Ekkehard Fehling, Paul Lorenz
Institute of Structural Engineering, Department of Concrete Structures, University of Kassel, Germany
1 Differential Equation of Bond
A mechanical model introduced in [1] describes the bond behaviour between steel and concrete under cyclic loading, taking into account energy dissipation. An analytical model introduced in [2] leads to the well known differential equation of bond:
ݏԢԢሺݔሻ ൌ ܷ
ܧݏή ܣݏ൫ͳ ߙܧߩݏǡ݂݂݁൯ ή ߬ሺݔሻ ൌܥ ή ߬ሺݔሻ
(1) The relationship between the second derivative of the slip and the bond stress tሺሻ is based on a constant which implies the contact area U between the two materials, the elastic moduli ratio (aE) and the cross sections ratio (reff) of both.
2 Energy-Based Description of Bond Behaviour
Following an idea of the first author it is possible to formulate the equation in terms of the first derivative of the slip and moving all parts of the equation tho the left side. Partial integration and separation of variables leads to the following expression.
ܹߝൌ ͳ
ʹܥሾݏԢሺݔʹሻʹെ ݏԢሺݔͳሻʹሿൌනݏሺݔʹሻ߬ሺݏሻ݀ݏ
ݏሺݔͳሻ ൌܩܾ
(2) The right side of this expression represents the area under the bond stress-slip law between two values x2 and x1 of slip at the sections ʹ and ͳ. This side can be seen as a length specific bond energy or work done when increasing the slip from ʹൌሺʹሻto ͳൌሺͳሻ. The left side of equation (2) represents the energy or work of strains e. The first derivative of (along the -axis) is the strain difference between the two materials. The difference between the squares of the strains at the positions ʹ and ͳ corresponds to the energy due to the stresses and strains for both materials at the mentioned positions. Figure 1 shows the mentioned relationships which, like equ. (2), imply linear elasticity for the two materials, and the same bond stress-slip law for each section along the reinforcement.
ݔൌ Ͳ ݔʹ
ݔ ܨͲǡݏͲ ܧݏǡܣݏ ܷ ܧܿǡܣܿǡ݂݂݁
ߝܾሺݔʹሻ ߝܾ
ݏሺݔʹሻ
߬ ߝܾሺݔͳሻ
ݏሺݔͳሻ
ݔͳ ݔൌ݈ܾ
නݏሺݔʹሻ߬݀ݏ
ݏሺݔͳሻ
ൌܩܾ
ͳ
=
ܥ නߝܾሺݔʹሻߝܾ݀ߝ
ߝܾሺݔͳሻ
ݏ ݔͳ ݔʹ ݔ
ߝݏെ ߝܿൌߝܾൌݏԢ
Figure 1: Strain and bond energy relationship between the concrete and the reinforcement for any given load 3 Energy-Based Determination of Maximum Force to be transferred by Bond The proposed energy-based approach enables to compute the maximum possible force to be transferred by bond without solving the differential equation explicitly. Besides the elastic and geometric properties expressed by the coefficient , it is sufficient to know the ultimate bond energy while the shape of the bond stress-slip law plays no role in this regard.
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At the loaded section of the reinforcement the strain difference has a maximum value and at the unloaded end section it is equal to zero, because here the strains of both materials are equal. Using the left side of equation (2) it is possible to calculate the strain energy which must be smaller than - or equal to - the ultimate bond energy.
If, however, the required transmission length or the distribution of the strains (or that of the slip) is the objective of the calculations, further considerations become necessary. This is also the case when the required bond length exceeds the possible length available in a structure. In this regard, it should be noted that the bond stress-slip law influences how “fast” the strain difference is reduced along the reinforcement position and how “fast” the slip is reduced from Ͳ (at the loaded section) to zero (at the unloaded section). Figure 2 shows the development of the bond length for three different bond stress-slip laws (decreasing, constant, increasing with the slip) for the same value of the bond energy . The relationships presented here can also be applied to treat bond of reinforcement in UHPC, reinforcement glued to concrete or masonry, long fillet welds, and long connections in steel structures with many bolts in one line after each other.
0 0,5 1 1,5 2 2,5 3 3,5
0 20 40 60 80 100 120
tau [MPa]
x [mm]
0 0,002 0,004 0,006 0,008
0 20 40 60 80 100 120
s' [-]
x [mm]
Bond stress at position x: t(x) Strain difference at
position x: s‘(x) 0
0,05 0,1 0,15 0,2 0,25 0,3 0,35
0 20 40 60 80 100 120
s [mm]
0 0,5 1 1,5 2 2,5 3
0 0,05 0,1 0,15 0,2 0,25 0,3
tau [MPa]
s [mm]
konst steig fallend fallend konst
Slip at position x: s(x) Bond stress-slip law: t(s)
konst steig fallend fallend konst
decreas.
constant increas.
s [mm]
Figure 2: Bond length, slip, strains and bond stresses for same bond energy but different bond stress-slip laws 4 Conclusions
In this paper, it has been concluded that the specific bond energy is the only parameter – and not the shape of the bond stress-slip law – which is decisive for the quantification of the maximum force to be transferred by bond. The proposed energy-based approach can help to find the slip, strain and stress development along the bonded length and the required (activated) bonded length by solving the differential equation of bond. In the case of short existing bonded length, where the slip at the unloaded end is not equal to zero, further considerations are needed. The proposed approach can be extended to any long connection, whatever the bonding materials are.
5 References
[1] Fehling, E.: Zur Energiedissipation und Steifigkeit von Stahlbetonbauteilen unter besonderer Berücksichtigung von Rißbildung und verschieblichem Verbund. Darmstadt, 1990.
[2] Rehm, G.: Über die Grundlagen des Verbundes zwischen Stahl und Beton. Deutscher Ausschuss für Stahlbeton, Heft 138, Verlag Wilhelm Ernst & Sohn, Berlin, 1961.
Session A7: Modelling
131
Experimental research on grouted connections for offshore wind turbine structures using UHPC
Attitou Aboubakr, Ekkehard Fehling, Jenny Thiemicke, Yuliarti Kusumawardaningsih Institute of Structural Engineering, Department of Concrete Structures, University of Kassel, Germany
1 Introduction and objectives
The effect of using different fibre ratio as well as the effect of increasing the thickness of the grout on the behaviour of the connection are analysed and discussed in this paper. Five specimens (SHC-20N-Q0, SHC-20N-Q1, SHC-20-N-Q2, SHC-30-N-Q1, and SHC-30-N-Q2) were tested for this purpose. UHPC M3Q mixture (appr. fc = 180 MPa) with a fibre ratio of 0, 1 and 2% by vol. for specimens SHC-20N-Q0, SHC-20N-Q1 and SHC-20-N-Q2 was used as a grout for these tests. To study the effect of increasing the thickness of the grout, specimens SHC-30-N-Q1 and SHC-30-N-Q2 were tested. The tests were performed at the Institute of Structural Engineering Laboratory at the University of Kassel.
2 Fabrication of test specimens
The trapezoidal shaped shear with a height of 3 mm and a width of 10 mm were created by drilling on the pile and sleeve. Figure 1 shows the shear keys shape, specimen with 20 and specimen with 30 mm.
Figure 1: The dimension of specimens with grout thickness of 20 and 30 mm.
UHPC M3Q mixture was elaborated at Kassel University during the work on the priority programme (SPP1182) of the German Research Foundation. The M3Q mix provided an average compressive strength of concrete cylinders at about 180 MPa . Only one type of steel fibres with a length of 13 mm and a diameter of 0.175 mm was used for all tests with equal tensile strength of 2500 MPa.To assure that the steel tubes remained concentric during the process of concrete pouring a wooden template was manufactured on an European pallet.
Figure 2 shows the specimens during the fabrication and after pouring the concrete.
Figure 2: The specimens during the fabrication and after pouring the concrete.
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Reference tests have been done to control the quality of the mix and to make a comparison with other previous studies. Reference tests referred to testing of fresh grout by slump cone test and testing of hardened grout by compressive strength test, direct tensile test, splitting tensile test and flexural strength test.
3 Loading and experimental test procedures
Thirty-six strain gauges were installed in both the axial and hoop direction and placed at 120°
spacing around the sleeve and pile perimeter. The relative axial displacement of the connection was recorded by using four Linear Variable Differential Transducers (LVDTs) which were located at 90° apart from each other. The specimens were tested in a universal hydraulic testing machine with a capacity of 6.3 MN. The specimens were loaded till the connection failed. The load was applied with a constant displacement rate of 0.002 mm/s. Mostly, it was observed that the performance of the specimens, after the peak at a displacement of 5 mm, had no significant change with time, therefore, the displacement rate then was increased up to 0.02 mm/s.
4 Test results and discussions
The results of the tests show that the stiffness of the grouted connection was somehow affected by the fibre ratio for specimens with grout thickness of 20 mm while there is not a significant increase in ultimate load or in ductility for connections with 20 mm grout thickness. However, for specimens with grout thickness of 30 mm, there was a significant increase in ultimate load and ductility while the stiffness of the grouted connection was not affected by the fibre ratio for these specimens. Furthermore, the specimens with grout thickness of 30 mm have the same general shape of axial load-displacement curves.
Figure 3: Load-displacement curve for the specimens.
5 Conclusions
Five specimens were tested to study the effect of using different fibre ratio and the effect of increasing the thickness of the grout on the behaviour of the connection The test result indicates that there is a fibre ratio effect for the grouted connection with 30 mm and 20 mm grout thickness, which enhances the behaviour of the grouted connection.
References
[1] Dallyn, P.; El-Hamalawi, A.; Palmeri, A.; Knight, R.: Experimental testing of grouted connections for offshore substructures: A critical review, 2352-0124/© 2015, The Institution of Structural Engineers.
Published by Elsevier Ltd
[2] Lamport, W.B.: Ultimate Strength of Grouted Pile-to-Sleeve Connections, PhD-Thesis, University of Texas at Houston, 1988.
[3] Aboubakr, A.: Behaviour Study of Grouted Connection for Offshore Wind Turbine Structures with Brittle Cement Based Grouts, PhD thesis, University Kassel (publication in preparation, 2020).
Session B7: Applications II
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