Use of Strut-and-Tie Models to Calculate the Strength of Deep Beams with Openings

Strut-and-tie modeling is a method applicable to almost every design situation in reinforced concrete. This is a behavioral theory proposed as a alternative to past design strategies utilizing empirical formulas and parameters. Since the original presentation of this method in the 60' s numerous experimental studies have been conducted, yet the topic of deep beams with large web openings has not been widely covered. Design codes and guidelines also do not commonly cover this topic. However empirical design equations have been proposed based on previous research in the field. An empirical method is presented and the relation to the beam geometry and behavior is discussed. A discussion of the strut-and-tie method is also given including the limited previous research and application of the method. These two methods are compared using previous experimental results of deep beams with openings. The comparison includes analysis of predicted loads and ultimate loads as well as predicted behavior using the strut-and-tie method for beams with and without web reinforcement. For beams with reinforcement a model was constructed to compare a realistic reinforcement detail. This generates a fairly accurate assessment of strength and behavior with the experimental results. In beams without reinforcement a model is presented using ties only where available. This general model was then adapted to three of the experimental beam geometries. This model gives consistent prediction of the ultimate load and beam behavior in each beam. The results presented reinforce the strut-and-tie method as a safe approach in structurally diverse situations where empirical methods may have a limited range of application

Performance of Reinforced Concrete Structures Subjected to Earthquake Motions

Two simplified methods for estimating the performance of reinforced concrete structures subjected to earthquake motions were evaluated. Both the Flat-Rate and Target Period methods characterize the expected level of performance in terms of the maximum estimated drift for a given intensity of ground motion. Drift estimates using the Flat-Rate method are based on the area of structural members, the total floor area of the structure, and the peak ground acceleration as a measure of earthquake intensity. The main parameters for the Target Period method are the initial period of the structures and the peak ground acceleration. The applicability of these methods to assess the expected level of performance of existing structures was investigated using experimental data. Drift values calculated with the Flat-Rate and Target Period methods were compared with measurements obtained from earthquake simulator tests performed on reduced-scale models of reinforced concrete structures. Results indicate that both methods provided an adequate assessment of performance

Design of Simply Supported Deep Beams Using Strut-and-Tie Models

A procedure to calculate the amount of reinforcement and the strength of deep beams based on strut-and-tie models is presented. The proposed design equations were calibrated using experimental results from 175 simply supported beams found in the literature with a maximum shear span-to-depth ratio of 3. The strength reduction coefficient for concrete in the main strut was found to decrease with the angle of inclination of the strut, resulting in lower values than those stated in Appendix A of the 2002 edition of the ACI 318 Building Code for beams with shear span-to-depth ratios greater than 1

Shear Strength of Reinforced Concrete Members Subjected to Monotonic and Cyclic Loads

The shear capacity of reinforced concrete members subjected to monotonic loads was investigated and used as the basis to formulate an expression to calculate the strength of members subjected to load reversals. The monotonic shear capacity of slender beams, deep beams, walls, and columns was calculated by superposition of components related to arch-action, trussaction, friction, and from a contribution of the uncracked compression zone, which is related to the tensile strength of concrete. A procedure to calculate the shear strength of members in the transition phase from deep to slender members was formulated, so that the proposed expression can be used for all geometries considered. The shear strength of members with and without web reinforcement was analyzed. The proposed model was calibrated using an extensive database of test results, and was found to give good results compared to other analysis models in an n-fold cross validation. The resistance to lateral load reversals was investigated for two failure modes: failure due to degradation of the flexural strength, and failure due to degradation of the shear strength. The degradation of flexural strength is expressed in terms of a linear slope derived from the displacement and load at yielding of the tensile reinforcement to the displacement at 80 percent of the yield load. Shear failure was defined by yielding of the transverse reinforcement. The degradation of shear strength was found to be non-linear with respect to the limiting displacement, and is formulated as a reduction factor for the initial shear strength. Degradation functions for the decrease in strength of the contributing arch and compression zone components, and for the truss mechanism are presented. The following key conclusions were drawn from this study: 1. The monotonic shear capacity can be modeled by the proposed superposition of contributing components for member geometries ranging from squat to deep members. Simply superimposing the individual components, however, does not reflect the actual member behavior. Functions transitioning between squat and slender members, as well as between reinforced members and members without web reinforcement, are necessary to model the member behavior accurately. 2. In the proposed model, the friction component is used to control the so-called "size effect." It was found that the "size effect" is not only an effect of the section depth, but is also influenced by the compressive strength of concrete, the tensile reinforcement ratio, and the average shear stress. ii 3. The shear strength degradation under cyclic lateral loads was found to be due to a reduction of the components related to friction and the compression zone, and to a reduction of the truss mechanism. 4. The shear analysis according to the proposed model gave more accurate results than the other models considered in the study at hand. Moreover, with the exception of the approach proposed by Watanabe, compared to other methods, it was the only model applicable to a wide range of member configurations

Effect of Preexisting Cracks on Lap Splice Strength of Reinforcing Bars

The effect of preexisting subsurface cracks on the strength of lap splices was investigated. Ten full-scale beams with No. 11 (No. 36) bars and lap splice lengths of 33, 79, and 120 in. (838, 2007, and 3048 mm) were tested. The beams had mitigating features that prevented catastrophic failure upon propagation of the preexisting cracks, such as staggered splices and the presence of some reinforcement crossing the plane of the cracks. The effect of preexisting cracks on the bar stress at failure was found to be most severe for the shortest splices and not significant for the two other splice lengths evaluated. The effect was found to be dependent on the amount of reinforcement crossing the plane of the cracks. Splice strength was unaffected in beams with the largest amount of reinforcement, and reduced on the order of 50% in beams without any reinforcement crossing the plane of the cracks

A Simplified Method to Estimate Nonlinear Response with an Approximate Linear Analysis for Reinforced Concrete Structures

A background on simple methods to estimate nonlinear response of multidegree- of-freedom (MDOF) systems currently in use is presented as an introduction to development of a new method. A series of nonlinear analyses of 105 concrete building structures with varying number of stories and structural configurations evaluated to determine the maximum drift demands imposed by a suite of 10 ground motions. The ground motions were selected and scaled to represent a smooth displacement spectrum. The combination of damping and effective stiffuess of equivalent single-degree-of-freedom (SDOF) linear systems that resulted in the most accurate estimates of the maximum nonlinear drift for high and moderate seismic demands is presented. The location and magnitude of the story drift ratio (SDR) for linear SDOF and nonlinear MDOF models of the building systems was also examined and compared. A primary conclusion of the study was that an equivalent SDOF system evaluated with an effective period of 2.3 and 2.0 times initial period in regions of high and moderate seismicity, respectively, and a 10% damped response spectrum produced the most consistent and accurate estimate of nonlinear building displacement for the frames and earthquakes considered. In general, the magnitude of SDR for the nonlinear MDOF systems were 1.5 time the SDR for linear SDOF systems

Drift-Dependent Confinement Requirements for Reinforced Concrete Columns Under Cyclic Loading

The influence of transverse reinforcement and axial load on the drift limit of rectangular columns was investigated using test results from 184 specimens subjected to cyclic loading. Columns within the set were selected to have shear span-to-depth ratios of at least equal to 2.5 so that truss action would be the primary mechanism of shear resistance and the deformation component related to shear would be small compared with that related to flexure. Expressions relating the limiting drift ratio to the axial load ratio and the amount of confining reinforcement were evaluated. Equations indicating the amount of confining reinforcement required to achieve a given limiting drift ratio for reinforced concrete columns in regions of moderate and high seismicity are proposed

Modeling Surface Deformations and Hinging Regions in Reinforced Concrete Bridge Columns

A high-resolution model of a bridge column was developed using the computer program ABAQUS and the accuracy of the model was evaluated for the displacement field and the rotations of a bridge system subjected to biaxial shake-table loading. The effect of simulation parameters (reinforcing bar slip within the joint and stiffness degradation of the concrete) was studied to determine the goodness-of-fit of the displacement and rotation fields recorded during the dynamic response. A Fourier Domain Error Index analyses showed that yield stress of the reinforcement and the boundary conditions of the column submodel were important parameters, and the damage and stiffness degradation parameters were not as important for the goodness-of-fit of the finite element model. The computed rotations at the plastic hinge regions near the beam caps had the best correlation