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Researchers
Abstract
Introduction
Results and discussion
Conclusion
Références |
BODOL MOMHA Merlin 1 , AMBA Jean Chills1,2*, DJOPKOP KOUANANG Landry1,2, ELAT Emmanuel 1,3, NKONGHO ANYI Joseph1,4, NZENGWA Robert1,2
1 Laboratory of Energy Modeling Materials and Methods (E3M)
2 National Higher Polytechnic School of Douala, Cameroon, PO Box 2701, Douala
3Laboratory of Mechanics and Materials of Civil Engineering (L2MGC), CY Cergy Paris University, 5 Mail Gay Lussac, Neuville sur Oise, F-95031 Cergy-Pontoise Cedex, France
4Department of Mechanical Engineering, Higher Technical Teachers Training College of the University of Buea in Kumba, Kumba, Cameroon
Corresponding author: jc.amba@enspd-udo.cm
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Abstract
The paper focuses on the influence of differential shrinkage linked by drying at the early age and stresses distribution of a concrete ring specimen. Depending on the gradient of dimension changes through the thickness, tensile stress occurs near the exposed surface where drying is greater and thus results in strains gradients development. An experimental design was carried out on a concrete ring cast in the laboratory conditions in order to monitor strains and displacements at different positions of the thickness. A multilayer approach in the drying conditions based on the displacement of the sheets and their interactions has been implemented in order to simulate the ring’s behaviour in drying conditions. The comparison of experimental results with numerical simulation shows that drying and tensile creep phenomena have the most important influence on the early age stress development in the walled ring.
Key words: multilayer shell; moisture; differential shrinkage; tensile creep; modeling
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Introduction
In order to lighten structures and improve aesthetic appeal, architects and engineers increasingly use curved structures. Material such as concrete, steel and composites are the most commonly used. In a particular case of concrete, these structures are sensitive to internal strains which could lead to failure [1]. It was noticed that the delayed strains have a preponderant part in the rupture of the shell structures in cementitious materials and this in the absence of apparent overloads. However little attention is paid to the distribution of the shrinkage in the thickness of the material. The distribution of the shrinkage in the core of the material can be important because it turns out that the concrete shells undergo to crack in the absence of any load. It is believed that the cracking of concrete pipes occurring a few days after casting is linked by drying since the amplitude of shrinkage increases with the rate of evaporation of the water contained in the mixture [2]. Bazant and Najjar [3] showed that the hydration of cement is proportional to the relative humidity of the pores. This suggests that, the hydration degree is a function of considered point position and consequently, the Young’s modulus may be defined by field points depending on the position and moisture content. The behaviour of the concrete is therefore found to be strongly non-linear and accentuated by the property gradients [4]. In this study, the water content distribution is obtained by solving the porous media transport equation.
With regard to the distribution of the Young's modulus in the thickness of the material, everything happens as if the shell is composed of several layers with different properties occupying various geometric positions. The multilayer approach introduced by Hand [5] seems suitable for solving this type of problem. In this work, the chemo-hydro-mechanical (CHM) approach is used to model displacements. There are many kinematic models of shells in the literature. All these formulations differ by the modeling of the transverse shears. In this paper, the Nzengwa and Tagne (N-T) [6] two dimensional model for linear elastic thick shells will be used. The numerical convergence and validation of this model can be found in the literature [7], [8].
Hygral-mechanical finite element modeling
All the calculation results presented in the context of this work were obtained using the Matlab software which is an interdisciplinary platform. The first order quadrangle elements are used for the calculation of the water content variation.
Moisture transport problem
The exposed concrete surface loses water by evaporation. The rate of evaporation depend of many criteria such as environment humidity, heat etc. [10]. We assume that just after casting (time ), the entire water contained in the material is the free water. The water likely to be evaporated is free water which is mobilised for cement hydration. At time t ( ), water contained in the material is the sum of free and bound water. Let the total water content at time t
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| The differential drying will cause differential shrinkage. If each layer could deformed independently from its neighboring layers, no stress can be generated. In reality, the layers in the element are monolithically interconnected [12]. The eccentric shells, constituting the layers, do not interact directly: it is combination of displacements at the level of the reference surface as well as the position of each layer an overall response is obtained [13], [14]. |
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Figure 2: (a) Multilayer shell representation; (b) Moisture variation in the cross section of multilayer shell
Consideration of strains
The strains recorded on the benchmark represent the global response of the material. The shrinkage strains can be expressed as follows:
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Figure 3 : Geometry and triangular shell element used for numerical simulation
Table 2: Geometry and mechanical data of the cylinder
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Length L = 0.5 m
radius R= 0.425 m
Thickness h = 0.1 m
Poisson’s Coefficient υ = 0.17
Conditions of symmetry Ux = 0 on AB; Uy = 0 on DC;
Limit Conditions Uz = 0 on BC
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Main data of tested shell
In this work, two cylindrical specimens (Figure 4& 5) of unreinforced concrete was cast, instrumented and subjected to drying in laboratory conditions (Temperature 22°C±1, relative humidity 70±5%). The shell is cylindrical, with an internal radius of R1 = 37.5 cm and an external radius of R2 = 47.5 cm. The ratio thickness over the smallest radius in absolute value is equal to where h = 10 cm. The aggregate used in concrete mixture has a maximum size of 1.5 cm. In this study, in order to considering granular inclusion, we subdivided the thickness into 3 layers of 3.3 cm each.
The cement used in this study is an ordinary Portland cement CEM I 42.5 N. The uniaxial compressive strength of concrete was determined on the cubic specimen of 10 cm side at 28 days. The mix proportions and the average compressive strength (f_c) are presented below:
Table 1: Composition of concrete for 1m3
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| For shrinkage and mass loss measurement, concrete specimens of size of and others prisms specimens of size of were cast. The measurements of shrinkage were started after form removal (24 hours after casting), then, the specimens were placed in the controlled room |
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Autogenous shrinkage was measured on 3 specimens of size of sealed with adiabatic waterproof sheet to prevent drying. Also the prisms used for monitoring the mass loss are sealed with waterproof paint and only one side is left free for drying.
The rings specimens were demolded 24 hours after pouring. After that period, the waterproof paint is applied to the inner and top faces of the ring and then the whole is covered with the polyan sheet for 22 hours. The instrumented ring specimens, data acquisition system and steel molds used are shown in Figure 5. We considered:
- The micrometer-sensitive digital comparators for displacements measurement. To monitor the displacements, the comparators are placed at different points of its thickness along the x-axis from the centre of ring, respectively at 39 cm (C3), 42.5 cm (C2), and 46 cm (C1);
- The strain gauges are bonded in the both inner face and outer face of the rings 22 hours after steel form removal. The adhesive used consists of two-component ‘M-Bond AE-15 Kit for strain gages’ proposed by MICRO MEASUREMENTS. This adhesive is highly resistant to moisture. For strains recovering, the strain gauges are fixed along the hoop direction of the rings;
- A data acquisition system in a quarter-bridge configuration for strains recording;
- A plastic film was placed below to prevent any suction of water from the test piece through the substrate and allow a free displacement at the base of specimen.
After preparation and instrumentation, the rings were exposed to drying after removal of the plastic film. This corresponds to a cured period of 48 hours after casting.
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Results and discussion
The experimental campaign carried out makes it possible on the one hand to determine the loss of mass in order to carry out the calibration of the coefficients of drying and on the other hand, the determination of the coefficients of water contraction. Based on the results of mass loss measurements, the diffusion coefficient is defined by parameter and we used to considering the exchange with environment. The numerical simulation carried out on the basis of the collected data gives the following results (Figure 6-9).
We note a low variability of the modulus in the thickness (Figure 9) with a maximum coefficient of variation estimated at 0.4% at 4 days. There is a slight variation in the modulus for the thickness h considered, the sheets drying almost uniformly. Globally, the non-uniform moisture distribution is observed. The internal moisture differs significantly according to the depth from exposed (h = 10 cm) surface where the free water content is small as compared to rest of the thickness. The results of radial displacements are given in Figure 10.
The results show that the radial displacement near the exposed surface is greater as compared to those measured near the inner face and midsurface. At the same time, the radial displacement measured near the inner surface is lower than the two others and can be considered practically constant. Also, the increase of displacements is globally observed at the half outer layer of the ring. The evolution of strains measured on benchmark at the outer and inner surfaces are presented in Figure 11. It should also be noted that the external deformation measurement gauge (DataR2ext) on ring 2 ceased to operate after approximately five days. The problem is believed to be related to the connection of said gauge.
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Figure 12: (a) Total strains versus time without relaxation effect; (b) Total strains versus time with relaxation effect
The variation of the deformations is similar to a signal comprising white noise. We believe that these noises or fluctuations are related to the adaptation of the material due to the fact that the outer sheets deform and move much more than the inner sheets which are relatively protected from desiccation. At the start of the drying process, all the layers are observed to compress. Then, at a given moment, the deformations of the outer sheet change sign, thus giving proof of the existence of a tensile tension in the body of the material as reported by Ranaivomanana [17].
Figure 12 show the variation of the simulated strains versus time. We note that the outer calculated strain continue to grow as if the outer strain are directly proportional to water loss. The simulated inner strain begin to grow with the same similitude than the measured strain after what it change the sense of its variation. When the stress is imposed on the concrete at a date t0, we show that the stress sigma is equal to the stress minus the relaxation [18]. If the relaxation phenomenon is taken into account, kinetic changes are observed on the evolution of the strains, although those simulated decrease with the lowering of the kinetics, compared to those recorded (Figure 12-b). However, this phenomenon allows us to simulate the change or even the attenuation of the kinetics of evolution of the strains in the thickness of the ring.
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Conclusion
The multilayer model was used with a view to highlighting the deformations of the extreme layers of the shell. The model would be likely to perform better with a thick shell favoring the great variation of the Young modulus and the increase of the number of sub-layers.
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Acknowledgement
The authors would like to thank the research team of the Civil Engineering laboratory of SJP- Polytechnic Institute of Douala as well as that of the L2MGC laboratory of CY-Cergy Paris University for providing necessary equipment for data acquisition during the course of this research. Our special thanks go to the following people: Prof. Albert Noumowé; Prof. Anne-Lise Beaucour; Prof. Javad Elsami; Msc Steve M. Nkomom; Msc. Hugues S. Taowe
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Références
[1] Zdeněk P. Bažant, RILEM draft recommendation: TC-242-MDC multi-decade creep and shrinkage of concrete: material model and structural analysis, Materials and Structures. 2015; 48:753-770.
[2] L. Zhang, X. Qian, J. Lai, K. Qian et M. Fang, Effect of different wind speeds and sealed curing time on early-age shrinkage of cement paste, Construction and Building Materials, 2020; 255, 119366.
[3] Z. P. Bažant &. L. J. Najjar, Nonlinear water diffusion in nonsaturated concrete, Matériaux et Construction. 1972 ; 5:3-20.
[4] J. C. Amba, J. P. Balayssac et C. H. Détriché, Characterization of differential shrinkage of bonded mortar overlays subjected to drying, Materials and Structures, 2010; 43:297-308.
[5] F. R. Hand and al., A layered finite element nonlinear analysis of reinforced concretep lates and shells, Urbana, Illinois: Structural research series N° 389, University of Illinois, 1972.
[6] R. Nzengwa et B. H. Tagne Simo, A two-dimensional model for linear elastic thick shells, International Journal of Solids and Structures, 1999; 36:5141-5176.
[7] J. N. Anyi, R. Nzengwa, J. C. Amba et C. V. Ngayihi Abbe, Approximation of Linear Elastic Shells by Curved Triangular Finite Elements Based on Elastic Thick Shells Theory, Mathematical Problems in Engineering, 2016, Article ID 8936075.
[8] J. N. Anyi, J. C. Amba, D. Essola, C. V. Ngahiyi Abbe , M. Bodol Momha et R. Nzengwa , Generalised assumed strain curved shell finite elements (CSFE-sh) with shifted-Lagrange and applications on N-T's shells theory, Curved and Layered Structures, 2020; 7:125-138.
[9] S. Wang, Thermal analysis of cylindrical concrete shell at transition boundary between regions with different reinforcement configurations, Engineering Structures, 2015; 84:279-286.
[10] M. S. Yatagan, The Investigation of the Relationship between Drying and Restrained Shrinkage In View of the Development of Micro Cracks, Journal of Engineering Technology. 2015; 3 (19):7-14.
[11] R. Mensi, P. Acker et A. Attoulou, Séchage du béton: analyse et modélisation, Materials and Structures, 1988; 21:3-12.
[12] G. Martinola, Hamid Sadouki et Folker H. Wittmann, Numerical model for minimizing risk of damage in repair system, Journal of Materials in Civil Engineering. 2001 ; 13 (2) :121-129.
[13] A. AOUAMEUR-MESBAH, Analyse non-linéaire matérielle et géométrique des structures coques en béton armé sous chargements statiques et dynamiques. 1998. Thèse de Doctorat de l'Ecole Nationale des Ponts et Chaussées.
[14] J. N. Anyi et al., Investigation of the macroscopic behaviour of laminates shells (MBLS) under varying loads using low order CSFE-sh FEM and the N-T’s 2-D shell equations, PLOS ONE 2022; 17(5): e0267480, 2022.
[15] G. De Schutter et L. Taerwe, Degree of hydration-based description of mechanical properties of early age concrete, Materials and Structures, 1996; 29:335-344.
[16] James A. Rhodest and al., Prediction of Creep, Shrinkage, and Temperature Effects in concrete structures, Comité ACI 209, 1997.
[17] N. Ranaivomanana, S. Multon et A. Turatsinze, Basic creep of concrete under compression, tension and bending, Construction and Building Materials, 2013; 38:173:180.
[18] J. COURBON, Fluage et relaxation du béton,» Techniques de l’Ingénieur, traité Construction, Doc. C 2 055.
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