resistance of the composite beam. The design bending resistance in the AISC Code is then obtained by multiplying the plastic bending resistance, Mpℓ by a performance factor of 0.9, which reflects the influence of material safety factors. The results of the elasto-plastic analyses to the two Codes are presented in Table 6. It is found that the design bending resistance to the AISC Code exceeds the plastic bending resistance to Eurocode 4 for shallower sections, but is less than Eurocode 4 for deeper sections. This difference increases with section size because the contribution of the concrete is proportionately less. When using the same stress– strain curve for concrete is in Eurocode 4, it was found that the bottom flange strain and the concrete strain at the design bending resistance to the AISC Code are less than to Eurocode 4. This comparison is presented in Table 6 in which the strains are calculated at a moment of 0.95Mpℓ to Eurocode 4 and at 0.9Mpℓ to the AISC Code (where Mpℓ is the plastic bending resistance calculated to the relevant Code). It is concluded that by using the performance factor of 0.9, the design to the AISC Code leads to lower steel and concrete strains at the design resistance of the composite section than to Eurocode 4. In
Table 5 Influence of cross-section shape on steel and concrete strains for 600 mm deep cellular un-propped beams. Method
No strain hardening in steel and with 0.85 factor on concrete strength With strain hardening in steel and with 0.85 factor on concrete strength No strain hardening in steel. Factor on concrete strength set to 1.0
Maximum Symmetric strains at non-cellular beam
Cellular beams
Steel strain, nεy Concrete strain, εc Steel strain, nεy Concrete strain, εc Steel strain, nεy Concrete strain, εc
9εy =0.0156
8.4εy =0.0145 10.4εy =0.0180
0.0050
0.0028
7.1εy =0.0122
7.1εy =0.0102 7.8εy =0.0135
0.0039
0.0026
Symmetric
1433
Fig. 15. Influence of propped and un-propped construction on the bending resistance of 800 mm deep symmetric composite cellular beam with 500 mm diameter openings.
particular, the concrete strains calculated according to the AISC design are approximately 40% lower than to Eurocode 4, which has the effect of also reducing the slip in the steel–concrete interface. This partly explains why the limiting degree of shear connection to the AISC Code is not as high as to Eurocode 4. 12.1. Simplified model for composite beams at high strains The results of the elasto-plastic analysis for various composite cross-sections are presented as a function of the multiple n of the yield strain εy in the bottom flange. An approximate relationship between the bending resistance of the composite section and the bottom flange strain nεy is given by: 0:5 M = Meℓ + Mpℓ −Meℓ ððn−1Þ=6Þ Meℓ is the elastic bending resistance of the composite section, which is approximately, Meℓ ≈ 0.75Mpℓ. In this formula, it follows that M = 0.95Mpℓ for n ≈ 5, which is typical of the theoretical results when strain hardening is included in the model. As first shown by Banfi [4], the end slip, s may be assumed to vary conservatively as the cube of the moment ratio, or: . 3 s = 6 M 0:95Mpℓ where s = 6 mm at a moment of 0.95 Mpℓ. This is very important when using the partial shear connection rules in Eurocode 4 for beams designed with low moment ratios. It is proposed that the minimum degree of shear connection for the particular beam span calculated according to Eurocode 4 may be multiplied by the ratio (M/Mpℓ)1.5, where M/Mpℓ is the moment ratio
Asymmetric
0.0035
0.0035
6.1εy = 0.0105 8.0εy =0.0138 9.2εy =0.0159 0.0026
0.0021
600 mm deep cellular beam has 400 mm diameter openings. Asymmetric section has 3:1 flange area ratio. All data for S355 Steel and C30 concrete. Slab depth = 130 mm with 60 mm deck profile depth. Bending moment = 0.95 Mpℓ.
0.0032
Fig. 16. Influence of propped and un-propped construction on the bending resistance of 800 mm deep asymmetric composite cellular beam with 500 mm diameter openings.
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Table 6 Comparison between AISC [8] and Eurocode 4 [2] for symmetric un-propped composite beams designed for full shear connection. Code
Eurocode 4
AISC LRFD
Parameter
Bending resistance, Mpℓ Bottom flange strain at 0.95 Mpℓ Concrete strain at 0.95 Mpℓ Design bending resistance, 0.9 Mpℓ Bottom flange strain at 0.9 Mpℓ Concrete strain at 0.9 Mpℓ
• Design according to the principles of the American AISC LRFD Code leads to lower steel and concrete strains at the design resistance of the composite beam that to Eurocode 4.
Beam depths 400 mm
600 mm
800 mm
1703 kNm
2603 kNm
3666 kNm
8.1εy = 0.0136
7.8εy = 0.0131
7.7εy = 0.0129
0.0049
0.0036
0.0031
1722 kNm
2540 kNm
3503 kNm
7.5εy = 0.0126
7.1εy = 0.0119
6.3εy = 0.0106
0.0029
0.0021
0.0018
Mpℓ is the plastic bending resistance calculated to the relevant Code. Data for 50 ksi steel (S345 equivalent strength) and 3 ksi concrete (C35 equivalent cylinder strength).
acting on the beam at the ultimate limit state. This corresponds to the variation of longitudinal shear force with moment ratio, as defined earlier. 13. Conclusions The following conclusions are made for elasto-plastic design of composite beams: • Strain hardening of steel and declining concrete strength at high strains should be considered in models of composite beams based on elasto-plastic behaviour. The declining concrete strength may be taken as linear from a strain of 0.0035 to zero at a strain of 0.025. • The development of 95% of the design plastic bending resistance of the composite beam, Mpℓ, is a reasonable limit when using elastoplastic models based on the parabolic–rectangular stress–strain relationship and partial factors for concrete. • The 0.85 factor on concrete strength in Eurocode 4 may be set to 1.0 in elasto-plastic methods when using the parabolic–rectangular stress–strain curve for concrete. • A moment 0.95Mpℓ is reached at a bottom flange strain of approximately 6×steel yield strain for un-propped composite beams. • For highly asymmetric or deep beams, unloading of the compression force in the concrete may occur due to crushing at high strains, in which case, the bending resistance reduces rapidly with increasing strain. • The effect of slip between the steel and concrete has a relatively small effect on the development of the bending resistance of unpropped composite beams at high strains. • Asymmetric cellular composite beams reach their plastic bending resistance at higher strains than un-perforated composite beams of the same dimensions. • Longitudinal shear forces in the slab may be assumed to vary in proportion to (M/Mpℓ)1.5 in un-propped construction, where M is the applied moment at the ultimate limit state. • The minimum degree of shear connection required to Eurocode 4 for a particular beam span and steel grade may be reduced according to the multiple of (M/Mpℓ)1.5 for the case of un-propped composite beams. However, in all cases, the minimum degree of shear connection should exceed 0.4.
Notation Beff effective slab width for composite beam design DLS dead load strain factor due to self-weight loads applied to the steel beam E elastic modulus of steel fcyℓ concrete cylinder strength steel yield strength fy steel ultimate tensile strength fu compression force in the concrete slab Fc compression force in the concrete slab at the design Fcpℓ resistance of the composite beam hc depth of concrete slab (above deck profile) depth of deck profile hp height of steel section hs L beam span M applied moment moment due to the compression force in the slab Mc plastic bending resistance of composite beam Mpℓ n multiple of steel yield strain, εy tf thickness of concrete flange (subscript b for bottom flange and t for top flange) ye neutral axis depth of composite section from top of the slab neutral axis depth of un-propped composite section from yeff top of the slab, taking account of strains in the beam after construction. ys elastic neutral axis depth of steel section concrete strain at the top of the slab εc limiting concrete strain (=0.0035) εcu slip strain at the steel–concrete interface εslip top flange strain εt steel yield strain (=fy/E) εy end slip due to slip strain, εslip δs γc partial factor on concrete strength (=1.5)
References [1] Aribert JM. Analyse et Formulation Pratique de l'Influence de la Nuance de l'Acier du Profile sur le Degré Minimum de Connexion Partielle d'une Poutre Mixte (Analysis and practical solutions for the influence of steel grade on the minimum degree of shear connection of a composite beam). Construction Metallique, No 3; 1997. p. 40–55. [2] EN 1994-1-1: Eurocode 4: design of composite steel and concrete structures part 1.1. General Rules and Rules for Buildings; 2004. [3] Johnson RP, Molenstra N. Partial shear connection in composite beams. Proc Inst Civil Engineers Part 2 December 1991;Vol. 91:679–704. [4] Banfi M. Slip in composite beams using typical material curves and the effect of changes in beam layout and loading engineering foundation conference V, South Africa; 2005. [5] EN 1992-1-1: Eurocode 2: design of concrete structures. Part 1.1 General–common rules for buildings and civil engineering structures; 2004. [6] EN 1993-1-5: Eurocode 3: design of steel structures. Part 1.5 Plated structural elements; 2006. [7] Byfield MP, Davies JM, Dhanalakshmi M. Calculation of the strain hardening behaviour of steel structures based on mill tests. J Const Steel Res 2005;61:133–50. [8] American Institute of Steel Construction Manual of steel construction, Load and resistance factor design. 3rd edition; 2001.
Contents lists available at ScienceDirect
Journal of Constructional Steel Research
Simplified elasto-plastic analysis of composite beams and cellular beams to Eurocode 4 R. Mark Lawson a,b,⁎, A.H. Anthony Saverirajan c a b c
Dept Civil Engineering, University of Surrey, Guildford GU 2 7XH, UK Steel Construction Institute, UK TEP Consultants Pte Ltd, Singapore
a r t i c l e
i n f o
Article history: Received 20 October 2010 Accepted 16 March 2011 Available online 24 May 2011 Keywords: Composite Beams Elasto-plastic Eurocodes Strain Bending resistance
a b s t r a c t The elasto-plastic analysis of composite beams is important when considering the increase in bending resistance of the beam and the end slip between the steel and concrete at higher strains. This paper provides a simplified method of elasto-plastic analysis by considering equilibrium of the composite cross-section as a function of its strain profile. A parabolic–rectangular stress block for concrete is used in this model with a declining concrete strength at strains exceeding 0.0035. The bending resistance of the composite beam is expressed as a function of the bottom flange strain, and is compared to fully plastic design to EN 1994-1-1: Eurocode 4 and the AISC LRFD Code. The effect of various parameters on the development of the plastic bending resistance of composite beams is investigated, such as asymmetry of the section, the steel strength, the influence of propped or un-propped construction, strain hardening in the steel and reducing concrete strength at high strains, interface slip, and the effect of openings in the web of the beams. It was found that a moment of 95% of the plastic bending resistance of a composite beam (0.95Mpℓ) is reached at a flange strain of 2 to 4 × yield strain for propped beams and 5 to 10 × yield strain for un-propped beams. When strain hardening in the steel is included in the analysis, bottom flange strains at a moment of 0.95Mpℓ are reduced by up to 30% relative to the case without strain hardening. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction The plastic design of composite beams is based on the development of idealised ‘plastic’ or rectangular stress blocks in the steel section and in the concrete slab. The transition from elastic to fully plastic behaviour occurs by the gradual development of plasticity in the composite cross-section as the steel strains increase above the yield strain, εy for the particular grade of steel. Theoretically, it is only possible to reach the full plastic resistance of the composite crosssection Mpℓ at infinitely high strains in the bottom flange. The principles of plastic design may be invalidated for composite sections of highly asymmetric shape or for composite cellular beams with large openings when the plastic neutral is low in the section depth because: • the strains in the concrete may exceed the limiting strain of 0.0035 before the plastic stress blocks are fully developed, which leads to a reducing bending resistance at high strains. • the influence of un-propped construction causes pre-existing strains in the steel section, which means that the plastic resistance of the composite beam is developed at higher strains than for propped beams. ⁎ Corresponding author at: Dept Civil Engineering, University of Surrey, Guildford GU 2 7XH, UK. Tel.: + 44 1483 686617. E-mail address: [email protected] (R.M. Lawson). 0143-974X/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jcsr.2011.03.016
• slip deformations at the interface between the concrete and steel may increase, leading to higher strains in the beam at a given bending moment. • these effects may be accentuated for asymmetric cellular beams with large web openings, where neutral axis depth may be lower in the section than in an unperforated beam. For practical design purposes when using the precise stress–strain models for steel and concrete, the plastic resistance of the composite beam, Mpℓ may be assumed to be developed at a steel strain corresponding to a bending resistance of 0.95Mpℓ. This is reached at a bottom flange strain of up to 10εy. This equivalent plastic moment of 0.95Mpℓ was also used by Aribert [1] in the calibration of the partial shear connection rules of EN 1994-1-1: Eurocode 4 [2]. However, in his calibration study, Aribert considered only propped beams with a span: depth ratio in the range of 30 to 33, which is higher than the typical ratio of 20 to 25 for un-propped beams. The beam proportions have an important effect on end slip. Also in the Eurocode 4 calibration study, Johnson and Molenstra [3] used different models for concrete strength and strain hardening in the steel, and also the beam performance was evaluated at Mpℓ rather than 0.95Mpℓ. In a summary by Banfi [4] of the calibration studies in Eurocode 4, the difference in end slip between un-propped and propped beams was identified, which was not considered in the development of the Eurocode 4 shear connection rules. Banfi also showed that end slip
R.M. Lawson, A.H.A. Saverirajan / Journal of Constructional Steel Research 67 (2011) 1426–1434
varies approximately as the cube of the moment ratio M/Mpℓ acting on the beam. The various assumptions in this paper and in previous analyses are summarised in Table 1. Therefore this paper presents an opportunity to examine the sensitivity of various geometric and material parameters on the development of the plastic resistance of composite beams. The following analysis considers the development of elasto-plastic behaviour of composite beams based on an exact equilibrium analysis. It allows for strain hardening in the steel, the declining strength of the concrete slab at high strains, and interface slip between the steel and the concrete. In un-propped construction, a pre-existing strain is applied to the steel section, which modifies the development of the plastic resistance, although the bending resistances of the propped and un-propped beams tend to the same plastic resistance. This type of behaviour would be extremely complex to model using Finite Element Analysis. The concrete properties are taken from the idealised stress–strain curve in BS EN 1992-1-1: Eurocode 2 [5]. A further unloading branch in the stress–strain relationship is considered in which it is assumed that the strength of concrete declines linearly from its design strength at a strain of 0.0035 to zero at a strain of 0.025. 2. Elasto-plastic analysis — propped beams In the elasto-plastic analysis of a propped composite beam with a solid slab, the bottom flange strain is given by nεy (where n ≥ 1, and defines the multiple of the elastic yield strain, εy). Various cases of neutral axis depth, ye (measured from the top of the slab) may be considered in the calculation of the elasto-plastic properties of the composite section. Ignoring the effects of slip, the strain εc at the top of the concrete slab is established from a linear strain distribution, as function of the bottom flange stain, nεy and neutral axis position, ye, as follows: εc = nεy
ye : hs + hc −ye
NA Fwb
hs
<εy
Fc Fft Fwt
hc Aft
1427
ye-hc
ye
(hs +hc -ye ) /n
tw
hs+hc-ye
εy
Fwp,b Ffb
nεy
Afb
Cross-section
εc
Stress blocks
Strain profile
Fig. 1. Elasto-plastic stress blocks in a composite section.
axis position, ye. This bending resistance may be compared to the idealised plastic bending resistance of the composite section, Mpℓ , which is developed at an infinitely high value of n (or in practice for n N 10). 3. Forces and moments in steel section Consider firstly the case where the neutral axis lies in the steel web, but the top flange is elastic (i.e. top flange strain is less than εy), as shown in Fig. 1. The forces in the steel section are established from the plastic and elastic stress blocks, and the moments of each stress block around the neutral axis position, are determined as follows: Tension force in bottom flange (plastic): Ffb = Afb fy Moment in bottom flange: Mfb = Afb fy hs + hc −ye −0:5tfb Tension force in bottom (plastic) part of web:
The forces in the steel elements are established from the particular strain profile, as illustrated in Fig. 1. For all cases, the extent of the plastic zone in the web is defined geometrically by the parameter, n. The compression force in the concrete Fc is calculated depending on the strains at the top and bottom of the slab — see later. Equilibrium is established at a value of ye at which the tension and compression forces in the composite section are equal. This is an iterative process. In all cases, ye is less than the elastic neutral axis depth measured from the top of the slab, which can be used as a starting point in the iteration. The elasto-plastic bending resistance of the composite beam is obtained by taking moments of each stress block around the neutral
ðn−1Þ −tfb Fwb;p = tw fy ðhs + hc −ye Þ n Moment in bottom (plastic) part of web: ðn−1Þ ðn + 1Þ −tfb −tfb Mwb = 0:5tw fy ðhs + hc −ye Þ ðhs + hc −ye Þ n n
Tension force in bottom (elastic) part of web: Fwb = 0:5tw fy ðhs + hc ye Þ = n
Table 1 Summary of various elasto-plastic analyses of composite beams. Properties used in models
Johnson and Molenstra [3]
Aribert [1]
This paper
Concrete
• Empirical polynomial stress–strain curve • Declining strength after εcu = 0.0035
• Parabolic–plastic stress–strain curve to EC2 • No declining strength after εcu = 0.0035
• Partial factor = 1.5 • Strain hardening immediately after yield • Strain hardening modulus of E/70 to E/200 • Partial factor = 1.0
• • • •
• Parabolic–plastic stress–strain curve to EC2 • Declining strength after εcu = 0.0035 to zero at 2.5% strain • Partial factor = 1.5 • Plastic deformation from εy to 6εy • Strain hardening after a strain of 6εy • Strain hardening modulus of 2.7 kN/mm2
Steel
Bending resistance Beam proportions studied
• Plastic bending resistance, Mpℓ • Existing composite beam tests
• • • • •
Partial factor = 1.0 Plastic deformation from εy to 10 εy Strain hardening after 10 εy Strain hardening modulus obtained from fy at 10 εy and fu at 25 εy Partial factor = 1.0 0.95 Mpℓ considered in parametric study Beams with span: depth ratio of 30 to 33 Propped beams Flange ratios of 1:1 and 1:3
• • • • •
Partial factor = 1.0 0.95 Mpℓ considered in parametric study Beams with span: depth ratio of 25 Propped and unpropped beams Flange ratios of 1:1 and 1:3
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Concrete strength
Moment in bottom (elastic) part of web: 2
0.85 fcyl/γc
2
Mwb = 0:33tw fy ðhs + hc −ye Þ =n
Assumed linear strength reduction from εc>0.0035
Compression force in top flange (elastic):
Fft = Aft fy n
0.002
2 ye −hc −0:5tft
0.0035
0.025
ðhs + hc −ye Þ
Fwt = 0:5tw fy n
2 ye −hc −tft ðhs + hc −ye Þ
Moment in the top (elastic) part of web:
Mwt = 0:33tw fy n
3 ye −hc −tft ðhs + hc −ye Þ
Equilibrium is satisfied for a value ye when: Ffb + Fwb;p + Fwb = Fft + Fwt + Fc For highly asymmetric sections, plasticity can occur in the top part of the web at high strains, as presented in Fig. 2. The top flange strain exceeds εy, when; hs n t : + hc + ð1 + nÞ ft ð1 + nÞ
There are two factors applied to the concrete strength to obtain its design compression resistance. The partial safety factor for the material, γc, is 1.5 for design at the ultimate limit state. In addition, the compression strength of concrete in a uniform stress field is taken as a factor of 0.85 times the concrete's cylinder strength. This factor also takes account of the difference between the area under the parabolic stress–strain curve and an equivalent rectangular stress block. In the design of concrete beams in bending to Eurocode 2 [5], this factor is taken as 1.0. However, in the design of composite beams to Eurocode 4, the 0.85 factor is retained. In this paper, the analyses are based on a concrete strength of 0.85 factor times the cylinder strength divided by a partial factor of 1.5, and the bending resistance at a given strain is compared to a notional bending resistance of 0.95Mpℓ, where Mpℓ is the design plastic bending resistance of the composite beam using the same material strengths. The sensitivity to the 0.85 factor is examined when considering the more precise concrete stress–strain relationship in Fig. 3. Consider firstly the case where the neutral axis lies in the steel section i.e. ye N hc. For a concrete strain of εc ≤ 0.002, the force in the concrete is given by the integral of the parabolic stress–strain curve, as follows:
y
Fc = 0:85fcyℓ = γc ⋅Be ∫y e−h e
Similar equations may be derived for the forces in the steel section and the moments of each stress block around the neutral axis position. 4. Forces and moment in the concrete slab The forces and moments in the concrete slab are calculated from the idealised stress–strain curve for concrete to EN 1992-1-1:Eurocode 2 [5], which is illustrated in Fig. 3. The first part of the stress–strain curve for concrete is parabolic up to a plateau, which corresponds to its design strength. Eurocode 2 gives no information on the performance of concrete after a limiting strain of 0.0035 when the concrete slab is assumed to have reached its maximum compression resistance. This was the case in the previous calibration studies [1,3].
Fc
hc Aft NA hs
Concrete strain
Fig. 3. Parabolic–linear stress–strain curve for concrete with declining strength at high strains.
Compression force in top (elastic) part of web:
ye ≥
Declining strength at high strain
ðhs + hc −ye Þ
Moment in top flange (elastic):
Mft = Aft fy n
Constant strength plateau
Parabolic stress-strain curve
ye −hc −0:5tft
Fwb
Cross-section
ye
1 εc y −hc 2 1 ye −hc 3 εc − e : × 1− + 3 0:002 3 ye ye 0:002
The bending resistance due to the parabolic concrete stress block is given by:
e
Stress blocks
c
2
= 0:566fcyℓ = γc ⋅Be ⋅ye × 1−0:375
2 2 εc 2y εc y − y⋅dy ye ye 0:002 0:002 ε
y
Mc = 0:85fcyℓ = γc ⋅Be ∫y e−h
c
0:002
εc y −hc 3 εc ye −hc 4 − e : + 0:375 0:002 ye 0:002 ye
For the case where the neutral axis lies in the concrete i.e. ye b hc, and εc ≤ 0.002, the force and moment in the concrete reduces to:
hs +hc -ye nεy
Afb
0:002
εc εy Fft Fwp,t Fwt = Fwb(hs +hc -ye)/n
εy
Fwp,b Ffb
2 2 2y εc y − dy ye ye 0:002
c
= 0:85fcyℓ = γc ⋅Be ⋅ye
(hs +hc -ye)/n
tw
εc c 0:002 ε
Fc = 0:85fcyℓ = γc ⋅Be ⋅ye
ε 1 εc c 1− 3 0:002 0:002
Strain profile 2
Fig. 2. Stress blocks with plasticity in the top and bottom flanges.
Mc = 0:566fcyℓ = γc ⋅Be ⋅ye
ε h ε i c c 1−0:375 : 0:002 0:002
R.M. Lawson, A.H.A. Saverirajan / Journal of Constructional Steel Research 67 (2011) 1426–1434
For εc N 0.002, but εc ≤ 0.0035, the concrete force is given by the integral over the complete parabolic stress–strain curve including the rectangular stress block, as follows; "
0:002 εc ye −hc 2 Fc = 0:85fcyℓ = γc ⋅Be ⋅ye 1−0:33 − εc 0:002 ye 2 3 # εc ye −hc + 0:33 : 0:002 ye The bending resistance of the concrete slab over the parabolic– rectangular stress–strain curve is given by: " Mc =
2 0:425fcyℓ=γc ⋅Be ⋅ye
+ 0:5
εc 0:002
2
0:002 2 εc ye −hc 3 −1:33 1−0:17 εc 0:002 ye
ye −hc ye
4 # :
Aft hs
5. Behaviour of concrete at high strains At high steel strains, concrete strains may exceed 0.0035 and theoretically the analysis should stop as the concrete has reached its limiting strain. However, by using the concrete stress–strain relationship in Fig. 3, it is possible to consider the effect of the declining concrete strength on the behaviour of the composite beam. For εc N 0.0035 but ≤ 0.025, the same formulae are used as for the case of a parabolic–rectangular stress block, except that the force and moment in the concrete slab are reduced according to the declining concrete strength. For the case where ye N hc, the force in the concrete, Fc is calculated using the same formula as for εc N 0.002, but is given by subtracting the last term in the following equation: Fc;ðεc N 0:0035Þ = Fc;ðεc N 0:002Þ −0:425fcyℓ = γc ⋅Be ⋅ye
ðεc −0:0035Þ2 : 0:0215εc
The bending resistance of the concrete slab, Mc, is calculated using the same formula as for εc N 0.002, but is given by subtracting the last term in the following equation: N 0:002Þ 2
−0:283fcyℓ = γc ⋅Be ⋅ye
2
ðεc −0:0035Þ 0:0215εc
0:0035 1+ 2εc
where Fc,(εc N 0.002) and Mc,(εc N 0.002) are the force and moment given by the previous equations for 0.002 b εc b 0.0035, but using the actual value of εc N 0.0035. A further case arises when the strain at the top of the slab is so high that the strain in the bottom of the concrete slab exceeds 0.002, which occurs when; ye N
hc : ð1−0:002 = εc Þ
ys
+
εt
yeff
=
tw
hs +hc -ye
εs
A fb
Cross-section
εc ye
Strains acting on steel beam
nεy-εs
Strains acting on composite section
nεy
Combined strains
Fig. 4. Strains in an un-propped beam combining the non-composite and composite strains.
In this case, the above equations are simplified as follows: Fc = 0:85fcyℓ = γc ⋅Be ⋅hc −0:425fcyℓ = γc ⋅Be ⋅ye
ðεc −0:0035Þ2 0:0215εc
2
−0:283fcyℓ = γc ⋅Be ⋅ye
ðεc −0:0035Þ2 0:0035 1+ : 2εc 0:0215εc
6. Influence of un-propped construction
1 0:002 2 2 Mc = 0:425fcyℓ = γc ⋅Be ⋅ye 1− : 6 εc
= Mc;ðεc
εc
Mc = 0:85fcyℓ = γc ⋅Be ⋅hc ⋅ðye −0:5hc Þ
1 0:002 Fc = 0:85fcyℓ = γc ⋅Be ⋅ye 1− 3 εc
N 0:0035Þ
εs ys (hs - ys)
hc
When the neutral axis lies in the slab i.e. ye b hc, and εc N 0.002 but εc ≤ 0.0035, the force and moment in the concrete slab reduce to:
Mc;ðεc
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In un-propped construction, the stresses due to the self-weight of the slab and beam act on the steel section. This has the effect of reducing the stress in the bottom flange that is available for composite action under the loads applied after construction. Conversely in unpropped construction, the slip at the ends of the beam is reduced for a given flange strain. The build-up of strains in an un-propped beam is illustrated in Fig. 4. Assume a pre-existing strain of DLS εY in the bottom flange, where DLS is the dead load strain factor due to the self weight of the beam and slab. DLS is in the range of 0.2 to 0.4 for a typical composite beam, equivalent to a stress = DLS fy in the bottom flange after construction. When the bottom flange strain is εb = nεy, the strain due to composite action of the un-propped beam is given by (n − DLS)εy.The top flange strain, εt, (compression positive), due to composite action combined with the initial stresses on the steel beam, is now given by: εt =
ye −hc ys ⋅ðn−DLSÞεy + DLS⋅εy h + hc −ye h−ys
where ys is the neutral axis depth of the steel section measured from the top of the steel beam — see Fig. 4. The concrete forces and moments are obtained as previously, but are calculated using a reduced concrete strain of: εc =
ye ⋅ðn−DLSÞεy : h + hc −ye
Also, for the case of a composite slab using profiled decking of height, hp, the same formulae for concrete force and moment are used provided hc is defined as the depth of solid concrete above the decking. In the previous formulae for the forces and moments in the steel section, and in formulae for εc and εt above, the total depth of the slab (hc + hp) replaces hc. 7. Slip at the interface between the steel and concrete For cases of partial shear connection, slip occurs between the steel and the concrete. This discontinuity in the strain profile is given by an interface strain εslip, as illustrated in Fig. 5. The integral of the slip strain, εslip over the half span of the beam defines the end slip. For a
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Fc
hc Aft NA
εc Fft Fwp,t Fwt = Fwb (hs +hc -ye) /n
Fwb
hs
hc
hc ye
ht
(hs +hc -ye) /n
tw
hs +hc -ye
εy
Fwp,b Ffb
n εy
Afb
Cross-section
εslip (ye/(ye- hc)) εslip
Stress blocks
hs
A ft
NA
εc
Fc Fft Fwt
ho
hc ht ye -hc -h t h s +hc -hb -ye
Fwp,b Ffb
hb
hb
Cross-section Fig. 5. Stress blocks for an interface slip, εslip between the steel and concrete.
sinusoidal deflected beam shape, the end slip is given approximately by εslipL/π, where L is the beam span. Often in cases of partial shear connection, and particularly in unpropped beams, the upper part of the steel section develops plastic blocks at a bottom flange strain that is less than for the case of full shear connection, as illustrated in Fig. 5.
ye
h s +hc -ye >εy
nεy
A fb
Strain profile
< εy
Stress blocks
Strain profile
Fig. 7. Elasto-plastic stress profile in a composite cellular beam (neutral axis lies within the opening depth).
opening depth is illustrated in Fig. 7. In this figure, the strain profile is such that the bottom Tee is fully plastic, but the top Tee is still elastic. Other design cases may be considered in which the top flange is plastic, as considered earlier for unperforated beams.
8. Strain hardening in steel 10. Results of elasto-plastic analyses Strain hardening in steel may be considered at high strains according to two alternative models. In the Annex to EN 1993-1-5 [6], the strain hardening modulus is taken as Es/100 = 2 kN/mm2. In the model by Byfield et al. [7], the steel strength is taken as its yield strength for strains of εy to 6εy and then increases according to a strain hardening modulus of 2.7 kN/mm2. The EN 1993-1-5 model leads to higher strengths in the normal range of plastic strains. In this study, the Byfield model will be used, and in this model shown in Fig. 6, the steel stress in the bottom flange increases by approximately 1.8% at a strain of 10εy. Although a relatively small effect, strain hardening can be important when the composite bending resistance at constant yield strength asymptotes to 0.95Mpℓ. The results for the two strain hardening models are considered later. 9. Application of elasto-plastic analysis to cellular beams Cellular beams are rolled or fabricated sections with regular circular openings for services distribution. They are designed as composite beams of 12 to 20 m span, and are most efficient as secondary beams with openings typically of 60 to 70% of the depth of the beam. Often cellular beams are highly asymmetric in shape, which are efficient when designed compositely. The partial shear connection rules in EN 1994-1-1: Eurocode 4 clause 6.6.1.2 were developed for I sections with solid webs, which are designed for their plastic bending resistance. For cellular beams, the bending resistance at an opening is the critical design case rather than for the unperforated section. The previous stress block approach may be modified for composite cellular beams by ignoring the contribution of the web within the opening depth. The case where the Neutral axis (NA) lies in the
The following parametric studies were carried out using the elasto-plastic analysis models presented above.
10.1. Bending resistance as a function of bottom flange strain The increase of bending resistance of a composite beam with bottom flange strain is illustrated in Fig. 8 for the case of an un-propped and propped symmetric beam of 800 mm depth. The flange thickness is 18 mm and the web thickness is 10 mm. The line corresponding to 0.95Mpℓ is the bending resistance that is sensibly achieved in plastic design when considering codified steel and concrete stress–stain relationships. It is apparent that the propped beam reaches 0.95Mpℓ at a strain of 3εy, whereas the same un-propped beam reaches this moment at a strain of 7εy. For an asymmetric beam with flange area ratio of 3:1 (bottom flange thickness of 36 mm and top flange thickness of 12 mm), Fig. 9 shows that the propped beam reaches 0.95Mpℓ at less than 2εy, whereas the same un-propped beam reaches this moment at 3εy. However, the rapid decline in bending resistance is apparent at higher strains due to the loss in strength of the concrete. The corresponding steel and concrete strains for a 600 mm deep symmetric composite beam are presented in Table 2 for moment ratios of 0.8 to 0.95Mpℓ. In comparison, the elastic bending resistance of the composite section is approximately 0.75Mpℓ. It is apparent that strains reduce rapidly with reducing moment ratio, M/Mpℓ, particularly in un-propped construction.
Stress fu BS EN 1993-1-5 fsh
Esh = 2.1 kN/mm2
Esh = 2.7 kN/mm2
fy Byfield et al
Es = 210 kN/mm2
εy
6εy
~22εy
Strain,ε
Fig. 6. Strain hardening models for steel (EN 1993-1-5 annex and Byfield).
Fig. 8. Influence of propped and un-propped construction on the bending resistance of 800 mm deep symmetric composite beam.
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Table 3 Influence of beam depth on steel and concrete strains for symmetric un-propped composite beams depending on analysis method. Method
Fig. 9. Influence of propped and un-propped construction on the bending resistance of 800 mm deep asymmetric composite beam.
10.2. Influence of steel strain hardening and the concrete strength factor The influence of strain hardening of the steel and of the 0.85 factor on concrete strength on the development of the plastic bending resistance of a composite beam are presented in Table 3 for the case of un-propped symmetric beams of 400 to 800 mm depth. This shows that the steel strain at a moment of 0.95Mpℓ is insensitive to the section depth but that the concrete strain is higher for shallower sections. Concrete strains can be well above 0.0035 before a moment of 0.95Mpℓ is reached. When strain hardening using the Byfield [7] model is included in the analysis, the maximum steel and concrete strains both reduce by 20 to 30% at a moment of 0.95Mpℓ, which shows the importance of strain hardening on the development of the plastic bending resistance of the composite beam. The effect of the two strain-hardening relationships for steel is presented in Fig. 10 for un-propped symmetric beams. At a moment of 0.95Mpℓ, the bottom flange strain is 5.5εy using the strain hardening model of BS EN 1993-1-5 [6] and 7.5εy using the Byfield [7] strain hardening model. This analysis shows that the strain at a moment close to Mpℓ is sensitive to the precise form of strain hardening considered in the model. The concrete strength factor of 0.85 in the previous formulae is taken directly from Eurocode 4 and takes account of the difference between the parabolic–rectangular stress strain curve for concrete and the equivalent plastic stress block. It may be argued that the 0.85 factor can be set to 1.0 when using the precise stress–strain curve for concrete rather than an idealised rectangular stress block. In Table 3, it is shown that the effect of the 0.85 factor on concrete cylinder strength is significant as it influences both the neutral axis position and also the contribution of the concrete slab to the bending resistance. When taking this factor as 1.0 rather than 0.85, the bending resistance of the composite beam is increased by up to 5%, as shown in Fig. 10, and at a moment of 0.95Mpℓ, the strains in the steel and the concrete are reduced by over 40%.
No strain hardening in steel and with 0.85 factor on concrete strength With strain hardening in steel and with 0.85 factor on concrete strength No strain hardening in steel. Factor on concrete strength set to 1.0
Max. strains
Beam depth 400 mm
600 mm
800 mm
Steel strain, nεy Concrete strain, εc Steel strain, nεy Concrete strain, εc Steel strain, nεy Concrete strain, εc
9.0εy =0.0156
9.0εy =0.0156
8.0εy =0.0138
0.0065
0.0050
0.0041
7.5εy =0.0130
7.1εy =0.0122
6.8εy =0.0118
0.0053
0.0039
0.0031
5.6εy =0.0097
6.1εy =0.0105
6.0εy =0.0104
0.0031
0.0026
0.0026
Data for S355 Steel and C30 concrete. Slab depth = 130 mm with 60 mm deck profile depth. Bending moment = 0.95 Mpℓ.
In Table 4, the effect of using S275 steel for a 600 mm deep asymmetric section is examined, based on the same parameters of strain hardening and concrete strength factor (Fig. 11). It is shown for S275 steel that the steel and concrete strains at a moment of 0.95Mpℓ are only slightly lower than for S355 steel, which indicates that the effect of steel grade on the strains close to the plastic resistance of the beam is not so significant. 10.3. Influence of declining concrete strength The effect of including the declining concrete strength at high strain is illustrated in Fig. 12 for 800 mm deep symmetric un-propped beam. It is apparent that the bending resistance begins to fall at a bottom flange strain of about 4εy, whereas without this declining strength, the bending resistance would otherwise increase gradually. It is important therefore that the effect of declining concrete strength is included for deeper or highly asymmetric beams, as the composite beam may ‘unload’ due to concrete crushing. 10.4. Influence of slip between the beam and the slab In this model, the interface slip between the steel beam and concrete slab is represented by a slip factor, k, multiplied by the concrete strain, εc at the top of the slab, such that the interface slip strain, εslip = k εc. The end slip, δs, may be obtained by integration of εslip along the span, and is given for a uniform loading case by the approximate formula: L δs = k⋅ ⋅εc π
Table 2 Influence of moment ratio on steel and concrete strains for a 600 mm deep symmetric composite section in S355 steel. Method
Maximum Moment ratio M/Mpl strains 0.95 0.9
Un-propped Steel beams strain, nεy Concrete strain, εc Propped Steel beams strain, nεy Concrete strain, εc
0.85
0.8
7.1εy =0.012 3.5εy =0.006 2.2εy =0.004 1.7εy =0.003 0.0039
0.0020
0.0014
0.0011
2.8εy =0.005 1.9εy =0.003 1.2εy =0.002 εy =0.0017 0.0022
0.0016
Data for S355 Steel and C30 concrete. Slab depth = 130 mm with 60 mm deck profile depth. Dead load strain factor = 0.3 in unpropped construction.
0.0011
0.0008
Fig. 10. Comparison of strain hardening relationships on the bending resistance of a 600 mm deep symmetric un-propped composite beam.
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Table 4 Influence of steel grade and section asymmetry for 600 mm deep un-propped composite beams. Method
No strain hardening in steel and with 0.85 factor on concrete strength With strain hardening in steel and with 0.85 factor on concrete strength No strain hardening in steel. Factor on concrete strength set to 1.0
Max. strains
Symmetric beams S275 steel
S355 steel
Asymmetric beam S355 steel
Steel strain, nεy Concrete strain, εc Steel strain, nεy Concrete strain,εc Steel strain, nεy Concrete strain,εc
11εy =0.0147
9εy =0.0156
4.0εy =0.0069
0.0052
0.0050
0.0048
8.1εy =0.0109
7.1εy =0.0122
4.0εy =0.0069
0.0030
0.0039
0.0048
8.6εy =0.0115
6.1εy =0.0105
3.6εy =0.0062
0.0025
0.0026
0.0023
Asymmetric section has 3:1 ratio of flange areas. Slab depth = 130 mm with 60 mm deck depth. Bending moment = 0.95 Mpℓ.
where L is the beam span, which may be taken as approximately 25 × beam depth, hs, for a uniformly loaded composite beam. The analyses were repeated for symmetric and asymmetric beams (with 1:1 and 3:1 flange asymmetry) and slip factors of k = 0, 0.3 and 0.5, as shown in Fig. 13. It is apparent that the influence of slip is relatively small at high steel strains as the bending resistance is dominated by the development plasticity in the steel section. For a uniformly loaded 600 mm deep symmetric beam of 15 m span, the end slip δs is 3.5 mm using the above formula with a slip factor of 0.3, and 6.5 mm for a slip factor of 0.5. In the Eurocode 4 calibration studies, a limiting slip of 6 mm was considered when developing the partial shear connection rules. It should be noted that Eurocode 4 would require a minimum degree of shear connection of 70% for a 15 m span beam of symmetric shape. 10.5. Longitudinal force in the slab as function of moment ratio At the plastic bending resistance of the composite beam, the longitudinal shear force transferred between the steel and the concrete is equal to the smaller of the compression resistance of the slab or the tensile resistance of the steel beam. Using the model based on the steel and concrete stress–strain relationships, the longitudinal forces in the slab reduce with moment ratio, as shown in Fig. 14 for a symmetric unpropped beam. Three slip factors are considered, and the highest longitudinal forces occur for the case of no slip. It may be assumed that the longitudinal forces in the slab reduce approximately linearly with moment ratio, M/Mpℓ. Fig. 13 shows that this linear assumption is conservative for lower moment ratios. As noted earlier, the design moment in the calibration studies is taken as 0.95Mpℓ,
Fig. 11. Influence of concrete strength factor on the bending resistance of a 600 mm deep symmetric composite beam.
Fig. 12. Influence of declining concrete strength on the bending resistance of an 800 mm deep composite beam.
at which point the force in the concrete reaches its design resistance (Table 5). It is proposed that the total longitudinal shear force Fc may be taken as a function of moment ratio as follows; 1:5 Fc = Fc ;pℓ = M =Mpℓ where Fc,pℓ is the compression force in the concrete slab at the plastic bending resistance of the composite beam. This reduction in longitudinal shear force with moment ratio is very important when considering the shear connector forces and requirements for transverse reinforcement in composite beams designed for serviceability limits of deflection or vibration, where moment ratios are generally below 0.8.
11. Results of elasto-plastic analyses for cellular beams The variation of bending resistance of cellular composite beams with bottom flange strain is considered for beams of 800 mm depth and with regular openings of 500 mm diameter. The results are presented in Fig. 15 for symmetric propped and un-propped cellular beams, and in Fig. 16 for asymmetric cellular beams with a bottom: top flange asymmetry ratio of 3:1. For a propped cellular beam, a bending resistance of 0.95Mpℓ is reached at a bottom flange strain of 2 to 3εy. For an un-propped cellular beam, this moment is reached at a bottom flange strain of 8 to 10εy. The corresponding concrete and steel strain are presented in Table 6 for both symmetric and asymmetric cellular beams. For asymmetric cellular beams, the concrete strain, εc does not exceed 0.0035, unlike the same composite section without openings.
Fig. 13. Influence of interface slip factor for a 600 mm deep symmetric un-propped composite beam.
R.M. Lawson, A.H.A. Saverirajan / Journal of Constructional Steel Research 67 (2011) 1426–1434
Fig. 14. Longitudinal force in the slab as a function of moment ratio and slip factor for a 600 mm deep symmetric composite beams.
12. Comparison of eurocode 4 with the American (AISC) code The AISC Limit State (LRFD) Code [8] is similar in general principles of composite beam design to Eurocode 4, but differs in terms of how the partial safety factors for materials are applied. The concrete strength is defined as 0.85fc, but partial safety factors for both concrete and steel are taken as unity to calculate the plastic bending resistance of the composite beam. The design bending resistance in the AISC Code is then obtained by multiplying the plastic bending resistance, Mpℓ by a performance factor of 0.9, which reflects the influence of material safety factors. The results of the elasto-plastic analyses to the two Codes are presented in Table 6. It is found that the design bending resistance to the AISC Code exceeds the plastic bending resistance to Eurocode 4 for shallower sections, but is less than Eurocode 4 for deeper sections. This difference increases with section size because the contribution of the concrete is proportionately less. When using the same stress– strain curve for concrete is in Eurocode 4, it was found that the bottom flange strain and the concrete strain at the design bending resistance to the AISC Code are less than to Eurocode 4. This comparison is presented in Table 6 in which the strains are calculated at a moment of 0.95Mpℓ to Eurocode 4 and at 0.9Mpℓ to the AISC Code (where Mpℓ is the plastic bending resistance calculated to the relevant Code). It is concluded that by using the performance factor of 0.9, the design to the AISC Code leads to lower steel and concrete strains at the design resistance of the composite section than to Eurocode 4. In
Table 5 Influence of cross-section shape on steel and concrete strains for 600 mm deep cellular un-propped beams. Method
No strain hardening in steel and with 0.85 factor on concrete strength With strain hardening in steel and with 0.85 factor on concrete strength No strain hardening in steel. Factor on concrete strength set to 1.0
Maximum Symmetric strains at non-cellular beam
Cellular beams
Steel strain, nεy Concrete strain, εc Steel strain, nεy Concrete strain, εc Steel strain, nεy Concrete strain, εc
9εy =0.0156
8.4εy =0.0145 10.4εy =0.0180
0.0050
0.0028
7.1εy =0.0122
7.1εy =0.0102 7.8εy =0.0135
0.0039
0.0026
Symmetric
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Fig. 15. Influence of propped and un-propped construction on the bending resistance of 800 mm deep symmetric composite cellular beam with 500 mm diameter openings.
particular, the concrete strains calculated according to the AISC design are approximately 40% lower than to Eurocode 4, which has the effect of also reducing the slip in the steel–concrete interface. This partly explains why the limiting degree of shear connection to the AISC Code is not as high as to Eurocode 4. 12.1. Simplified model for composite beams at high strains The results of the elasto-plastic analysis for various composite cross-sections are presented as a function of the multiple n of the yield strain εy in the bottom flange. An approximate relationship between the bending resistance of the composite section and the bottom flange strain nεy is given by: 0:5 M = Meℓ + Mpℓ −Meℓ ððn−1Þ=6Þ Meℓ is the elastic bending resistance of the composite section, which is approximately, Meℓ ≈ 0.75Mpℓ. In this formula, it follows that M = 0.95Mpℓ for n ≈ 5, which is typical of the theoretical results when strain hardening is included in the model. As first shown by Banfi [4], the end slip, s may be assumed to vary conservatively as the cube of the moment ratio, or: . 3 s = 6 M 0:95Mpℓ where s = 6 mm at a moment of 0.95 Mpℓ. This is very important when using the partial shear connection rules in Eurocode 4 for beams designed with low moment ratios. It is proposed that the minimum degree of shear connection for the particular beam span calculated according to Eurocode 4 may be multiplied by the ratio (M/Mpℓ)1.5, where M/Mpℓ is the moment ratio
Asymmetric
0.0035
0.0035
6.1εy = 0.0105 8.0εy =0.0138 9.2εy =0.0159 0.0026
0.0021
600 mm deep cellular beam has 400 mm diameter openings. Asymmetric section has 3:1 flange area ratio. All data for S355 Steel and C30 concrete. Slab depth = 130 mm with 60 mm deck profile depth. Bending moment = 0.95 Mpℓ.
0.0032
Fig. 16. Influence of propped and un-propped construction on the bending resistance of 800 mm deep asymmetric composite cellular beam with 500 mm diameter openings.
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Table 6 Comparison between AISC [8] and Eurocode 4 [2] for symmetric un-propped composite beams designed for full shear connection. Code
Eurocode 4
AISC LRFD
Parameter
Bending resistance, Mpℓ Bottom flange strain at 0.95 Mpℓ Concrete strain at 0.95 Mpℓ Design bending resistance, 0.9 Mpℓ Bottom flange strain at 0.9 Mpℓ Concrete strain at 0.9 Mpℓ
• Design according to the principles of the American AISC LRFD Code leads to lower steel and concrete strains at the design resistance of the composite beam that to Eurocode 4.
Beam depths 400 mm
600 mm
800 mm
1703 kNm
2603 kNm
3666 kNm
8.1εy = 0.0136
7.8εy = 0.0131
7.7εy = 0.0129
0.0049
0.0036
0.0031
1722 kNm
2540 kNm
3503 kNm
7.5εy = 0.0126
7.1εy = 0.0119
6.3εy = 0.0106
0.0029
0.0021
0.0018
Mpℓ is the plastic bending resistance calculated to the relevant Code. Data for 50 ksi steel (S345 equivalent strength) and 3 ksi concrete (C35 equivalent cylinder strength).
acting on the beam at the ultimate limit state. This corresponds to the variation of longitudinal shear force with moment ratio, as defined earlier. 13. Conclusions The following conclusions are made for elasto-plastic design of composite beams: • Strain hardening of steel and declining concrete strength at high strains should be considered in models of composite beams based on elasto-plastic behaviour. The declining concrete strength may be taken as linear from a strain of 0.0035 to zero at a strain of 0.025. • The development of 95% of the design plastic bending resistance of the composite beam, Mpℓ, is a reasonable limit when using elastoplastic models based on the parabolic–rectangular stress–strain relationship and partial factors for concrete. • The 0.85 factor on concrete strength in Eurocode 4 may be set to 1.0 in elasto-plastic methods when using the parabolic–rectangular stress–strain curve for concrete. • A moment 0.95Mpℓ is reached at a bottom flange strain of approximately 6×steel yield strain for un-propped composite beams. • For highly asymmetric or deep beams, unloading of the compression force in the concrete may occur due to crushing at high strains, in which case, the bending resistance reduces rapidly with increasing strain. • The effect of slip between the steel and concrete has a relatively small effect on the development of the bending resistance of unpropped composite beams at high strains. • Asymmetric cellular composite beams reach their plastic bending resistance at higher strains than un-perforated composite beams of the same dimensions. • Longitudinal shear forces in the slab may be assumed to vary in proportion to (M/Mpℓ)1.5 in un-propped construction, where M is the applied moment at the ultimate limit state. • The minimum degree of shear connection required to Eurocode 4 for a particular beam span and steel grade may be reduced according to the multiple of (M/Mpℓ)1.5 for the case of un-propped composite beams. However, in all cases, the minimum degree of shear connection should exceed 0.4.
Notation Beff effective slab width for composite beam design DLS dead load strain factor due to self-weight loads applied to the steel beam E elastic modulus of steel fcyℓ concrete cylinder strength steel yield strength fy steel ultimate tensile strength fu compression force in the concrete slab Fc compression force in the concrete slab at the design Fcpℓ resistance of the composite beam hc depth of concrete slab (above deck profile) depth of deck profile hp height of steel section hs L beam span M applied moment moment due to the compression force in the slab Mc plastic bending resistance of composite beam Mpℓ n multiple of steel yield strain, εy tf thickness of concrete flange (subscript b for bottom flange and t for top flange) ye neutral axis depth of composite section from top of the slab neutral axis depth of un-propped composite section from yeff top of the slab, taking account of strains in the beam after construction. ys elastic neutral axis depth of steel section concrete strain at the top of the slab εc limiting concrete strain (=0.0035) εcu slip strain at the steel–concrete interface εslip top flange strain εt steel yield strain (=fy/E) εy end slip due to slip strain, εslip δs γc partial factor on concrete strength (=1.5)
References [1] Aribert JM. Analyse et Formulation Pratique de l'Influence de la Nuance de l'Acier du Profile sur le Degré Minimum de Connexion Partielle d'une Poutre Mixte (Analysis and practical solutions for the influence of steel grade on the minimum degree of shear connection of a composite beam). Construction Metallique, No 3; 1997. p. 40–55. [2] EN 1994-1-1: Eurocode 4: design of composite steel and concrete structures part 1.1. General Rules and Rules for Buildings; 2004. [3] Johnson RP, Molenstra N. Partial shear connection in composite beams. Proc Inst Civil Engineers Part 2 December 1991;Vol. 91:679–704. [4] Banfi M. Slip in composite beams using typical material curves and the effect of changes in beam layout and loading engineering foundation conference V, South Africa; 2005. [5] EN 1992-1-1: Eurocode 2: design of concrete structures. Part 1.1 General–common rules for buildings and civil engineering structures; 2004. [6] EN 1993-1-5: Eurocode 3: design of steel structures. Part 1.5 Plated structural elements; 2006. [7] Byfield MP, Davies JM, Dhanalakshmi M. Calculation of the strain hardening behaviour of steel structures based on mill tests. J Const Steel Res 2005;61:133–50. [8] American Institute of Steel Construction Manual of steel construction, Load and resistance factor design. 3rd edition; 2001.