Corrosion of reinforcing steel is a major problem facing infrastructures owners with billions of dollars spent in repairing our aging infrastructure. One of the first steps in the repair process is to quantify the strength degradation in a reinforced concrete element caused by the corrosion of reinforcing steel. An understanding of the forces involved in the load carrying mechanisms is imperative; the transfer of shear forces in reinforced concrete beams is one of these load carrying mechanisms. The shear transfer mechanism is different near the end of beams, adjacent to point loads, and near changes in cross section. These regions are known as disturbed regions. Structural engineers have a good understanding of the shear transfer mechanism in disturbed regions. However, the effects of corroded shear reinforcement in these regions have not been widely investigated. The current study is comprised of an experimental program and analytical strut and tie modeling aimed at quantifying the strength reduction that occurs in disturbed regions of reinforced concrete beams with corroded shear reinforcement. The feasibility of strengthening a beam with dry lay-up carbon fibre reinforced polymer (CFRP) to repair the damage caused by corrosion of the shear reinforcement was also investigated. In the experimental study, a total of 16 reinforced concrete beams were cast. The specimens were 350 mm deep, 125 mm wide and 1850 mm long. Three shear-span to depth ratios (1.0, 1.5, 2.0) were selected. Each specimen was reinforced in flexure with two 25M bars and the shear reinforcement was 10M spaced at 150 mm on centre. The specimens were corroded for 21 days, 60 days, and 120 days corresponding to low, medium, and high corrosion levels. In addition, three specimens were constructed without shear reinforcement in the shear-span in order to compare the results from the corroded specimens. One specimen was also corroded to a high level and repaired with dry lay-up CFRP. The specimens were corroded using an accelerated corrosion technique. There was evidence of cracking of the cover concrete in all specimens, and in the more severely corroded specimens delamination of the cover concrete was recorded. The stiffness of the corroded specimens was less than their corresponding control specimen, and a strength reduction was evident in most specimens. The maximum recorded strength reduction was 52% compared to the companion uncorroded specimen. It was revealed that a more critical case occurs when the corroded shear reinforcement was shifted during placement or was inclined closer to the direction of the compressive force flow. Also, it was observed that the corroded shear reinforcement still provides limited ductility in comparison to the un-corroded reinforcement. A strut and tie model was developed based on the experiments to explain the behaviour of disturbed regions with corroded shear reinforcement. The model consisted of direct and indirect struts. The effects of corrosion were expressed in terms of a reduction in the stirrup cross-section, a reduction of compressive strength due to corrosion cracking, and a reduction in the concrete cross section width. It was hypothesized that the corrosion crack width influences the concrete compressive strength in the strut; consequently, a mathematical model was developed that related the reduction in concrete compressive strength with corrosion crack width. Also, a relationship between reinforcing steel mass loss and corrosion crack width was utilized from the published literature. An effective cross section width was obtained by reducing the width by the damaged concrete cover. The results from these models were input into a strut and tie model as a reduction in concrete compressive strength. The output from the strut and tie model was the ultimate shear strength of the specimen. The developed models were compared with a model from the literature and compared with the experimental results. The major contribution of this research is to allow designers to analyze disturbed regions with corroded shear reinforcement and determine the strength degradation; subsequently, one can determine what strengthening procedure would be most appropriate.

Corrosion of reinforcing steel is a major problem facing infrastructures owners with billions of dollars spent in repairing our aging infrastructure. One of the first steps in the repair process is to quantify the strength degradation in a reinforced concrete element caused by the corrosion of reinforcing steel. An understanding of the forces involved in the load carrying mechanisms is imperative; the transfer of shear forces in reinforced concrete beams is one of these load carrying mechanisms. The shear transfer mechanism is different near the end of beams, adjacent to point loads, and near changes in cross section. These regions are known as disturbed regions. Structural engineers have a good understanding of the shear transfer mechanism in disturbed regions. However, the effects of corroded shear reinforcement in these regions have not been widely investigated. The current study is comprised of an experimental program and analytical strut and tie modeling aimed at quantifying the strength reduction that occurs in disturbed regions of reinforced concrete beams with corroded shear reinforcement. The feasibility of strengthening a beam with dry lay-up carbon fibre reinforced polymer (CFRP) to repair the damage caused by corrosion of the shear reinforcement was also investigated. In the experimental study, a total of 16 reinforced concrete beams were cast. The specimens were 350 mm deep, 125 mm wide and 1850 mm long. Three shear-span to depth ratios (1.0, 1.5, 2.0) were selected. Each specimen was reinforced in flexure with two 25M bars and the shear reinforcement was 10M spaced at 150 mm on centre. The specimens were corroded for 21 days, 60 days, and 120 days corresponding to low, medium, and high corrosion levels. In addition, three specimens were constructed without shear reinforcement in the shear-span in order to compare the results from the corroded specimens. One specimen was also corroded to a high level and repaired with dry lay-up CFRP. The specimens were corroded using an accelerated corrosion technique. There was evidence of cracking of the cover concrete in all specimens, and in the more severely corroded specimens delamination of the cover concrete was recorded. The stiffness of the corroded specimens was less than their corresponding control specimen, and a strength reduction was evident in most specimens. The maximum recorded strength reduction was 52% compared to the companion uncorroded specimen. It was revealed that a more critical case occurs when the corroded shear reinforcement was shifted during placement or was inclined closer to the direction of the compressive force flow. Also, it was observed that the corroded shear reinforcement still provides limited ductility in comparison to the un-corroded reinforcement. A strut and tie model was developed based on the experiments to explain the behaviour of disturbed regions with corroded shear reinforcement. The model consisted of direct and indirect struts. The effects of corrosion were expressed in terms of a reduction in the stirrup cross-section, a reduction of compressive strength due to corrosion cracking, and a reduction in the concrete cross section width. It was hypothesized that the corrosion crack width influences the concrete compressive strength in the strut; consequently, a mathematical model was developed that related the reduction in concrete compressive strength with corrosion crack width. Also, a relationship between reinforcing steel mass loss and corrosion crack width was utilized from the published literature. An effective cross section width was obtained by reducing the width by the damaged concrete cover. The results from these models were input into a strut and tie model as a reduction in concrete compressive strength. The output from the strut and tie model was the ultimate shear strength of the specimen. The developed models were compared with a model from the literature and compared with the experimental results. The major contribution of this research is to allow designers to analyze disturbed regions with corroded shear reinforcement and determine the strength degradation; subsequently, one can determine what strengthening procedure would be most appropriate.

A compatibility-based strut-and-tie model C-STM is proposed for analyzing deep beams and disturbed regions with particular emphasis on reinforced concrete bridge piers. In addition to the normal strut-and-tie force equilibrium requirements the model accounts for non-linear behavior through displacement compatibility using inelastic constitutive laws of cracked reinforced concrete. The model is implemented into widely used commercial structural analysis software and validated against results from previously conducted large scale experiments. A near full-scale experiment on a reinforced concrete sub-assemblage that represents cantilevered and straddle pier bents is conducted to investigate the shear-flexure performance of deep (disturbed) regions. Insights into the development of nonlinear behavior and the final collapse failure mechanism are then evaluated and accurately modeled using the C-STM. It is concluded that the proposed C-STM serves as an advanced method of analysis that can predict with suitable accuracy the force-deformation response of both D- and B- regions, deep beams, and beam-columns. This provides engineers with a supplementary analysis tool that can be used to assess the nonlinear behavior of bridge piers with stocky members and/or large disturbed regions.

[pt] No presente trabalho se estuda um elemento finito subparamétrico que aproxima o campo de deslocamentos com funções spline, implementando um programa que pode ser utilizado para calculo estático, dinâmico e de instabilidade de estruturas compostas de placas, vigas de paredes finas, vigas caixão e em geral em elementos alongados (pontes e perfis metálicos). O grau de liberdade de rotação perpendicular ao plano do elemento é introduzido na formulação para possibilitar uma análise tridimensional. Apresenta-se um método que serve como base para determinar a constante de rigidez correspondente. Nos exemplos apresentados avalia-se a precisão obtida utilizando pouco número de divisões longitudinais do continuo, vantagem que justifica o uso desses elementos em estudos de pré-projeto ou otimização de estruturas. Comparam-se os resultados com soluções teóricas ou resultados de outros programas estruturais, permitindo apreciar as possibilidades e limitações da modelagem usando elementos finitos com funções spline. As diferenças observadas, que surgem principalmente em placas espessas, são explicadas pela aproximação da deformação de cisalhamento encontrada na literatura para os elementos utilizados na comparação. Mostra-se também exemplos de instabilidade analisados em três dimensões que permitem considerar diferentes condições de apoio e discutir os resultados de fórmulas conhecidas. / [en] The present work presents a subparametric finite element model with spline displacement functions, implemented for static, dynamic and instability analysis of folded plates, thin-walled beams, box girders, and elongated structures such as bridges and structural shapes. A drilling degree of freedom (rotation about an axis perpendicular to the plane of the element) is introduced in the formulation to allow for three-dimensional analysis. A method for determining the corresponding contribution to the stiffness matrix is presented. The examples presented evaluate the accuracy obtained using a small number of longitudinal subdivisions of the continuum, convenient in the case of analyses for preliminary design and optimization. The results obtained are compared to theoretical solutions or results of standard structural analysis programs, allowing for an appraisal of the advantages and limitations of modeling with use of spline functions. The differences in the results, observed specially in the case of thick plates, are explained by the approximations for the shear strain in the elements used for comparison. From the examples it is possible to comment results of threedimensional modeling of instability problems with different boundary conditions.

yes / Twenty two reinforced concrete continuous deep beams with openings and two companion solid deep beams were tested to failure. The main variables investigated were the configuration of web reinforcement around openings, location of openings, and shear span-to-overall depth ratio. The influence of web reinforcement on controlling diagonal crack width and load capacity of continuous deep beams with openings was significantly dependent on the location of openings. The development of diagonal crack width and load capacity of beams having openings within exterior shear spans were insensitive to the configuration of web reinforcement. However, for beams having openings within interior shear spans, inclined web reinforcement was the most effective type for controlling diagonal crack width and increasing load capacity. It has also observed that higher load and shear capacities were exhibited by beams with web reinforcement above and below openings than those with web reinforcement only above openings. The shear capacity at failed shear span of continuous beams tested is overestimated using Kong et al’s formula developed for simple deep beams with openings.

Yes / The test results on eight two-span deep beams made of self-compacting concrete (SCC) are presented and discussed in this paper. The main parameters investigated were the shear span-to-depth ratio, and the amount and configuration of steel reinforcement. All beams failed due to a major diagonal crack formed between the applied mid-span load and the intermediate support separating the beam into two blocks: the first one rotated around the end support leaving the other block resting on the other two supports. Both concrete compressive strength and web reinforcement had a major effect in controlling the shear capacity of the beams tested. For the shear span-to-depth ratio considered, the vertical web reinforcement had more influence on the shear capacity of the specimens than the horizontal web reinforcement. The shear provisions of the ACI 318M-11 are unconservative for most of the beams tested. Comparisons of test results with the strut-and-tie model (STM) suggested by ACI 318M-11, EC2 and CSA23.4-04 showed that the predictions are reasonable for continuous deep beams made with low and medium compressive strength. Although the equation suggested by ACI 318M-11 is very simple, its prediction is more accurate than the STM suggested by different design codes. / This research investigation was funded by the Higher Education Ministry in The Libyan Government.

Yes / Ten reinforced-concrete continuous deep beams with openings were tested to failure. The main variables investigated were the shear span-to-overall depth ratio, and the size and location of openings. Two failure modes influenced by the size and location of web openings regardless of the shear span-to-overall depth ratio were observed. The normalised load capacity of beams having a web opening area ratio of 0·025 within exterior shear spans was approximately similar to that of their companion solid beams. Continuous deep beams having web openings within interior shear spans exhibited a higher load capacity reduction with the increase of the opening size, similar to simply supported deep beams with web openings. Formulae based on the upper bound analysis of the plasticity theory were proposed to predict the load capacity of continuous deep beams with web openings. Comparisons between the measured and predicted load capacities showed a good agreement.

Yes / Nine continuous concrete deep beams reinforced with glass fibre reinforced polymer (GFRP) bars were experimentally tested to failure. Three main parameters were investigated, namely, shear span-to-overall depth ratio, web reinforcement and size effect. The experimental results confirmed the impacts of web reinforcement and size effect that were not considered by the strut-and-tie method (STM) of the only code provision, the Canadian S806-12, that addressed such elements. The experimental results were employed to evaluate the applicability of the methods suggested by the American, European and Canadian codes as well as the previous studies to predict the load capacities of continuous deep beams reinforced with GFRP bars. It was found that these methods were unable to reflect the influences of size effect and/or web reinforcement, the impact of which has been confirmed by the current experimental investigation. Therefore, a new effectiveness factor was recommended to be used with the STM. Additionally, an upper-bound analysis was developed to predict the load capacity of the tested specimens considering a reduced bond strength of GFRP bars. A good agreement between the predicted results and the experimental ones was obtained with the mean and coefficient of variation values of 1.02 and 5.9%, respectively, for the STM and 1.03 and 8.6%, respectively, for the upper-bound analysis. / Higher Committee of Education Development in Iraq (HCED)

Several reinforced concrete bent caps (deep beams) in Texas have developed significant diagonal cracks in service. The cracking in two bent caps was so extensive that costly retrofits were implemented to strengthen the structures. Strut-and-tie modeling is currently recommended in most U.S. design specifications for the design of reinforced concrete bent caps and deep beams. Designers have expressed concerns with the lack of clarity and serviceability-related considerations in strut-and-tie model design provisions. Due to concerns with strut-and-tie modeling design provisions and field problems of in-service bent caps, TxDOT Project 5253 was funded. Several tasks conducted within Project 5253 are addressed in this dissertation. The effects of minimum web reinforcement and member depth on the strength and serviceability behavior of deep beams are presented. The transition between deep beam shear capacity and sectional shear capacity near a shear-span-to-depth (a/d) ratio of 2 is addressed. A service-load shear check to limit diagonal cracking in service is outlined. Lastly, a simple chart that correlates the maximum width of diagonal cracks in a deep beam to its residual capacity is developed. To accomplish the objectives of Project 5253, thirty-seven tests were conducted on reinforced concrete beams with the following cross-sectional dimensions: 21”x23”, 21”x42”, 21”x44”, 21”x75”, and 36”x48.” The specimens were loaded with a/d ratios of 1.2, 1.85, and 2.5. The test specimens are among the largest reinforced concrete deep beams in the literature. To supplement the findings of the experimental program, a database of deep beam test results was compiled. Entries in the database that lacked sufficient information and that did not meet established cross-sectional size or web reinforcement criteria were filtered from the database. The use of the database in conjunction with the experimental program enabled each objective to be addressed from both broad and specific viewpoints. Several recommendations for improving the strength and serviceability design of deep beams are presented including a minimum web reinforcement requirement, provisions to ease the transition between calculated deep beam and sectional shear capacity, and a design check to limit diagonal cracking in service. / text

The present research is conducted to investigate the structural behaviour of continuously supported deep beams made with SCC. A series of tests on eight reinforced two-span continuous deep beams made with SCC was performed. The main parameters investigated were the shear span-to-depth ratio, the amount and configuration of web reinforcement and the main longitudinal reinforcement ratio. All beams failed due to a major diagonal crack formed between the applied mid-span load and the intermediate support separating the beam into two blocks: the first one rotated around the end support leaving the rest of the beam fixed on the other two supports. The amount and configuration of web reinforcement had a major effect in controlling the shear capacity of SCC continuous deep beams. The shear provisions of the ACI 318M-11 reasonably predicted the load capacity of SCC continuous deep beams. The strut-and-tie model recommended by different design codes showed conservative results for all SCC continuous deep beams. The ACI Building Code (ACI 318M-11) predictions were more accurate than those of the EC2 and Canadian Code (CSA23.3-04). The proposed effectiveness factor equations for the strut-and-tie model showed accurate predictions compared to the experimental results. The different equations of the effectiveness factor used in upper-bound analysis can reasonably be applied to the prediction of the load capacity of continuously supported SCC deep beams although they were proposed for normal concrete (NC). The proposed three dimensional FE model accurately predicted the failure modes, the load capacity and the load-deflection response of the beams tested.