A model for prediction of time from corrosion initiation to corrosion cracking

Prediction of time to corrosion cracking is a key element in evaluating the service life of corroded reinforced concrete (RC) structures. This paper presents a mathematical model that predicts the time from corrosion initiation to corrosion cracking. In the present model a relationship between the steel mass loss and the internal radial pressure caused by the expansion of corrosion products is developed. The concrete around a corroding steel reinforcing bar is modeled as a thick-walled cylinder with a wall thickness equal to the thinnest concrete cover. The concrete ring is assumed to crack when the tensile stresses in the circumferential direction at every part of the ring have reached the tensile strength of concrete. The internal radial pressure at cracking is then determined and related to the steel mass loss. Faraday’s law is then utilized to predict the time from corrosion initiation to corrosion cracking. The model accounts for the time required for corrosion products to fill a porous zone before they start inducing expansive pressure on the concrete surrounding the steel reinforcing bar. The accuracy of the model is demonstrated by comparing the model’s predictions with experimental data published in the literature.     

An element enriched formulation for simulation of splitting failure

Radial cracking propagation is often related to the bond transfer mechanism induced by slippage of a deformed bar. However, this failure pattern can also develop in other situations, namely: (i) concrete pipes submitted to an excessive inner pressure or (ii) concrete structures exposed to adverse environmental conditions under which corrosion or frost develops.In this paper a new contribution for the simulation of radial splitting failure is given. A discrete strong discontinuity formulation is presented which is fully capable of embedding radial discontinuities into axisymmetric finite elements. Numerical examples are used to show: (i) the capability of fully softening the applied inner pressure and (ii) mesh independence. Comparison with two published analytical approaches is performed for varying brittleness numbers. Finally, the model is applied to the simulation of both plain and reinforced concrete cylinders subjected to increasing inner pressure. A good agreement with experimental data is obtained.     

Investigation on the cracking behavior of concrete cover induced by corner located rebar corrosion

Non-uniform corrosion of reinforcement causes concrete cracking in chloride contaminated RC structures. Due to the special boundary conditions, the concrete cover with corner located rebar is often subjected to the attack of chloride ions in a marine environment from two directions, and thus the corresponding non-uniform corrosion distribution should be different from the one for side-located rebar. The aim of the work is to explore the effect of corner located rebar corrosion on the cracking of cover concrete. For corner located rebar, an improved non-uniform corrosion distribution model was established based on the analysis results of two-dimensional chloride diffusion in concrete. Considering the heterogeneities of concrete, a meso-scale mechanical model and method for the study on the failure behavior of concrete cover was built. In the analysis model and method, the non-uniform radial displacement distribution was adopted to simulate the corrosion expansion behavior of the rebar. The cracking of concrete cover with corner located rebar was simulated and studied. The present approach was verified by the available experimental observations. The influences of concrete heterogeneity, corrosion distribution types, rebar diameter and concrete cover thickness on the failure patterns of concrete cover and the expansive pressure were investigated. The simulation results indicate that the developed approach can well describe the cracking behavior of cover concrete and the corrosion-expansion behavior of steel rebar.     

Influence of crack width,cover depth and concrete quality on corrosion of steel in HPC containing corrosion inhibiting admixtures and fly ash

In this study the influence of crack width, cover depth and concrete quality on corrosion of steel in high performance concrete was investigated. Three mixtures, one control and two more containing corrosion inhibiting admixtures, Calcium Nitrite and Disodium Tetrapropenyl Succinate, in combination with 20% fly ash replacement with respect to the cement weight were prepared. Specimens were concrete cylinders measuring 100 mm in diameter and 65 and 105 mm in length, with a 16-mm steel bar centrally placed at two concrete covers of 25 and 45 mm. Before being exposed to a simulated marine environment, the specimens were pre-cracked under a controlled splitting test with crack widths ranging from 90 to 330 μm formed perpendicularly to the reinforcing bars. During a 16-month exposure, the corrosion risk of the reinforcing bars was evaluated by half-cell corrosion potentials and the corrosion rate by linear polarization method. Also, the total integrated corrosion current was estimated. Results show that, albeit to different degrees, cracking was found to be an influencing factor in promoting corrosion of the steel in concrete with either 25 mm or 45 mm concrete cover; nevertheless, the effectiveness of the concrete cover depended greatly on the crack thickness. Results also revealed that corrosion inhibitors and fly ash were effective in delaying corrosion even in cracked concrete.     

Meso-scale numerical investigation on cracking of cover concrete induced by corrosion of reinforcing steel

Concrete cover cracking induced by corrosion of steel reinforcement is a major influencing factor for durability and serviceability of reinforced concrete structures. Here in this study, the influence of concrete meso-structure on the failure pattern of concrete cover is accounted for. The concrete is assumed to be a three-phase composite composed of aggregate, mortar matrix and the interfacial transition zone (ITZ). And a concrete random aggregate structure is established for the study on the mechanical behavior of radial corrosion expansion. In the present simulations, the plasticity damaged model is used to describe the mechanical behavior of the mortar matrix and the ITZ, and it is assumed that the corrosion of steel reinforcement is uniform. The cracking of concrete cover due to steel reinforcement corrosion is numerically simulated. The simulation results have a good agreement with the available test data and they are between the two analytical results. The failure patterns obtained from the macro-scale homogeneous model and the meso-scale heterogeneous model are compared. Furthermore, the influences of ratio of cover thickness and reinforcement diameter (i.e. c/d), the location of the steel reinforcement (i.e., placed at the middle and corner zones) and the concrete tensile strength on the steel corrosion rate when the concrete cover cracks are investigated. Finally, some useful conclusions are drawn.     

Hydrodynamic stability of viscous flow between curved porous channel with radial flow

A linear stability analysis has been presented for the flow between long concentric stationary porous cylinders driven by constant azimuthal pressure gradient, when a radial flow through the permeable walls of the cylinders is present. The radial Reynolds number, based on the radial velocity at the inner cylinder and the inner radius is varied from −100 to 30. The linearized stability equations form an eigenvalue problem which are solved using a numerical technique based on classical Runge-Kutta scheme combined with a shooting method, termed as unit disturbance method. It is observed that radially outward flow and strong inward flow have a stabilizing effect, while weak inward flow has a destabilizing effect on the stability. Profiles of the relative amplitude of the perturbed radial velocities show that radially outward flow shifts the vortices toward the outer cylinder, while radially inward flow shifts the vortices toward the inner cylinder.     

预应力钢绞线锈胀及混凝土开裂预测

针对钢绞线锈蚀导致混凝土开裂现象,开展了不同应力状态下混凝土的锈胀开裂试验,基于红外和热重分析研究了预应力钢绞线锈蚀产物的膨胀率,分析了预应力对保护层临界开裂时间和裂缝宽度的影响,综合考虑预应力、铁锈膨胀率和混凝土开裂损伤等因素,建立了开裂初始和发展全过程的锈胀裂缝预测模型,并通过试验结果进行了验证。结果表明:预应力会加速混凝土的锈胀开裂,在钢绞线抗拉强度75%的预应力水平下,保护层初始开裂时间降低了22%,裂缝扩展速率增加了9%;建立的模型具有较好的精度,可以合理地预测预应力混凝土的锈胀开裂。     

Viscous vortex core generation

Summary The unsteady flow of a viscous incompressible fluid between two porous co-axial circular cylinders is analysed when the outer cylinder is impulsively set into rotation. When there is radial inflow, vorticity is transferred from an unsteady boundary layer initially over the outer cylinder to a steady boundary layer over the inner cylinder.With 1 Figure     

Estimation of critical free expansions related to surface cracking in ASR-affected concretes

This paper proposes a simple procedure to determine the critical free expansion for macro-surface cracking in ASR-affected model concrete cylinders. It is assumed that the non-reactive near-surface concrete regions act as restraint against uniform expansion of the inner core due to ASR. The critical free expansion is calculated assuming that a longitudinal crack forms when the maximum circumferential tensile stress in the concrete cylinder reaches the tensile strength of the material. The expansive pressure that causes the tensile stress depends on degree of restraint, as determined from measured expansive pressures and free expansions. The calculated critical free expansions for concretes with various strengths at various depths of the non-reactive concrete cylinders can be used to qualitatively interpret the significance of suppression of ASR macro-cracking by drying and relatively lower probability of surface cracking in structures in indoor rather than outdoor environments. It appears the proposed procedure can be used for estimating critical free expansions independently of temperature.     

Analyses of the multiple cracking behaviour of brittle hollow cylinders under internal pressure

In order to elucidate the multiple cracking behaviour of brittle hollow cylinders under static internal pressure, two-dimensional dynamic finite element analyses have been performed firstly for graphite hollow cylinders with inner and outer diameter of 16 and 22 mm, respectively, under internal gas pressure. In the analyses, propagation speed of the primary crack was set to be extremely high by instantaneously releasing the nodes that defined the path of the primary crack, and internal pressure was preserved after the primary cracking. The analyses showed that the stress was enhanced due to stress waves generated by the primary cracking. The initial stress enhancement was observed at the side position of the cylinder, which was located at approximately ±90° with respect to the primary cracking site. This implied that secondary cracking could occur at the side positions. Fracture modes of the cylinders might depend on the following parameters: (1) propagation speed of the primary crack, (2) pressure drop rate after the primary cracking, (3) medium to generate internal pressure, (4) geometry of a cylinder, (5) mechanical properties of brittle materials, and (6) presence of a notch. Thus, the effect of the above parameters on the behaviour of the multiple cracking was also analysed. It was found that secondary cracking would still occur at the side positions if (i) the crack propagation speed was between 70% and 100% of the theoretical crack propagation speed, (ii) the pressure drop rate was below 10⁷ Pa/s, (iii) wall thickness of the cylinder was changed, and (iv) other brittle materials were employed. Also, it was found that multiple cracking would not be observed if liquid pressure was employed instead of gas pressure, because of fluid-structure interaction. In addition, the position of the secondary cracking would be shifted by introducing a notch on the outer surface of the cylinder. These results were in good accordance with experiments formerly reported.     

Stress intensity factors for slanted through-wall cracks based on elastic finite element analyses

Based on detailed 3‐dimensional (3‐D) elastic finite element (FE) analyses, the present paper provides stress intensity factors (SIFs) for plates with slanted through‐wall crack (TWC) and cylinders with slanted circumferential TWC. Regarding loading conditions, axial tension was considered for the plates, whereas axial tension, global bending and internal pressure were considered for the cylinders. To cover a practical range, the geometric variables affecting the SIF were systematically varied. Based on FE results, SIFs along the crack front, including the inner and outer surface points, were provided. The present results can be used to evaluate the fatigue crack growth or stress corrosion cracking behaviour of a slanted TWC and furthermore to perform detailed Leak‐Before‐Break analysis considering a more realistic crack shape.     

Design of a compliant-cylinder-type fiber-optic accelerometer: theory and experiment

Experimental and theoretical research was carried out in order to establish the dependence of the performance of a compliant-cylinder-based fiber-optic accelerometer on the geometry and elastic properties of the transducer cylinders. The sensitivity and the natural frequency of the sensor were measured as a function of the ratio ε = (inner cylinder diameter)/(outer cylinder diameter). Two transducer materials with different elastic properties, a silicone rubber (Ecosil) and a polyetheretherketone polymer (PEEK 450G), were examined. It was found that with decreasing ε the sensitivity increases in the case of Ecosil and decreases in the case of PEEK. In both cases the natural frequency increases with decreasing ε. A simple analytical model was developed in order to explain this behavior qualitatively. The model takes into account the contributions to the effective stiffness from both the cylinder material and the fiber wrapped around the cylinder. The model is useful for the design of such types of accelerometer.     

Short-term corrosion rate measurement of OPC and HPC reinforced concrete specimens by electrochemical techniques

This paper presents the results of a laboratory investigation in which the applicability of Tafel plot and linear polarization techniques in short-term corrosion rate measurement of reinforcing bar in concrete has been evaluated. One hundred and four OPC and HPC concrete cylinders, fifty-two of each kind and each cylinder with a single embedded reinforcing bar, have been subjected to various controlled conditions, and the corrosion rate of each specimen has been monitored. Results indicate that sodium chloride concentration plays an effective role on the propagation of corrosion,i.e., the higher the NaCl concentration, the higher the corrosion rate.     

Cover cracking as a function of bar corrosion: Part I-Experimental test

The appraisal of concrete structures suffering rebar corrosion is one of the most urgent needs regarding the selection of the technical and economical optimum time for repair. Up to now this appraisal has been mainly based on empirical and subjective considerations. Among the different distressing consequences of rebar corrosion the best known is the cracking of concrete cover. However, very few data have been reported in the literature on the amount of corrosion needed to induce this cracking. In the present paper, some preliminary experiments are reported in which small reinforced beams are artificially corroded by an impressed current, and the amount of current (and loss of bar cross-section) needed to induce the crack at the surface are monitored, together with the evolution of crack width, by the use of strain gauges applied to the surface of the specimens. In a companion paper, a numerical model to relate the decrease in rebar cross-section to the cover cracking will be developed. That model is based on the orderly imposition of corrosion to finite elements of the rebar by a fictitious temperature increment that produces analogous effects, while concrete cracking is introduced by a standard smeared-crack model. The experimental results indicate that only a few micrometres of loss in rebar cross-section are needed to induce visible cover cracks (0.1 mm width) in the conditions of the test.     

Stress intensity factors and crack opening displacements for slanted axial through-wall cracks in pressurized pipes

This paper proposes elastic stress intensity factors and crack opening displacements (CODs) for a slanted axial through-wall cracked cylinder under an internal pressure based on detailed three-dimensional (3D) elastic finite element (FE) analyses. The FE model and analysis procedure were validated against existing solutions for both elastic stress intensity factor and COD of an idealized axial through-wall cracked cylinder. To cover a practical range, four different values of the ratio of the mean radius of cylinder to the thickness ( R _m/ t ) were selected. Furthermore, four different values of the normalized crack length and five different values of the ratio of the crack length at the inner surface to the crack length at the outer surface representing the slant angle were selected. Based on the elastic FE results, the stress intensity factors along the crack front and CODs through the thickness at the centre of the crack were provided. These values were also tabulated for three selected points, that is, the inner and outer surfaces and at the mid-thickness. The present results can be used to evaluate the crack growth rate and leak rate of a slanted axial through-wall crack due to stress corrosion cracking and fatigue. Moreover, the present results can be used to perform a detailed leak-before-break analysis considering more realistic crack shape development.     

Effect of corrosion layer of steel bar in concrete on time-variant corrosion rate

Corrosion current density of steel bar in concrete was measured in active corrosion process under a designed artificially controlled climate environment. The active corrosion process shows the characteristics of the time-variant corrosion rate, and the three phases of the corrosion process are presented. The corrosion rate decreases at first; this is followed by a steady state phase; finally after concrete cover cracking caused by corrosion, an ascending phase of the corrosion rate is observed. The mechanism of the time-variation characteristics is discussed based on the microstructure of the interfacial transition zone (ITZ) between steel bars and concrete at different corrosion levels. The microstructure shows that the porous interfacial transition zone gradually transforms into a dense corrosion layer composed of concrete and corrosion products due to expansion of the corrosion products. The layer is called as corrosion layer for short in this paper. The main reason for the descent of the corrosion rate is that transportation of oxygen and moisture is retarded due to the dense corrosion layer. When the equilibrium between rates of consumption and transportation of oxygen is reached, the corrosion rate tends to be steady. The concrete cover cracking offers new access for transporting oxygen and the corrosion rate speeds up.     

Optimum design method for double-layer thick-walled concrete cylinder with different modulus

The stress state of thick-walled cylinder will be improved when it is composed of multi-layer concrete with different Young’s modulus of elasticity. The optimum design method for double-layer concrete cylinders is discussed in this paper. When the modulus of the inner-layer concrete is less than the outer one, the stress concentration on the cylinder inner surface will be relieved and the maximum tangential stress may be transferred from the inner surface of inner-layer to the inner surface of outer-layer, where is in triaxial stress state. The elastic limit bearing capacity of double-layer concrete cylinder will be improved due to the high compressive strength of concrete in triaxial stress state. Compared with the conditional single-layer thick-walled cylinder, the thickness of the double-layer cylinder can be reduced evidently. On the basis of stress solution of double-layer cylinder and by using the mixed penalty function method, the minimum wall thickness required, the best thickness ratio and the modulus ratio of inner layer to outer layer are calculated for cases in which the external load and uniaxial compressive strength are known already. If the thickness and uniaxial compressive strength is given, the best thickness ratio and modulus ratio of inner layer to outer layer can also be proposed by using the mixed penalty function method.     

Analytical solutions for functionally graded incompressible eccentric and non-axisymmetrically loaded circular cylinders

We study analytically plane strain static deformations of functionally graded eccentric and non-axisymmetrically loaded circular cylinders comprised of isotropic and incompressible linear elastic materials. Normal and tangential surface tractions on the inner and the outer surfaces of a cylinder may vary in the circumferential direction. The shear modulus is taken to vary either as an exponential function or as a power law function of the radius only. The radial and the circumferential displacements, and the hydrostatic pressure are expanded in Fourier series in the angular coordinate, and expressions for their coefficients are derived from equations expressing the balance of mass (or the continuity equation) and the balance of linear momentum. Boundary conditions are satisfied in the sense of Fourier series. For the exponential variation of the shear modulus, the method of Frobenius series is used to solve 4th-order ordinary differential equations for coefficients of the Fourier series. It is shown that the series solutions for displacements and the hydrostatic pressure converge rapidly. Results for eccentric cylinders and non-axisymmetrically loaded circular cylinders are computed and exhibited graphically. Effects on stress distributions of the eccentricity in the cylinders and of the gradation in the shear modulus are illuminated. It is found that in a thin cylinder subjected to cosinusoidally varying pressure on the inner surface, segments of the cylinder between two adjacent cusps in the pressure deform due to bending rather than stretching.     

On the determination of the radiation shape factor of a system of two coaxial cylinders

Calculation of radiation shape factors is presented for a system of two coaxial cylinders of equal height. The shape factor for self-irradiation of the outer cylinder is plotted versus the ratio of the diameters of the inner and outer cylinders for various ratios of the outer cylinder diameter to its height. Similar plots are also presented for the shape factors for radiation from the outer cylinder to the inner one and from the inner cylinder to the bases. The maximum error in the predicted shape factors does not exceed 7 percent.