Longitudinal Moment Research Articles

Determination of internal efforts in the base finite elements of SAFEM

The principles of calculating the internal efforts of a circular finite element in the semi-analytical finite element method (FEM) based on the obtained components of the stress tensor and the peculiarities of the approach associated with the use of the moment scheme of the finite element (FEM) are considered. Formulas for determining longitudinal, shear forces, bending and torque moments have been obtained.
 A special place, among the variety of objects considered with the help of analytical and numerical methods, is occupied by bodies of revolution of complex shape and cross-sectional structure, formed by the movement of some creative surface along a closed or opened line without breaks. The selected geometric class is used as natural structures of nodes and details in construction of mechanical engineering. The sufficiently wide distribution of the specified forms in the construction and machine-building industries, on the one hand, and the possibility of significantly simplifying the solving relationships by taking into account their geometric features, on the other hand, provide a basis for the development and use of various modifications of the finite element method (FEM). The semi-analytical finite element method (SAFEM) is one such approach that has gained widespread use for solving problems whose objects are prismatic bodies and bodies of revolution of complex shape and cross-sectional structure. Due to the introduction of additional hypotheses that do not reduce the accuracy of the approximation, the representation of deformations and stresses in physical terms and in accordance with the moment scheme of the finite element (MSFE), on the one hand, it is possible to avoid the time-consuming procedure of numerical integration over the cross-sectional area of the finite element (FE), on the other hand - maintain the high efficiency of 3D discretization.
 Despite the large number of publications devoted to the semi-analytical method of finite elements, the question of determining internal forces, which are often component factors of the strength criteria laid down in state building codes, is inappropriately neglected. The use of SAFEM in combination with МSFЕ creates some mathematical features of calculating internal longitudinal, shearing forces and moments.

Seismic fragility analysis of water supply pipelines retrofitted with corrosion-protection liner buried in non-uniform site

The corrosion-protection liner (CPL) technology consists of installing flexible plastic liners with anchoring studs inside existing pipelines and subsequently filling cement mortar to the gap between the waterproof liner and the pipeline. With the excellent chemical resistance, impermeability and fast construction speed, CPL provides an economical and environmentally friendly alternative for pipeline rehabilitation without large-scale excavation. However, the seismic performance of water supply pipelines after being retrofitted with CPL has not been well studied yet. In this study, a series of full-scale quasi-static experiments were firstly conducted on ductile-iron push-on joints before and after reinforcement with CPL to investigate the nonlinear behavior of the joints under longitudinal load and transverse bending. Simplified numerical models of straight pipelines with joints before and after retrofitting in the non-uniform site were then developed in the OpenSees platform, and incremental dynamic analysis (IDA) were performed with consideration of nonlinear soil-pipeline interaction. Twenty-eight pairs of ground motions with the peak ground velocities (PGV) collectively scaled from 200 mm/s to 3000 mm/s were used as inputs. Seismic fragility curves of the pipelines before and after retrofitting were developed with respect to the optimum intensity measure, PGV. It can be seen from the experimental results that the longitudinal load and bending moment capacity of the push-on joint increased about 400% and 20% after CPL reinforcement, respectively, and the joint opening and rotation decreased about 20% and 18% after retrofitting. Numerical results show that the CPL reinforced pipelines exhibit better seismic performance and CPL effectively reduces the failure probability of segmented pipelines under earthquake ground excitations.

Shear Force and Bending Moment Tuning Algorithm of Shuttle Tanker Model for Global Structural Analysis

Global ship analysis is conducted using a finite element model (FE model) for ship design and construction, which involves structural, motion, and vibration analyses. It is crucial to examine the structural safety of the hull and motion response. In the ship FE model used in global ship analysis, weight distribution is employed to adjust the light weight and center of gravity (COG), which are required to perform the analysis. Further, the FE model needs to satisfy the required longitudinal shear force (SF) and bending moment (BM) under the loading conditions of the ship. Moreover, the SF and BM in the ship Trim and Stability data are utilized to perform shear force tuning (SFT) and bending moment tuning (BMT) for the ship FE model. This ensures the ship model exhibits curves of the SF and BM that coincide with those of the ship. The SFT and BMT for the ship FE model are time-consuming and costly. Thus, to address these limitations, we propose an effective and accurate algorithm and program for SFT and BMT. Accordingly, we developed a C#-based algorithm to tune the weight, SF, BM, and COG of the ship FE model to the required target value. Finally, the accuracy of the newly developed algorithm was analyzed and compared by applying it to the shuttle tanker FE model under the ballast and full load conditions. Accuracy was within tolerance in both loading conditions. The average errors of SF and BM were smaller in the ballast condition than in the full load condition, and the errors were smaller at the bow than at the stern.

The Optimization of Secondary Lining Construction Time for Shield Tunnels Based on Longitudinal Mechanical Properties

In the field of shield tunnels, the occurrence of uneven longitudinal settlement in segment linings has presented persistent challenges, including heightened risks of localized damage and water leakage. While the adoption of a secondary lining has been proposed as a viable solution to these issues, the question of how to select an appropriate construction time for the secondary lining, one that enables it to fully harness its load-bearing capacity while optimizing the tunnel’s overall stress and deformation characteristics, continues to be a pressing concern. To address this issue, this study established a three-dimensional longitudinal refined numerical model of double-layer-lined shield tunnel. In addition, the deformation degree of the segment lining was used as a time indicator to define the construction time for the secondary lining. Subsequently, an analysis of the impact of the construction time of the secondary lining on the longitudinal mechanical properties of the double-layer-lined shield tunnel is conducted through an assessment of tunnel longitudinal deformation and structural stress. The research findings indicated that the construction of the secondary lining improved the longitudinal deformation resistance of shield tunnels. Simultaneously, it led to a significant increase in the longitudinal shear forces within the segment lining and a notable reduction in longitudinal bending moments. Moreover, the construction time of the secondary lining played a pivotal role in these alterations. Considering the longitudinal force situations and load-bearing characteristics of the double-layer lining structure, it was determined that the optimal construction time for the secondary lining fell within the range of 20% to 40% of the total construction duration. In this scenario, the deformation and internal forces within the segment lining remained within permissible limits. Additionally, both the segment lining and the secondary lining were able to fully utilize their load-bearing capacities, ensuring the economic and safety aspects of the tunnel.

Coupled analysis between catenary mooring and VLFS with structural hydroelasticity in waves

The rapid growth of marine renewables has led to the development of very large floating structures (VLFS) that are designed to operate in deep seas. It is significant to understand the mechanism of the coupled effects between deformable VLFS and catenary mooring system. This paper presents a time-domain hydro-elastic-moored model developed by integrating a quasi-static mooring module into a fully coupled Computational Fluid Dynamics (CFD) - discrete-module-beam (DMB) approach. The model is used to investigate the coupled effects between structural hydroelasticity and loose-type mooring systems on a deformable VLFS in waves. The mooring and hydroelasticity codes are validated separately and show favourable agreement with other numerical and experimental results. Then the coupled effects between the mooring system and structural hydroelasticity are evaluated by assigning various design parameters, i.e., VLFS structural stiffness and mooring stiffness. The numerical results, including dynamic motions, longitudinal vertical bending moments (VBMs) and mooring tension forces are presented and analysed. These results can be used to design a VLFS with mooring in medium-deep sea, and help with the conventional mooring design for a less-stiffness VLFS due to hydroelastic response.

Theoretical modeling of the pre-cutting system performance for the tunnel face stability in very weak rock masses

By forming continuous pre-lining around the tunnel's perimeter, pre-cutting is a recognized tunneling technique that enables full-face excavation of a tunnel even in unfavorable ground conditions. In this study, a theoretical method is put out to simulate the performance of this kind of pre-support in squeezing conditions. The loading distribution on the pre-cutting shells is determined for this purpose using the LDP of an unsupported tunnel, and the non-linear stiffness of the pre-cutting shell is then determined using the shell theory. The convergence confinement method is used to calculate how much the pre-supported tunnel isdeformed. The non-uniform loading distribution also results in the tangential and longitudinal bending moments on the pre-cutting shell. To show the effectiveness of pre-cutting shells in extremesqueezingconditions, several examples are solved. As a result, a few graphs for the preliminarydesign of a tunnel using this technique are given. The graphs show the overall tunnel displacement for various pre-cutting shells under various geological conditions. The outcomes demonstrate good agreement between FALC 3D resultsand the ones by thesuggested analytical model. A preliminary design can be made using the assessment of the proposed methodonthe loads on the pre-cutting system.

Longitudinal Parameter Estimation Using Wind Tunnel and Simulated Flight Data of Missile Configuration

The present study makes an attempt to estimate aerodynamic parameters from a typical flight data of a short range missile. Exhaustive wind tunnel testing was conducted to generate longitudinal force and moment coefficients at low speed. Identification methods were applied on the selected wind tunnel data to capture the general form of the aerodynamic model. During the application of Maximum likelihood (ML) method, the estimation algorithm assumed this wind tunnel identified aerodynamic model to be exact. To avoid any requirement of postulation of aerodynamic model, the Delta method was applied to estimate the aerodynamic parameters. The Delta method used measured aircraft motion and control variables as the inputs to the Feed Forward Neural Network and the aerodynamic force or moment coefficient was the output for training the Feed Forward Neural Network. The application of the Delta method results in large scatters in the estimated parameters. To overcome this problem of large scatter, the Delta method was modified by changing the training strategy. The Delta method with new training strategy will be referred as the Modified Delta method. It is expected that the proposed Modified Delta method would result in estimates with less uncertainties. Further to check the robustness of the ML, Delta and the Modified delta methods, the estimation was also carried out with flight data having known measurement noise. The effect of control input form in the accuracy of estimates obtained by ML, the Delta and the Modified Delta methods are also studied. It is observed that the Modified Delta method can advantageously be applied on the flight data of a tactical missile to estimate aerodynamic parameters. The paper progresses with the description of the generation of wind tunnel data and aerodynamic model identification using selected wind tunnel data. Finally it concludes by demonstrating applicability of ML, the Delta and the Modified Delta methods on simulated flight data of a typical short range tactical missile configuration.

Investigate the elastoplastic deformation behaviour of a motorcycle frame under different mechanical load configurations

This study investigates the elastoplastic deformation behaviour of a motorcycle frame under quasi-static mechanical loading. We present a novel test rig to analyse the mechanical integrity of a motorcycle frame under four different load configurations. These four load configurations reproduce longitudinal forces, lateral forces, torques and bending moments that act on the frame in normal operation as braking, acceleration or cornering. Additionally, the four load configurations represent loads that act on the frame in the most frequent crash scenarios in urban areas, such as a motorcycle’s frontal or inclined collision against a car. We conducted three repetitions of each load configuration with a frame from a motorcycle, especially for the urban area. The compliance of the measured load–displacement curves is rated by an objective assessment method. All tested frames show a similar elastoplastic deformation pattern; specifically, the longitudinal tubes are pressed outside, and fractures occur predominantly at areas of welding spots while the inner part remains scarcely deformed. The load–displacement curves, mechanical failures, weak points and deformation behaviour of the frames can be used in future to validate a numerical model of a motorcycle frame and improve the crashworthiness of a motorcycle frame. Highlights Novel experiments on the elastoplastic characterisation of motorcycle frames Multidirection deformation and failure behaviour Motorcycle frame as protection for sensitive components such as batteries Approach to validate deformation patterns and mechanical resistance of simulation models Validation of the test reproducibility by compliance assessment of measurement data

Analytical and numerical analysis on the torsional performance of elliptically deformed tunnels in the longitudinal direction

Segmental lining structures constructed by shields/tunnel boring machines are often subjected to uneven longitudinal cross-section torsion as a result of eccentric external loads, which is extremely adverse to tunnel safety but has not received sufficient attention for a long time. To figure out the torsional performance of segmental tunnels, it is essential to assess the longitudinal torsional stiffness and active-torsion-rejection capability of a segmented tunnel. The aim of the paper is to derive an analytical solution to the longitudinal torsional stiffness of a segmental tunnel with existing elliptical deformation. The longitudinal torsional stiffness under different internal force combinations is deduced considering the longitudinal axial force and bending moment based on the equivalent continuous model and force balance equation. The validity of the analytical solution is verified by comparing it with finite element method results. Then, a parametric analysis, using the new analytical solution, is included to investigate the effect of the key parameters on torsional behaviors, including the segment size, the bolt size and the transverse bending stiffness, etc. It is found that: (1) the longitudinal torsional stiffness efficiency (LTSE) of the segmental tunnel decreases with the rise of segment thickness to diameter ratio but increases with the ring width to diameter ratio; (2) the LTSE reduces with the increase of the effective shear length but rises with the diameter of bolts; (3) the LTSE increases rapidly with the ratio of compression-torsion or bending-torsion. Furthermore, the envelope curve of the critical load (N0, M0) for a tunnel to actively resist a certain internal torque is given. The proposed solution can be easily utilized to determine the longitudinal torsional stiffness of segmental tunnels and is an effective tool for tunnel design and maintenance.

Spatially nonlocal instability modeling of torsionaly loaded nanobeams

Various aspects of buckling of nanobeams have been explored; nevertheless, instability and spatial buckling of these tiny elements under the concurrent action of longitudinal load and torsional moment have not yet been displayed using nonlocal continuum mechanics. To fill this crucial research gap, herein, we are interested in methodically examining this critical phenomenon. Aiming at this goal, suitable nonlocal beam models are established and the deformation equations are constructed accounting for the nonlocality. In order to assess the critical buckling response, a Galerkin-based meshless approach is developed and spatially nonlocal buckling behavior of torsionally loaded nanobeams is inclusively addressed. The nonlocal unstable regions associated with the spatially buckled nanobeam for various nonlocality and geometry data are specified for the first time and the critical roles of nonlocality and shear deformation on them are carefully revealed and discussed. In the presence of applied torsional moments, the concept of the critical lateral and longitudinal dimensions of the nanostructure is also introduced.

Emergence of universal scaling in isotropic turbulence.

Universal properties of turbulence have been associated traditionally with very high Reynolds numbers, but recent work has shown that the onset of the power laws in derivative statistics occurs at modest microscale Reynolds numbers of the order of 10, with the corresponding exponents being consistent with those for the inertial range structure functions at very high Reynolds numbers. In this paper we use well-resolved direct numerical simulations of homogeneous and isotropic turbulence to establish this result for a range of initial conditions with different forcing mechanisms. We also show that the moments of transverse velocity gradients possess larger scaling exponents than those of the longitudinal moments, confirming past results that the former are more intermittent than the latter.

Fatigue Behaviors of Joints between Steel Girders with Corrugated Webs and Top RC Slabs under Transverse Bending Moments

Steel-concrete composite box beams are widely used in bridge engineering, which might bear transverse and longitudinal bending moments simultaneously under vehicle loads. To investigate the fatigue performance of joints between the steel girders and the top reinforced concrete (RC) slabs under transverse bending moments, a reduced scale joint between the weathering steel girder with the corrugated steel web (CSW) and the top RC slab was designed and tested under constant amplitude fatigue loads. Test results show that the joint initially cracked in the weld metal connecting the CSW with the bottom girder flange during the fatigue loading process. The initial crack propagated from the longitudinal fold to the adjacent inclined folds after the specimen was subjected to 7.63 × 105 loading cycles and caused the final fatigue failure. Compared with the calculated fatigue lives in the methods recommended by EC3 and AASHTO, the fatigue performance of the details involved in the joint satisfied the demands of fatigue design. Meanwhile, finite element (FE) models of joints with different parameters were established to determine their effect on the stress ranges at the hot spot regions of the joints. Numerical results show that improving the bending radius or the thickness of the CSW helps to reduce the stress ranges in the hot spot regions, which is beneficial to enhance the fatigue resistance of the investigated fatigue details accordingly.

A Heuristic Method for Solving Ship Floating Problems Based on Improved Particle Swarm Optimization

The study of ship floating balance plays a very important role in the safety of ships. According to certain conditions of ship statics, the nonlinear equations of ship balance can be established with relevant parameters such as ship weight, draft, heeling angle, longitudinal inclination angle and static moment. In this paper, motivated by the least square method, the nonlinear equations of floating equilibrium are transformed to a nonlinear programming problem. The draught, heel and trim of the ship are taken as variables, and inequality constrained by the actual situation. The penalty function for nonlinear programming is considered and minimized by particle swarm optimization. The proposed optimization method simplifies the calculation process by avoiding the calculation of parameters. Numerical results verify the efficiency of the proposed method.

МЕТОД ПІДТВЕРДЖЕННЯ РЕСУРСНИХ ХАРАКТЕРИСТИК МЕТАЛЕВОЇ ЛОПАТІ НЕСУЧОГО ГВИНТА ВЕРТОЛЬОТА ЗА РЕЗУЛЬТАТАМИ ВИПРОБУВАНЬ

The helicopter main rotor blade is the basic product that determines the reliability and service life of the helicopter as a whole. The problem of predicting and ensuring the specified blade life is an urgent problem considered at the stage of its design. The analysis of the design, structural materials and design and technological solutions of the main rotor blade (RB) of the Mi-8 helicopter has been carried out. A brief description of the main rotor blade of the Mi-8 helicopter is presented. The analysis was carried out and a standard flight cycle (SFC) of the helicopter was developed. The type of bench equipment for carrying out bench fatigue tests of the blade has been selected and justified. The loads on the main rotor blade for the SFC are determined. To determine the fatigue life of a blade, it is necessary to know the characteristics of the stress-strain state. The calculation of the stress-strain state (SSS) of a blade by the finite element method (FEM) using the ANSYS system is presented. The characteristics of the stress-strain state of the spar of the regular and irregular parts of the rotor blade of a helicopter are determined using the ANSYS system. The use of numerical methods for calculating the characteristics of the stress-strain state can significantly reduce the time and cost of designing a blade. The paper presents the results of calculating the regular part of the main rotor blade of the Mi-8 helicopter in the hover mode in the case of its loading with aerodynamic and inertial load from rotation, as well as the force from its own weight. With the help of the ANSYS system, a finite element model of the regular part of the blade was developed, consisting of a set of beam elements of variable section, a calculation was carried out taking into account the geometric nonlinearity of the structure's behavior, and an analysis of the results obtained was carried out. To describe the response of materials to an external action, a model of an elastically deformable isotropic body was used with the assignment of the corresponding elastic constants of the material. The analysis of the calculation results includes the determination of reactions at the attachment points, the values of the maximum displacements of structural elements and stresses in dangerous sections. Dangerous sections are determined and the values of the longitudinal force and bending moment in these sections are calculated. The assessment of the static strength of the blade by the safety factor was carried out. When evaluating the static strength, the equivalent stresses according to Mises were considered as the maximum design stresses. To assess the fatigue strength, we analyzed the distribution of the main tensile stresses in the power elements over typical stress concentrators. The maximum level of the main tensile stresses in the dangerous section indicates that the blade material operates in the zone of high-cycle fatigue. A technique for calibrating strain gauge channels has been developed. The calculation of the characteristics of the rotor blade of a helicopter is based on the requirements set forth in the technical literature, regulatory documents. When performing work, the requirements of the Aviation Rules, Part 29 (AP - 29) were taken into account. These studies were the basis for the development of a method for confirming the resource characteristics of a helicopter main rotor blade based on the results of flight and bench tests.

Evaluation of the CHBDC-simplified method of analysis for transverse wood deck on wood girder bridges

Between the 2006 and 2014 editions of CSA S6, significant changes were made to the simplified method of analysis (SMA) for all girder bridges, including transverse wood deck on wood girder bridges. The load-distribution predictions became more conservative, resulting in both load-limit posting of existing bridges and increased cost for bridge construction, repair, and replacement. In this paper, we examine the load distribution for the timber bridge inventory in NS. Finite-element models are developed and rigorously validated; a parametric study, consisting of 204 bridges, is performed; current SMA equations are evaluated; and new equations for the truck-load distribution width are derived. The new equations bring SMA predictions of the longitudinal bending moment for transverse wood deck on wood girder bridges into line with currently expected safety-index levels. Corrected bias factors and coefficients of variation for live-load analysis/evaluation in conjunction with CSA S6 are presented.

Effects of Local Denting and Fracture Damage on the Residual Longitudinal Strength of Box Girders

The residual strength of denting- and fracture-damaged box girders were experimentally and numerically investigated. The experiments were conducted under a pure bending moment using the four-point bending test method. The load, deflection, and strain were measured. The strains of the extension structures were also measured, and the frictional forces between the model and the supported round bar were estimated. Test models consisting of two groups were fabricated. These groups were designed to estimate the residual strength of denting- and fracture-damaged models. The damage was induced by releasing a striker using a drop-testing machine to consider the dynamic effect. Additionally, numerical analyses were performed via a nonlinear finite element analysis, where the measured initial imperfection data and welding residual stresses were considered. The ultimate longitudinal moments that considered the frictional force of the round bars were reduced by 12% (on average) compared with those obtained by neglecting the frictional forces.

Концепция многофункционального высокоскоростного магистрального экранолёта наземно-эстакадного исполнени

The paper formulates the main concept for an ekranolet designing on a flyover, an analysis for the Held of the efficient use as well as basic information on the advanced design projects for various operating conditions, including the Far North, Siberia and the Far East climatic conditions. Keywords: concept, ekranolet, ellipse, overpass, ekranoplan, longitudinal and lateral stability, longitudinal moment, angle of attack.

Design & Fabrication of Quad Processing Machine

In this project, the quad processing machine for cutting, shaping, grinding and drilling machines is kept to be in same attachment. This eliminates the risk of material handling and to the machine table. This unit is driven by 0.25 H.P. three phase motor of 1440 rpm. This is linked with the cast-iron (CI) slider. This enables the slide to get the rotation movement into reciprocating motion having a stroke length of 75mm. The head of the slotting can be tilted to be getting the shaping machine at a desired angle. The head is welded to the slider at the bottom edge. In this machine there can be one operation will be done which is slotting or shaping operation. In the tool head, a slot & tapped holes are provided for holding the tool. According to the height of the job the head can be moved up or down by means of column and can be fixed using a clamp provided. If a table with a machine vice having longitudinal moment is attached to the machine, the setup can be used for machining slots and shaper in small jobs like pulleys etc. Instead of MS sheet, if CI is used for body and other parts it will have longer life, less friction, better lubrication and the production cost will also be low. This machine can also be used for production of small washers by punching if suitable dies and brake arrangement are fixed.

Исследование проектных параметров бортовой системы изменения массовоинерционных характеристик летательного аппарата

The paper presents the results of a study of a new concept of a system for changing the mass-inertial characteristics of nominally axisymmetric rotor structures, in particular, aircraft of an axisymmetric shape. This system can be used for static and dynamic balancing of the product directly during operation and is a balancing weight installed on the movable platform of the mechanism with parallel kinematics of the hexapod type. It can be applied, including for nominally axisymmetric aircraft, to parry disturbances from asymmetries of various kinds by purposefully creating a mass-inertial asymmetry of the structure. The paper analyzes the ways to bring the system to the required state (by linearly changing the lengths of the drive rods, as well as by moving and rotating the balancing weight according to linear laws). The ratios of the mass-inertia characteristics (longitudinal and transverse moment of inertia of the balancing weight) of the “body of the aircraft – balancing weight” system and their influence on the kinematic parameters of the mechanism, in particular on the required displacements and rotation angles of the balancing weight, the magnitude of the change in the lengths of the drive rods were evaluated. Based on the original model as a system of two bodies “body of the aircraft – balancing weight”, an extended model was created taking into account the mass-inertial characteristics of the drive rods. The applicability of the two-body model is analyzed depending on the mass-inertia characteristics of the drive rods, in particular, on the mass of the rods. The influence of the accuracy of changing the lengths of the drive rods on the final mass-inertia characteristics of the system, such as the axial moments of inertia of the system and the residual uncompensated centrifugal moment of inertia, is analyzed.

Analysis of Skew Bridge-Slab Under IRC Vehicle Loading

Skew slab bridges are essential in mountainous areas because topographic restrictions prevented changing the alignment of the road also, crossings of roads and railroads. In simple supported bridges, the effect of skew may typically be ignored up to 15 degrees of skew, and the bridge can be constructed as a right-angled bridge. The behaviour of skew slab bridges more than 15 degrees is complicated hence the study on the behaviour of skew-bridge slab under IRC vehicle loading is carried out. About 70 different deck slab models are analyzed in STAAD Pro software with varying width from 1 to 4 lanes, and span lengths of 7.5m and 12m with for skew angles 0° to 50° in increment of 10°.The vehicle loads and positioning of vehicles are done as per IRC-6:2017 standard specifications. The results show that the bending moment and deflection in near edge beams decreases with increase in skew angle. In farer edge beam the bending moment increase till skew of 30° and deflection increases after skew of 30°. The longitudinal bending moment decreases with increase in skew angle. Transverse bending moments are prominent at near the corners that at the centre. The torsion moments are more concerned in slabs with small width and large span. Deflection decreases at all the parts of the slab as skew increases. The maximum deflection in free edges of the slab shifts towards the obtuse corners.