3D fracture modelling and limit state analysis of prestressed composite concrete pipes

Abstract

In this manuscript, we study fracture of prestressed cylindrical concrete pipes. Such concrete pipes play a major role in tunneling and underground engineering. The structure is modelled fully in 3D using three-dimensional continuum elements for the concrete structure which beam elements are employed to model the reinforcement. This allows the method to capture important phenomena compared to a pure shell model of concrete. A continuous approach to fracture is chosen when concrete is subjected to compressive loading while a combined continuous-discrete fracture method is employed in tension. The model is validated through comparisons with experimental data.

Keywords

cylindrical concrete structures / limit state analysis / 3D fracture modelling / prestressed composite pipes / reinforced concrete / three-point bending test

Introduction

Pipe systems for water supplies in cities are commonly made of steel. However, such pipes are expensive and sensitive to corrosion. A good alternative are pipes fabricated from reinforced concrete which belongs to the most important materials in Civil and Structural Engineering due to its low cost. However, in contrast to steel, concrete has a very low tensile strength and cracking cannot be avoided. In order to increase the load carrying capacity and to control cracking, concrete structures are often prestressed. The prestressing creates compressive stresses predominantly in areas which are subjected to tensile stresses when the concrete structure is in service. Leakage is one concern in such structures and commonly crack openings are controlled. To gain a better understanding of prestressed concrete pipes, computational studies are carried out which are drastically cheaper and less time consuming than experimental testing. Furthermore, important phenomena occurring inside the structure can be studied. This is barely possible in experiments.

Computational modeling of prestressed concrete pipes, especially modeling the fracture process, is challenging. Computational models for fracture need to account for different failure mechanisms such as multi-axial tensile and compressive failure. When crack openings should be determined, it is also not possible to use the ‘simpler’ continuous approaches to fracture such as non-local model [¹–⁸], gradient models [⁹], phase-field models [¹⁰–¹⁴] or models based on the screened-poisson equation [¹⁵–¹⁷]. It requires discrete crack methods such as cohesive elements [¹⁸], boundary element method [¹⁹], the extended finite element method (XFEM) [²⁰–²²], smoothed extended finite element methods [²³,²⁴], isogeometric analysis and extended isogeometric analysis [²⁵–³³], the phantom node method [³⁴–³⁸], numerical manifold method [³⁹], efficient remeshing techniques [⁴⁰–⁴⁵], meshfree methods [⁴⁶,⁴⁷] and extended meshfree methods [⁴⁷–⁵⁷], cracking particles method [⁵⁸–⁶⁰], multiscale methods for fracture [⁶¹–⁶⁸], peridynamics [⁶⁹,⁷⁰] and dual-horizon peridynamics [⁷¹,⁷²]. However, most of those approaches are still not accessible in commercial software and when we study our problem with the XFEM version in ABAQUS, it gave very poor results not matching the experimental data at all. Another difficulty of modeling reinforced concrete structures is to capture various failure mechanisms which can be classified in compressive failure and tensile failure of the concrete, which finally includes also failure of the reinforcement, and bond failure. Bond failure can further be distinguished into anchorage or pullout failure, splitting and splitting tensile failure. The literature on computational approaches which can capture a wide range of such failure modes is scarce. Rabczuk et al. presented impressive results on fracture of reinforced and prestressed concrete structures with excellent agreement to experimental results [⁷³–⁷⁵] capturing a wide range of failure mechanisms of reinforced concrete structures. They used cohesive zone models to account for the energy dissipation at postlocalization. Unfortunately, such models are still not available in commercial software.

In this manuscript, we are interested to predict the structural response of pre-stressed concrete pipes under axial and lateral compression. We also carried out experiments for validation of our model. It employs a non-local damage approach in compression and a smeared crack-element deletion technique under tensile loading. In the next section, we will state the weak form of the underlying governing equation describe how we model failure. Then, we will present numerical results before concluding our manuscript.

Weak forms and constitutive model

Weak form

The strong form of the boundary value problem is given by

(1)\begin{array}{l}\nabla \cdot P+b=\mathrm{0}\forall x\in \mathrm{\Omega }\\ E=(\nabla u+u\nabla )\forall x\in \mathrm{\Omega }\end{array},

where the first equation is the equilibrium equation and the second equation the kinematic equation, i.e., the relation between the linear strain tensor and the displacement vector. These equations have 15 unknowns, i.e., the Cauchy stress tensor P(6 unknowns), the linear strain tensor E(6 unknowns) and the displacement field u(3 unknowns). Therefore, the above 9 equations need to be extended by 6 additional equations, the constitutive equations which will be described in the next subsection. Since these equations cannot be solved analytically, they are solved numerically taking advantage of the finite element method. The other symbols in Eq. (1) are the body force vector b, coordinate vectorxand the domain \mathrm{\Omega }. The above equations in the domain \mathrm{\Omega } are supplemented with essential and natural boundary conditions:

(2)\begin{array}{l}u=\widehat{u}\mathrm{on}{\Gamma }_u\\ P\cdot N=t\mathrm{on}{\Gamma }_t\end{array},

where N indicates the normal vector, t are the tractions and the imposed ‘hat’ denotes superimposed values. Note that essential boundary conditions are necessary in statics to suppress rigid body motion ensuring regularity of the global system stiffness matrix. The essential and natural boundaries {\Gamma }_u and {\Gamma }_t need to fulfil the following conditions: \Gamma ={\Gamma }_u\cup {\Gamma }_t and {\Gamma }_u\cap {\Gamma }_t=\varnothing. It is easy to show that the strong form can be transformed into a weak form which is well known and defined by

(3){\displaystyle {\int }_{\Omega }\delta E:P\mathrm{d}\mathrm{\Omega }}-{\displaystyle {\int }_{{\Gamma }_t}\delta u\cdot \widehat{t}}\mathrm{d}\mathrm{\Gamma }+{\displaystyle {\int }_{\Omega }\delta u\cdot b\mathrm{d}\mathrm{\Omega }=\mathrm{0}},

\delta denoting the test functions.

We took advantage of a scalar damage model in compression exploiting non-local regularization. A simple smeared crack model is used in tension and the anisoptropy at fracture is introduced through element deletion. It also avoids stress locking which is a well-known short coming of smeared crack models. Let us first decompose the incremental linear strain tensor into an elastic part and plastic part:

(4)\mathrm{\Delta }E=\mathrm{\Delta }{E}_e+\mathrm{\Delta }{E}_p,

where the subscripts e and p stand for ‘elastic’ and ‘plastic’, respectively. The constitutive equations can then be written as

(5)P_e^{\nabla }-P^{\nabla }=C:\Delta E-C:\Delta E_e,

the superscript \nabla indicating an objective stress rate and the elasticity tensor is denoted byC. Decoupling the loading surface for F_d is and plasticity F_p results in the damage and plasticity increments given by

(6)\begin{array}{l}\mathrm{\Delta }{P}_p=\mathrm{\Delta }{\lambda }_p{m}_p\mathrm{\Delta }{E}_p=-\mathrm{\Delta }{\lambda }_p{C}^{-\mathrm{1}}:m_p\\ \mathrm{\Delta }{P}_d=\mathrm{\Delta }{\lambda }_d{m}_d\mathrm{\Delta }{E}_d=-\mathrm{\Delta }{\lambda }_d{C}^{-\mathrm{1}}:m_d\end{array},

subscript d denoting ‘damage’;\mathrm{\Delta }{\lambda }_p and \mathrm{\Delta }{\lambda }_d are the plastic and damage multipliers, respectively, and m_p and m_d govern the orientation of the stress increment associated to plasticity and damage, respectively. The Kuhn Tucker conditions

(7)F\le \mathrm{0},\mathrm{\Delta }\lambda \ge \mathrm{0},F\mathrm{\Delta }\lambda =\mathrm{0},

together with the incremental consistency conditions

(8)\begin{array}{l}\mathrm{\Delta }{F}_p={\displaystyle \frac{\partial F_p}{\partial P}}:\mathrm{\Delta }P+{\displaystyle \frac{\partial F_p}{\partial {\lambda }_p}}\mathrm{\Delta }{\lambda }_p=m_p:\mathrm{\Delta }P-H_P\mathrm{\Delta }{\lambda }_p=\mathrm{0}\\ \mathrm{\Delta }{F}_d={\displaystyle \frac{\partial F_d}{\partial P_e}}:\mathrm{\Delta }{P}_e+{\displaystyle \frac{\partial F_d}{\partial {\lambda }_d}}\mathrm{\Delta }{\lambda }_d=m_d:\mathrm{\Delta }P-H_d\mathrm{\Delta }{\lambda }_d=\mathrm{0}\end{array},

finally determine the multipliers that can be derived from Eq. (6):

(9)\begin{array}{l}\mathrm{\Delta }{P}_p={\displaystyle \frac{m_p\otimes n_p}{H_p}}:\mathrm{\Delta }P\\ \mathrm{\Delta }{P}_d={\displaystyle \frac{m_d\otimes n_d}{H_d}}:\mathrm{\Delta }{P}_d\end{array}\mathrm{.}

Note that we assume that the plastic part does not cause any damage. To account for failure, we employ the well know Drucker-Prager model:

(10)F(\overline{P},E_p)={\displaystyle \frac{\mathrm{1}}{\mathrm{1}-A}}(q-\mathrm{3}Ap+B(E_p)\langle P_{\max }\rangle -\langle C(P_{\max })\rangle )-P_C(E_p),

with

(11)\begin{array}{l}B(E_p)={\displaystyle \frac{P_C(E_p)}{P_t(E_p)}}(\mathrm{1}-A)-(\mathrm{1}+A)\\ A={\displaystyle \frac{P_{b\mathrm{0}}-P_{c\mathrm{0}}}{\mathrm{2}{P}_{b\mathrm{0}}-P_{c\mathrm{0}}}}\end{array},

and

(12)C={\displaystyle \frac{\mathrm{3}(\mathrm{1}-K_c)}{\mathrm{2}{K}_c-\mathrm{1}}},

where P_{c\mathrm{0}} and P_{b\mathrm{0}} standing for the uniaxial and biaxial compressive strength and K_c is another material constant. To obtain the final form of the constitutive relation, we define the plastic potential as follows:

(13)F_p=\sqrt{{(E{P}_{t\mathrm{0}}\tan \mathrm{\Psi })}^{\mathrm{2}}+q^{\mathrm{2}}}-p\tan \mathrm{\Psi },

so that the constitutive relation will take the following form:

(14)\mathrm{\Delta }P=(\mathrm{1}-D)C:E_e\mathrm{.}

As mentioned above, we have two scalar damage variable, one in tension denoted by D_t and one in compression, i.e., D_c. The evolution of both variables obey the following relationship

(15)D_t=\{\begin{array}{ll}\mathrm{1}-{\rho }_t(\mathrm{1.2}-\mathrm{0.2}{\displaystyle \frac{E}{E_t}})& {\displaystyle \frac{E}{E_t}}\le \mathrm{1}\\ \mathrm{1}-{\displaystyle \frac{{\rho }_t}{A_t{({\displaystyle \frac{E}{E_t}}-\mathrm{1})}^{\mathrm{1.7}}+{\displaystyle \frac{E}{E_t}}}}& {\displaystyle \frac{E}{E_t}}>\mathrm{1}\end{array},

and

(16)D_c=\{\begin{array}{ll}\mathrm{1}-{\displaystyle \frac{n{\displaystyle \frac{E}{E_c}}}{n-\mathrm{1}+{\left({\displaystyle \frac{E}{E_c}}\right)}^{\mathrm{2}}+{\displaystyle \frac{E}{E_c}}}}& {\displaystyle \frac{E}{E_c}}\le \mathrm{1}\\ \mathrm{1}-{\displaystyle \frac{{\rho }_c}{A_c{({\displaystyle \frac{E}{E_c}}-\mathrm{1})}^{\mathrm{2}}+{\displaystyle \frac{E}{E_c}}}}& {\displaystyle \frac{E}{E_c}}>\mathrm{1}\end{array},

with

(17){\rho }_t={\displaystyle \frac{f_t}{C_t{E}_t}}{\rho }_c={\displaystyle \frac{f_c}{C_c{E}_c}},

where C_t and C_c indicate the secant modulus of the equivalent stress-strain curve; A_t, A_c, f_t and f_c are material constants. It is well known that such local damage models cannot lead mesh independent results and therefore, we introduce a non-local regularization in compression which is defined as

(18)D_c^{NL}={\displaystyle \frac{\mathrm{1}}{V}}{\displaystyle {\int }_V{w}^{NL}(x-\widehat{x}})D_c(\widehat{x})\mathrm{d}V\mathrm{.}

A smeared crack model is employed in tension which accounts for the correct energy dissipation at post-localization. Therefore, the stress is not expressed in terms of the strain but displacement which can be related to a strain ‘due to cracking’. Smeared crack models can be implemented with ease as they simply scale the constitutive model in horizontal direction. As mentioned earlier, we employ element deletion when the damage variable D_t reaches a value of 1.

An elasto-plasticity model with isotropic hardening is used for the steel and a bond model described in [⁷⁶] is employed as well.

Models for the composite pipe

Let us consider the cylindrical concrete pipe as depicted in Fig. 1. The associated meshes showing the concrete and reinforcement is illustrated in Fig. 2. Note that the outer layer is the ‘standard’ (meaning non-prestressed) reinforcement while the inner layer contains the pre-stressed tendons. 3D Euler Bernoulli beam elements are used for both the reinforcement and the pre-stressed tendons as the bending stiffness cannot be neglected. The concrete is discretized with tri-linear hexahedra elements. Note that a thin of elements with a very small stiffness are include between the inner layer of concrete and the external layer of concrete as indicated in Fig. 3. We also tested the influence of different mesh refinements in order to ensure that our results are convergent though we only show the solution of one mesh size.

Let us first describe the cylindrical concrete specimen under three-point bending as depicted in Fig. 4. As suggested in [⁷⁴‒⁷⁶], prestressing of the tendons is modelled by cooling down the tendons

(19)\mathrm{\Delta }T={\alpha }_T{E}_T,

where {\alpha }_T= 1×10^‒5 denoting the thermal coefficient for steel and the strain E_T is computed by E_T=P_s/E_swhere P_s and E_s indicate stresses in longitudinal direction of the prestressed tendons and the their Young’s modulus, respectively. Note that it is an iterative procedure to create the prestressing since the flexibility of the concrete needs to be taken into account. The load in the experiment was applied slowly in three stages justifying the static conditions: The load was first increased to 1.25 MN. Then the specimen was completely unloaded and subsequently reloaded until failure. Several quantities were measured in the experiment

1. The stresses in the tendons at the end of the prestressing (1000 MPa).

2. The vertical load-circumferential strain curve.

3. The load carrying capacity (2.6 MN).

4. The crack pattern (at the surface only) though the crack width were not determined.

We also observed debonding between the exterior and intermediate concrete layer.

The second set of experiments were hydraulic fracture tests since it is a significant loading case for in-service conditions. In these experiments, radial and axial strains were measured. All specimen were made of the same type of concrete and steel and the material constants used in our model can be found in Table 1.

Numerical and experimental results

Three-point bending test

The vertical load-circumferential strain plot illustrated in Figs. 5 and 6 at three different positions according to Fig. 4(b), i.e., at 0°, 180°, and 270°compares the computational predictions with the experimental data. The best agreement is observed for 0° though the agreement between experiments and simulations is good considering the amount of uncertainties involved in both experiments and simulation. We could have done a more detailed uncertainty analysis as in our previous work [⁷⁷–⁷⁹] in order to identify the key input parameters with respect to a certain output but that would leave the scope of this manuscript. We will focus on it in the future. We were also able to accurately capture the maximum vertical load of the experiment. Figures 7 and 8 depict different stress values in the steel tendons and in the concrete specimen. They provide valuable insight which can barely obtained by experiments.

The Figs. 9, 10, and 11 compare the damage or crack pattern of the simulation with the experimental ones and a good agreement is found. The first figure illustrates the damage near the spigot while the second figure shows the lateral fracture pattern of the concrete cylinder. The third figure focuses on the inner ceiling of the structure. Except for the cracks at the bottom nucelating from the interior, lateral cracks at the outer surface occur in a 90° distance from the major inner crack. In conclusions, all major fracture patterns were captured by our model.

Figure 12 shows the anchorage failure between the intermediate and exterior concrete layer. Such failure occurs either in tangential direction or in normal direction depending if shear or tension is the dominant failure mechanism. These causes are difficult to extract experimentally but they can be easily obtained from numerical simulations. Our results show that the maximum shear stress of 3.3 MPa in our simulation is significantly smaller than the shear strength (12.88 MPa) which is 4 times the value of the tensile strength. This can be clearly seen from the stress distribution in radial and circumferential direction, respectively, in Fig. 13 which in turn suggests that the debonding is caused due to tension.

Conclusions

In this papers, we studied the structural failure of prestressed cylindrical concrete pipes with a new computational method. The computational approach is able to capture different failure mechanisms important for such structures. It takes advantage of a non-local damage model under compressive loading while a combined smeared/discrete crack approach is employed under tensile loading. The smeared crack approach can be easily implemented in any software and accounts for the correct energy dissipation at postlocalization. In order to avoid stress locking and the associated artificial transfer of stresses over wide opening cracks, a simple element deletion technique is introduced which finally can be considered as discrete crack model. Since the discrete crack is introduced after the stresses have decayed to zero in the smeared crack model, the approach does not suffer from classical problems of element deletion. Our approach is completely three-dimensional and carefully validated with experiments which we carried out ourselves, i.e, a three-point bending test and hydraulic fracture test due to axial loading conditions. We show that we accurately capture load-deflection curves, load carrying capacity and different failure mechanisms including experimentally observed failure patterns.

Furthermore, we obtained similar strains at different loads and locations of the experiments.

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[68] Talebi H, Silani M, Bordas S P A, Kerfriden P, Rabczuk T. Molecular dynamics/XFEM coupling by a three-dimensional extended bridging domain with applications to dynamic brittle fracture. International Journal for Multiscale Computational Engineering, 2013, 11(6): 527–541

[69] Silling S A. Reformulation of elasticity theory for discontinuities and long-range forces. Journal of the Mechanics and Physics of Solids, 2000, 48(1): 175–209

[70] Rabczuk T, Ren H. A peridynamics formulation for quasi-static fracture and contact in rock. Engineering Geology, 2017, 225: 42–48

[71] Ren H, Zhuang X, Rabczuk T. Dual-horizon peridynamics: a stable solution to varying horizons. Computer Methods in Applied Mechanics and Engineering, 2017, 318: 762–782

[72] Ren H, Zhuang X, Cai Y, Rabczuk T. Dual-horizon peridynamics. International Journal for Numerical Methods in Engineering, 2017, 318: 768–782

[73] Rabczuk T, Eibl J. Modelling dynamic failure of concrete with meshfree methods. International Journal of Impact Engineering, 2006, 32(11): 1878–1897

[74] Rabczuk T, Belytschko T. Application of meshfree particle methods to static fracture of reinforced concrete structures. International Journal of Fracture, 2006, 137(1‒4): 19–49

[75] Rabczuk T, Zi G, Bordas S, Nguyen-Xuan H. A geometrically nonlinear three-dimensional cohesive crack method for reinforced concrete structures. Engineering Fracture Mechanics, 2008, 75(16): 4740–4758

[76] Rabczuk T, Akkermann J, Eibl J. A numerical model for reinforced concrete structures. International Journal of Solids and Structures, 2005, 42(5‒6): 1327–1354

[77] Vu-Bac N, Lahmer T, Zhang Y, Zhuang X, Rabczuk T. Stochastic predictions of interfacial characteristic of polymeric nanocomposites (PNCs). Composites. Part B, Engineering, 2014, 59: 80–95

[78] Vu-Bac N, Lahmer T, Zhuang X, Nguyen-Thoi T, Rabczuk T. A software framework for probabilistic sensitivity analysis for computationally expensive models. Advances in Engineering Software, 2016, 100: 19–31

[79] Hamdia K, Silani M, Zhuang X, He P, Rabczuk T. Stochastic analysis of the fracture toughness of polymeric nanoparticle composites using polynomial chaos expansions. International Journal of Fracture, 2017, 206(2): 215–227

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