The theoretical stiffness of the undamaged UHPFRC slab was derived based on the elastic theory of thin plates with small deflections. The bending strain energy in a slab undergoing an elastic deformation is given by
4 ∫ ∫ ∫ (1 2𝐸 ∈𝑥𝑥 2 + 𝐸 2(1 + 𝜐)∈𝑥𝑦 2 + 1 2𝐸 ∈𝑦𝑦 2 ) 𝑑𝑥 𝑑𝑦 𝑑𝑧. 𝑧 −𝑧 𝑌 0 𝑋 0 (4.40)
Similarly, the bending strains are given by
∈𝑥𝑥= 𝑧 (𝜕 2𝑤 𝜕𝑥2) ∈𝑥𝑦= 2𝑧 ( 𝜕2𝑤 𝜕𝑥𝜕𝑦) ∈𝑦𝑦= 𝑧 ( 𝜕2𝑤 𝜕𝑦2) , (4.41) where
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X,Y = half span of the slab in x-axis and y-axis, respectively,
x,y = any point along the half span in x-axis and y-axis, respectively, z = half slab thickness,
E = Modulus of Elasticity, υ = Poisson Ratio, and
w = shape function (see Table 4.5)
Assuming that the energy loss in bending is negligible, the potential energy (U) stored in the slab is taken as
U = Bending strain energy – Work done
U = Bending strain energy – Fδ . (4.42)
At equilibrium,
𝑑𝑈
𝑑𝛿 = 0 . (4.43)
The slab stiffness is determined by solving equations (4.41) to (4.43) and equating the solution with the Hooke’s Law
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The Modulus of Elasticity (E) in bending was estimated from a static 3-point bending test by re-arranging the equation
𝛿 = 𝐹𝐿 3 48𝐸𝐼 to be (4.45) 𝐸 = 𝐹𝐿 3 48𝛿𝐼 , (4.46) where F = Force (N),
L = support to support distance (mm), I = second moment of area (mm4), and δ = deflection (mm).
The results of the 3-point bending test on a UHPFRC beam with the same mix design conducted by Mahmud et al. [22] were used. The test involved three notched beams measuring 150 mm × 150 mm section and 500 mm support to support distance. The experimental work measured the load versus crack mouth opening displacement (CMOD) and was transformed into load versus deflection values using the following relationship as recommended by BS EN 14651 [23]:
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Figure 4.21 shows the load-CMOD curve from Mahmud et al. [22] together with the equivalent load-deformation curve using equation (4.47). The elastic load-deformation relationship was considered up to 32.23 kN with a deflection of 0.0925 mm. Incorporating these values in equation (4.46) provides the value for E = 21.51 × 103 MPa.
Figure 4.21: Load-CMOD curve [22] and the equivalent load-deformation curve. 0 10 20 30 40 50 60 70 0 1 2 3 4 5 F o rc e (k N) CMOD/Deformation (mm) CMOD Deflection This text box is where the unabridged thesis included the following third party copyrighted material:
Mahmud GH, Yang Z, Hassan AM. Experimental and numerical studies of size effects of Ultra High Performance Steel Fibre Reinforced Concrete (UHPFRC) beams. Construction and Building Materials. 2013;48:1027-1034.
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Finally, the maximum resistance force (RM) was calculated based on the assumption that the crack lines follow the concept of the yield-line theory for a fully fixed slab subjected to a concentrated load as shown in Figure 4.22.
Figure 4.22: Yield-lines for a fully fixed square slab subjected to a concentrated load.
By using the Work Method and considering the segment ABC:-
Work done = Internal Energy
𝑅𝑀 4 × 𝐿 = 2𝐿 × (𝑚𝑢+ 𝑚𝑢′) 𝑅𝑀 = 8 (𝑚𝑢+ 𝑚𝑢′) ; 𝑚𝑢 = 𝑚𝑢′ = 𝑀𝑢𝑙𝑡 𝑅𝑀 = 16 𝑀𝑢𝑙𝑡 (4.48) Pult 2L
negative moment yield line
positive moment yield line
2L B A C mu mu’
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Mult = ultimate moment of resistant,
Pult = ultimate load at collapse = RM ,
mu & mu’ = positive and negative moment of resistance, respectively.
The ultimate moment of resistant (Mult) was then calculated based on the plastic solution proposed by Spasojevic [24]: 𝑀𝑢𝑙𝑡= ( 3 − 2√2𝑓𝑐𝑡 √𝑓𝑐𝑡(𝐸𝑐𝜀𝑓𝑐𝑡 + √𝑓𝑐𝑡(2𝐸𝑐𝜀𝑓𝑐𝑡− 𝑓𝑐𝑡) ) ) . 𝑓𝑐𝑡 . 𝑡2 6 , (4.49) where,
𝑓𝑐𝑡 = tensile strength (MPa),
𝐸𝑐 = Young modulus of elasticity (MPa),
𝜀𝑓𝑐𝑡 = maximum tensile strain at the end of the pseudo-plastic tensile plateau, taken as 0.025 (as in Figure 4.17), and
t = slab thickness (mm). .
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4.8 Summary of Chapter 4
This chapter presented the numerical simulation procedures to predict the response of the UHPFRC slabs under low velocity impact. ANSYS Explicit Dynamics R13.0 software package were used for the FE simulations involving a single impact event. The modifications adopted in generating the 3-D models from the actual test set up were shown. Such modifications involve reducing the length of the impactor, simplifying the support condition and reducing the mesh quantity by utilising half model. This chapter also presented the concept of developing the pre-damaged condition to the slab prior activating the impact loading. The modelling of the pre-damaged effect in the dynamic simulation was carried out by selecting a higher pressure loading with shorter load duration compared to the actual static condition. RHT concrete model was used to represent the dynamic properties of the UHPFRC materials. The input parameters for using RHT concrete model were identified and several parameters were calibrated using available test data. The results from the calibration exercise were discussed. Finally, in the analytical modelling, the methods to derive the SDOF parameters from a static test, theoretical equations and other published work were also presented.
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4.9 References
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[3] J. Leppanen, “Concrete subjected to projectile and fragment impacts: Modelling of crack softening and strain rate dependency in tension,” International Journal of Impact Engineering, vol. 32, no. 11, pp. 1828-1841, 2006.
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[7] Z. Tu and Y. Lu, “Evaluation of typical concrete material models used in hydrocodes for high dynamic response simulations,” International Journal of Impact Engineering, vol. 36, pp. 132-146, 2009.
[8] A. M. T. Hassan, S. W. Jones and G. H. Mahmud, “Experimental test methods to determine the uniaxial tensile and compressive behaviour of ultra high performance fibre reinforced concrete (UHPFRC),” Construction and Building Materials, vol. 37, pp. 874-882, 2012.
[9] A. M. Neville, Properties of concrete, 5th ed. London: Prentice Hall/Pearson Education Limited, 2011.
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[10] Y. Farnam, M. Moosavi, M. Shekarchi, S. K. Babanajad and A. Bagherzadeh, “Behaviour of slurry infiltrated fibre concrete (SIFCON) under triaxial compression,” Cement and Concrete Research, vol. 40, no. 11, pp. 1571-1581, 2010.
[11] T. H. Tan, “Effects of triaxial stress on concrete,” presented at the 30th Conference on Our World in Concrete and Structures, Singapore, Aug. 23-24, 2005.
[12] J. M. Magallanes, Y. Wu, K. B. Morrill and J. E. Crawford, “Feasibility studies for a plasticity-based constitutive model for ultra-high performance fiber-reinforced concrete,” presented at the 21st International Symposium on Military Aspects of Blast and Shock, Jerusalem, Israel, Oct. 4-8, 2010.
[13] K. Habel and P. Gauvreau, “Response of ultra-high performance fiber reinforced concrete (UHPFRC) to impact and static loading,” Cement and Concrete Composites, vol. 30, no. 10, pp. 938-946, 2008.
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[17] S. G. Millard, T. C. K. Molyneaux, S. J. Barnett and X. Gao, “Dynamic enhancement of blast-resistant ultra high performance fibre-reinforced concrete under flexural and shear loading,” International Journal of Impact Engineering, 37(4), pp. 405-413, 2010.
[18] T. Ngo and P. Mendis, “Modelling reinforced concrete structures subjected to impulsive loading using concrete lattice model,” Electronic Journal of Structural Engineering, vol. 8, pp. 80-89, 2008.
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[19] A. M. T. Hassan and S. W. Jones, “Non-destructive testing of ultra high performance fibre reinforced concrete (UHPFRC): A feasibility study for using ultrasonic and resonant frequency testing techniques,” Construction and Building Materials, vol. 35, pp. 361-367, 2012.
[20] J. M. Biggs, Introduction to Structural Dynamics. New York: McGraw-Hill Book Company, 1964.
[21] K. Fujikake, B. Li and S. Soeun, “Impact response of reinforced concrete beam and its analytical evaluation,” Journal of Structural Engineering, vol. 135, no. 8, pp. 938- 950, 2009.
[22] G. H. Mahmud, Z. Yang and A. M. Hassan, “Experimental and numerical studies of size effects of Ultra High Performance Steel Fibre Reinforced Concrete (UHPFRC) beams,” Construction and Building Materials, vol. 48, pp. 1027-1034, 2013.
[23] British Standards Institution. Test method for metallic fibre concrete – Measuring the flexural tensile strength (limit of proportionality (LOP), residual), BS EN 14651:2005+A1:2007, 2007.
[24] A. Spasojevic, D. Redaelli and A. Muttoni, “Thin UHPFRC slabs without conventional reinforcement as light-weight structural elements,” presented at the fib Symposium, London, June 22-24, 2009.
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