Probabilistic model for steel corrosion in reinforced concrete structures of large dimensions considering crack effects

Non-uniform rust expansion for chloride-induced pitting corrosion in RC structures

1 Complex Numbers Sums and Products Basic Algebraic Properties Further Properties Vectors and Moduli Complex Conjugates Exponential Form Products and Powers in Exponential Form Arguments of Products and Quotients Roots of Complex Numbers Examples Regions in the Complex Plane 2 Analytic Functions Functions of a Complex Variable Mappings Mappings by the Exponential Function Limits Theorems on Limits Limits Involving the Point at Infinity Continuity Derivatives Differentiation Formulas Cauchy-Riemann Equations Sufficient Conditions for Differentiability Polar Coordinates Analytic Functions Examples Harmonic Functions Uniquely Determined Analytic Functions Reflection Principle 3 Elementary Functions The Exponential Function The Logarithmic Function Branches and Derivatives of Logarithms Some Identities Involving Logarithms Complex Exponents Trigonometric Functions Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions 4 Integrals Derivatives of Functions w(t) Definite Integrals of Functions w(t) Contours Contour Integrals Some Examples Examples with Branch Cuts Upper Bounds for Moduli of Contour Integrals Antiderivatives Proof of the Theorem Cauchy-Goursat Theorem Proof of the Theorem Simply Connected Domains Multiply Connected Domains Cauchy Integral Formula An Extension of the Cauchy Integral Formula Some Consequences of the Extension Liouville's Theorem and the Fundamental Theorem of Algebra Maximum Modulus Principle 5 Series Convergence of Sequences Convergence of Series Taylor Series Proof of Taylor's Theorem Examples Laurent Series Proof of Laurent's Theorem Examples Absolute and Uniform Convergence of Power Series Continuity of Sums of Power Series Integration and Differentiation of Power Series Uniqueness of Series Representations Multiplication and Division of Power Series 6 Residues and Poles Isolated Singular Points Residues Cauchy's Residue Theorem Residue at Infinity The Three Types of Isolated Singular Points Residues at Poles Examples Zeros of Analytic Functions Zeros and Poles Behavior of Functions Near Isolated Singular Points 7 Applications of Residues Evaluation of Improper Integrals Example Improper Integrals from Fourier Analysis Jordan's Lemma Indented Paths An Indentation Around a Branch Point Integration Along a Branch Cut Definite Integrals Involving Sines and Cosines Argument Principle Rouche's Theorem Inverse Laplace Transforms Examples 8 Mapping by Elementary Functions Linear Transformations The Transformation w = 1/z Mappings by 1/z Linear Fractional Transformations An Implicit Form Mappings of the Upper Half Plane The Transformation w = sin z Mappings by z2 and Branches of z1/2 Square Roots of Polynomials Riemann Surfaces Surfaces for Related Functions 9 Conformal Mapping Preservation of Angles Scale Factors Local Inverses Harmonic Conjugates Transformations of Harmonic Functions Transformations of Boundary Conditions 10 Applications of Conformal Mapping Steady Temperatures Steady Temperatures in a Half Plane A Related Problem Temperatures in a Quadrant Electrostatic Potential Potential in a Cylindrical Space Two-Dimensional Fluid Flow The Stream Function Flows Around a Corner and Around a Cylinder 11 The Schwarz-Christoffel Transformation Mapping the Real Axis onto a Polygon Schwarz-Christoffel Transformation Triangles and Rectangles Degenerate Polygons Fluid Flow in a Channel Through a Slit Flow in a Channel with an Offset Electrostatic Potential about an Edge of a Conducting Plate 12 Integral Formulas of the Poisson Type Poisson Integral Formula Dirichlet Problem for a Disk Related Boundary Value Problems Schwarz Integral Formula Dirichlet Problem for a Half Plane Neumann Problems Appendixes Bibliography Table of Transformations of Regions Index

Complex variables and applications

A non-uniform corrosion model and meso-scale fracture modelling of concrete

Non-uniform time-to-corrosion initiation in steel reinforced concrete under chloride environment

Understanding of cracking behavior of cover concrete is important for assessment of the remaining service life of corrosion-affected concrete structures. Predicting initiation of concrete cracking is commonly based on the evolution of stresses caused by the volume expansion of corrosion products on reinforcing bars. This paper presents an analytical method that is capable of providing the solution to two-dimensional (2D) elastic stress field caused by nonuniform volume expansion around the corroding reinforcing bars. The developed method features the formulation of a displacement model, which is applied as a boundary condition to simulate the effect of nonuniform corrosion on the deformation of the surrounding concrete. A closed-form solution to the stress field is obtained using the complex variable method of Muskhelishvili. Validation of the solution is supported by a comparison with finite-element-based simulation results. The solution reveals the evolution of stresses surrounding two typical locations...

Nonuniform Corrosion-Induced Stresses in Steel-Reinforced Concrete

Developing predictive models for quantitative assessment of the deterioration of rock bolts exposed to corrosive environments is essential for rational planning of maintenance activities for anchorage structures. This article presents a probability-based computational model for predicting the time-dependent deterioration of bond capacity of corroding rock bolts due to the attack of chlorides. The inherent stochastic nature involved in the corrosion and degradation process is identified and simulated. A method is developed based on fundamental Mohr–Coulomb theory to evaluate the effect of corrosion on the bond strength due to a failure mode involving splitting of grout cover. Parameters affecting both strength and stress state at the bolt–grout interface are quantitatively related to the degree of corrosion. Integrating the realised corrosion stochastic process and the developed bond loss evaluation method, a computational algorithm based on Monte Carlo simulation is presented for evaluating the bond deter...

Probabilistic modelling of the bond deterioration of fully-grouted rock bolts subject to spatiotemporally stochastic corrosion

One important factor affecting crevice corrosion of passive metals is the crevice geometry, e.g., the gap of the crevice. The crevice geometry can be affected by the properties of the materials used as the crevice formers, i.e., a polymeric crevice former can conform to the surface roughness of a metal specimen, which helps the creation of a tighter crevice gap. The objective of this work was to investigate the effects of crevice former materials on the evolution of crevice corrosion damage of Alloy 22 (UNS N06022). The factors that may limit the initiation and slow or stop the propagation of crevice corrosion are addressed. Crevice formers made from ceramic, polymers, and polytetrafluoroethylene (PTFE) tape-covered ceramic are compared in controlled crevice corrosion tests, which are performed under highly aggressive, accelerated conditions. Results show that crevice corrosion of Alloy 22 is affected by the crevice former materials and by the surface finishes of the crevice former and specimen. ...

Effect of Polymer and Ceramic Crevice Formers on the Crevice Corrosion of Ni-Cr-Mo Alloy 22

Over twenty six percent of the bridges in the United States are structurally deficient or functionally obsolete. Corrosion of steel used in structures like bridges and buildings is a problem that has gained increased interest and focused concern. Steel is often the metal that is preferred for use in such applications due to a synergism of ease of availability, acceptable mechanical properties and cost effectiveness. Through the years, titanium has grown in strength, stature and significance to be recognized as an emerging high performance metal that is both stronger and lighter than steel. A distinctive property of titanium and its alloys is its non-corrosive nature. However, a major drawback in the selection and use of pure titanium or its alloy counterpart is the prohibitively high cost. Therefore, it may be possible to combine steel and pure titanium and/or its alloy in structures by restricting steel for bulk of the structure and selectively using titanium and its alloys for the critical but low volume elements, such as, gusset plates and bearings. A hybrid use of titanium in conjunction with steel for structural members will result in better performance while concurrently proving to be both cost-effective and economically affordable. The synergistic use of structural steel and titanium in close proximity with each other could result in accelerated corrosion of steel in the immediate vicinity of titanium. The corrosion performance of titanium plates coupled with steel members is presented. A few viable strategies for minimizing galvanic coupling effects between steel and titanium are discussed. Corrosion experiments were conducted to measure the severity of corrosion when titanium and steel form a galvanic couple, and copper and steel was a comparative system. The study revealed that adequate precautions are needed to minimize localized corrosion when titanium gusset plates are coupled with structural steel members.

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Isolating Corrosion of Steel Plates Coupled with Titanium

A new type of enamel coated dowel bars for use as concrete pavements has been recently developed and implemented. A gradual deterioration of such bars can occur due to corrosion occurring at locations of the fine microscopic cracks and other defects in the coating. The results of a study aimed at evaluating and understanding the corrosion resistance of such dowel bars is presented and comparisons made with corresponding epoxy-coated dowel bars. Identical defects were introduced onto the surface of the dowel bars with the two types of coatings. The extrinsic influence of environment and resultant corrosion based on type, location and severity of the defects was carefully examined. Test results reveal the corrosion resistance of enamel-coated dowel bars to be as good as the corrosion resistance of the corresponding epoxy-coated dowel bars. The presence of corrosion products was essentially confined to locations of the defects in the test specimens when the steel was exposed to concrete and subject to induce...

Joe Payer

Papers

Nonuniform Corrosion-Induced Stresses in Steel-Reinforced Concrete

Probabilistic modelling of the bond deterioration of fully-grouted rock bolts subject to spatiotemporally stochastic corrosion

Effect of Polymer and Ceramic Crevice Formers on the Crevice Corrosion of Ni-Cr-Mo Alloy 22

Isolating Corrosion of Steel Plates Coupled with Titanium

Extrinsic influence of environment on the corrosion behaviour of enamel-coated steel dowel bars