Analytical Solution for Piles Supporting Combined Lateral Loads

Analytical solutions of normalized maximum deflection and normalized maximum moment for laterally loaded long piles in homogeneous elastoplastic soil under combined loads are presented in this paper. Both the normalized deflection surface and normalized moment surface are continuous and increasing constantly with normalized applied force and moment with various slopes. It shows that the normalized applied force and moment have different contributions to the deflection and moment. In general, the variation of normalized moment surface is relatively moderate compared to normalized maximum deflection. Due to the nonlinear effect, the deflections and moments using superposition approach will be on the unsafe side. The analytical solutions can be used for any elastic materials, any type of soil, and any shapes of pile cross section. Above all, the analytical solutions may be easily applied to calculate the maximum deflection or moment of lateral long piles subject to combined loads accurately using a calculator.     

电梯导轨矫直压弯挠度理论计算与数值模拟

为使导轨的平直度达到高速电梯的要求,通过应用弹塑性理论,建立了电梯导轨在矫直过程中的数学模型,模型给出了电梯导轨在反弯矫直过程中挠度、弯矩、曲率以及中性层之间的关系,并获得了矫直压下量的取值范围.通过有限元软件ANSYS分析验证了理论计算得到的矫直压下量取值的可靠性,并证明能够使矫直后的导轨符合使用要求.     

Equivalent Moment of Inertia Based on Integration of Curvature

This paper evaluates the benefits of computing deflection with an equivalent moment of inertia based on integration of curvature to account for changes in member stiffness along the span. Results are evaluated for steel and fiber-reinforced polymer reinforced (FRP-reinforced) concrete flexural members with different loading arrangements and support conditions. Closed-form solutions of integrated expressions for deflection are expressed in terms of an equivalent moment of inertia Ie′ and compared to deflection computed with an effective moment of inertia Ie based on the stiffness at the critical section. Results from this comparison are validated with measured deflections from an experimental database for FRP-reinforced concrete. Current code-related approaches are also compared to the experimental database. It is shown herein that the use of an integration-based expression for the moment of inertia can lead to improved prediction of deflection, though the use of an effective moment of inertia based on member stiffness at the critical section gives a reasonably conservative estimate of deflection in many cases. The benefits of taking account of changes in stiffness along the member span are more evident when low reinforcing ratios are used in combination with FRP reinforcement, and use of the integration-based expression Ie′ may be warranted when deflection control is critical in such cases.     

Numerical Solution for Laterally Loaded Piles in a Two-Layer Soil Profile

Piles are often embedded in a layered soil profile, such as sand or clay layer underlain by rock. Several existing solutions are available for laterally loaded piles in a layered soil system. However, these solutions are only applicable to constant soil stiffness for each layer. In this paper, a variational approach is employed to numerically solve the problem of laterally loaded piles in layered soils using beam on an elastic foundation model. The soil stiffness can be either constant with depth or linearly varying with depth. The numerical solution is validated against an existing solution for linearly varying soil stiffness in a single soil layer system and an existing solution for a two-layer soil system with constant soil stiffness. Case studies using the proposed solution for field lateral load tests on full size drilled shafts embedded in weak rock with an overlying sand layer are presented. The simplicity and the relative ease of using the solution make it a good alternative approach for estimating the deflection and moment responses of a laterally loaded pile in a two-layer soil profile.     

Buckling and Postbuckling of Antisymmetrically Laminated Cross-ply Shear-Deformable Cylindrical Shells under Axial Compression

The generalized Donnell-type equations governing large deflection of antisymmetrically laminated cross-ply cylindrical shells counting for transverse shear deformations are derived and presented. An asymptotic series solution is constructed by regular perturbation technique for postbuckling behaviors of the cylindrical shells with simply supported edges subjected to axial compression. Boundary layer influence at both ends of the shells on overall buckling and postbuckling are considered, and for consistency of the boundary valued problem, the boundary layer solutions are also designed to match the out-of-plane edge conditions by singular perturbation approach. Effects of transverse shear deformation, Batdorf’s parameter, elastic moduli ratio, and initial geometric imperfection on buckling and postbuckling performance of the shells are examined. Some numerical examples are taken for comparison of the present results of buckling loads and load–deflection curves of the shells with corresponding theoretical predictions to show effectiveness and accuracy of the present asymptotic perturbation solution.     

Design Approach for Calculating Deflection of FRP-Reinforced Concrete

Deflection of reinforced concrete is typically computed with an effective moment of inertia Ie that accounts for nonlinear behavior after the concrete cracks. Existing expressions for Ie tend to overpredict the member stiffness of concrete reinforced with fiber-reinforced polymer (FRP) bars, and an alternative expression is used as the basis for developing a practical design approach to compute deflection. The proposed expression has a rational basis that incorporates basic concepts of tension stiffening to provide a reasonable estimate of deflection for both steel and FRP-reinforced concrete without the need for empirically derived correction factors. Calculation of deflection with the proposed expression for Ie is recommended using the code value for the elastic modulus Ec of concrete because computed values of deflection are relatively insensitive to variations in Ec, and shrinkage restraint is taken into account by using a reduced cracking moment less than the code-based value of the cracking moment Mcr. Ie is conservatively based on the moment at the critical section (where the member stiffness is lowest), unless more accuracy is required with an integration-based expression that gives an equivalent moment of inertia Ie′ to account for the variation in stiffness along the member length. Recommendations are validated by comparison with a database of deflection test results for FRP-reinforced concrete.     

Behavior of Pile Subject to Excavation-Induced Soil Movement

This paper presents the results of centrifuge model tests on unstrutted deep excavation in dense sand and its influence on an adjacent single pile foundation behind the retaining wall. It is found that, in the case of a stable wall, the induced pile bending moment and deflection decrease exponentially with increasing distance between the pile and the wall. Pile head boundary condition plays an important role in affecting the pile responses due to an adjacent excavation. In the case of retaining wall collapse, the failure pattern of the soil behind the wall features a slip plane projecting from near the wall toe to the ground surface. Soil within the failure zone demonstrates large lateral movement and induces significant bending moment and deflection on pile located within the zone. Soil movement and pile responses outside this zone are noted to be significantly less. A comparison between the experimental results and the theoretical predictions by an existing numerical method shows good agreement, provided that appropriate assumptions are made on the soil parameters and conditions, especially in the case of retaining wall collapse.     

Closed-Form Solutions for Flexural Response of Fiber-Reinforced Concrete Beams

A constitutive law for fiber-reinforced concrete materials consisting of an elastic perfectly plastic model for compression and an elastic-constant postpeak response for tension is presented. The material parameters are described by using Young’s modulus and first cracking strain in addition to four nondimensional parameters to define postpeak tensile strength, compressive strength, and ultimate strain levels in tension and compression. The closed-form solutions for moment-curvature response are derived and normalized with respect to their values at the cracking moment. Further simplification of the moment-curvature response to a bilinear model, and the use of the moment-area method results in another set of closed-form solutions to calculate midspan deflection of a beam under three- and four-point bending tests. Model simulations are correlated with a variety of test results available in literature. The simulation of a three- and four-point bending test reveals that the direct use of uniaxial tensile response underpredicts the flexural response.     

Analysis of Thick Circular Plates Undergoing Large Deflections

This investigation considers the effect of transverse shear deformation on bending of the axisymmetrically loaded isotropic and orthotropic circular and annular plates undergoing large deflection. The analysis treats the nonlinear terms of lateral displacement as fictitious loads acting on the plate. The solution of a von Kármán‐type plate is, therefore, reduced to a plane problem in elasticity and a linear plate‐bending problem. Results are presented for simply supported and clamped plates and are in good agreement with the available solutions. For plates considered in this study, the influence of shear deformation on lateral displacement becomes more significant as the orthotropic parameter increases. The linear and nonlinear solutions for orthotropic plates deviate at a low value of the maximum deflection‐to‐thickness ratio (w/h). Consequently, the extent of w/h within which the small‐deflection theory is applicable to orthotropic plates is much lower than the value of about 0.4 typically used for isotropic plates, and it depends, in general, on the degree of orthotropy. The technique employed in this study is well suited for the analysis of nonlinear plate problems.     

Moment/Rotation Effects on Laterally Loaded Drilled Shaft Group Response

Drilled shaft groups are often designed to resist lateral loads for transportation structures. The shaft group capacity usually corresponds to a load being applied at the shaft cap level. However, in abutment wall applications, the lateral load is, in fact, applied well above the cap elevation. Thus, the load is transferred to the cap with an additional moment, causing the cap to deflect and rotate more than if this added moment were absent. As a result, the lateral capacity for a given allowable deflection of the group should be reduced because of this effect. Design engineers usually select or approve the allowable deflection at the top of the abutment wall. However, deflection at the cap level is needed to design the group capacity. The main objective of this paper is to report the results from a series of finite-element analyses on abutment wall cap configurations to study the effect of moment on the capacity of the shaft group under lateral load. A scaling factor is defined as the ratio between the group capacity for load applied at a given height above the cap and the group capacity for load applied at the bottom of the cap, and it was found to be dependent on the wall height, the spacing between shafts, and the cap deflection level, and more or less independent of the soil type, the cap thickness, and the shaft diameter. The ratio between the deflection at the top of the abutment wall and the deflection at the cap was found to be dependent on the wall stiffness (wall thickness to wall height ratio).     

Pile Behavior due to Excavation-Induced Soil Movement in Clay. I: Stable Wall

A series of centrifuge model tests has been conducted to investigate the behavior of a single pile subjected to excavation-induced soil movements behind a stable retaining wall in clay. The results reveal that after the completion of soil excavation, the wall and the soil continue to move and such movement induces further bending moment and deflection on an adjacent pile. For a pile located within 3?m behind the wall where the soil experiences large shear strain (>2%) due to stress relief as a result of the excavation, the induced pile bending moment and deflection reach their maximum values sometime after soil excavation and thereafter decrease slightly with time. For a pile located 3?m beyond the wall, the induced pile bending moment and deflection continue to increase slightly with time after excavation until the end of the test. A numerical model developed at the National University of Singapore is used to back-analyze the centrifuge test data. The method gives a reasonably good prediction of the induced bending moment and deflection on a pile located at 3?m or beyond the wall. For a pile located at 1?m behind the wall where the soil experiences large shear strain (>2%) due to stress relief resulting from the excavation, the calculated pile response is in good agreement with the measured data if the correct soil shear strength obtained from postexcavation is used in the analysis. However, if the original soil shear strength prior to excavation is used in the analysis, this leads to an overestimation of the maximum bending moment of about 25%. The practical implications of the findings are also discussed in this paper.     

Crack deflection: Implications for the growth of long and short fatigue cracks

The influences of crack deflection on the growth rates ofnominally Mode I fatigue cracks are examined. Previous theoretical analyses of stress intensity solutions for kinked elastic cracks are reviewed. Simple elastic deflection models are developed to estimate the growth rates of nonlinear fatigue cracks subjected to various degrees of deflection, by incorporating changes in the effective driving force and in the apparent propagation rates. Experimental data are presented for intermediate-quenched and step-quenched conditions of Fe/2Si/0.1C ferrite-martensite dual phase steel, where variations in crack morphology alone influence considerably the fatigue crack propagation rates and threshold stress intensity range values. Such results are found to be in good quantitative agreement with the deflection model predictions of propagation rates for nonlinear cracks. Experimental information on crack deflection, induced by variable amplitude loading, is also provided for 2020-T651 aluminum alloy. It is demonstrated with the aid of elastic analyses and experiments that crack deflection models offer a physically-appealing rationale for the apparently slower growth rates of long fatigue cracks subjected to constant and variable amplitude loading and for the apparent deceleration and/or arrest of short cracks. The changes in the propagation rates of deflected fatigue cracks are discussed in terms of thelocal mode of crack advance, microstructure, effective driving force, growth mechanisms, mean stress, slip characteristics, and crack closure.     

Flexural Behavior and Deformability of Fiber Reinforced Polymer Prestressed Concrete Beams

After a brief review of the ductility and deformability indices currently used in the design of concrete beams reinforced or prestressed with steel or fiber reinforced polymer (FRP) tendons, a new definition of a deformability index (factor) for prestressed concrete beams is proposed. The new factor is defined in terms of both a deflection factor and a strength factor. The deflection factor is the ratio of the deflection at failure to the deflection at first cracking, while the strength factor is the ratio of the ultimate moment (or load) to the cracking moment (or load). The proposed deformability factor is verified not only by test results obtained by the writer, but also by other test results available in the literature and it appears to be a suitable measurement of the deformability of concrete beams prestressed with either FRP tendons or steel tendons.     

Latex allergy

The purpose of this study was to assess the mechanical properties--torsional moment, maximum angular deflection, maximum bending moment, and permanent angular deflection--of four brands of nickel-titanium (NiTi) endodontic file, and compare them with a conventional stainless-steel instrument, both in the presence and absence of sodium hypochlorite (NaOCl). NiTi instruments from four manufacturers were randomly selected and subjected to NaOCl treatment for 12 or 48 h, or not at all. The mechanical properties under test were then measured automatically by a digital torque memocouple. Torsional moment and maximum angular deflection indicate the resistance to torsional fracture of an instrument, maximum bending moment the stiffness of the instrument, and permanent angular deflection the strength of the base alloy. All instruments evaluated complied with or exceeded ADA/ANSI Specification No. 28, with the sole exception of the Maillefer ISO size 40 for torsional moment. JS Dental and McSpadden NiTi files were the most resistant to torsional fracture, but all NiTi files were inferior when compared with stainless-steel files from a previous study. However, NiTi files were superior in flexibility, and Maillefer and Brasseler instruments were the best of the instruments tested. NiTi files also had negligible permanent deformation angles. Furthermore, for all properties tested, NaOCl had no statistically significant effect.     

Deflection Prediction of Steel and FRP-Reinforced Concrete-Filled FRP Tube Beams

This paper presents the results of experimental and theoretical investigations that study the flexural behavior of reinforced concrete-filled fiber-reinforced polymer (FRP) tubes (RCFFTs) beams. The experimental program consists of 10 circular beams [6 RCFFT and 4 control reinforced concrete (RC) beams] with a total length of 2,000?mm, tested under four-point bending load. The experimental results were used to review and verify the applicability of various North American code provisions and some available equations in the literature to predict deflection of RCFFT beams. The measured deflections and the experimental values of the effective moment of inertia were analyzed and compared with those predicted using available models. The results of the analysis indicated that the behavior of steel and FRP-RCFFT beams under the flexural load was significantly different than that of steel and FRP-RC members. This is attributed to the confining effect of the FRP tubes and their axial contribution. This confining behavior in turn enhanced the overall flexural behavior and improved the tension stiffening of RCFFT beams. For that, the predicted tension stiffening of steel and FRP-RCFFT beams using the conventional equations (steel or FRP-RC member) underestimates the flexural response; therefore, the predicted deflections are overestimated. Based on the analysis of the test results, the Branson’s equation for the effective moment of inertia of RC structures is modified, and new equations are developed to accurately predict the deflection of concrete-filled FRP tube (CFFT) beams reinforced with steel or FRP bars.     

Deflection Calculation of FRP Reinforced Concrete Beams Based on Modifications to the Existing Branson Equation

Fundamental concepts of tension stiffening are used to explain why Branson’s equation for the effective moment of inertia Ie does not predict deflection well for fiber reinforced polymer (FRP) reinforced concrete beams. The tension stiffening component in Branson’s equation is shown to depend on the ratio of gross-to-cracked moment of inertia (Ig/Icr), and gives too much tension stiffening for beams with an Ig/Icr ratio greater than 3. FRP beams typically have an Ig/Icr ratio greater than 5, leading to a much stiffer response and underprediction of computed deflections as observed by others in the past. One common approach to computing deflection of FRP reinforced concrete beams has been to use a modified form of the Branson equation. This paper presents a rational development of appropriate modification factors needed to reduce the tension stiffening component in Branson’s original expression to realistic levels. Computed deflections using this approach give reasonable results with the right modification factor, and compare well with a more general unified approach that incorporates a realistic tension stiffening model. Comparison is made with the existing and past correction factors recommended by ACI 440 for predicting deflection of FRP beams. The method presently used by ACI 440 gives reasonable estimates of deflection for glass and carbon FRP reinforced beams. However, this method underestimates deflection of aramid FRP reinforced beams and is restricted to rectangular sections. A proposal is made for adoption of a simple modification factor that works well for all types of FRP bar and beam cross-sectional shape.     

Interactive Failure of Two Impacting Beams

A complete analysis of an inelastic beam-to-beam impact is presented. Both beams are of solid, rectangular cross-sections. The problem is formulated based on the momentum conservation and the kinematic and dynamic continuity conditions at the moving wavefront. Closed-form solutions are obtained for transient transverse velocities, deflection profiles, and tensile strains based on the moderately large deflection theory of the beam. Three different regimes of the solution are distinguished, depending on relative values of mass ratios and wave speed ratios. Critical impact velocities to break either of the beams are determined for both the striking and struck beams by assuming both beams fail by tensile necking. However, location of the fracture point depends on relative values of various parameters. It can be right in the contact zone or away from it. It is also shown that after one beam breaks, the other beam will deform further without breaking.     

Closed-Form Solution for Reinforced Timoshenko Beam on Elastic Foundation

This paper suggests a method for obtaining closed-form solutions for a reinforced Timoshenko beam on an elastic foundation subjected to any pressure loading. A particular solution is obtained for uniform pressure loading at any location of the beam. This solution can be used to calculate settlement, rotation, tension, bending moment, and shear force of the beam. A parametric study is carried out to investigate the effects of geosynthetic shear stiffness and tension modulus and the location of the pressure loading. Results are presented and discussed.     

Prestressing of compression springs

A scheme of force variation during the compression of a spring in both elastic and elastic-plastic state is presented. For the elastic state it has been found that equivalent load increments occur with equivalent deflection increments. In the elastic-plastic state equivalent load increments correspond to higher deflection increments. Prestressing the spring increases the range of the elastic deformation of a spring. Equations for the determination of torsional moment during prestressing and relationships of total non-dilatational strain versus elastic strain and radii of elastic zone are also presented. The effect of characteristic spring dimensions and non-dilatational strain on spring deflection has been determined. A nomogram for the determination of the resultant equivalent strain as a function of external load, also considering residual stress distribution, has as well been elaborated.     

Inverse optimization: functional and physiological considerations related to the force-sharing problem

This paper is a review of the optimization techniques used for the solution of the force-sharing problem in biomechanics; that is, the distribution of the net joint moment to the force generating structures such as muscles and ligaments. The solution to this problem is achieved by the minimization (or maximization) of an objective function that includes the design variable (usually muscle forces) that are subject to certain constraints, and it is generally related to physiological or mechanical properties such as muscle stress, maximum force or moment, activation level, etc. The usual constraints require the sum of the exerted moments to be equal to the net joint moment and certain boundary conditions restrict the force solutions within physiologically acceptable limits. Linear optimization (objective and constraint functions are both linear relationships) has limited capabilities for the solution of the force sharing problem, although the use of appropriate constraints and physiologically realistic boundary conditions can improve the solution and lead to reasonable and functionally acceptable muscle force predictions. Nonlinear optimization provides more physiologically acceptable results, especially when the criteria used are related to the dynamics of the movement (e.g., instantaneous maximum force derived from muscle modeling based on length and velocity histories). The evaluation of predicted forces can be performed using direct measurements of forces (usually in animals), relationship with EMG patterns, comparisons with forces obtained from optimized forward dynamics, and by evaluating the results using analytical solutions of the optimal problem to highlight muscle synergism for example. Global objective functions are more restricting compared to local ones that are related to the specific objective of the movement at its different phases (e.g., maximize speed or minimize pain). In complex dynamic activities multiobjective optimization is likely to produce more realistic results.