lechnical RE
n of C rete
"
k,
Acknowledgements The work of preparing this Report was funded by the following organisations: Arcelor, Bekaert, Propex Concrete Systems, The Highways Agency The Concrete Society is grateful to the following for providing photographs for inclusion in the Report: Arcelor (Figures 3, 4, 7, 8,10, 11 and 36) Bekaert (Figures 6, 9,12,16,17, 20, 22, 24 and 38) Halcrow (Figure 23) Kingspan (Figures 13 and 14) Propex Concrete Systems (Figures 5,15,21 and 37)
Published by The Concrete Society
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Guidance for the Design of
~
~
1
Contents Members of the Project Steering Group List of figures List of tables Notation 1.
2.
3.
Introduction 1.1 Background 1.2 Aims and scope of document 1.3 Terminology Fibres and their behaviour 2.1 Types of fibre 2.2 How steel fibres work 2.21 Enhancement of concrete properties 2.2.2 Comparison with concrete reinforced with conventional steel bars or fabric 2.3 Combination with micro synthetic fibres 2.4 5 usta inabilityhecycli ng Overview of typical applications 3.1 Ground-supported slabs 3.1.1 Industrial floors 3.1.2 Roads and external paving 3.1.3 Overlays 31.4 Railways 3.2 Suspended slabs
V
vi vi i ...
Vlll
1 1 1 2 4 4 5 5
6 7 7 8
8 8 8
9 10 10
3.2.1 General
10
3.2.2 Pile-supported slabs
10
3.2.3 Elevated suspended slabs
11
12
3.3 In-situ concrete 3.4 Composite slabs on steel decking 3.5 Precast units 3.51 Tunnel lining segments 3.5.2 Storage tanks, pipes, etc. 3.5.3 Precast beams and panels 3.6 Sprayed concrete (shotcrete) 3.6.1 General 3.6.2 Tunnel linings 3.6.3 Slope and rock stabilisation 3.6.4 Repairs 3.7 Structures subjected to blast and ballistic loading
18 19 19
4.
Test methods to establish material properties of SFRC 4.1 General 4.1.1 Axial tensile strength of SFRC 41.2 Flexural strength of SFRC 4.2 Beam tests to determine residual flexural strength 4.2.1 Introduction 4.2.2 BS EN 14651: 2005 4.2.3 RlLEM TC-162 TDF 4.2.4 JCI-SF4test 4.2.5 EFNARC beam test 4.2.6 BS EN 14488 beam test 4.2.7 DIN beam test 4.2.8 ASTM beam tests 4.3 Slab tests to determine toughness 4.31 EFNARC 4.3.2 BS EN 14488 plate test 4.3.3 ASTM tests 4.3.4 Statically indeterminate slab tests not covered by Standards 4.3.5 Discussion
20 20 20 21 21 21 22 23 23 24 24 24 24 2s 2s 25 25 26 26
5.
Overview of design processes 5.1 General 5.2 Design on the basis of material properties 5.3 Design assisted by testing 5.4 Design on the basis of performance
28 28 28 28 29
ii
13 14 14 15 15 16 16 17
~
6.
7.
General design approaches 61 Background 611 General 61.2 Elastic design 61.3 Yield line design 6.2 Design recommendations for flexure 6.21 Design in terms of Re,3: Sections without conventional steel reinforcement 6.2.2 Sections with conventional steel reinforcement 6.2.3 Design in terms of BS EN 14651 - Moment:crack width response 6.2.4 Flexural size effects 6.3 Shear strength 6.31 Beam shear 6.3.2 Punching shear 6.4 Serviceability limit state 6.41 Deflections 6.4.2 Cracking 6.4.3 Causes of cracking 6.4.4 Crack control 6.4.5 Minimum reinforcement 6.5 Durability
30 30 30 33 33
Design for specific applications 7l. Fire design 7.2 Ground-supported slabs 7.21 Industrial floors 7.2.2 External paving 7.3 Pile-supported slabs 7.3.1 Background 7.3.2 Elastic design methods
45 45 45 45 46 46 46 47
7.3.3 Yield line design of piled rafts 7.3.4 Serviceability limit state check 7.3.5 Construction details 7.4 Composite floors on steel decking 7.5 Sprayed concrete for rock support
48 50 51 51 53
7.51 7.5.2 7.5.3 7.5.4 7.5.5
I nt roduc t io n Semi-empirical approach Use of toughness characterisation Deterministic design Use of ASTM C 1550 round panel tests
34 34 36 37 39 39 39 41 42 42 42 42 43 43 44
53 54 55 57 60
... 111
8.
9.
7.6 Precast products 7.6.1 General design approach 7.6.2 Tunnel lining segments
62 62 62
7.6.3 Pipes and ancillary products Construction aspects 81 Cast in-situ or precast concrete 811 Specification 81.2 Adding fibres to the concrete 81.3 Pumping 81.4 Placing 8.1.5 Compacting and finishing 81.6 Health and safety 81.7 Testing for fibre quantity and distribution 8.2 Sprayed concrete 8.21 General guidance
64 65 65 65
8.2.2 Testing for fibre quantity and distribution 8.2.3 Health and safety
70 71
In-service performance 91 Durability 9.2 Inspection and repair 9.3 Surface appearance 9.4 Demolition and recycling
I
69 70 70
72 72 72 73 73
References
74
Appendices A. Design of ground-supported slabs A 1 Bending A.2 Punching shear A.3 Other design considerations
79 79 79 81 82
A.4 Comparison of test results with the design approach in TR 34 I
65 66 67 67 69
B.
iv
Design B1 Design for flexure 8.2 Flexural size effects B.21 Size effects in plain concrete beams B.2.2 Size effects in fibre-reinforced concrete 8.2.3 Size effects in RILEM B--E method
82 84 84 85 85 88 90
Members of the Project Steering Group Full members
'
* Lead author for general chapters
** Lead author for sections dealing with sprayed concrete *** Lead author for design chapters
Corresponding members
Neil Loudon Derrick Beckett John Clarke Xavier Destree David Dibb-Fuller Simon Evans JohnGreenhalgh Anne Hoekstra Tilo Hoelzel Nary Narayanan Paul Noble Chris Peaston Tony Rice David S t Quinton lan Simms Nick Swannell Tim Viney Robert Vollum
Highways Agency (Chairman) Consultant The Concrete Society* (Secretary) Arcelor Bissen Cifford Consulting Propex Concrete Systems Be kaert Bekaert Burks Green Clark Smith Partnership Abbey Pynford Arup Arcelor Sheffield Ltd Kingspan Steel Construction Institute Halcrow** Bekaert Imperial College***
Kevin Baker Brian Bell Tom Clasby Phil Rhodes Phi1 Ridge Copal Sangarapillai Marios Soutsos
Jordan Pritchard Corman Network Rail Cement, Concrete and Aggregates Australia Waterman Group WA Fairhurst & Partners NHBC Technical Liverpool University
V
List of figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 Figure 11 Figure 12 Figure 13 Figure 14 Figure 15 Figure 16 Figure 17 Figure 18 Figure 19 Figure 20 Figure 21 Figure 22 Figure 23 Figure 24 Figure 25 Figure 26 Figure 27 Figure 28 Figure 29 Figure 30 Figure 31 Figure 32 Figure 33 Figure 34 Figure 35
Derivation of equivalent flexural stress. Types of steel fibre. Casting industrial floor. Industrial floor in service. Storage area in Southampton Docks. Aircraft hard-standing. Testing elevated suspended slab. Construction of steel-free elevated suspended slab in Walmley, Birmingham. Construction of basement using Insulating Concrete Formwork. In-situ concrete wall. Motorway barrier in Austria. Lining water supply channel. Casting concrete on steel decking. Retail development incorporating composite slabs. Precast tunnel lining units for the ChannelTunnel Rail Link. Trial assembly of tunnel lining ring. Completed tunnel lining. Long-term test on precast beam with fibre reinforcement. Precast roof beams for distribution centre in Erfurt, Germany. Precast wall panels. Sprayed concrete using hand-held ‘gun’. Spraying concrete using robotic arm. Stockholm Metro with permanent shotcrete linings. Stabilised rock face. Typical graph of load against CMOD for SFRC. Typical load: deflection response for SFRC in a beam test. Idealised moment:curvature diagram for strain hardening material. Simplified stress block for SFRC. Simplified stress block for SFRC with supplementary reinforcement. Simplified stress block for deriving M-w response for SFRC. Typical M-w response for SFRC section. EFNARC residual strength and deformation classes. Rock block or zone of loose rock loading sprayed concrete. Potential modes of sprayed concrete failure. Stress block for steel-fibre-reinforced sprayed concrete tunnel linings derived from DBV guidelines. Figure 36 Blast machine adding fibres on site. Figure 37 Adding fibres on site via conveyor. Figure 38 Pumping steel fibre concrete for slabs of 10 storey building.
Figure B1 Structural model for Oleson o-w method. Figure B2 Fictitious crack model. Figure 83 Idealised o-w relationship for plain concrete. Figure 84 Load displacement response of plain concrete beam and associated stress blocks. Figure B5 Influence of doubling beam length and depth on flexural strength. Figure B6 Relative contributions of fibre and concrete t o o-w response of SFRC. Figure 87 Influence of beam depth and crack width on flexural resistance for Oleson o-w relationship with b, = 0.75. Figure 88 Influence of beam depth and crack width on flexural resistance for Oleson o-w relationship with b, = 0.5. Figure B9 RlLEM o-E design method. Figure B10 Stress blocks used in derivation of RlLEM o-E method.
List of tables Table 1 Table 2 Table 3 Table 4 Table 5 Table 6
Properties of steel-fibre-reinforced concrete in relation t o unreinforced concrete. Examples of fibre-only pile-supported slabs.
Table 9 Table AI
Current state of the art. EFNARC residual strength class definition points. Toughness Performance Levels for different tunnel conditions. Correlation of Toughness Performance Level (TPL), Q System rock support classes and fibre reinforced spayed concrete performance. Correlation between EFNARC beam tests and ASTM C 1550 round panel tests by energy equivalence. Correlation between equivalent flexural strength and energy absorption for low deflection situations. Precast concrete segment loading history. Comparison between measured and predicted strengths of ground-supported
Table B1
slabs. Material properties assumed in Figures 87 and 88.
Table 7 Table 8
vii
Notation [Note that only terms that are used a t a number of locations throughout the Report are defined here.] fibre cross-sectional area area of tensile reinforcement total area under the load deflection relationship t o a deflection of 3.0mm secant modulus of elasticity of concrete span peak moment in bending test or peak moment capacity flexural resistance plastic moment of resistance (Section 4.2.2) peak negative moment (used for sections a t which cracking leads t o immediate failure in TR34) peak positive moment (used for sections in which plastic redistribution of moment occurs after cracking in TR34) residual moment at the support when the section reaches M, axial force due t o load or prestress test load, or peak post-cracking load or average load t o a deflection of span/l50 equivalent flexural ratio, maximum deflection 1.5mm equivalent flexural ratio, maximum deflection 3 m m toughness from JCl SF4 test critical fibre fraction volume fraction of fibres
f, and ,f fh 0.2. Linear interpolation can be used for values of a / I between 0 and 0.2.
79
For an internal load with:
all=O P,
= 2n
(MP+ M,,)
(Equation A2a)
a l l 2 0.2
:j
(Equation A2b)
+ ZM,
(Equation A3a)
[
P,=4n(Mp+M,)/
1- -
For an edge load with: a/l=0
P,
+ M,,)
121
= [IT (MP
a I I 2 0.2
[
.?I
PU=[~(Mp+Mn)+4M,,]/ 1 - -
(Equation A3b)
For a free corner load with: a/l=0 P,
=
2Mn
(Equation A4a)
a l l 2 0.2 P, = 4Mn I [I -
(U
I I)]
(Equation A4b)
It should be noted that:
These are ultimate limit state equations and, hence, P, is equal to the applied load times the appropriate partial safety factor for actions. These equations deal with flexure; it is essential to check for punching shear. In the above equations:
MP = =
Mn = =
80
ultimate positive (sagging) resistance moment of the slab (fctk."
1 Yc ) %.3 h2 1 6
(Equation ASa)
ultimate negative (hogging) resistance moment of slab (fctk,fl
1 Y, 1 h2 1 6
(Equation ASb)
where: fCtk," = =
h
=
Re,3 =
characteristic tensile strength of concrete (MPa)
'
1' + (200 )"'I fctk(005) fctk(0.05) slabdepth (mm) equivalent flexural strength ratio.
(Equation A6)
The use of EquationA6 is discussed further in Appendix B, where it appears as Equation B6).
A.2 Punching shear
Design against punching shear a t concentrated loads is based on the approach in the Eurocode for suspended slabs. This means that designs are conservative because no account is taken of transfer of load directly through the slab to the ground beneath. Punching shear capacity is determined in accordance with Eurocode 2 by checking the shear at the face of the contact area and at the critical perimeter situated at a distance 2d from the face of the contact area, where d is the effective depth of the slab, taken as 0.75h for fibre-reinforced slabs, where h is the overall slab depth. Generally, it is the shear on the critical perimeter that governs load-carrying capacity. The shear stress a t the face of the contact area should not exceed vmaxirrespective of the amount of reinforcement in the slab. The value of vmaxis given by: (Equation A7)
vmaX= 0.5 k, fcd
where:
fcd=
k,
=
design concrete compressive strength (cylinder) 0.6 (1 - fck/ 250)
= fck
vc
where:
fck=
characteristic compressive strength (cylinder)
is given by: Hence, maximum load capacity in punching, Pp,max, (Equation A8) where: uo =
perimeter of the loaded area
The shear stress is checked on the critical shear perimeter a t a distance 2dfrom the edge of the loaded area. The slab ultimate load capacity for punching shear (plain concrete only) is given by: Pp= (0.035k3/*fC~/* )u,d
(Equation A9)
where: k I 1+(200/d)05 length of the critical shear perimeter. U, =
81
RlLEM guidance@)suggests that the presence of steel fibres will increase the design shear capacity over that of plain concrete by an amount vf given by: Vf = 0.12 %3
(Equation A10)
fctk fl
where:
fctk," =
characteristic flexural strength of plain concrete
The guidance depends on the presence of conventional reinforcement, but tests on largescale steel fibre-only reinforced ground-supported slabs have shown that application of the above guidance gives conservative results. Thus, for steel fibre-reinforced concrete the slab load capacity, Pp, is given by: Pp= (0.035k3"fC,"*
+ 0.12Re,,fc,k,fl)u,d
(Equation A l l )
Note that the above are ultimate limit state equations and, hence, Pp should be greater than the applied load times the appropriate partial safety factor for actions.
A.3 Other design considerations
A.4 Comparison Of test results with the design approach in TR 34
Equations are also given in TR 3 4 for combinations of loads, line loads and uniformly distributed loads. It also gives guidance on serviceability considerations.
In order t o check the validity of the design equations in Technical Report 34(68), They tested comparisons have been made with the results of tests by Roesler four ground-supported slabs, two with synthetic fibres and two with steel fibres. Only the ones with steel fibres are considered here. The comparisons are made on the basis of the quoted cylinder compressive strengths and flexural strengths, with partial safety factors set initially t o 1.0. Note that the values were measured at 56 days so will be higher than the 28 day values used in the TR 34 design approach . Both slabs were 131.8mm thick and were loaded through a 203mm square steel plate. Thus, a = 114.5mm Average modulus of sub-grade reaction quoted as k = 0.1. Following the procedure outlined above, the comparisons were carried out as summarised in Table Al.
82
Table A1 barison betweenI rneasuiPed and predicted strengths of ground,-supported slabs.
Thus, using the quoted actual cylinder strengths and flexural strengths, with partial safety factors set to 1.0, all failures occurred below the characteristic values. This may be due t o an overestimate of flexural tensile concrete strength (see Section B.2). Nonetheless, if comparisons are made including a materials partial safety factor o f 1.5, i.e. comparing with the predicted design strengths, the slabs failed at 25% above the predicted value, indicating that the approach is safe.
83
Appendix B. Design B.1 Design for flexure
Various methods have been proposed for calculating the moment of resistance of SFRC sections, which can be classified as follows: 1. Discrete crack non-linear finite element analysis 2. Smeared crack non-linear finite element analysis 3. The stress-crack width method o-w 4. The stress-strain method o-E 5. Plastic analysis. The first two of these methods are specialised and will not be discussed further. Of the remainder, the o-w method is the most rigorous but is overly complex for normal use. Methods 4 and 5 are widely used in practice and are described in Section 6.2. Only the o-w method is discussed in this Appendix. The o-w method is a simplified method for modelling the hinge rotation at a crack in SFRC.The RILEM o-w is a useful introduction to the method. Figure B1 shows an idealised o-w structural model and corresponding o-w relationship for SFRC proposed by 0leson(lo6).In SFRC, stress is transferred across cracks by the combined contribution of the concrete and fibres.The crack can be modelled using the fictitious crack model (see Figure 82) that Hillerborg er al.(lo7)developed to model fracture in plain concrete. The tensile stress at the top of the crack is assumed to equal the concrete tensile strength. The crack is subdivided into a fictitious crack of length a with width I w, (see Figure B2) across which tensile stress is transferred in accordance with the o-w diagram. The length of crack below the fictitious crack is referred to as a true crack since it is stress free.
Figure B1 Structural modo1for Olrron
U-w mrthod.
crack a) Geometry, loading and deformation of crack
b) Geometry, loadingand deformation of cracked incremental horizontal strip of hinge
- w Wl w2 c) Definition of parameters of bilinear stress-crack opening relationship;parametersa, and a, are negative slopes of left and right-line segments respectively
84
Appendix B - Design
Figure B2 Fictitious crack model.
I
4 . -x-
4
traction free crack
- -b
aggregate interlock
'
process zone
I
aggregate interlock
process zone
b
4
fibre bridging
IIIII 4
fictitious crack
b
I' Concrete crack
fictitious crack Crack in SFRC
The essence of the o-w method is shown in Figure B1, which shows the geometry, loading and deformation of the cracked hinge proposed by Oleson('O'j). The non-linear response is assumed to be concentrated within the cracked hinge. The moment capacity of the hinge is related to the hinge rotation, which, in turn, is related to the crack width. It is possible to derive the moment-rotation response of a hinge directly from the stress-crack opening relationship, which can be idealised as shown in Figure B1. It should be noted that it is necessary to assume the length of the hinge in the o-w method.The length of the hinge can be derived from calibratingthe method with non-linear finite element analysis. Ulfkjaer have shown that the length of the hinge is approximately half the depth of the beam. Oleson gives closed form equations that define the moment-rotation response correspondingto his cracked hinge model with the o-w relationship shown in Figure B1.
8.2 Flexural size effects 8.23 Size effects in plain
concrete beams
Flexural size effects need to be considered in the design of plain and fibre-reinforced concrete. The flexural strength of concrete is defined as:
fd=6M,,l(bh2) where: M,=
(Equation BI)
peak moment in a 4 point bending test.
85
~-
I-
--?#
Tests show that the flexural strength of concrete is size dependent. The size dependency of fctflis widely recognised but there are significant variations in the concrete flexural strengths adopted in published design guidance as noted below:
Eurocode 2 (ENV) (Equation B2) where:
fctk(0051 = lower characteristic tensile strength of concrete = 0.21fc;'3
(Equation B3)
Dramix Guideline and RlLEM o-E method (Equation B4)
Eurocode 2 (Equation 85)
TR 34 (Equation B6) The size effect terms used in Eurocode 2 and TR 34 are compared in Figure B2, which shows that Eurocode 2 gives significantly lower strengths than TR 34. The TR 34 equation was taken from a draft of Eurocode 2 that was subsequently amended to give the current Eurocode 2 equation. The draft equation was used in an extensive calibration process during the drafting of TR34, which set appropriate safety factors, etc. It is recommended that theTR 34 equation is only used in designs for ground-bearing slabs in strict accordance with TR 34. The size dependency of the flexural strength can be explained with non-linear fracture mechanics. Experimental research shows that concrete is a strain-softening material in tension. In plain concrete, stress is transferred across micro-cracks by crack bridging for crack widths up to around 0.05mrn. The strain-softening behaviour can be characterised by plotting the residual tensile strength against the crack width in a displacement controlled tensile test. Figure B3 shows an idealised o-w diagram for plain concrete.The area under the o-w diagram is equal to the fracture energy, C,.The strain-softening behaviour of plain concrete was first explained by Hillerborg using the fictitious crack concept.The tensile stress a t the top of the crack is assumed to equal the concrete tensile strength. The fictitious crack model can be used to explain why the flexural strength of concrete is greater than its strength in direct tension. Section analysis shows that the peak moment corresponding to the o-w relationship in Figure B3 occurs after the stress reaches the tensile strength a t the extreme tension fibre. Figure 84 shows a load displacement curve for a plain concrete beam in a displacement controlled test and the corresponding stress blocks that change shape after cracking due to the strain-softeningresponse of the concrete in tension.
86
Appendix B - Design
Figure B3 Stress o
Idealised U-w relationshipfor plain concrete.
Crack width w
Figure 84 load displacement response of plain concrete beam and associated stress blocks. Q
(-1
a(+)
-3
0
o(-1
+3
0
-3
0
(+I
+3
I
2
n 4
3
F/b (MN/rn)
E =30,00OMPa
50
100
150
Deflection (pm)
87
Appendix E3 - Design
The presence of a size effect is also explained by the fictitious crack model. Consider the two cracked beams shown in Figure B5.The larger beam has the same width but twice the length and depth of the smaller.The crack in the larger beam is twice as long as that in the smaller. Assume that the stress distribution a t the crack in the smaller beam corresponds to the peak moment and, furthermore, that the crack in the larger beam is twice as wide as that in the smaller beam. Assume that the o-w response is as shown in Figure B3. Therefore, the stress a t the bottom of the fictitious crack is reduced by a factor of 0.5 in the larger beam. It follows that the flexural strength is size dependent. The magnitude of the size effect can be quantified experimentally or numerically using non-linear Finite Element Analysis.
Figure B5 Influence of doubling beam length and depth on flexural strength.
1
t
B.2.2 Size effects in fibrereinforced concrete
88
0.5
aft
t
There is little consensus on size effects in fibre-reinforced concrete beams in flexure. For example, size effects are included in some design methods such asTR 34 and the RlLEM o-E design recommendationsbut not others.The influence of size effects on SFRC varies with loading due to the relative contributions of the concrete and the fibres. The concrete contribution is the stress-crack opening relationship for plain concrete while the fibre contribution consists of an ascending part followed by a softening part as illustrated in Figure B6, which is adapted from the RlLEM o-w design guideline. Figure B6 shows that the concrete contribution dominates a t very small crack widths but is insignificant a t crack widths greater than 0.1-0.2mm.
Appendix B - Design
Figure 86 R e l a t h contributionsof fibre and concrete to a-w responseof SFRC. Stress (MPa) 4.00
i
3.00
2.00
-
Concrete contribution
1.00
0.00 0.01
0.10
1.00
10.00
w (mm)
The influence of fibres on the moment-rotation response of the hinge a t a crack can be modelled using the 0-w method described in Appendix BI. The ratio between the peak moment and the first cracking moment (M, = {bh2/6}fc,)depends significantly on the G-w relationship, which is fibre and concrete dependent. Figures 87 and 88 show the results of a parametric study on the series of beams described in Table 81. The analysis was carried out using the cracked hinge model of Oleson(’06)with the material properties he adopted. A bilinear G-w relationship was used as shown in Figure B1 with b, = 0.75 and 0.5. The moment of resistance was normalised with respect to M, = (bh2/6)fc,.
Figure 87 Influenceof beamd q t h a d crack width on flexural resistancefor Oleson e w d d o n s h i p with ba P 0.75.
-
1.0 . I
--C
0.5
~
Nofibres
+ w=3.Smm b2=0.75
Ir
*
Maxb2=0.75
4
w=l.Smm
+ w=0.5mm b2=0.75 01
0
I
I
I
I
I
I
100
200
300
400
500
600
700
Section depth (mm)
89
Hppenaix tj - Design I
Figure B 8 Influence of b a r n depth and crack width on flexural resistancefor Oleson 0-w relationshipwith ba= 0.5.
2.0 1.8
1.6
1.4
$
-
$ 1.2 \
r,
1.0 0.8
0.6
m
=
I
-f Max b2pO.5
0.4
Max no fibres 0.2 0
+wr3.5mmb24.5 I
0
I
I
I
I
I
I
100
200
300
400
500
600
1
Table B1 Material properties assumed in Figures 87 and B8.
L88I20
3
15
0.2
1
=I
0.75 0.5
Figures 87 and 88 show that the influence of beam depth on the peak moment depends significantly on the U-w relationship. The Re,3value was calculated for the o-w relationship assumed in Figure B1 by analysing a beam with the same loading arrangement and geometry as used in the Japanese beam test. The hinge was assumed to form a t mid-span. The corresponding Re,3value was 0.75. Figures B7 and 88 show that the influence of size effects on the residual moment of resistance after cracking reduces with increasing crack width. This suggests that size effects arise since the design flexural resistance is defined typically as an average flexural resistance up to a prescribed crack width.
8.2.3 Size effects in RlLEM a-E method
The stress block used in the RILEM U--E design method is shown in Figure B9.The design ultimate moment is the least corresponding to either a maximum crack width of 3.5mm or an extreme fibre strain of 25%. In practice, the 3.5mm crack width usually governs for section depths greater than 150mm. The stress block used in the RILEM U--E design method can be simplified as shown in Figure 6.5 with little error. Figures 87 and B8 suggest that size effects are likely to be most significant at CMOD of 0.5mm (corresponding to f R , ) and least significant at CMOD of 3.5mm (corresponding to fR4). The origin of the size effect in the RILEM U--E method (which is almost equal to k,,=I.O - 0.6(h - 12.5)/47.5 for all CMOD where h is in cm) appears to be related to the definition of u2,which is defined in terms of fR,as illustrated in Figure BIO. The background to the size effect is given in Section 3.1.1 of the RlLEM guideline, which states:
~~
Appendix B - Design
factors. A comparison of the predictions of the design method and of the experimental results of structuralelements of various sizes revealeda severe overestimation of the carrying capacity by the design method. In order to compensatethis effect, size dependent safety factors have been introduced. It should be noted that the origin of this apparent size effect is not yet fully understood.”
I
Figure B9
tb
RILEM a-r design method.
. Ec[%O]
2.0A7+
E3
E2
I
3[
E1
I
(d in m)
o1 = 0.7 fctm,fe (1.6 -d) 02 = 0.45 fR1 kh
(N/mm2) (N/mm2)
03 = 0.37 fR4 kh
(N/mm2)
E, = 9500 ( f c t m ) ” 3 kh : size factor
(N/mm2)
I
kh
= 1.0 - 0.6
h [cm] - 12.5 47.5
=ol /E, e2 = E, 0.1 % = 25 %o
+
I 12.5 s h 460 [cm]
I
-
0.2
I I
I
10
I
I
I
20
30
I
40
I
I
1
50
60
70
.
h [cml
1
I
I
t
t
I I
I 1
I
I I I
I
I
4 ,
I
I
1
I
I
I E%
I
I
25
0
-2.0 -3.5
91
I
Figure B10 Stress Mocks used in derivation of RlLEM 0-a
method.
bh'
Ml
=Y f R . 1
M7=
ybh2 f R . 4
Mz= b 0.66h, 0.5h, Of7
OSh,
OSh,
B.2.4Size effects in TR 34
M2=b0.9hq OSh,
Of4
A size effect is incorporated in both design methods given in TR 34 since the design strength is assumed to be proportional to the flexural strength given by Equation 12. TR 34 defines
the design moment of resistance as an average moment of resistance related to Re,, in Equation A5a. Figures 87 and B8 suggest that it is conservative to apply the size factor for plain concrete to the equivalent flexural strength fctkeq300 derived in the Japanesebeam test.
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CONCRETE SYSTEMS
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Guidancefor the Design of Steel-Fibre-Reinforced Concrete
1
This Report summarises the wide range of current applications for steel-fibre-reinforced concrete, including ground-supporteo and pile-supported slabs, sprayed concrete, composite slabs on steel decking and precast units. Practical aspect productionand quality control are also con the material has beem used for a number of years, t agreed design approaches for many of tk- *la----+ . The Report reviews the methods currently L promoting an understanding of the technic, information provided will allow designers tc area of evolving technology
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