Buntara Sthenly Gan

Large Deflection Analysis of Functionally Graded Beams Resting on a Two-Parameter Elastic Foundation

Abstract This paper presents a finite element procedure for the large deflection analysis of functionally graded (FG) beams resting on a two-parameter elastic foundation. The material properties of the FG beams are assumed to vary continuously in the thickness direction by a power-law distribution. Based on the strain energy expression, a shear deformable beam element, taking the effect of the material non-homogeneity and the foundation support into account, is formulated and employed in the analysis. An incremental/iterative procedure in combination with the arc-length control method is used for solving nonlinear equilibrium equations. The numerical results show that the convergence of the formulated element is fast, and the large displacements of the beams can be accurately assessed by using a few numbers of the elements. A parametric study is carried out to highlight the effect of the material non-homogeneity and the foundation support on the large deflection behavior of the beams. The influence of the aspect ratio on the large deflection response of the beam is also examined and highlighted.

Effect of Intermediate Elastic Support on Vibration of Functionally Graded Euler-Bernoulli Beams Excited by a Moving Point Load

The effect of intermediate elastic support on the vibration of functionally graded Euler-Bernoulli beams excited by a moving point load is studied. The material properties of the beams are assumed to vary in the thickness direction of the beams by a power function. Governing equations of motion that take into account the shift in the physically neutral surface position, are constructed using Hamilton′s principle. A finite element model is developed and employed in computing the dynamic response of the beams. The influence of the stiffness and position of the elastic support on the dynamic characteristics of the FGM beams is examined and highlighted.

Finite Element Formulation of Beam Elements

In this chapter, various types of beams on a plane are formulated in the context of finite element method. The formulation of the beam elements is based on the Euler-Bernoulli and Timoshenko theories. The kinematic assumptions, governing equations via Hamilton’s principle and matrix formulations by using shape functions, are described in detail. In constructing the beam element formulations, the shape functions which are derived from the homogeneous governing equations lead to high-accuracy beam analyses. The theories discussed and derived herewith will be used in the subsequent chapters when we deal with the Isogeometric approach to beams.

Effect of concrete strength gradation to the compressive strength of graded concrete, a numerical approach

Concrete casting, compacting method, and characteristic of the concrete material determine the performance of concrete as building element due to the material uniformity issue. Previous studies show that gradation in strength exists on building member by nature and negatively influence the load carrying capacity of the member. A pilot research had modeled the concrete gradation in strength with controllable variable and observed that the weakest material determines the strength of graded concrete through uniaxial compressive loading test. This research intends to confirm the recent finding by a numerical approach with extensive variables of strength disparity. The finite element analysis was conducted using the Strand7 nonlinear program. The results displayed that the increase of strength disparity in graded concrete models leads to the slight reduction of models strength. A substantial difference in displacement response is encountered on the models for the small disparity of concrete strength. However, the higher strength of concrete mix in the graded concrete models contributes to the rise of material stiffness that provides a beneficial purpose for serviceability of building members.Concrete casting, compacting method, and characteristic of the concrete material determine the performance of concrete as building element due to the material uniformity issue. Previous studies show that gradation in strength exists on building member by nature and negatively influence the load carrying capacity of the member. A pilot research had modeled the concrete gradation in strength with controllable variable and observed that the weakest material determines the strength of graded concrete through uniaxial compressive loading test. This research intends to confirm the recent finding by a numerical approach with extensive variables of strength disparity. The finite element analysis was conducted using the Strand7 nonlinear program. The results displayed that the increase of strength disparity in graded concrete models leads to the slight reduction of models strength. A substantial difference in displacement response is encountered on the models for the small disparity of concrete strength. However, ...

Inclusion-to-specimen Volume Ratio Influence on the Strength and Stiffness Behaviors of Concrete: an Experimental Study

The stress-strain response of the basic concrete making material, i.e. the mortar and aggregates, are well known. In general, the aggregate behaves linearly up till failure, possessing a very high ultimate compression strength and stiffness. The behavior of mortar is non-linear, even at low loading levels. The resulting composite material, the concrete, exhibits a less stiff response, in combination with degradation in strength. This study looked into the influence of the inclusion-to-specimen volume ratio of a 100x100x50 mm mortar specimen. Two inclusion configurations were considered, parallel and diagonal to the line of loading, while the ratio varied from zero to 0.66. It was shown that the inclusion-to-specimen volume ratio strongly influenced the strength, the stiffness, and failure mode. The strength behavior had a minimum and a maximum bifurcation point, while the stiffness response increased, as a function of an increase in the inclusion-to-specimen volume ratio. Visual observation of the cracking pattern revealed that the initial cracking was always situated at the interface between the aggregate and mortar in tension and propagated through the mortar matrix. It was also perceived that the crack propagation path of the very dense, diagonally arranged inclusions deviated from the columnar configuration observed from the parallel inclusion formation. These densely diagonally arranged aggregates also resulted in spalling in the lateral direction.

Isogeometric Approach to Beam Element

This chapter presents the formulation of beam elements in the finite element method by using the isogeometric approach. The implementation of the NURBS functions as either the geometry or the shape functions to various types of beam formulations on a plane is highlighted in detail and accompanied by MATLAB program lists. This chapter will use the program lists and concept of the NURBS from Chap. 1. The numerical integration introduced in Chap. 2 will be adopted in constructing the beam element matrices from the NURBS functions. Based on the beam theories developed in Chap. 3, the implementation of the isogeometric approach to beam element will be discussed.

Representation of Curves on a Plane

In this chapter, we will discuss in detail several methods of representing curves in geometric modeling. Unlike the other books which usually start from the standard basic theories behind the equations, this chapter will lead us directly on how to find equations or functions from a given set of data points. This following and connected subsection will reveal the techniques of representing the curves in real applications of beam geometric modeling. The objective of this chapter is to bring the reader to understand the concept of the nonuniform rational B-spline (NURBS) which is the basis foundation for the construction of beam element formulations in the Isogeometric approach.

Straight Beam Element Examples

In this chapter, the isogeometric approach is applied to straight beam element. The numerical solutions for some examples for static analysis and free vibration analysis are presented and compared with the exact solutions.

General Curved Beam Element Examples

In this chapter, the isogeometric approach is applied to general curved beam element. The general curved beams considered in this section include parabolic, elliptical, and sinusoidal shapes. The numerical solutions for some examples in free vibration problems are presented and compared with the results of published studies.

Free Curved Beam Element Examples

In this chapter, the Isogeometric approach is applied to free curved beam element. The free curved beams considered in this section were modeled by the least element number which is necessary. The numerical solutions for the examples in static and free vibration problems are presented to show the effectiveness of the NURBS functions in modeling free curved beams.