Modeling Beams on Elastic Foundations Using Plate Elements in Finite Element Method

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Modeling Beams on Elastic Foundations Using Plate Elements in Finite Element Method"

Transcription

1 Modeling Beams on Elastic Foundations Using Plate Elements in Finite Element Method Yun-gang Zhan School of Naval Architecture and Ocean Engineering, Jiangsu University of Science and Technology, Zhenjiang, Jiangsu, China ABSTRACT Beams on elastic foundations are often encountered in the regions of civil engineering, harbour engineering and railway engineering. Analytical equations for this kind of problem are not complex to be used to analyze regular beams and plates, but they are not suitable for abnormal beams or plates. Finite element code incorporating beam elements on elastic foundation is a better choice for this problem. General-purpose finite element package ABAQUS provide an option of FOUNDATION to model elastic foundation, but it does not support beam elements. In this paper, the application of using plate (shell) element to represent behaviors of beams was verified firstly, and then two numerical examples of beams on elastic foundations were examined by combining plate (shell) element with FOUNDATION option. It is found that this method could be used to analyze this kind of problem without losing accuracy in comparison with analytical results. KEYWORDS: beams on elastic foundations; plate (shell) element; finite element analysis. INTRODUCTION The model of beams on elastic foundations is a simplified method to analyze forces of beam foundations or mat foundations, in which the reaction forces of subsoil are proportional to the deflection of the beam or mat at each point. That is to say, the foundations seem to lie on identical, independent, closely spaced, and linearly elastic springs. The stiffness of these springs, known as modulus of subsoil reaction k 0 [kn/m 2 /m], links intensity of reaction forces on foundation (pressure p) and deflection w, as p = k0 w. This postulate is presented first by Winkler in 1867 and still used in geoengineering designs due to its simplicity. Many simple calculation software or diagrams are formulated to analyze this kind of problems, but most of them are based on thin beam or thin plate theory and can only deal with beams or plates with simple outline, such as straight beam, regular flat plate. Some commercial finite element packages, e.g., SAP, MARC, also incorporate element type of beam on elastic foundation, then beams or plates with complex outline on elastic foundations can be analyzed

2 Vol. 17 [2012], Bund. N 2064 easily and quickly by using them. General-purpose finite element package ABAQUS provides an option of FOUNDATION, which acts like springs to ground and is a simple way of including the stiffness effects of a support (such as the soil under a building) without modeling the details of the support (Abaqus, 2010). This option can be used to simulate elastic foundation by prescribing the foundation stiffness per area. However, this elastic foundation option does not support beam elements, a number of spring elements are often laid under beam to accomplish this type of simulation (Charlton, 2007). This paper presents an alternative method to analysis beams on elastic foundation based on calculation framework of ABAQUS, in which the beam is modeled by plate (shell) element. Its reliableness is verified by two numerical examples comparing with the numerical solutions. BEAM AND PLATE THEORY It is well known that there are two kinds of beam theories: Euler-Bernoulli beam (thin beam) theory and Timoshenko beam (thick beam) theory (Hetenyi, 1950; Hughes, and Cohen, 1978). Figure 1 can illustrate the difference between them. For a beam under transverse loads, Euler- Bernoulli beam assumes that the cross-sectional plane being perpendicular to the neutral axis of the beam remains plane and is still perpendicular to the neutral axis after deformation. Namely, no transverse shearing deformation is taken into account, ϕ = d w/dx. For beams with relatively high cross-sections compared with the beam length subjected to larger shear forces, it is inapplicable to neglect the transverse shear deformation. Timoshenko beam (thick beam) theory relaxes the perpendicular assumption but remains the plane assumption of cross-section plane to consider shear deformation γ xz, i.e. ϕ d w/dx. d w/dx γ xz Z ϕ Neutral axis dx dw X Figure 1: Bending of beam Plate theory, including thin plate theory and thick plate theory, can be viewed as extensions of one-dimensional beam theory to two-dimension space. For thin plate, an assumption referred to as hypothesis of straight normal, which is like what adopted in thin beam theory, is assumed. The normal to the middle plane of plate before bending remains straight and normal to the middle surface during the deformation, and the length of such elements is not altered. This also means that the vertical shear strains at cross section, γ xz andγ yz, are negligible and the normal strain may also be omitted. Thick plate theory takes the assumption which is analogous to that associated with the Timoshenko beam theory, i.e., γ xz 0 and γ 0. yz

3 Vol. 17 [2012], Bund. N 2065 From the above mentioned, one can use plate elements to substitute beam elements in finite element analysis, if the loads and supports are equivalent to that of beams. For example, line loads along cross section of the plate can substitute concentric forces acting on beams, and line loads acting on longitudinal axis of beams can be substituted by surface tractions on the plate. In the following sections, a cantilever beam subjected to concentric force was taken as an example to examine the applicable substitution of beams with plates, and examples of beam on elastic foundation were analyzed and compared with analytical solutions. NUMERICAL EXAMPLE OF BEAM SUBSTITUTED BY PLATE To illustrate that the plate elements reduces to the same behavior as beam elements, a cantilever beam subjected to concentric force, F = 10000kN, shown in Figure 2 was modeled. The elastic modulus of material of beam is 20GPa and Poisson's ratio is 0.3. The numerical model of the beam was established using beam element, plate (shell) element, and continuum element, as presented in Figure 3. F 6.0m 1.6m Figure 2: Cantilever beam 0.8m (a) Beam element (b) Shell element (c) Continuum element Figure 3: Numerical models of cantilever beam The end displacements calculated by analytical and numerical methods are presented in Table 1. It is obviously that the vertical displacements predicted by thick beam or thick plate theory are larger than that calculated by thin beam or thin plate theory, for the effect of shear deformation on displacement is considered. This example clearly indicates that plate (shell) element provided by ABAQUS can model a one-dimensional beam without losing accuracy. Table 1: Vertical end displacement of cantilever beam Analytical result Beam element Plate (shell) element Continuum Euler-Bernoulli Timoshenko Euler-Bernoulli Timoshenko Thin Thick Generalelement beam Beam beam Beam shell shell Purpose shell 0.132m 0.137m 0.132m 0.139m 0.129m 0.138m 0.138m 0.137m

4 Vol. 17 [2012], Bund. N 2066 NUMERICAL EXAMPLES OF BEAMS ON ELASTIC FOUNDATIONS In this section, the option of FOUNDATION and plate (shell) element was used to model beams on elastic foundations. Two examples were analyzed and compared with analytical solutions. Consider a beam on elastic foundation with free ends. The geometrical dimensions loadings are shown in Figure 4. The modulus of elasticity of material of the beam is 20GPa, and the 4 Poisson's ratio is 0.2. The modulus of subsoil is k 0 = kn/m 2 /m. Figure 5 presents the deformed figuration of the beam modeled by plate (shell) elements, which is consistent with analytical prediction (Huang and He, 2005). The maximum vertical displacement occurred at the middle of the beam is 6.44mm, with corresponding bending moment kN m. There are only slight differences between the numerical results and the analytical ones, which are 6.46mm and kN m, separately. 4 q = kpa Example 1 F = 400kN 0.3m 5.0m 5.0m 1.0m Figure 4: Numerical example 1 Figure 5: Deflection of f example 1 Magnified by factor: 100 Example 2 Example 2 is a beam laid on elastic foundations withh 10m in length, 1.1m in width and 0.5m in depth (Parvanova, 2011). Loads that are more complexx were applied on the beam, as shown in Figure 6. The elastic modulus of material of the beam is 30GPa, and the Poisson's ratio is The modulus of subsoil is k 0 = 5 10 kn/m /m. 200kN 100kN m 200kN/ /m 0.5m 1.0m 3.0mm 1.0m 5.0m 1.1m Figure 6: Numerical example 2

5 Vol. 17 [2012], Bund. N 2067 Numerical models based on plate (shell) elements were established to analyze this problem, with thin plate element and thick plate element used separately. However, the results analyzed with these two types of elements have no remarkable difference and the results of thick plate element are presented in the following. The vertical displacements, EIv( x ) obtained by numerical method are compared with analytical results in Figure 7 and they match very well. Figure 8 shows the distribution of bending moment which is consistence with the analytical result. The maximum bending moment predicted by numerical analysis is 163.4kN m, which is 3.3percent lower that of analytical result, 169.1kN m. The bending moments at the 1-m cross section and 5- m cross section away from the left end of beam are kN m and kN m, which are very close to that of analytical solutions, kN m and kN m. Length (m) EIv(x) Analytical solution Numerical solution 1600 Figure 7: Deflection along axis of the beam Figure 8: Distribution of bending moments along axis of the beam SUMMARY In this paper, reduction of thick and thin plate (shell) element to Euler-Bernoulli beam element and Timoshenko beam element was validated by an example of cantilever beam. Then plate (shell) element in conjugated with FOUNDATION option was used to analyze two numerical examples of beams on elastic foundations. Through comparing with numerical solutions, it can be drawn that this method could be used to analyze these kinds of problems without losing accuracy.

6 Vol. 17 [2012], Bund. N 2068 REFERENCES 1. Abaqus, (2010). User s Manual. Version Charlton,Z. I. (2007) Innovative Design Concepts for Insulated Joints, Master thesis, Virginia Polytechnic Institute and State University. 3. Hetenyi, M. (1950) A General Solution for The Bending of Beams on an Elastic Foundation of Arbitrary Continuity, Journal of Applied Physics, Vol. 21(1), Huang, Y. and He, F. S. (2005) Beams, Plates and Shells on Elastic Foundations (in chinese), Science Press of China, Beijing. 5. Hughes,T.J.R. and Cohen, M. (1978) The Heterosis Finite Element for Plate Bending, Computers and Structures, Vol. 9, Parvanova,S. (2011) Lecture notes: Structural Analysis II, University of Architecture, Civil Engineering and Geodesy, Sofia, Bulgaria ejge

METHODS FOR ACHIEVEMENT UNIFORM STRESSES DISTRIBUTION UNDER THE FOUNDATION

METHODS FOR ACHIEVEMENT UNIFORM STRESSES DISTRIBUTION UNDER THE FOUNDATION International Journal of Civil Engineering and Technology (IJCIET) Volume 7, Issue 2, March-April 2016, pp. 45-66, Article ID: IJCIET_07_02_004 Available online at http://www.iaeme.com/ijciet/issues.asp?jtype=ijciet&vtype=7&itype=2

More information

Finite Element Formulation for Plates - Handout 3 -

Finite Element Formulation for Plates - Handout 3 - Finite Element Formulation for Plates - Handout 3 - Dr Fehmi Cirak (fc286@) Completed Version Definitions A plate is a three dimensional solid body with one of the plate dimensions much smaller than the

More information

Finite Element Formulation for Beams - Handout 2 -

Finite Element Formulation for Beams - Handout 2 - Finite Element Formulation for Beams - Handout 2 - Dr Fehmi Cirak (fc286@) Completed Version Review of Euler-Bernoulli Beam Physical beam model midline Beam domain in three-dimensions Midline, also called

More information

EML 5526 FEA Project 1 Alexander, Dylan. Project 1 Finite Element Analysis and Design of a Plane Truss

EML 5526 FEA Project 1 Alexander, Dylan. Project 1 Finite Element Analysis and Design of a Plane Truss Problem Statement: Project 1 Finite Element Analysis and Design of a Plane Truss The plane truss in Figure 1 is analyzed using finite element analysis (FEA) for three load cases: A) Axial load: 10,000

More information

BEAM THEORIES The difference between Euler-Bernoulli and Timoschenko

BEAM THEORIES The difference between Euler-Bernoulli and Timoschenko BEAM THEORIES The difference between Euler-Bernoulli and Timoschenko Uemuet Goerguelue Two mathematical models, namely the shear-deformable (Timoshenko) model and the shearindeformable (Euler-Bernoulli)

More information

Optimum proportions for the design of suspension bridge

Optimum proportions for the design of suspension bridge Journal of Civil Engineering (IEB), 34 (1) (26) 1-14 Optimum proportions for the design of suspension bridge Tanvir Manzur and Alamgir Habib Department of Civil Engineering Bangladesh University of Engineering

More information

Strength of Materials Prof: S.K.Bhattacharya Dept of Civil Engineering, IIT, Kharagpur Lecture no 23 Bending of Beams- II

Strength of Materials Prof: S.K.Bhattacharya Dept of Civil Engineering, IIT, Kharagpur Lecture no 23 Bending of Beams- II Strength of Materials Prof: S.K.Bhattacharya Dept of Civil Engineering, IIT, Kharagpur Lecture no 23 Bending of Beams- II Welcome to the second lesson of the fifth module which is on Bending of Beams part

More information

1. a) Discuss how finite element is evolved in engineering field. (8) b) Explain the finite element idealization of structures with examples.

1. a) Discuss how finite element is evolved in engineering field. (8) b) Explain the finite element idealization of structures with examples. M.TECH. DEGREE EXAMINATION Branch: Civil Engineering Specialization Geomechanics and structures Model Question Paper - I MCEGS 106-2 FINITE ELEMENT ANALYSIS Time: 3 hours Maximum: 100 Marks Answer ALL

More information

New approaches in Eurocode 3 efficient global structural design

New approaches in Eurocode 3 efficient global structural design New approaches in Eurocode 3 efficient global structural design Part 1: 3D model based analysis using general beam-column FEM Ferenc Papp* and József Szalai ** * Associate Professor, Department of Structural

More information

EFFICIENT NUMERICAL SIMULATION OF INDUSTRIAL SHEET METAL BENDING PROCESSES

EFFICIENT NUMERICAL SIMULATION OF INDUSTRIAL SHEET METAL BENDING PROCESSES ECCOMAS Congress 06 VII European Congress on Computational Methods in Applied Sciences and Engineering M. Papadrakakis, V. Papadopoulos, G. Stefanou, V. Plevris (eds.) Crete Island, Greece, 5 0 June 06

More information

Shear Forces and Bending Moments

Shear Forces and Bending Moments Chapter 4 Shear Forces and Bending Moments 4.1 Introduction Consider a beam subjected to transverse loads as shown in figure, the deflections occur in the plane same as the loading plane, is called the

More information

The elements used in commercial codes can be classified in two basic categories:

The elements used in commercial codes can be classified in two basic categories: CHAPTER 3 Truss Element 3.1 Introduction The single most important concept in understanding FEA, is the basic understanding of various finite elements that we employ in an analysis. Elements are used for

More information

8.2 Elastic Strain Energy

8.2 Elastic Strain Energy Section 8. 8. Elastic Strain Energy The strain energy stored in an elastic material upon deformation is calculated below for a number of different geometries and loading conditions. These expressions for

More information

ENGI 8673 Subsea Pipeline Engineering Faculty of Engineering and Applied Science

ENGI 8673 Subsea Pipeline Engineering Faculty of Engineering and Applied Science GUIDANCE NOTE LECTURE 12 THERMAL EXPANSION ANALYSIS OVERVIEW The calculation procedure for determining the longitudinal pipeline response can be formulated on the basis of strain. The longitudinal strain

More information

Course 1 Laboratory. Second Semester. Experiment: Young s Modulus

Course 1 Laboratory. Second Semester. Experiment: Young s Modulus Course 1 Laboratory Second Semester Experiment: Young s Modulus 1 Elasticity Measurements: Young Modulus Of Brass 1 Aims of the Experiment The aim of this experiment is to measure the elastic modulus with

More information

Interaction between plate and column buckling

Interaction between plate and column buckling Delft, University of Technology Engineering office of Public works Rotterdam Interaction between plate and column buckling Master Thesis Name: Alex van Ham Student number: 1306138 Email: vanham.alex@gmail.com

More information

Calculation and Analysis of Tunnel Longitudinal Structure under Effect of Uneven Settlement of Weak Layer

Calculation and Analysis of Tunnel Longitudinal Structure under Effect of Uneven Settlement of Weak Layer Calculation and Analysis of Tunnel Longitudinal Structure under Effect of Uneven Settlement of Weak Layer 1,2 Li Zhong, 2Chen Si-yang, 3Yan Pei-wu, 1Zhu Yan-peng School of Civil Engineering, Lanzhou University

More information

MCE380: Measurements and Instrumentation Lab. Chapter 9: Force, Torque and Strain Measurements

MCE380: Measurements and Instrumentation Lab. Chapter 9: Force, Torque and Strain Measurements MCE380: Measurements and Instrumentation Lab Chapter 9: Force, Torque and Strain Measurements Topics: Elastic Elements for Force Measurement Dynamometers and Brakes Resistance Strain Gages Holman, Ch.

More information

Deflections. Question: What are Structural Deflections?

Deflections. Question: What are Structural Deflections? Question: What are Structural Deflections? Answer: The deformations or movements of a structure and its components, such as beams and trusses, from their original positions. It is as important for the

More information

Unit 6 Plane Stress and Plane Strain

Unit 6 Plane Stress and Plane Strain Unit 6 Plane Stress and Plane Strain Readings: T & G 8, 9, 10, 11, 12, 14, 15, 16 Paul A. Lagace, Ph.D. Professor of Aeronautics & Astronautics and Engineering Systems There are many structural configurations

More information

Universal Format of Shape Function for Numerical Analysis using Multiple Element Forms

Universal Format of Shape Function for Numerical Analysis using Multiple Element Forms , pp.63-67 http://dx.doi.org/0.4257/astl.206. Universal Format of Shape Function for Numerical Analysis using Multiple Element Forms Yuting Zhang, Yujie Li, Xiuli Ding Yangtze River Scientific Research

More information

MECHANICS OF SOLIDS - BEAMS TUTORIAL 1 STRESSES IN BEAMS DUE TO BENDING. On completion of this tutorial you should be able to do the following.

MECHANICS OF SOLIDS - BEAMS TUTORIAL 1 STRESSES IN BEAMS DUE TO BENDING. On completion of this tutorial you should be able to do the following. MECHANICS OF SOLIDS - BEAMS TUTOIAL 1 STESSES IN BEAMS DUE TO BENDING This is the first tutorial on bending of beams designed for anyone wishing to study it at a fairly advanced level. You should judge

More information

Stress and Deformation Analysis. Representing Stresses on a Stress Element. Representing Stresses on a Stress Element con t

Stress and Deformation Analysis. Representing Stresses on a Stress Element. Representing Stresses on a Stress Element con t Stress and Deformation Analysis Material in this lecture was taken from chapter 3 of Representing Stresses on a Stress Element One main goals of stress analysis is to determine the point within a load-carrying

More information

Estimation of Adjacent Building Settlement During Drilling of Urban Tunnels

Estimation of Adjacent Building Settlement During Drilling of Urban Tunnels Estimation of Adjacent Building During Drilling of Urban Tunnels Shahram Pourakbar 1, Mohammad Azadi 2, Bujang B. K. Huat 1, Afshin Asadi 1 1 Department of Civil Engineering, University Putra Malaysia

More information

Finite Element Simulation of Simple Bending Problem and Code Development in C++

Finite Element Simulation of Simple Bending Problem and Code Development in C++ EUROPEAN ACADEMIC RESEARCH, VOL. I, ISSUE 6/ SEPEMBER 013 ISSN 86-48, www.euacademic.org IMPACT FACTOR: 0.485 (GIF) Finite Element Simulation of Simple Bending Problem and Code Development in C++ ABDUL

More information

THE STUDY OF LATERAL TORSIONAL BUCKLING BEHAVIOUR OF BEAM WITH TRAPEZOID WEB STEEL SECTION BY EXPERIMENTAL AND FINITE ELEMENT ANALYSIS

THE STUDY OF LATERAL TORSIONAL BUCKLING BEHAVIOUR OF BEAM WITH TRAPEZOID WEB STEEL SECTION BY EXPERIMENTAL AND FINITE ELEMENT ANALYSIS THE STUDY OF LATERAL TORSIONAL BUCKLING BEHAVIOUR OF BEAM WITH TRAPEZOID WEB STEEL SECTION BY EXPERIMENTAL AND FINITE ELEMENT ANALYSIS Fatimah Denan 1, Mohd Hanim Osman 2 & Sariffuddin Saad 3 1 School

More information

DYNAMIC ANALYSIS OF THICK PLATES SUBJECTED TO EARTQUAKE

DYNAMIC ANALYSIS OF THICK PLATES SUBJECTED TO EARTQUAKE DYNAMIC ANALYSIS OF THICK PLATES SUBJECTED TO EARTQUAKE ÖZDEMİR Y. I, AYVAZ Y. Posta Adresi: Department of Civil Engineering, Karadeniz Technical University, 68 Trabzon, TURKEY E-posta: yaprakozdemir@hotmail.com

More information

INTRODUCTION TO BEAMS

INTRODUCTION TO BEAMS CHAPTER Structural Steel Design LRFD Method INTRODUCTION TO BEAMS Third Edition A. J. Clark School of Engineering Department of Civil and Environmental Engineering Part II Structural Steel Design and Analysis

More information

STRESS-STRAIN RELATIONS

STRESS-STRAIN RELATIONS STRSS-STRAIN RLATIONS Strain Strain is related to change in dimensions and shape of a material. The most elementar definition of strain is when the deformation is along one ais: change in length strain

More information

Section 16: Neutral Axis and Parallel Axis Theorem 16-1

Section 16: Neutral Axis and Parallel Axis Theorem 16-1 Section 16: Neutral Axis and Parallel Axis Theorem 16-1 Geometry of deformation We will consider the deformation of an ideal, isotropic prismatic beam the cross section is symmetric about y-axis All parts

More information

DEVELOPMENT OF A NEW TEST FOR DETERMINATION OF TENSILE STRENGTH OF CONCRETE BLOCKS

DEVELOPMENT OF A NEW TEST FOR DETERMINATION OF TENSILE STRENGTH OF CONCRETE BLOCKS 1 th Canadian Masonry Symposium Vancouver, British Columbia, June -5, 013 DEVELOPMENT OF A NEW TEST FOR DETERMINATION OF TENSILE STRENGTH OF CONCRETE BLOCKS Vladimir G. Haach 1, Graça Vasconcelos and Paulo

More information

Objectives. Experimentally determine the yield strength, tensile strength, and modules of elasticity and ductility of given materials.

Objectives. Experimentally determine the yield strength, tensile strength, and modules of elasticity and ductility of given materials. Lab 3 Tension Test Objectives Concepts Background Experimental Procedure Report Requirements Discussion Objectives Experimentally determine the yield strength, tensile strength, and modules of elasticity

More information

MECHANICS OF SOLIDS - BEAMS TUTORIAL 2 SHEAR FORCE AND BENDING MOMENTS IN BEAMS

MECHANICS OF SOLIDS - BEAMS TUTORIAL 2 SHEAR FORCE AND BENDING MOMENTS IN BEAMS MECHANICS OF SOLIDS - BEAMS TUTORIAL 2 SHEAR FORCE AND BENDING MOMENTS IN BEAMS This is the second tutorial on bending of beams. You should judge your progress by completing the self assessment exercises.

More information

MATERIALS SELECTION FOR SPECIFIC USE

MATERIALS SELECTION FOR SPECIFIC USE MATERIALS SELECTION FOR SPECIFIC USE-1 Sub-topics 1 Density What determines density and stiffness? Material properties chart Design problems LOADING 2 STRENGTH AND STIFFNESS Stress is applied to a material

More information

Stresses in Beam (Basic Topics)

Stresses in Beam (Basic Topics) Chapter 5 Stresses in Beam (Basic Topics) 5.1 Introduction Beam : loads acting transversely to the longitudinal axis the loads create shear forces and bending moments, stresses and strains due to V and

More information

Recitation #5. Understanding Shear Force and Bending Moment Diagrams

Recitation #5. Understanding Shear Force and Bending Moment Diagrams Recitation #5 Understanding Shear Force and Bending Moment Diagrams Shear force and bending moment are examples of interanl forces that are induced in a structure when loads are applied to that structure.

More information

Plates and Shells: Theory and Computation - 4D9 - Dr Fehmi Cirak (fc286@) Office: Inglis building mezzanine level (INO 31)

Plates and Shells: Theory and Computation - 4D9 - Dr Fehmi Cirak (fc286@) Office: Inglis building mezzanine level (INO 31) Plates and Shells: Theory and Computation - 4D9 - Dr Fehmi Cirak (fc286@) Office: Inglis building mezzanine level (INO 31) Outline -1-! This part of the module consists of seven lectures and will focus

More information

Nonlinear Models of Reinforced and Post-tensioned Concrete Beams

Nonlinear Models of Reinforced and Post-tensioned Concrete Beams 111 Nonlinear Models of Reinforced and Post-tensioned Concrete Beams ABSTRACT P. Fanning Lecturer, Department of Civil Engineering, University College Dublin Earlsfort Terrace, Dublin 2, Ireland. Email:

More information

Chapter 2: Load, Stress and Strain

Chapter 2: Load, Stress and Strain Chapter 2: Load, Stress and Strain The careful text- books measure (Let all who build beware!) The load, the shock, the pressure Material can bear. So when the buckled girder Lets down the grinding span,

More information

MATHEMATICAL MODEL FOR VIBRATIONS OF NON-UNIFORM FLEXURAL BEAMS

MATHEMATICAL MODEL FOR VIBRATIONS OF NON-UNIFORM FLEXURAL BEAMS Engineering MECHANICS, Vol. 15, 2008, No. 1, p. 3 11 3 MATHEMATICAL MODEL FOR VIBRATIONS OF NON-UNIFORM FLEXURAL BEAMS Mohamed Hussien Taha, Samir Abohadima* A simplified mathematical model for free vibrations

More information

Foundation Size Effect on Modulus of Subgrade Reaction on Sandy Soils

Foundation Size Effect on Modulus of Subgrade Reaction on Sandy Soils Foundation Size Effect on Modulus of Subgrade Reaction on Sandy Soils Aminaton Marto Professor of Faculty of Civil Engineering, University Technology Malaysia (UTM) 81310 Skudai, Johor Bahru, Malaysia;

More information

MECHANICS OF SOLIDS COMPRESSION MEMBERS TUTORIAL 1 STRUTS. On completion of this tutorial you should be able to do the following.

MECHANICS OF SOLIDS COMPRESSION MEMBERS TUTORIAL 1 STRUTS. On completion of this tutorial you should be able to do the following. MECHANICS OF SOLIDS COMPRESSION MEMBERS TUTORIAL 1 STRUTS You should judge your progress by completing the self assessment exercises. On completion of this tutorial you should be able to do the following.

More information

COMPUTATIONAL ENGINEERING OF FINITE ELEMENT MODELLING FOR AUTOMOTIVE APPLICATION USING ABAQUS

COMPUTATIONAL ENGINEERING OF FINITE ELEMENT MODELLING FOR AUTOMOTIVE APPLICATION USING ABAQUS International Journal of Advanced Research in Engineering and Technology (IJARET) Volume 7, Issue 2, March-April 2016, pp. 30 52, Article ID: IJARET_07_02_004 Available online at http://www.iaeme.com/ijaret/issues.asp?jtype=ijaret&vtype=7&itype=2

More information

CAD and Finite Element Analysis

CAD and Finite Element Analysis CAD and Finite Element Analysis Most ME CAD applications require a FEA in one or more areas: Stress Analysis Thermal Analysis Structural Dynamics Computational Fluid Dynamics (CFD) Electromagnetics Analysis...

More information

Design of diaphragm and sheet pile walls. D-Sheet Piling. Verification Report

Design of diaphragm and sheet pile walls. D-Sheet Piling. Verification Report Design of diaphragm and sheet pile walls DR AF T D-Sheet Piling Verification Report D-SHEET PILING Design of diaphragm and sheet pile walls Verification Report Version: 16.1 Revision: 00 16 September

More information

A refined shear deformation theory for flexure of thick beams

A refined shear deformation theory for flexure of thick beams 8(011) 183 195 A refined shear deformation theory for flexure of thick beams Abstract A Hyperbolic Shear Deformation Theory (HPSDT) taking into account transverse shear deformation effects, is used for

More information

Neural network for constitutive modelling in finite element analysis

Neural network for constitutive modelling in finite element analysis Neural network for constitutive modelling in finite element analysis A. A. Javadi, T. P. Tan Department of Engineering, School of Engineering and Computer Science University of Exeter, Exeter, Devon, EX4

More information

Aluminium systems profile selection

Aluminium systems profile selection Aluminium systems profile selection The purpose of this document is to summarise the way that aluminium profile selection should be made, based on the strength requirements for each application. Curtain

More information

Impact Load Factors for Static Analysis

Impact Load Factors for Static Analysis Impact Load Factors for Static Analysis Often a designer has a mass, with a known velocity, hitting an object and thereby causing a suddenly applied impact load. Rather than conduct a dynamic analysis

More information

Problem 1: Computation of Reactions. Problem 2: Computation of Reactions. Problem 3: Computation of Reactions

Problem 1: Computation of Reactions. Problem 2: Computation of Reactions. Problem 3: Computation of Reactions Problem 1: Computation of Reactions Problem 2: Computation of Reactions Problem 3: Computation of Reactions Problem 4: Computation of forces and moments Problem 5: Bending Moment and Shear force Problem

More information

EVERYDAY ENGINEERING EXAMPLES FOR SIMPLE CONCEPTS

EVERYDAY ENGINEERING EXAMPLES FOR SIMPLE CONCEPTS EVERYDAY ENGINEERING EXAMPLES FOR SIMPLE CONCEPTS Simple Method to Calculate Young s Modulus Elasticity MECN 4600 - Mechanical Measurement and Instrumentation Dr. Omar Meza Copyright 2015 MSEIP Engineering

More information

Dispersion diagrams of a water-loaded cylindrical shell obtained from the structural and acoustic responses of the sensor array along the shell

Dispersion diagrams of a water-loaded cylindrical shell obtained from the structural and acoustic responses of the sensor array along the shell Dispersion diagrams of a water-loaded cylindrical shell obtained from the structural and acoustic responses of the sensor array along the shell B.K. Jung ; J. Ryue ; C.S. Hong 3 ; W.B. Jeong ; K.K. Shin

More information

MATERIALS AND SCIENCE IN SPORTS. Edited by: EH. (Sam) Froes and S.J. Haake. Dynamics

MATERIALS AND SCIENCE IN SPORTS. Edited by: EH. (Sam) Froes and S.J. Haake. Dynamics MATERIALS AND SCIENCE IN SPORTS Edited by: EH. (Sam) Froes and S.J. Haake Dynamics Analysis of the Characteristics of Fishing Rods Based on the Large-Deformation Theory Atsumi Ohtsuki, Prof, Ph.D. Pgs.

More information

MENG 302L Lab 4: Modulus of Elasticity and Poisson s Ratio

MENG 302L Lab 4: Modulus of Elasticity and Poisson s Ratio MENG 302L Lab 4: Modulus of Elasticity and Poisson s Ratio Introduction: In Lab 4 we will measure the two fundamental elastic constants relating stress to strain: Modulus of Elasticity and Poisson s Ratio.

More information

The kinematic interaction of a single pile with heterogeneous soil

The kinematic interaction of a single pile with heterogeneous soil Earthquake Ground Motion: Input Definition for Aseismic Design 135 The kinematic interaction of a single pile with heterogeneous soil M. Maugeri, E. Motta, E. Raciti & D. Ardita Department of Civil and

More information

Bending Stress in Beams

Bending Stress in Beams 936-73-600 Bending Stress in Beams Derive a relationship for bending stress in a beam: Basic Assumptions:. Deflections are very small with respect to the depth of the beam. Plane sections before bending

More information

Module 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method. Version 2 CE IIT, Kharagpur

Module 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method. Version 2 CE IIT, Kharagpur Module Analysis of Statically Indeterminate Structures by the Matrix Force Method esson 11 The Force Method of Analysis: Frames Instructional Objectives After reading this chapter the student will be able

More information

Impacts of Tunnelling on Ground and Groundwater and Control Measures Part 1: Estimation Methods

Impacts of Tunnelling on Ground and Groundwater and Control Measures Part 1: Estimation Methods Impacts of Tunnelling on Ground and Groundwater and Control Measures Part 1: Estimation Methods Steve Macklin Principal Engineering Geologist GHD Melbourne 1. Introduction, scope of Part 1 2. Terminology

More information

The Basics of FEA Procedure

The Basics of FEA Procedure CHAPTER 2 The Basics of FEA Procedure 2.1 Introduction This chapter discusses the spring element, especially for the purpose of introducing various concepts involved in use of the FEA technique. A spring

More information

FUNDAMENTAL FINITE ELEMENT ANALYSIS AND APPLICATIONS

FUNDAMENTAL FINITE ELEMENT ANALYSIS AND APPLICATIONS FUNDAMENTAL FINITE ELEMENT ANALYSIS AND APPLICATIONS With Mathematica and MATLAB Computations M. ASGHAR BHATTI WILEY JOHN WILEY & SONS, INC. CONTENTS OF THE BOOK WEB SITE PREFACE xi xiii 1 FINITE ELEMENT

More information

The Bernoulli- Euler Beam

The Bernoulli- Euler Beam 8 The ernoulli- Euler eam 8 1 Chapter 8: THE ERNOULLI- EULER E TLE OF CONTENTS Page 8.1. Introduction 8 3 8.2. The eam odel 8 3 8.2.1. Field Equations................. 8 3 8.2.2. oundary Conditions..............

More information

STRESS AND DEFORMATION ANALYSIS OF LINEAR ELASTIC BARS IN TENSION

STRESS AND DEFORMATION ANALYSIS OF LINEAR ELASTIC BARS IN TENSION Chapter 11 STRESS AND DEFORMATION ANALYSIS OF LINEAR ELASTIC BARS IN TENSION Figure 11.1: In Chapter10, the equilibrium, kinematic and constitutive equations for a general three-dimensional solid deformable

More information

CE-632 Foundation Analysis and Design

CE-632 Foundation Analysis and Design CE-63 Foundation Analysis and Design Shallow Foundations 1 SUMMARY of Terminology Gross Loading Intensity Total pressure at the level of foundation including the weight of superstructure, foundation, and

More information

Development of Membrane, Plate and Flat Shell Elements in Java

Development of Membrane, Plate and Flat Shell Elements in Java Development of Membrane, Plate and Flat Shell Elements in Java by Kaushalkumar Kansara Thesis submitted to the Faculty of the Virginia Polytechnic Institute & State University In partial fulfillment of

More information

Experiment 3 Modulus of Elasticity

Experiment 3 Modulus of Elasticity Related topics Young s modulus, modulus of elasticity, stress, deformation, Poisson s ratio, Hooke s Law. Principle and task A flat bar is supported at two points. It is bent by the action a force acting

More information

Chapter 4: Summary and Conclusions

Chapter 4: Summary and Conclusions Chapter 4: Summary and Conclusions 4.1 Summary Three different models are presented and analyzed in this research for the purpose of studying the potential of using post-buckled or pre-bent elastic struts

More information

Unit 48: Structural Behaviour and Detailing for Construction. Chapter 13. Reinforced Concrete Beams

Unit 48: Structural Behaviour and Detailing for Construction. Chapter 13. Reinforced Concrete Beams Chapter 13 Reinforced Concrete Beams Concrete is a material strong in its resistance to compression, but very weak indeed in tension. good concrete will safely take a stress upwards of 7 N/mm 2 in compression,

More information

SIMPLE ANALYSIS OF FRAMED-TUBE STRUCTURES WITH MULTIPLE INTERNAL TUBES

SIMPLE ANALYSIS OF FRAMED-TUBE STRUCTURES WITH MULTIPLE INTERNAL TUBES SIMPLE ANALYSIS OF FRAMED-TUBE STRUCTURES WITH MULTIPLE INTERNAL TUBES By Kang-Kun Lee, Yew-Chaye Loo 2, and Hong Guan 3 ABSTRACT: Framed-tube system with multiple internal tubes is analysed using an orthotropic

More information

Shell Elements in ABAQUS/Explicit

Shell Elements in ABAQUS/Explicit ABAQUS/Explicit: Advanced Topics Appendix 2 Shell Elements in ABAQUS/Explicit ABAQUS/Explicit: Advanced Topics A2.2 Overview ABAQUS/Explicit: Advanced Topics ABAQUS/Explicit: Advanced Topics A2.4 Triangular

More information

AN AD OPTIMIZED METHOD FOR THE STRUCTURAL ANALYSIS OF PRESSURE VESSELS

AN AD OPTIMIZED METHOD FOR THE STRUCTURAL ANALYSIS OF PRESSURE VESSELS Proceedings of ICAD2014 ICAD-2014-24 AN AD OPTIMIZED METHOD FOR THE STRUCTURAL ANALYSIS OF PRESSURE VESSELS ABSTRACT Andrea Girgenti a.girgenti@unimarconi.it Alessandro Giorgetti a.giorgetti@unimarconi.it

More information

Optimising plate girder design

Optimising plate girder design Optimising plate girder design NSCC29 R. Abspoel 1 1 Division of structural engineering, Delft University of Technology, Delft, The Netherlands ABSTRACT: In the design of steel plate girders a high degree

More information

INTERACTION BETWEEN MOVING VEHICLES AND RAILWAY TRACK AT HIGH SPEED

INTERACTION BETWEEN MOVING VEHICLES AND RAILWAY TRACK AT HIGH SPEED INTERACTION BETWEEN MOVING VEHICLES AND RAILWAY TRACK AT HIGH SPEED Prof.Dr.Ir. C. Esveld Professor of Railway Engineering TU Delft, The Netherlands Dr.Ir. A.W.M. Kok Associate Professor of Railway Engineering

More information

INVESTIGATION ON VIBRATION ENERGY FLOW CHARACTERISTICS IN COUPLED PLATES BY VISUALIZAITON TECHNIQUES

INVESTIGATION ON VIBRATION ENERGY FLOW CHARACTERISTICS IN COUPLED PLATES BY VISUALIZAITON TECHNIQUES Journal of Marine Science and Technology, Vol. 8, No. 6, pp. 907-94 (00) 907 INVESTIGATION ON VIBRATION ENERG FLOW CHARACTERISTICS IN COUPLED PLATES B VISUALIAITON TECHNIQUES Kai Li*, Sheng Li*, and De-you

More information

Group Members: Blake Maloney, Khaled Alroumi, Robert Mitchell

Group Members: Blake Maloney, Khaled Alroumi, Robert Mitchell ME 461: Finite Element Analysis Fall 2015 A Semester Report on: The Development and Analysis of a Nose Landing Gear Group Members: Blake Maloney, Khaled Alroumi, Robert Mitchell Department of Mechanical

More information

Higher order beam finite elements with only displacement degrees of freedom

Higher order beam finite elements with only displacement degrees of freedom Higher order beam finite elements with only displacement degrees of freedom Erasmo Carrera 1, Marco Petrolo 1, Christian Wenzel 1, Gaetano Giunta 1,2, Salim Belouettar 2 1 Aeronautics and Space Engineering

More information

Module 1 : Conduction. Lecture 5 : 1D conduction example problems. 2D conduction

Module 1 : Conduction. Lecture 5 : 1D conduction example problems. 2D conduction Module 1 : Conduction Lecture 5 : 1D conduction example problems. 2D conduction Objectives In this class: An example of optimization for insulation thickness is solved. The 1D conduction is considered

More information

NonLinear Materials. Introduction. Preprocessing: Defining the Problem

NonLinear Materials. Introduction. Preprocessing: Defining the Problem NonLinear Materials Introduction This tutorial was completed using ANSYS 7.0 The purpose of the tutorial is to describe how to include material nonlinearities in an ANSYS model. For instance, the case

More information

3 Concepts of Stress Analysis

3 Concepts of Stress Analysis 3 Concepts of Stress Analysis 3.1 Introduction Here the concepts of stress analysis will be stated in a finite element context. That means that the primary unknown will be the (generalized) displacements.

More information

Response to Harmonic Excitation Part 2: Damped Systems

Response to Harmonic Excitation Part 2: Damped Systems Response to Harmonic Excitation Part 2: Damped Systems Part 1 covered the response of a single degree of freedom system to harmonic excitation without considering the effects of damping. However, almost

More information

In-situ Load Testing to Evaluate New Repair Techniques

In-situ Load Testing to Evaluate New Repair Techniques In-situ Load Testing to Evaluate New Repair Techniques W.J. Gold 1 and A. Nanni 2 1 Assistant Research Engineer, Univ. of Missouri Rolla, Dept. of Civil Engineering 2 V&M Jones Professor, Univ. of Missouri

More information

Bending, Forming and Flexing Printed Circuits

Bending, Forming and Flexing Printed Circuits Bending, Forming and Flexing Printed Circuits John Coonrod Rogers Corporation Introduction: In the printed circuit board industry there are generally two main types of circuit boards; there are rigid printed

More information

Distribution of Forces in Lateral Load Resisting Systems

Distribution of Forces in Lateral Load Resisting Systems Distribution of Forces in Lateral Load Resisting Systems Part 2. Horizontal Distribution and Torsion IITGN Short Course Gregory MacRae Many slides from 2009 Myanmar Slides of Profs Jain and Rai 1 Reinforced

More information

SHEAR IN SKEWED MULTI-BEAM BRIDGES

SHEAR IN SKEWED MULTI-BEAM BRIDGES 20-7/Task 107 COPY NO. SHEAR IN SKEWED MULTI-BEAM BRIDGES FINAL REPORT Prepared for National Cooperative Highway Research Program Transportation Research Board National Research Council Modjeski and Masters,

More information

6 1. Draw the shear and moment diagrams for the shaft. The bearings at A and B exert only vertical reactions on the shaft.

6 1. Draw the shear and moment diagrams for the shaft. The bearings at A and B exert only vertical reactions on the shaft. 06 Solutions 46060_Part1 5/27/10 3:51 PM Page 329 6 1. Draw the shear and moment diagrams for the shaft. The bearings at and exert only vertical reactions on the shaft. 250 mm 800 mm 24 kn 6 2. Draw the

More information

Design of Steel Structures Prof. S.R.Satish Kumar and Prof. A.R.Santha Kumar

Design of Steel Structures Prof. S.R.Satish Kumar and Prof. A.R.Santha Kumar Problem 1 Design a hand operated overhead crane, which is provided in a shed, whose details are: Capacity of crane = 50 kn Longitudinal spacing of column = 6m Center to center distance of gantry girder

More information

Introduction, Method of Sections

Introduction, Method of Sections Lecture #1 Introduction, Method of Sections Reading: 1:1-2 Mechanics of Materials is the study of the relationship between external, applied forces and internal effects (stress & deformation). An understanding

More information

Numerical modelling of shear connection between concrete slab and sheeting deck

Numerical modelling of shear connection between concrete slab and sheeting deck 7th fib International PhD Symposium in Civil Engineering 2008 September 10-13, Universität Stuttgart, Germany Numerical modelling of shear connection between concrete slab and sheeting deck Noémi Seres

More information

SEISMIC MODEL TEST AND ANALYSIS OF MULTI-TOWER HIGH-RISE BUILDINGS

SEISMIC MODEL TEST AND ANALYSIS OF MULTI-TOWER HIGH-RISE BUILDINGS SEISMIC MODEL TEST AND ANALYSIS OF MULTI-TOWER HIGH-RISE BUILDINGS Wensheng LU 1 And Xilin LU 2 SUMMARY This paper summarizes tests of several scaled multi-tower high-rise building models on the shaking

More information

Examples of New version for Designing members of Reinforced Concrete, Steel or Timber according to Eurocode 2, Eurocode 3 and Eurocode 5

Examples of New version for Designing members of Reinforced Concrete, Steel or Timber according to Eurocode 2, Eurocode 3 and Eurocode 5 Examples of New version for Designing members of Reinforced Concrete, Steel or Timber according to Eurocode 2, Eurocode 3 and Eurocode 5 Copyright RUNET Software www.runet-software.com 1 1. Examples 1.1

More information

EDEXCEL NATIONAL CERTIFICATE/DIPLOMA MECHANICAL PRINCIPLES AND APPLICATIONS NQF LEVEL 3 OUTCOME 1 - LOADING SYSTEMS TUTORIAL 3 LOADED COMPONENTS

EDEXCEL NATIONAL CERTIFICATE/DIPLOMA MECHANICAL PRINCIPLES AND APPLICATIONS NQF LEVEL 3 OUTCOME 1 - LOADING SYSTEMS TUTORIAL 3 LOADED COMPONENTS EDEXCEL NATIONAL CERTIICATE/DIPLOMA MECHANICAL PRINCIPLES AND APPLICATIONS NQ LEVEL 3 OUTCOME 1 - LOADING SYSTEMS TUTORIAL 3 LOADED COMPONENTS 1. Be able to determine the effects of loading in static engineering

More information

Module 3. Analysis of Statically Indeterminate Structures by the Displacement Method. Version 2 CE IIT, Kharagpur

Module 3. Analysis of Statically Indeterminate Structures by the Displacement Method. Version 2 CE IIT, Kharagpur odule 3 Analysis of Statically Indeterminate Structures by the Displacement ethod Lesson 21 The oment- Distribution ethod: rames with Sidesway Instructional Objectives After reading this chapter the student

More information

ENGINEERING SCIENCE H1 OUTCOME 1 - TUTORIAL 3 BENDING MOMENTS EDEXCEL HNC/D ENGINEERING SCIENCE LEVEL 4 H1 FORMERLY UNIT 21718P

ENGINEERING SCIENCE H1 OUTCOME 1 - TUTORIAL 3 BENDING MOMENTS EDEXCEL HNC/D ENGINEERING SCIENCE LEVEL 4 H1 FORMERLY UNIT 21718P ENGINEERING SCIENCE H1 OUTCOME 1 - TUTORIAL 3 BENDING MOMENTS EDEXCEL HNC/D ENGINEERING SCIENCE LEVEL 4 H1 FORMERLY UNIT 21718P This material is duplicated in the Mechanical Principles module H2 and those

More information

Back to Elements - Tetrahedra vs. Hexahedra

Back to Elements - Tetrahedra vs. Hexahedra Back to Elements - Tetrahedra vs. Hexahedra Erke Wang, Thomas Nelson, Rainer Rauch CAD-FEM GmbH, Munich, Germany Abstract This paper presents some analytical results and some test results for different

More information

Improving FE models of a long-span flat concrete floor using natural frequency measurements

Improving FE models of a long-span flat concrete floor using natural frequency measurements Computers and Structures 80 (2002) 2145 2156 www.elsevier.com/locate/compstruc Improving FE models of a long-span flat concrete floor using natural frequency measurements Emad El-Dardiry a, Endah Wahyuni

More information

Unit 21 Influence Coefficients

Unit 21 Influence Coefficients Unit 21 Influence Coefficients Readings: Rivello 6.6, 6.13 (again), 10.5 Paul A. Lagace, Ph.D. Professor of Aeronautics & Astronautics and Engineering Systems Have considered the vibrational behavior of

More information

DESIGN AND NON-LINEAR ANALYSIS OF A PARABOLIC LEAF SPRING

DESIGN AND NON-LINEAR ANALYSIS OF A PARABOLIC LEAF SPRING 47 DESIGN AND NON-LINEAR ANALYSIS OF A PARABOLIC LEAF SPRING Muhammad Ashiqur Rahman*, Muhammad Tareq Siddiqui and Muhammad Arefin Kowser Department of Mechanical Engineering, Bangladesh University of

More information

A beam is a structural member that is subjected primarily to transverse loads and negligible

A beam is a structural member that is subjected primarily to transverse loads and negligible Chapter. Design of Beams Flexure and Shear.1 Section force-deformation response & Plastic Moment (M p ) A beam is a structural member that is subjected primarily to transverse loads and negligible axial

More information

Laterally Loaded Piles

Laterally Loaded Piles Laterally Loaded Piles 1 Soil Response Modelled by p-y Curves In order to properly analyze a laterally loaded pile foundation in soil/rock, a nonlinear relationship needs to be applied that provides soil

More information

Numerical and Experimental Analysis of a Cantilever Beam: a Laboratory Project to Introduce Geometric Nonlinearity in Mechanics of Materials*

Numerical and Experimental Analysis of a Cantilever Beam: a Laboratory Project to Introduce Geometric Nonlinearity in Mechanics of Materials* Int. J. Engng Ed. Vol. 19, No. 6, pp. 885±892, 2003 0949-149X/91 $3.00+0.00 Printed in Great Britain. # 2003 TEMPUS Publications. Numerical and Experimental Analysis of a Cantilever Beam: a Laboratory

More information

Chapter 9 Deflections of Beams

Chapter 9 Deflections of Beams Chapter 9 Deflections of Beams 9.1 Introduction in this chapter, we describe methods for determining the equation of the deflection curve of beams and finding deflection and slope at specific points along

More information