Nonlinear Structural Analysis

The literature regarding the free vibration analysis of Bernoulli-Euler and Timoshenko beams on elastic soil is plenty, but the free vibration analysis of Reddy-Bickford beams on elastic soil with/without axial force effect using the Differential Transform Method (DTM) has not been investigated by any of the studies in open literature so far. In this study, the free vibration analysis of axially loaded Reddy-Bickford beam on elastic soil is carried out by using DTM. The model has six degrees of freedom at the two ends, one transverse displacement and two rotations, and the end forces are a shear force and two end moments in this study. The governing differential equations of motion of the rectangular beam in free vibration are derived using Hamilton's principle and considering rotatory inertia. Parameters for the relative stiffness, stiffness ratio and nondimensionalized multiplication factor for the axial compressive force are incorporated into the equations of motion in order to investigate their effects on the natural frequencies. At first, the terms are found directly from the analytical solutions of the differential equations that describe the deformations of the cross-section according to the high-order theory. After the analytical solution, an efficient and easy mathematical technique called DTM is used to solve the governing differential equations of the motion. The calculated natural frequencies of one end fixed and the other end simply supported Reddy-Bickford beam on elastic soil using DTM are tabulated in several tables and figures and are compared with the results of the analytical solution where a very good agreement is observed and the mode shapes are presented in graphs.

ABSTRACTThis paper reports the results of a series of full-scale drilled shaft load tests subjected to combined axial and lateral loading and lateral loading only. The tested shafts, 1.4m in diameter, were embedded 37m in sandy silt. All tested shafts were installed using reverse circulation method. The test results indicated, given the same lateral loading, 63% of pile head displacement resulted from combined load corresponded with the case of lateral loading only. The test results were compared to the numerical results of the software LPILE as well as the analytical solutions proposed by the senior author and his co-workers. The analytical results of the pile bending moments along shaft showed better results than that of LPILE.

In engineering, many pressure pipes are made of steels with good plasticity, which are subject to internal pressure, axial force, shear force, bending moment, torsion moment or their combined loads. The plastic limit load is an important indicator of the load capacity of pressure pipe. According to Hill yield function, the theoretical solutions of limit load of orthotropic cylindrical pipe under various combined loads under internal pressure, axial force, shear force, torsion moment, and bending moment have been derived on the basis of elastic perfectly plastic constitutive model. The effects of radial stress on different combined limit loads of cylindrical pipe are explored and these results show that the radial stress should be considered about the limit load calculation especially for thick-walled cylindrical pipe. The interactions of various load combination are analyzed in detail and drawn with the interaction curves. For isotropic cylindrical pipe, the limit load increases with the yield strength. For the orthotropic cylindrical pipe, the limit loads of cylindrical pipe under axial force, bending moment, shear force, and torsion moment without internal pressure are only related to the axial yield strength. The limit bending moment is mainly dependent on the axial yield strength when internal pressure is lower, while the impact of the circumferential yield strength of orthotropic cylindrical pipe is obvious when internal pressure is some higher. When the axial yield strength of orthotropic cylindrical pipe is the same, the circumferential yield strength can enhance the limit axial load, limit torsion moment, and limit shear load. Under the different load conditions including internal pressure, bending moment, axial force, shear force, and torsion moment or their combined loads, the relation of limit bending moment with yield strength ratio is diverse, which is decide by the load combination, the circumferential yield strength, and the axial yield strength.

Stochastic free and forced vibrations of layered beams are analyzed that result from a single bounded random stiffness parameter whose probability density function is considered to be zero outside of a given interval, i.e., it is a member of a fuzzy set with bounded uncertainty. The relevant properties of natural vibrations of an ensemble of sandwich beams with three perfectly bonded layers under hard hinged support conditions are worked out in detail when a bounded random shear stiffness of the core material is assigned by employing interval mathematics. The main structure of a compound single-span railway bridge, effectively modeled as a two-layer beam, is subjected to a single moving load as well as to a series of repetitive moving loads traveling with constant speed. It serves as a complex example for the resulting forced random vibrations and resonances under the severe condition of an elastic interface slip of bounded random stiffness. In both cases exact homogenization yields a stochastic sixth-order partial differential equation of motion of the layered beam. Light modal damping is considered. The analysis of the illustrative problems is based on the interval representation with a triangular membership function of the stiffness modulus assigned. A short comment provides information on the limits of such triangular membership functions. Membership functions in the form of envelopes of the random natural frequencies, the dynamic magnification factors, and the phase angles in free vibrations are determined. Both, fuzzy peak deflection and acceleration are derived for the forced single-span compound railway bridge subjected to the moving loads. Approximating superposition of modal maxima is considered by standard routines of reliability analysis.

Free flexural vibration of symmetric beams with inertia induced cross section deformations

The literature on free vibration analysis of Bernoulli-Euler and Timoshenko piles embedded in elastic soil is plenty, but that of Reddy-Bickford piles partially embedded in elastic soil with/without axial force effect is fewer. The soil that the pile partially embedded in is idealized by Winkler model and is assumed to be two-layered. The pile part above the soil is called the first region and the parts embedded in the soil are called the second and the third region, respectively. It is assumed that the behaviour of the material is linear-elastic, that axial force along the pile length to be constant and the upper end of the pile that is semi-rigid supported against rotation is modelled by an elastic spring. The governing differential equations of motion of the rectangular pile in free vibration are derived using Hamilton’s principle and Winkler hypothesis. The terms are found directly from the solutions of the differential equations that describe the deformations of the cross-section according to the high-order theory. The models have six degrees of freedom at the two ends, one transverse displacement and two rotations, and the end forces are a shear force and two end moments. Natural frequencies of the pile are calculated using transfer matrix and the secant method for non-trivial solution of linear homogeneous system of equations obtained due to values of axial forces acting on the pile, total and embedded lengths of the pile, the linear-elastic rotational restraining stiffness at the upper end of the pile and to the boundary conditions of the pile. Two different boundary conditions are considered in the study. For the first boundary condition, the pile’s end at the first region is semi-rigid connected and not restricted for horizontal displacement and the end at the third region is free and for the second boundary condition, the pile’s end at the first region is semi-rigid connected and restricted for horizontal displacement and the end at the third region is fixed supported. The calculated natural frequencies of semi-rigid connected Reddy-Bickford pile embedded in elastic soil are given in tables and compared with results of Timoshenko pile model.

Free vibration and dynamic stability of functionally graded composite microtubes reinforced with graphene platelets

A novel setup for testing of stiffened panels in ship decks subjected to axial and lateral loads is developed in the Department of Ocean Engineering, IITMadras. The entire setup consists of self-straining test rig, imperfection measuring device, displacement controlled hydraulic jacks, rigid grillage, inflatable air balloon with and without opening, and a data acquisition system. The fabricated deformable test rig replicates similar boundary conditions along the loaded and unloaded edges of stiffened panels in between transverse stiffeners of a ship deck. An imperfection measuring device located on the test rig is designed and fabricated to measure geometrical initial imperfections on the plate and stiffener. A unique displacement controlled twin hydraulic jack system is developed to apply axial load on the panel. A rigid grillage connected to the bottom of the test rig to act as reactive support for the application of lateral load on the panel is fabricated by performing static nonlinear analysis using ANSYS. An inflatable air balloon with and without opening is fabricated to simulate constant cargo (lateral) load acting on the panels. The applied axial load and produced axial deformation measured by load cells and LVDT, respectively, is plotted simultaneously while testing using an integrated data acquisition system. Nonlinear finite element analysis of tested specimens is performed using ANSYS to compare the ultimate load obtained from experiments. A close agreement between the experimental data and finite element analysis results indicates realistic simulation of truly combined axial and lateral loads with deformable supports acting on stiffened panels as in ship decks achieved in the laboratory and numerically.

Non-symmetric bifurcation and collapse of elastic-plastic thick-walled cylinders under combined radial and axial loading

Structural analysis and design for a warehouse using STAAD PRO

The foundation has a function to hold the load in the form of axial load, lateral load and moment. The axial load on the pile foundation is supported by the pole end resistance and pile friction, while the lateral load on the pile foundation is supported by the relation of the blanket area along with the pile with the ground in the lateral direction. Lateral load causes lateral deflection in which the magnitude of deflection must not exceed the permissible lateral deflection limit of 2.54 cm. The determining factors in lateral deflection are the type of pile foundation, soil type, and the magnitude of the force that occurs. The foundation of a single pile in which the number of one pile, when embedded in clay soil, needs to be investigated for the magnitude of lateral deflection that occurs, this is due to the changing nature of the clay when under the influence of high or low water content. The foundation of the pile uses a concrete head with a condition of a free end with a cylindrical shape with a diameter of 60 cm that is not solid with a wall thickness of 10 cm and a compressive strength of concrete of 60 MPa. Clay soil data uses soil drilling test data in Wirosari, Grobogan, Central Java. The planning method used is the finite element method (PLAXIS program) and the finite difference method (py curve and the ALLPILE program) with lateral load variations of 10 kN, 15 kN and 20 kN given to the top end of the pole with the principle of giving trial and error loads. The results of the calculation of the lateral deflection of the pile on the py curve method with a finite-difference resolution, the ALLPILE program, and the PLAXIS program respectively with a lateral load of 10 kN are 0.0629 cm; 1.21 cm; 0.27 cm, lateral loads of 15 kN are 0.0943 cm; 2.13 cm; 0.4051 cm, and a lateral load of 20 kN is 0.1258 cm; 3.14 cm; 0.5402 cm. Thus the lateral deflection load limit is 15 kN, so as not to exceed the permissible lateral deflection limit of 2.54 cm. The recommended method used is the ALLPILE program to better get the level of security from the lateral deflection of the pole when applied in the field

A method for computer-aided design and analysis of beam-columns under static axial and lateral loads is presented. Axial and lateral support reactions may be described as linear or nonlinear functions of beam-column displacement. The method utilizes discrete-element mechanical models of continuous members to facilitate numerical solutions of complex structural members. The axial and lateral solutions are coupled to include the effects of axial forces on the behavior of the member in bending. The method may be applied to a wide variety of problems involving beam-column type structural members. The nonlinear support curves are normally used to represent reactions developed by soil foundations. The support curves may be shifted with respect to the beam-column to simulate horizontal or vertical support movement Typical applications of the method in offshore construction include the analysis and design of axially and laterally loaded piles, the consideration of forces due to soil movement such as mudslides or settlement, the bending and buckling of soil-supported columns, and the analysis of members with varied cross-section or having complex load Or support conditions. Example problems are included to illustrate the versatility of the computer program. Introduction A variety of problems involving beam-column type members arise for which there are no convenient mathematical solutions. Included in these are axially and laterally loaded piles with nonlinear supports, beams with varying cross-section or resting on closely-spaced distributed supports, and the elastic buckling of columns with varying cross-section or with distributed nonlinear supports. A computer program has been developed for the analysis of beam-column members under combined axial and lateral loads with linear or nonlinear supports. The effects of internal axial forces on the behavior in bending are considered. The basic axial solution process was developed as AXCOL 3 (Refs 1 and 2). The basic lateral solution process was developed as BMCOL 28 (Ref 3). The recently developed computer program, BMCOL 76, transfers the axial thrust computed in the axial solution for consideration of the effects in the lateral solution. The nonlinear support curves, used primarily for the description of reactions developed by soil foundations, are input as sets of resistance displacement coordinates. Offsets to the origin of the curves may be input to simulate support movement, such as that encountered in down drag 'or mudslide problems. All data, including the support curve offsets, are linearly interpolated between input values along the length of the beam-column. The computer program utilizes discrete-element mechanical models of continuous members to facilitate numerical solutions for complex structural members. The development of the equations based on mechanical models and the input arrangements allow complete freedom in the variation of member properties, loads, and supports along the beam-column. Axial Model The discrete-element mechanical model of a short segment of a beam-column subjected to axial loads and restraints is shown in Fig 1. The beam-column is divided into a number of equal increments of length h.

To encounter the growing energy demand in the offshore environment, there is a necessity of a huge deployment of energy devices like wind turbines. Upscaled screw piles are gaining popularity as an effective alternative for the offshore wind turbine foundations in deeper waters. The foundations of the offshore wind turbines are typically subjected to axial compressive load from the superstructure and lateral load from the wind and waves. Therefore, the combined load-carrying capacity of screw piles needs to be evaluated for their efficient use as a foundation structure for the offshore wind turbines. In the present study, a three-dimensional finite element analysis is carried out using ABAQUS 2018 to determine the combined load-carrying capacity (under axial compression and lateral) of the screw pile installed in sand. The soil is modelled as Mohr–Coulomb material and the shaft and helix as linear elastic materials. The effect of the number of helixes and the placement of helix on the combined compressive and lateral load-carrying capacity of the screw pile is studied. It is found that the screw piles can significantly increase the combined lateral and vertical load-carrying capacity provided the helixes are placed at a suitable location. The present study will be useful in the design of screw pile for offshore application where they undergo combined lateral (from winds and waves) and dead loads (from superstructure).

This study investigates time-dependent lateral response of piles in elastoplastic soils under combined axial and lateral loadings. Mohr Coulomb model representing the soil and the finite element formulation were described and verified. The transient analysis of pile problem was then carried out and the results of the analysis were presented which demonstrate the effects of the consolidation process on the lateral response of the pile under combined loadings. Lower vertical to horizontal load ratio gives lower differences in deformations with time and becomes more significant with increasing load ratio. Also lateral soil pressure and shear stress in soil increased with time and depth but in a rate lowered as the load ratio decreased.

Behavior of encased steel-high strength concrete columns against axial and cyclic loading