Bending Moment Distribution Research Articles

Model Tests on Y-Shaped Piles under Compressive and Lateral Loading in Saturated Sand

Y-shaped piles are a new type of pile whose cross-section is like the letter Y: they are often used in ground improvement for road or train subgrades in the eastern coastal region of China. To investigate the bearing behaviour of Y-shaped piles in saturated sand, a series of model tests under compressive and horizontal load for Y-shaped piles, C 1 circular pile (the same cross-sectional area of Y-shaped pile), and C 2 circular pile (the same perimeter of Y-shaped pile) were carried out. Comparative analysis was conducted on bearing capacity, axial force and side resistance distribution, load sharing ratio, bending moment, and lateral soil pressure distribution along the embedded length. The results show that the bearing capacity of a Y-shaped pile does not increase in proportion, and the shaft resistance is weakened to some extent in saturated sand; Y-shaped pile can effectively improve the compressive bearing capacity for the same amount of concrete. The lateral bearing capacity of a Y-shaped pile has directionality, and hanging a circular section into a Y-shaped section may improve the horizontal bearing capacity for the same amount concrete, but cannot give full play to the advantage of the larger side area for horizontal bearing capacity in saturated sand.

Assessment of SSI interaction effects for deep diaphragm walls and adjacent buildings considering spatial variability of parameters, post-peak behavior and concrete cracking

Recent research works into the design and response evaluation of deep diaphragm walls and induced settlements to adjacent buildings demonstrated that the nonlinear behavior of surrounding soil, cracking effects affecting the structural members’ flexural rigidity and spatial variability of soil parameters are a prerequisite for a rigorous analysis of serviceability performance. However, as a result of the complexity of the mechanisms developed in the case of deep multi-propped retaining structures, most of the proposed Soil Structure Interaction (SSI) design methodologies are biased towards geotechnical or structural behavior by adequately capturing the nonlinearities of the one field and considering a rather simplified behavior of the other. With the aim to bridge the gap between the above design methodologies, a fully coupled analysis of a multi-propped diaphragm wall of an underground station with a supporting grout slab at the excavation bottom has been carried out. Further to the nonlinear behavior of the soil, the spatial variability of the grout treated zone strength and stiffness together with the post-peak behavior have been considered. In addition, the concrete cracking effects, associated with substantial decrease of structural members’ stiffness, have been taken into account. Comparative simplified empirical approaches disregarding cracking effects by adopting constant diaphragm wall flexural rigidity instead, have been carried out as well. Interesting conclusions were drawn relating the results of the above simplified approaches and the Serviceability Limit State (SLS) analyses with regard to the response of the diaphragm wall, the bending moment distribution along the wall and the displacement field in the surrounding soil and adjacent buildings.

Structural parameters analysis for tapered steel section with perforation: effect of shear buckling behaviour

Purpose Due to economic development, tapered members are commonly applied in steel frames, namely, industrial halls, warehouses, exhibition centres, etc. In the design of cantilever steel beam structures in cities building design, tapering is introduced at the web profile to achieve utmost economy and suit the bending moment distributions. The cross-sectional shape of the beam is varied linearly to the moment gradient to achieve the target of higher efficiency with lower cost. Design/methodology/approach The shear deformation pattern and efficiency of the tapered steel section with perforation were investigated using finite element analysis. In addition, I-beam with web opening is studied numerically via LUSAS software for different parameters of tapering ratio, perforation shape and perforation size and perforation layout. Findings The highest contributing parameters for the highest shear buckling capacity and efficiency of the section were due to the small opening size and tapering ratio. Whilst the variation of perforation layout and spacing give a major effect on the shear strength and efficiency of the tapered steel section with perforation. Besides that, the highest efficiency model is found when the section is designed with 0.4 D diamond perforation in Layout 3 under a tapering ratio of 0.3. The critical shear buckling load and efficiency is reduced 14.39% and 13.91%, respectively, when perforations are added onto the tapered steel sections. Originality/value The tapered steel section with perforation has lower critical shear buckling load and efficiency compared to the tapered section without perforation but obtains a higher critical shear buckling load and efficiency compared to the uniform section without perforation.

Energy-balance based plastic design and analysis of hybrid coupled wall system

The hybrid coupled wall (HCW) system consists of steel coupling beams that connect adjacent reinforced concrete (RC) wall piers. In order to minimize or even eliminate the lengthy design iterations of conventional performance-based seismic design method and directly account for inelastic structural behavior, the modified energy-balance based plastic design (EBPD) method is proposed and developed for the seismic design of HCW systems with the consideration of influences of the degradation of hysteretic behavior and pinching effects. The design base shear can be obtained by solving the modified energy balance equation analytically. Twelve HCW prototype structures with different structural heights and coupling ratio (CR) were designed and evaluated by conducting a series of push-over analyses and nonlinear dynamic time-history analyses with the proved-reliable program of PERFORM-3D. The analysis results indicate that all HCW prototype structures exhibit the favorable yield mechanism and satisfactory seismic performance. Particularly, the influences of CR on the deformation characteristics, bending moment distribution among wall piers and vertical distribution of overturning moment are revealed. Suggestions are made in terms of the appropriate selection of CR for different structural heights.

Observed Performance and FEM-Based Parametric Analysis of a Top-Down Deep Excavation in Soil-Rock Composite Stratum

This paper investigates the performance of a top-down deep excavation in soil-rock composite stratum. The behavior of the excavation bracing system, consisting of ground anchors and end-suspended piles, has not been well understood due to the lack of relevant research. Based on the observed data of a typical deep excavation case history for the May Fourth Square Station in Tsingtao, China, the characteristics of the horizontal and vertical pile displacements, ground surface settlements, building settlements, axial forces in ground anchors, earth pressure, and pore water pressure during excavation were analysed. Two-dimensional finite element simulations were carried out to further explore the deformation and internal force responses of end-suspended piles and to capture the effects of pile diameter, embedded depth, and rock-socketed depth on the horizontal displacement and bending moment distributions along the pile shaft. It was found that the pattern of the vertical pile displacements could be categorized into three types: rapid settlement, slow settlement, and rapid heave. The magnitudes of the ground and building responses can be well controlled within allowable limits by combining the top-down method with the adopted bracing system. Among the investigated parameters, pile diameter is dominant in affecting the horizontal pile displacement. The primary influence zone for pile bending moment varies, depending on the parameters. It is recommended that a combination of top-down method, ground anchors, and end-suspended piles be adopted for restraining excavation deformation and lowering construction costs of similar deep excavations in soil-rock composite stratum.

Impact of subgrade and backfill stiffness on values and distribution of bending moments in integral box bridge

Abstract The article presents parametric analysis regarding the impact of subgrade and backfill stiffness on values and distribution of bending moments in the structural elements of a small integral box bridge made of cast in situ reinforced concrete. The analyzed parameters are the modulus of subgrade reaction under and behind the bridge structure (kv kh). At the beginning, the author presents the integral box bridge and selected parts of the bridge design. In particular, the author focuses on the method of modeling of the subgrade stiffness parameters under and behind the bridge structure, as well as their impact on the values and distribution of bending moments in the bridge structural elements. The bridge was designed by the author and built on the M9 motorway between the towns of Waterford and Kilcullen in Ireland. In conclusions, the author shares his knowledge and experience relating to the design of small integral bridges and culverts and puts forward recommendations as to further research on these type of structures in Poland.

Analysis of the effect of reinforcement of diaphragm spans on their spatial work

A large number of bridges in Ukraine have been built in the 60–70 s (about 54 % of the total number). In that regard, the number of cases of detecting various types of damage on the elements of bridge structures is increasing, which can cause emergencies in the future. It also suggests that most of the bridges and overpasses were built according to the old technical standards and now do not meet the modern standards of DBN V.2.3-14: 2006 and DBN V.2.3-22: 2009, both in terms of carrying capacity and dimensions of the roadway. Therefore, the issues of strengthening the existing spans are relevant. Goal. The work analyzes the effect of reinforcement of diaphragm spans on their spatial work in order to determine the effectiveness of reinforcement by gluing additional rigid reinforcement and installing transverse metal beams. The tasks that have been solved in the work are as follows: creating an adequate finite element model of the diaphragm span; limiting the number of options for design schemes to the optimal number; construction and analysis of the graphs of bending moments and deflections in the cross-section in the middle of the span of beams. Results. The graphs of the distribution of bending moments in the beams of the superstructure for different dimensions show that the presence of reinforcement does not affect the value of the bending moment, and the distribution of forces between the beams of the superstructure becomes less uniform with an increase in their number. The deflections for all load cases and dimensions are greater than the deflections of beams with reinforcement; however, the installation of an additional transverse metal beam does not affect the values and distribution of deflections. Beam deflection graphs become less uniform with an increase in their number. Practical value. Analysis of the effect of reinforcement on the spatial work of span structures has shown that only reinforcement of beams with longitudinal rigid reinforcement without installing transverse beams is expedient in practice.

Yield interaction model of jack-up considering nonlinear load-displacement relationship before yield

Offshore structures with jack-up systems can be operated at depths up to 150 m and are used not only as drilling rigs and production rigs but also as support and accommodation units. Jack-up operation is carried out under environmental loads such as wind and wave, for which it is essential to understand jack-up behavior and structural response. As the boundary condition, the foundation model of offshore structures affects the vibration mode of a structure and, consequently, the behavioral and structural analysis results as well. Typical simple foundation models such as pinned and linear spring do not reflect soil-structure interaction (SSI) in jack-up analysis. As an alternative, the International Organization for Standardization (ISO) guidelines have suggested this SSI model as a simple secant model, a yield interaction model, and a time-consuming but accurate soil continuum model. In the present study, a structural analysis of jack-up is performed using the yield interaction model and the soil continuum model. The yield interaction model, Model B for clay, derived for consideration of the nonlinear behavior of soil, has been studied and continuously improved until recently. The existing model has generally assumed a linear load-displacement relationship in the elastic region; but this relationship may overestimate the load, as soil plasticity occurs gradually in practice. In this paper, the Hyperbolic Model B is proposed, based on which the horizontal and rotational load-displacement material curves in the region before yield are assumed to have a hyperbolic relationship. The regression equation for the initial stiffness accompanying the model is also suggested for single clay and soft-over-stiff clay. A fully coupled SSI analysis with a soil continuum model is performed to validate the proposed model. Large soil deformation accompanied by deep penetration is considered simultaneously in the structural analysis of jack-up. As a result, inside the yield envelope, the existing model may overestimate the moment acting on the soil, thereby underestimating the bending moment at the hull-leg joint. The Model B developed and proposed in this paper well predicts the soil response and bending moment distribution of the leg, which results are validated in comparison with those of the soil continuum model. The proposed model can also simulate the hysteresis soil response for a sinusoidal load like in the soil continuum model. These effects make that behavior of soil and structure before yield is well simulated, therefore, efficient design is available in terms of the fatigue strength assessment.

Stiffness amplification coefficient for composite frame beams considering slab spatial composite effect

The slab spatial composite effect arises due to compatible deformation of the reinforced concrete slab with the steel beam in composite frames. This effect can significantly amplify the stiffness of composite frame beams. Current Chinese design code adopts the stiffness amplification coefficient method for the design of composite frame beams, and the equivalent moment of inertia of the composite frame beam can be calculated to conduct the global analysis. However, the current formulae are only suitable for lateral load cases and symmetric I-shaped steel section. Therefore, this study aims to propose unified design equations of the stiffness amplification coefficient considering both vertical and lateral load cases and both symmetric and asymmetric I-shaped steel sections to address the aforementioned limitations. A series of parametric finite element analyses using the beam-shell mixed element model are conducted to derive the unified design equations. The reasonability of the proposed equations is verified by comparing the critical structural indices, such as the natural vibration period, inter-storey shear, lateral displacement, bending moment, and deflection distribution of numerical examples of multi-storey composite frames. Good accuracy is obtained through the comparison between the linear beam element model modified by the proposed equations and the beam-shell mixed element model.

Nonlinear flexural-gravity waves due to a body submerged in the uniform stream

The two-dimensional nonlinear problem of steady flow past a body submerged beneath an elastic sheet is considered. The mathematical model is based on the velocity potential theory with fully nonlinear boundary conditions on the fluid boundary and the elastic sheet, which are coupled throughout the numerical procedure. The integral hodograph method is employed to derive the complex velocity potential of the flow which contains the velocity magnitude on the interface in explicit form. The coupled problem has been reduced to a system of nonlinear equations with respect to the unknown magnitude of the velocity on the interface, which is solved using a collocation method. Case studies are undertaken for both subcritical and supercritical flow regimes. Results for interface shape, bending moment, and pressure distribution are presented for the wide ranges of Froude numbers and depths of submergence. According to the dispersion equation, two waves on the interface may exist. The first longest wave is that caused by gravity, and the second shorter wave is that caused by the elastic sheet. The obtained solution exhibits a strongly nonlinear interaction of these waves above the submerged body. It is found that near the critical Froude number, there is a range of submergences in which the solution does not converge.

Full-scale performance testing of bored piles with retaining walls in high cutting slope

This study aims to verify the suitability and reinforcement behavior of newly designed bored piles with retaining walls (BPRWs) for cutting-slope stabilization in the Tibet Plateau. As the first BPRWs prevention works in the alpine and hypoxic areas, full-scale performance testing is conducted of the high cutting slopes along the Lhasa-Nyingchi Railway under construction. Amid the construction processes of slope excavation and BPRWs installation, the overall structural performance, including lateral displacement, bending moment and lateral earth pressure, was determined to evaluate the cutting-slope stability and retaining mechanisms of BPRWs. The monitoring parameter distribution rules were thoroughly analysed. The relationship between the lateral earth pressure and pile displacement was studied, and a soil resistance calculation model was proposed. Finally, the BPRWs suitability was assessed. The results revealed that the retaining behaviour of BPRWs is closely related to the construction process. The displacement rate over time is characterized by alternating peaks and valleys, and the maximum displacement rate and bending moment occurs at the retaining wall pouring stage. The bow-shaped bending moment distribution with the pile depth exhibits low values on both ends and a high value in the middle. The ground reaction coefficient is allometrically attenuated with increasing lateral displacement. The proposed soil resistance model can satisfactorily predict the lateral earth pressure. The BPRWs deformation satisfies the service requirements at the construction and service stages. This case study provides a reinforcement reference for BPRW in alpine and hypoxic areas.

Bending moment distribution estimation of an actual steel building structure by microstrain measurement under small earthquakes

Most structural health monitoring systems estimate the overall behavior by measuring the acceleration response, which cannot directly measure the stress or damage state of individual structural members. An alternative approach is to use strain measurements; however, methods for analyzing and utilizing strain data for actual steel buildings have not been established. In this study, highly precise semiconductor strain gauges were applied to an actual building. The accelerations and strains measured during earthquake loading were used to calculate the ratio of the bending moment at the beam or column sections to the displacement at the top of the building, which was defined as the “local stiffness.” This physical index represents the stiffness of structural elements near the measurement location and can be easily predicted through simple static frame analysis. The measured local stiffness was comparable to the analytical local stiffness values for the beams but was larger than that for the columns. This indicates that nonstructural members may exhibit a certain degree of restoring force and that the measured local stiffness may be strongly affected by nonstructural elements that are not considered during the structural design stage. Conversely, the measured local stiffness can be used to estimate the behavior of nonstructural components. The measured dominant frequency and local stiffness of the beams and columns showed a dependency on amplitude, but opposite trends were observed for the beams and columns. This indicates that the amplitude dependency of the dominant frequency is not due to the behavior of the beams and columns but to other reasons such as nonstructural components or changes in mass.

Numerical Analysis of the Beam-Column Resistance Compared to Methods by European Standards

The optimisation of the design method for verification of slender steel beam-columns is still a current issue not only from scientific point of view, but also for design practice. Therefore, the main objective of this paper is comparison of the suitability of established design approaches, according to the European standards for steel and aluminium structures, on the basis of numerical simulations. Thus, a finite element model was validated on the basis of experimental analysis available in the scientific literature. To perform the comparison of accuracy of design approaches according to European standards, a commercial software program ANSYS was used for observation of the resistances of beam-columns. The resistance of european I beams with parallel flanges (IPE) and a rectangular hollow cross-section (RHS) were investigated for four load cases on a simply supported member and also on a pinned-fixed beam column with linear bending moment distribution, where the resistance of the cross-section governs. Finally, the conclusions for suitability of the respective design approaches are discussed, together with some findings that arose from this work.

Dynamic responses of hybrid FRP-concrete-steel double-skin tubular column (DSTC) under lateral impact

Hybrid fibre-reinforced polymer (FRP)-concrete-steel double-skin tubular column (DSTC) is a new form of composite column that consists of an outer FRP tube and an inner steel tube, with the space between them filled with concrete. Although many studies have been conducted on the hybrid DSTC, studies on its lateral impact behaviour are very limited. This study numerically investigates the dynamic responses of hybrid DSTC under lateral impact. Numerical models of the hybrid DSTCs were developed and validated with existing test results. Afterwards, the dynamic responses, including the deflection-time histories and the force–time histories, were analysed. Specifically, the bending moment and shear force distributions at typical cross sections were discussed, and the impact resistance (shear and flexure) from each component (FRP tube, concrete, and steel tube) was investigated. The results indicate that the hybrid DSTC exhibits ductile behaviour under lateral impact. Even though the FRP tube cannot contribute to the impact resistance directly, it can greatly improve the impact resistance of the concrete. Moreover, parametric analyses were conducted to investigate the influences of different parameters on the lateral impact resistance of hybrid DSTC. Finally, simplified design-oriented equations were developed to predict the dynamic flexural capacity of hybrid DSTC.

Analytical approach for membrane action in laterally-restrained reinforced concrete square slabs under uniformly distributed loads

Laterally-restrained reinforced concrete slabs can mobilise compressive membrane action and subsequent tensile membrane action under extreme loading conditions, thereby enhancing the load resistance under uniformly distributed loads. Previous analytical study focuses primarily on tensile membrane action in simply-supported slabs. This paper describes an analytical approach for membrane action in laterally-restrained square slabs. In the model, a yield line pattern is presumed for slabs, and the whole slab is divided into a strip system in the orthogonal directions. Each strip is analysed by establishing compatibility and equilibrium. The model is validated against test data of reinforced concrete square slabs, and reasonably good agreement is obtained in terms of the load capacity. The distributions of membrane force, bending moment and neutral axis depth along yield lines are also obtained through the analytical approach, and propagation of the tension zone from the centre towards the edge of slabs is also demonstrated by using the contour of membrane forces in the slab. Contributions of bending moment and membrane force to the load resistance of slabs are quantified by decomposing the total resistance according to equilibrium. The contribution of each slab strip is also quantified to gain insight on the distribution of resistance in the whole slab. Finally, a design method is proposed to calculate the capacity of compressive membrane action in square slabs.

A Symmetric Three-Layer Plate with Two Coaxial Cracks under Pure Bending

The purpose of this research was to investigate the effect of mechanical features and geometrical parameters on the stress–strain state of a cracked layered plate under pure bending (bending moments are uniformly distributed at infinity). The sixth-order bending problem of an infinite, symmetric, three-layer plate with two coaxial through cracks is considered under the assumption of no crack closure. By using complex potentials and methods of the theory of functions of a complex variable, the solution to the problem was obtained in the form of a singular integral equation. It is reduced to the system of linear algebraic equations and solved in a numerical manner by the mechanical quadrature method. The distributions of stresses and bending moments near the crack tips are shown. Numerical results are presented as graphical dependences of the reduced moment intensity factor on various problem parameters. In this particular case, the optimum ratio of layer thicknesses is determined.

Study on horizontal bearing characteristics of pile foundations in coral sand

This paper presents the horizontal bearing characteristics of piles in coral sand and silica sand from comparative experimental studies. Six model piles with different diameters were tested. The horizontal bearing capacity, deformation characteristic, bending moment, p–y curve, change in soil horizontal pressure, as well as particle breakage behaviour of coral sand were investigated. The results show that, in the coral sand foundation, the horizontal bearing capacities of piles and the increments in soil horizontal pressures are obviously greater than those in silica sand. Accordingly, the lateral displacement, rotation of the pile head, bending moment, and corresponding distribution depth in coral sand are significantly smaller than those in silica sand. The p–y curves indicate that the horizontal stiffness of coral sand is greater than that of silica sand. Remarkably, the breakage behaviour of coral sand is mainly distributed in the range of 10 times the pile diameter depth and 5 times the pile diameter width on the side where the sand is squeezed by the pile. Furthermore, in coral sand, the influence of pile size is more pronounced, the squeezing force generated by the pile spreads further, and its influence range is larger compared to those in silica sand.

Centrifuge modelling on monotonic and cyclic lateral behaviour of monopiles in kaolin clay

This paper details a programme of centrifuge tests (100g) on the monotonic and cyclic lateral behaviour of large-diameter semi-rigid monopiles in kaolin clay. Piles of different diameters were used, with a maximum prototype value of 5·9 m. The pile–soil interaction was investigated in terms of the overall load–displacement behaviour, distribution of bending moment and pile deflection, p–y curves and excess pore pressure response. For monotonic loading, procedures were developed to fit the monotonic p–y curves obtained from different pile and soil properties using a hyperbolic tangent model. In addition, an explicit model was derived to assess the development of excess pore pressure within the surficial soil, which was found to approach the vertical effective soil stress at large displacement. For cyclic loading, the accumulation of lateral pile displacement could be accompanied by either increase or decrease in soil resistance, which strongly depends on the loading rate. Strong correlation between the development of peak soil resistance and a dimensionless velocity group was obtained, and was supported by the measured excess pore pressure. Under undrained conditions, the peak soil resistance decreases almost linearly with the accumulation of excess pore pressure.

Combined Action of Bending and Tension of an Isotropic Plate with Through Crack in the Absence of Contact between the Faces and with Regard for the Plastic Zones and Hardening of Material at the Tips

We study the problem of combined bending and tension of an isotropic plate containing a through crack by distributed bending moments and forces at infinity in the absence of contact between the crack faces and external loads applied to the crack but in the presence of plastic zones at its tips. We also take into account the phenomenon of hardening of the material satisfying the Tresca plasticity conditions in the form of a surface layer or a plastic hinge. By using complex potentials of plane problem and the classical theory of bending of plates, we obtain an analytic solution of the problem in the class of functions bounded at the tips of the plastic zones. We also present the dependences for the lengths of the plastic zones and the opening displacements of the crack faces at the tips. The numerical analysis of these dependences is performed for different parameters of the problem.

The effect of spatial correlation of sea states on extreme wave loads of ships

ABSTRACT The purpose of the study is to include a spatial statistical correlation between sea states that ship encounters during a voyage to improve wave scatter diagram for the analysis of ship structures. The methodology consists of computation of system reliability of a series of partially correlated events, representing sea states that ship encounters along the sailing route. The North Atlantic wave scatter diagram is modified to account for such spatial correlation and new scatter diagrams for partially correlated sea states are constructed. Based on such “correlated scatter diagrams”, long-term distribution of vertical wave bending moments is performed and compared to the results following from an “uncorrelated scatter diagram”. It was found that a spatial correlation may considerably reduce extreme vertical wave bending moments. Verification of the methodology is performed by simulating shipping route in the North Atlantic using ERA5 wave database.