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Charles Moore
Charles Moore

Gergely Lutz Equation [HOT] Crack Width Comparator



The width of flexural cracks in a concrete member internally reinforced with fiber reinforced polymer (FRP) bars depends on the characteristic of interfacial bond between the reinforcement and concrete. This paper reports the findings of a task group formed under the aegis of ACI Sub-committee 440 H (Reinforced Concrete) to evaluate and compare the effects of bond characteristics in previous and proposed equations for predicting crack widths. Based on the experimental data assembled by the task group, it is shown that the bond coefficient in the proposed equation by Frosch is 19% greater than the bond coefficient in the previous equation attributed to Gergely and Lutz.




Gergely Lutz Equation Crack Width Comparator



N2 - The width of flexural cracks in a concrete member internally reinforced with fiber reinforced polymer (FRP) bars depends on the characteristic of interfacial bond between the reinforcement and concrete. This paper reports the findings of a task group formed under the aegis of ACI Sub-committee 440 H (Reinforced Concrete) to evaluate and compare the effects of bond characteristics in previous and proposed equations for predicting crack widths. Based on the experimental data assembled by the task group, it is shown that the bond coefficient in the proposed equation by Frosch is 19% greater than the bond coefficient in the previous equation attributed to Gergely and Lutz.


AB - The width of flexural cracks in a concrete member internally reinforced with fiber reinforced polymer (FRP) bars depends on the characteristic of interfacial bond between the reinforcement and concrete. This paper reports the findings of a task group formed under the aegis of ACI Sub-committee 440 H (Reinforced Concrete) to evaluate and compare the effects of bond characteristics in previous and proposed equations for predicting crack widths. Based on the experimental data assembled by the task group, it is shown that the bond coefficient in the proposed equation by Frosch is 19% greater than the bond coefficient in the previous equation attributed to Gergely and Lutz.


Abstract:This study deploys a hybrid Grey Wolf Optimizer Neural Network Model for predicting the crack width in reinforced concrete slabs strengthened with carbon fiber-reinforced polymers (CFRP). Reinforced concrete (RC) one-way slabs (1800 400 120 mm in size) were strengthened with CFRP with various lengths (1800, 1100, and 700 mm) and subjected to four-point bending. The experimental results were compared to corresponding values for conventional RC slabs. The observed crack width results were recorded, and subsequently examined against the expression recommended by Eurocode 2. To estimate the crack width of CFRP-reinforced slabs, ANN combined with the Grey Wolf Optimizer algorithm was employed whereby the applied load, CFRP width/length, X/Y crack positions, and stress in steel reinforcement and concrete were defined as the input parameters. Experimental results showed that the larger the length and width of the carbon fiber, the smaller the maximum crack width in the tensile area of the slab at the final load step. On average, the crack width in slabs retrofitted with CFRP laminates increased by around 80% compared to a slab without CFRP. The results confirm that the equation provided by Eurocode 2 provides an unconservative estimation of crack widths for RC slabs strengthened with CFRP laminates. On the other hand, the results also confirm that the proposed informational model could be used as a reliable tool for estimating the crack width in RC slabs. The findings provide valuable insight into the design approaches for RC slabs and rehabilitation strategies for existing deficient RC slabs using CFRP.Keywords: crack width; CFRP; artificial intelligence; neural networks; concrete slab


This research paper presents and comments on analytical models for calculating the widths of cracks formed as a result of imposed deformations generating tensile stresses in reinforced concrete base-restrained members. This issue regarding the mechanics of concrete structures has been presented on the basis of calculation models since 1968. In accordance with the current regulations of the European standard, the mechanics of the cracking of base-restrained members have been presented in a very simplified way, which was justified by a limited number of research studies performed on such members as well as in a few subject publications. The main purpose of this work was to present especially those models that had the greatest practical significance within a specific period of time or formed the basis for further studies of other authors. In addition, future trends in the development of computational tools are presented. The chronologically presented development of design ideas, which takes into account varying degrees of advancement of the mechanics of cracking due to the distinctly different design consequences, is a valuable source of information and an inspiration for subsequent researchers. In the second part of the paper, a few of the most important issues connected with the calculation of the crack width in base-restrained walls are presented. It is shown that currently, on the basis of the up-to-date knowledge, there are possibilities to create more complementary standard guidelines, which is already taking place in the case of European guidelines.


One of the first studies of base-restrained members was conducted by Stoffers (1978) and analysed the influence of reinforcement, wall geometry and restraint conditions on the morphology of cracks, their spacing and their widths, which enabled the introduction of such factors in the mechanism of cracking which would enable the calculation of crack width depending on the diameter and spacing of the reinforcement. Subsequent studies of various authors focused on the formulation of the computational model, assuming such a relationship of height to length for which exceeding the tensile strength of concrete led to the formation of dilatation cracks. As a result of making significant progress in determining the development of the heat of hydration and its influence on the development of physical properties, as well as the formal description of these phenomena, further research studies enabled attempts to combine and expand the problem of the mechanics of cracking by taking the development of thermal stresses increased by restraints along the edges of the member into consideration: Van Breugel (1982, 1995), Emborg (1989), Rostásy and Onken (1994). As far as analyses using FEM (fine element method) are concerned, attention should be paid to the research studies of the team of Pettersson and Thelandesson (2001a, 2001b) and Pettersson et al. (2002). These studies present a wide parametric analysis of the influence of the properties of concrete, the amount of reinforcement and the boundary conditions on the maximum crack width. The issue was simplified to 2D, i.e., only average strains were considered on the wall thickness. The intensive development of numerical methods enabled further refinements to the models, as exemplified by research studies performed by the team of Flaga and Klemczak (2016), Flaga (2011), which contain the proposal of an advanced numerical model and an engineering model which, from the point of view of designers who do not have access to advanced computer programs, allowing both the size of the deformation and its effects to be determined. The problem of the mechanics of cracking under the influence of imposed deformations takes multiple forms. For example, Klemczak and Knoppik-Wróbel (2015) and Knoppik-Wróbel (2015) presented a significant influence of the support conditions on the degree of restraint. If wall rotation is considered, the degree of restraint in the structural joint increases, but it decreases in the upper part of the wall. This effect is more visible in the case of longer walls and it is almost imperceptible in the case of shorter walls.


This research paper attempts to comment on some of the most common models (since 1968) to calculate crack widths in base-restrained members. Their development followed the progress in the research performed on these members and the conducted parametric analyses. The activities performed in various scientific centres were finalised with the issuance of the first European standard EN 1992-3 (2006), which proposes, for example, an approach to determine crack widths in base-restrained members. Much of the information contained in EN 1992-3 (2006) is quoted from the British Standard BS 8007 (1987) regarding the design of tanks for liquids. Separate provisions in this regard also apply in the United States (ACI 207.2R-95, 1995) and in Japan (JCI, 2008).


First, this paper presents the development of the approach to design in the field of the fundamental issue of the calculation of the width of the cracks in maturing concrete and it thus provides the inspiration for the improvement of the current design guidelines and also for the creation of new computational models.


Evans and Hughes (1968) were among the first to perform studies on imposed strains in a real structure. The results of their research confirmed a larger scale of strains caused by a change of concrete hardening than by its shrinkage. They proposed the following equation for cracks spacing in a wall restrained along the bottom edge:


Evans and Hughes (1968) assumed that the initial crack spacing halved when further cracking formed. By contrast, stresses in concrete increase linearly from zero in the cracked cross section to the maximum value at the distance of smin from cracked section. It follows that with a degree of reinforcement greater than ρcrit (i.e., the degree of reinforcement at which the reinforcing steel does not become plastic), the maximum mean tensile strain in the uncracked cross section along the length smin adjacent to the crack is εctu. If the next crack is formed at a distance of s (usually greater than smin), then the mean strain at the length s/2 is equal to εctu(s/2smin). Thus, Evans and Hughes (1968) proposed a formula for calculating the crack width in the following form:


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