Lecture 7: The Beam Element Equations.

Size: px
Start display at page:

Download "Lecture 7: The Beam Element Equations."

Transcription

1 4.1 Beam Stiffness. A Beam: A long slender structural component generally subjected to transverse loading that produces significant bending effects as opposed to twisting or axial effects. MECH 40: Finite Element Applications

2 MECH 0 and 0 leave us with... The bending moment, shear force, and distributed load are internal loads. Their sense is defined according to specific sign conventions ŷ 1 M = EI = EIκ ρ dm V = dv w = Based on the assumption that plane sections remain plane M V x MECH 40: Finite Element Applications

3 Governing kinematic relationships (displacement and motion knowledge irrespective of causal forces). d v x 1 M( x) ( ) = κ( x) = ; κ( x) = beam curvature. ρ( x ) EI Give the differential relationships that exist between bending moment, shear force, and distributed transverse load (from static equilibrium) M V w EI EI = EI ( ) ( ) ( ) dvx dvx 4 dvx 4 MECH 40: Finite Element Applications = = By definition, these quantities must to be interpreted according to the civil engineering sign convention.

4 So to form our element equations we must form our own approximation to the beam mechanics. This approximate beam (or beam element) should be applicable in all types of structural problems involving beams. f f f ix iy iz i = 1.. N K (the stiffness matrix) d d d ix iy iz i = 1.. N Defined in terms of a Cartesian reference frame MECH 40: Finite Element Applications

5 Step 1: Set the element type: An important step as we are identifying the state variables of the beam element. M M MECH 40: Finite Element Applications

6 We have introduced some limitations by requiring that only nodal loads exist. Comparing Figures 4.1 and 4.: ( ) dvx 1y =+ = f V EI ( ) dvx 1z = = m M EI ( ) dvx y = = f V EI ( ) dvx z =+ = m M EI x= 0 x= 0 x = x = Since we have not included distributed loads in the analysis, we only see up to third order differentials of the displacement function v. MECH 40: Finite Element Applications

7 Step : Select a displacement function for vx ( ). Given our assumption/limitation that 4 dvx ( ) w= EI = 4 d x 0 A conforming element is one that will ensure: Compatibility/Continuity: a continuous displacement and rotation exist within a single beam element and across element boundaries. That is, a smooth first derivative of v(x) exists. Completeness: a constant shear force can exist within the element. That is v(x) can be differentiated up to times. MECH 40: Finite Element Applications

8 So we choose a cubic function to approximate vx ( ) : vx ( ) = a + ax+ ax + ax 4 1 The displacement function must interpolate the nodal generalized displacements. This ensures compatibility. v(0) = a = d 4 1y dv = a = tan( φ 1) φ1 x= 0 v ( ) = a+ a+ a+ a = d dv x = 1 4 y = a + a+ a = tan( φ ) φ 1 1 Assuming small transverse displacements MECH 40: Finite Element Applications

9 In a matrix form [ ] vx ( ) N N N N d 1y φ d y φ 1 = N1 = ( x x + ) 1 N = x x + x 1 N = ( x + x ) 1 N4 = ( x x ) ( ) Shape functions that blend the nodal displacements over the element domain. Eq. (4.1.7) of ogan. MECH 40: Finite Element Applications

10 Step : Define the stress-strain relationships. Also a means to recover any stress-strain knowledge after obtaining the state variables of the model. dv ux ( ) = y dx du ε ( x x) = dv ε ( x x) = y dx dv σ ( x x) = Ey dx MECH 40: Finite Element Applications

11 Step 4: Derive the element equations Can we express the nodal forces in terms of the displacement function? ( φ φ ) f V EI d d x= 0 dvx ( ) EI 1y =+ = = 1 1y y + 6 ( φ φ ) m M EI 6d 4 6d x= 0 dvx ( ) EI 1z = = = 1y + 1 y + ( φ φ ) f V EI d d x = dvx ( ) EI y = = = 1 1y y 6 dvx ( ) EI z = = = ( 6d 1y + φ1 6dy + 4 φ) m M EI x = MECH 40: Finite Element Applications

12 And in matrix form f 1y d 1y m 1z EI 4 6 φ1 = f 1 6 d y y m SYM 4 z φ k MECH 40: Finite Element Applications

Lecture 8: Assembly of beam elements.

Lecture 8: Assembly of beam elements. ecture 8: Assembly of beam elements. 4. Example of Assemblage of Beam Stiffness Matrices. Place nodes at the load application points. Assembling the two sets of element equations (note the common elemental

More information

Chapter 5 Structural Elements: The truss & beam elements

Chapter 5 Structural Elements: The truss & beam elements Institute of Structural Engineering Page 1 Chapter 5 Structural Elements: The truss & beam elements Institute of Structural Engineering Page 2 Chapter Goals Learn how to formulate the Finite Element Equations

More information

Chapter 5 Elastic Strain, Deflection, and Stability 1. Elastic Stress-Strain Relationship

Chapter 5 Elastic Strain, Deflection, and Stability 1. Elastic Stress-Strain Relationship Chapter 5 Elastic Strain, Deflection, and Stability Elastic Stress-Strain Relationship A stress in the x-direction causes a strain in the x-direction by σ x also causes a strain in the y-direction & z-direction

More information

CHAPTER 5. Beam Theory

CHAPTER 5. Beam Theory CHPTER 5. Beam Theory SangJoon Shin School of Mechanical and erospace Engineering Seoul National University ctive eroelasticity and Rotorcraft Lab. 5. The Euler-Bernoulli assumptions One of its dimensions

More information

Institute of Structural Engineering Page 1. Method of Finite Elements I. Chapter 2. The Direct Stiffness Method. Method of Finite Elements I

Institute of Structural Engineering Page 1. Method of Finite Elements I. Chapter 2. The Direct Stiffness Method. Method of Finite Elements I Institute of Structural Engineering Page 1 Chapter 2 The Direct Stiffness Method Institute of Structural Engineering Page 2 Direct Stiffness Method (DSM) Computational method for structural analysis Matrix

More information

Variational Formulation of Plane Beam Element

Variational Formulation of Plane Beam Element 13 Variational Formulation of Plane Beam Element IFEM Ch 13 Slide 1 Beams Resist Primarily Transverse Loads IFEM Ch 13 Slide 2 Transverse Loads are Transported to Supports by Flexural Action Neutral surface

More information

Lecture 8. Stress Strain in Multi-dimension

Lecture 8. Stress Strain in Multi-dimension Lecture 8. Stress Strain in Multi-dimension Module. General Field Equations General Field Equations [] Equilibrium Equations in Elastic bodies xx x y z yx zx f x 0, etc [2] Kinematics xx u x x,etc. [3]

More information

Review of Strain Energy Methods and Introduction to Stiffness Matrix Methods of Structural Analysis

Review of Strain Energy Methods and Introduction to Stiffness Matrix Methods of Structural Analysis uke University epartment of Civil and Environmental Engineering CEE 42L. Matrix Structural Analysis Henri P. Gavin Fall, 22 Review of Strain Energy Methods and Introduction to Stiffness Matrix Methods

More information

Internal Internal Forces Forces

Internal Internal Forces Forces Internal Forces ENGR 221 March 19, 2003 Lecture Goals Internal Force in Structures Shear Forces Bending Moment Shear and Bending moment Diagrams Internal Forces and Bending The bending moment, M. Moment

More information

Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras. Module - 01 Lecture - 11

Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras. Module - 01 Lecture - 11 Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras Module - 01 Lecture - 11 Last class, what we did is, we looked at a method called superposition

More information

Chapter 12 Plate Bending Elements. Chapter 12 Plate Bending Elements

Chapter 12 Plate Bending Elements. Chapter 12 Plate Bending Elements CIVL 7/8117 Chapter 12 - Plate Bending Elements 1/34 Chapter 12 Plate Bending Elements Learning Objectives To introduce basic concepts of plate bending. To derive a common plate bending element stiffness

More information

Lecture 15 Strain and stress in beams

Lecture 15 Strain and stress in beams Spring, 2019 ME 323 Mechanics of Materials Lecture 15 Strain and stress in beams Reading assignment: 6.1 6.2 News: Instructor: Prof. Marcial Gonzalez Last modified: 1/6/19 9:42:38 PM Beam theory (@ ME

More information

Institute of Structural Engineering Page 1. Method of Finite Elements I. Chapter 2. The Direct Stiffness Method. Method of Finite Elements I

Institute of Structural Engineering Page 1. Method of Finite Elements I. Chapter 2. The Direct Stiffness Method. Method of Finite Elements I Institute of Structural Engineering Page 1 Chapter 2 The Direct Stiffness Method Institute of Structural Engineering Page 2 Direct Stiffness Method (DSM) Computational method for structural analysis Matrix

More information

[8] Bending and Shear Loading of Beams

[8] Bending and Shear Loading of Beams [8] Bending and Shear Loading of Beams Page 1 of 28 [8] Bending and Shear Loading of Beams [8.1] Bending of Beams (will not be covered in class) [8.2] Bending Strain and Stress [8.3] Shear in Straight

More information

Lecture 27: Structural Dynamics - Beams.

Lecture 27: Structural Dynamics - Beams. Chapter #16: Structural Dynamics and Time Dependent Heat Transfer. Lectures #1-6 have discussed only steady systems. There has been no time dependence in any problems. We will investigate beam dynamics

More information

Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras. Module - 01 Lecture - 13

Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras. Module - 01 Lecture - 13 Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras (Refer Slide Time: 00:25) Module - 01 Lecture - 13 In the last class, we have seen how

More information

Slender Structures Load carrying principles

Slender Structures Load carrying principles Slender Structures Load carrying principles Basic cases: Extension, Shear, Torsion, Cable Bending (Euler) v017-1 Hans Welleman 1 Content (preliminary schedule) Basic cases Extension, shear, torsion, cable

More information

6. Bending CHAPTER OBJECTIVES

6. Bending CHAPTER OBJECTIVES CHAPTER OBJECTIVES Determine stress in members caused by bending Discuss how to establish shear and moment diagrams for a beam or shaft Determine largest shear and moment in a member, and specify where

More information

Stress analysis of a stepped bar

Stress analysis of a stepped bar Stress analysis of a stepped bar Problem Find the stresses induced in the axially loaded stepped bar shown in Figure. The bar has cross-sectional areas of A ) and A ) over the lengths l ) and l ), respectively.

More information

Bending Stress. Sign convention. Centroid of an area

Bending Stress. Sign convention. Centroid of an area Bending Stress Sign convention The positive shear force and bending moments are as shown in the figure. Centroid of an area Figure 40: Sign convention followed. If the area can be divided into n parts

More information

General elastic beam with an elastic foundation

General elastic beam with an elastic foundation General elastic beam with an elastic foundation Figure 1 shows a beam-column on an elastic foundation. The beam is connected to a continuous series of foundation springs. The other end of the foundation

More information

Module 4 : Deflection of Structures Lecture 4 : Strain Energy Method

Module 4 : Deflection of Structures Lecture 4 : Strain Energy Method Module 4 : Deflection of Structures Lecture 4 : Strain Energy Method Objectives In this course you will learn the following Deflection by strain energy method. Evaluation of strain energy in member under

More information

Indeterminate Analysis Force Method 1

Indeterminate Analysis Force Method 1 Indeterminate Analysis Force Method 1 The force (flexibility) method expresses the relationships between displacements and forces that exist in a structure. Primary objective of the force method is to

More information

Chapter 4 Deflection and Stiffness

Chapter 4 Deflection and Stiffness Chapter 4 Deflection and Stiffness Asst. Prof. Dr. Supakit Rooppakhun Chapter Outline Deflection and Stiffness 4-1 Spring Rates 4-2 Tension, Compression, and Torsion 4-3 Deflection Due to Bending 4-4 Beam

More information

Consider an elastic spring as shown in the Fig.2.4. When the spring is slowly

Consider an elastic spring as shown in the Fig.2.4. When the spring is slowly .3 Strain Energy Consider an elastic spring as shown in the Fig..4. When the spring is slowly pulled, it deflects by a small amount u 1. When the load is removed from the spring, it goes back to the original

More information

Module 3. Analysis of Statically Indeterminate Structures by the Displacement Method

Module 3. Analysis of Statically Indeterminate Structures by the Displacement Method odule 3 Analysis of Statically Indeterminate Structures by the Displacement ethod Lesson 16 The Slope-Deflection ethod: rames Without Sidesway Instructional Objectives After reading this chapter the student

More information

Unit 13 Review of Simple Beam Theory

Unit 13 Review of Simple Beam Theory MIT - 16.0 Fall, 00 Unit 13 Review of Simple Beam Theory Readings: Review Unified Engineering notes on Beam Theory BMP 3.8, 3.9, 3.10 T & G 10-15 Paul A. Lagace, Ph.D. Professor of Aeronautics & Astronautics

More information

3D Elasticity Theory

3D Elasticity Theory 3D lasticity Theory Many structural analysis problems are analysed using the theory of elasticity in which Hooke s law is used to enforce proportionality between stress and strain at any deformation level.

More information

Vibration analysis of circular arch element using curvature

Vibration analysis of circular arch element using curvature Shock and Vibration 15 (28) 481 492 481 IOS Press Vibration analysis of circular arch element using curvature H. Saffari a,. Tabatabaei a, and S.H. Mansouri b a Civil Engineering Department, University

More information

Types of Structures & Loads

Types of Structures & Loads Structure Analysis I Chapter 4 1 Types of Structures & Loads 1Chapter Chapter 4 Internal lloading Developed in Structural Members Internal loading at a specified Point In General The loading for coplanar

More information

Finite Element Method in Geotechnical Engineering

Finite Element Method in Geotechnical Engineering Finite Element Method in Geotechnical Engineering Short Course on + Dynamics Boulder, Colorado January 5-8, 2004 Stein Sture Professor of Civil Engineering University of Colorado at Boulder Contents Steps

More information

BEAM DEFLECTION THE ELASTIC CURVE

BEAM DEFLECTION THE ELASTIC CURVE BEAM DEFLECTION Samantha Ramirez THE ELASTIC CURVE The deflection diagram of the longitudinal axis that passes through the centroid of each cross-sectional area of a beam. Supports that apply a moment

More information

FREE VIBRATION OF AXIALLY LOADED FUNCTIONALLY GRADED SANDWICH BEAMS USING REFINED SHEAR DEFORMATION THEORY

FREE VIBRATION OF AXIALLY LOADED FUNCTIONALLY GRADED SANDWICH BEAMS USING REFINED SHEAR DEFORMATION THEORY FREE VIBRATION OF AXIALLY LOADED FUNCTIONALLY GRADED SANDWICH BEAMS USING REFINED SHEAR DEFORMATION THEORY Thuc P. Vo 1, Adelaja Israel Osofero 1, Marco Corradi 1, Fawad Inam 1 1 Faculty of Engineering

More information

ENG2000 Chapter 7 Beams. ENG2000: R.I. Hornsey Beam: 1

ENG2000 Chapter 7 Beams. ENG2000: R.I. Hornsey Beam: 1 ENG2000 Chapter 7 Beams ENG2000: R.I. Hornsey Beam: 1 Overview In this chapter, we consider the stresses and moments present in loaded beams shear stress and bending moment diagrams We will also look at

More information

Quintic beam closed form matrices (revised 2/21, 2/23/12) General elastic beam with an elastic foundation

Quintic beam closed form matrices (revised 2/21, 2/23/12) General elastic beam with an elastic foundation General elastic beam with an elastic foundation Figure 1 shows a beam-column on an elastic foundation. The beam is connected to a continuous series of foundation springs. The other end of the foundation

More information

Finite element modelling of structural mechanics problems

Finite element modelling of structural mechanics problems 1 Finite element modelling of structural mechanics problems Kjell Magne Mathisen Department of Structural Engineering Norwegian University of Science and Technology Lecture 10: Geilo Winter School - January,

More information

Symmetric Bending of Beams

Symmetric Bending of Beams Symmetric Bending of Beams beam is any long structural member on which loads act perpendicular to the longitudinal axis. Learning objectives Understand the theory, its limitations and its applications

More information

3. BEAMS: STRAIN, STRESS, DEFLECTIONS

3. BEAMS: STRAIN, STRESS, DEFLECTIONS 3. BEAMS: STRAIN, STRESS, DEFLECTIONS The beam, or flexural member, is frequently encountered in structures and machines, and its elementary stress analysis constitutes one of the more interesting facets

More information

ε t increases from the compressioncontrolled Figure 9.15: Adjusted interaction diagram

ε t increases from the compressioncontrolled Figure 9.15: Adjusted interaction diagram CHAPTER NINE COLUMNS 4 b. The modified axial strength in compression is reduced to account for accidental eccentricity. The magnitude of axial force evaluated in step (a) is multiplied by 0.80 in case

More information

Lecture Pure Twist

Lecture Pure Twist Lecture 4-2003 Pure Twist pure twist around center of rotation D => neither axial (σ) nor bending forces (Mx, My) act on section; as previously, D is fixed, but (for now) arbitrary point. as before: a)

More information

Virtual Work and Variational Principles

Virtual Work and Variational Principles Virtual Work and Principles Mathematically, the structural analysis problem is a boundary value problem (BVP). Forces, displacements, stresses, and strains are connected and computed within the framework

More information

UNIT IV FLEXIBILTY AND STIFFNESS METHOD

UNIT IV FLEXIBILTY AND STIFFNESS METHOD SIDDHARTH GROUP OF INSTITUTIONS :: PUTTUR Siddharth Nagar, Narayanavanam Road 517583 QUESTION BANK (DESCRIPTIVE) Subject with Code : SA-II (13A01505) Year & Sem: III-B.Tech & I-Sem Course & Branch: B.Tech

More information

Geometry-dependent MITC method for a 2-node iso-beam element

Geometry-dependent MITC method for a 2-node iso-beam element Structural Engineering and Mechanics, Vol. 9, No. (8) 3-3 Geometry-dependent MITC method for a -node iso-beam element Phill-Seung Lee Samsung Heavy Industries, Seocho, Seoul 37-857, Korea Hyu-Chun Noh

More information

FINAL EXAMINATION. (CE130-2 Mechanics of Materials)

FINAL EXAMINATION. (CE130-2 Mechanics of Materials) UNIVERSITY OF CLIFORNI, ERKELEY FLL SEMESTER 001 FINL EXMINTION (CE130- Mechanics of Materials) Problem 1: (15 points) pinned -bar structure is shown in Figure 1. There is an external force, W = 5000N,

More information

CHAPTER -6- BENDING Part -1-

CHAPTER -6- BENDING Part -1- Ishik University / Sulaimani Civil Engineering Department Mechanics of Materials CE 211 CHAPTER -6- BENDING Part -1-1 CHAPTER -6- Bending Outlines of this chapter: 6.1. Chapter Objectives 6.2. Shear and

More information

3 2 6 Solve the initial value problem u ( t) 3. a- If A has eigenvalues λ =, λ = 1 and corresponding eigenvectors 1

3 2 6 Solve the initial value problem u ( t) 3. a- If A has eigenvalues λ =, λ = 1 and corresponding eigenvectors 1 Math Problem a- If A has eigenvalues λ =, λ = 1 and corresponding eigenvectors 1 3 6 Solve the initial value problem u ( t) = Au( t) with u (0) =. 3 1 u 1 =, u 1 3 = b- True or false and why 1. if A is

More information

Development of Beam Equations

Development of Beam Equations CHAPTER 4 Development of Beam Equations Introduction We begin this chapter by developing the stiffness matrix for the bending of a beam element, the most common of all structural elements as evidenced

More information

MECHANICS OF MATERIALS Design of a Transmission Shaft

MECHANICS OF MATERIALS Design of a Transmission Shaft Design of a Transmission Shaft If power is transferred to and from the shaft by gears or sprocket wheels, the shaft is subjected to transverse loading as well as shear loading. Normal stresses due to transverse

More information

PLAXIS. Scientific Manual

PLAXIS. Scientific Manual PLAXIS Scientific Manual 2016 Build 8122 TABLE OF CONTENTS TABLE OF CONTENTS 1 Introduction 5 2 Deformation theory 7 2.1 Basic equations of continuum deformation 7 2.2 Finite element discretisation 8 2.3

More information

2C9 Design for seismic and climate changes. Jiří Máca

2C9 Design for seismic and climate changes. Jiří Máca 2C9 Design for seismic and climate changes Jiří Máca List of lectures 1. Elements of seismology and seismicity I 2. Elements of seismology and seismicity II 3. Dynamic analysis of single-degree-of-freedom

More information

CIVL 7/8117 Chapter 4 - Development of Beam Equations - Part 2 1/34. Chapter 4b Development of Beam Equations. Learning Objectives

CIVL 7/8117 Chapter 4 - Development of Beam Equations - Part 2 1/34. Chapter 4b Development of Beam Equations. Learning Objectives CIV 7/87 Chapter 4 - Development of Beam Equations - Part /4 Chapter 4b Development of Beam Equations earning Objectives To introduce the work-equivalence method for replacing distributed loading by a

More information

Bilinear Quadrilateral (Q4): CQUAD4 in GENESIS

Bilinear Quadrilateral (Q4): CQUAD4 in GENESIS Bilinear Quadrilateral (Q4): CQUAD4 in GENESIS The Q4 element has four nodes and eight nodal dof. The shape can be any quadrilateral; we ll concentrate on a rectangle now. The displacement field in terms

More information

ME FINITE ELEMENT ANALYSIS FORMULAS

ME FINITE ELEMENT ANALYSIS FORMULAS ME 2353 - FINITE ELEMENT ANALYSIS FORMULAS UNIT I FINITE ELEMENT FORMULATION OF BOUNDARY VALUE PROBLEMS 01. Global Equation for Force Vector, {F} = [K] {u} {F} = Global Force Vector [K] = Global Stiffness

More information

Mechanics of Inflatable Fabric Beams

Mechanics of Inflatable Fabric Beams Copyright c 2008 ICCES ICCES, vol.5, no.2, pp.93-98 Mechanics of Inflatable Fabric Beams C. Wielgosz 1,J.C.Thomas 1,A.LeVan 1 Summary In this paper we present a summary of the behaviour of inflatable fabric

More information

SEMM Mechanics PhD Preliminary Exam Spring Consider a two-dimensional rigid motion, whose displacement field is given by

SEMM Mechanics PhD Preliminary Exam Spring Consider a two-dimensional rigid motion, whose displacement field is given by SEMM Mechanics PhD Preliminary Exam Spring 2014 1. Consider a two-dimensional rigid motion, whose displacement field is given by u(x) = [cos(β)x 1 + sin(β)x 2 X 1 ]e 1 + [ sin(β)x 1 + cos(β)x 2 X 2 ]e

More information

A short review of continuum mechanics

A short review of continuum mechanics A short review of continuum mechanics Professor Anette M. Karlsson, Department of Mechanical ngineering, UD September, 006 This is a short and arbitrary review of continuum mechanics. Most of this material

More information

ENG202 Statics Lecture 16, Section 7.1

ENG202 Statics Lecture 16, Section 7.1 ENG202 Statics Lecture 16, Section 7.1 Internal Forces Developed in Structural Members - Design of any structural member requires an investigation of the loading acting within the member in order to be

More information

ENGN2340 Final Project: Implementation of a Euler-Bernuolli Beam Element Michael Monn

ENGN2340 Final Project: Implementation of a Euler-Bernuolli Beam Element Michael Monn ENGN234 Final Project: Implementation of a Euler-Bernuolli Beam Element Michael Monn 12/11/13 Problem Definition and Shape Functions Although there exist many analytical solutions to the Euler-Bernuolli

More information

Slender Structures Load carrying principles

Slender Structures Load carrying principles Slender Structures Load carrying principles Continuously Elastic Supported (basic) Cases: Etension, shear Euler-Bernoulli beam (Winkler 1867) v2017-2 Hans Welleman 1 Content (preliminary schedule) Basic

More information

M.S Comprehensive Examination Analysis

M.S Comprehensive Examination Analysis UNIVERSITY OF CALIFORNIA, BERKELEY Spring Semester 2014 Dept. of Civil and Environmental Engineering Structural Engineering, Mechanics and Materials Name:......................................... M.S Comprehensive

More information

Basic concepts to start Mechanics of Materials

Basic concepts to start Mechanics of Materials Basic concepts to start Mechanics of Materials Georges Cailletaud Centre des Matériaux Ecole des Mines de Paris/CNRS Notations Notations (maths) (1/2) A vector v (element of a vectorial space) can be seen

More information

Geometric Stiffness Effects in 2D and 3D Frames

Geometric Stiffness Effects in 2D and 3D Frames Geometric Stiffness Effects in D and 3D Frames CEE 41. Matrix Structural Analsis Department of Civil and Environmental Engineering Duke Universit Henri Gavin Fall, 1 In situations in which deformations

More information

FLEXIBILITY METHOD FOR INDETERMINATE FRAMES

FLEXIBILITY METHOD FOR INDETERMINATE FRAMES UNIT - I FLEXIBILITY METHOD FOR INDETERMINATE FRAMES 1. What is meant by indeterminate structures? Structures that do not satisfy the conditions of equilibrium are called indeterminate structure. These

More information

a x Questions on Classical Solutions 1. Consider an infinite linear elastic plate with a hole as shown. Uniform shear stress

a x Questions on Classical Solutions 1. Consider an infinite linear elastic plate with a hole as shown. Uniform shear stress Questions on Classical Solutions. Consider an infinite linear elastic plate with a hole as shown. Uniform shear stress σ xy = T is applied at infinity. Determine the value of the stress σ θθ on the edge

More information

A HIGHER-ORDER BEAM THEORY FOR COMPOSITE BOX BEAMS

A HIGHER-ORDER BEAM THEORY FOR COMPOSITE BOX BEAMS A HIGHER-ORDER BEAM THEORY FOR COMPOSITE BOX BEAMS A. Kroker, W. Becker TU Darmstadt, Department of Mechanical Engineering, Chair of Structural Mechanics Hochschulstr. 1, D-64289 Darmstadt, Germany kroker@mechanik.tu-darmstadt.de,

More information

FIXED BEAMS IN BENDING

FIXED BEAMS IN BENDING FIXED BEAMS IN BENDING INTRODUCTION Fixed or built-in beams are commonly used in building construction because they possess high rigidity in comparison to simply supported beams. When a simply supported

More information

Introduction to Finite Element Method. Dr. Aamer Haque

Introduction to Finite Element Method. Dr. Aamer Haque Introduction to Finite Element Method 4 th Order Beam Equation Dr. Aamer Haque http://math.iit.edu/~ahaque6 ahaque7@iit.edu Illinois Institute of Technology July 1, 009 Outline Euler-Bernoulli Beams Assumptions

More information

M5 Simple Beam Theory (continued)

M5 Simple Beam Theory (continued) M5 Simple Beam Theory (continued) Reading: Crandall, Dahl and Lardner 7.-7.6 In the previous lecture we had reached the point of obtaining 5 equations, 5 unknowns by application of equations of elasticity

More information

Methods of Analysis. Force or Flexibility Method

Methods of Analysis. Force or Flexibility Method INTRODUCTION: The structural analysis is a mathematical process by which the response of a structure to specified loads is determined. This response is measured by determining the internal forces or stresses

More information

Shafts: Torsion of Circular Shafts Reading: Crandall, Dahl and Lardner 6.2, 6.3

Shafts: Torsion of Circular Shafts Reading: Crandall, Dahl and Lardner 6.2, 6.3 M9 Shafts: Torsion of Circular Shafts Reading: Crandall, Dahl and Lardner 6., 6.3 A shaft is a structural member which is long and slender and subject to a torque (moment) acting about its long axis. We

More information

Lecture Kollbruner Section 5.2 Characteristics of Thin Walled Sections and.. Kollbruner Section 5.3 Bending without Twist

Lecture Kollbruner Section 5.2 Characteristics of Thin Walled Sections and.. Kollbruner Section 5.3 Bending without Twist Lecture 3-003 Kollbruner Section 5. Characteristics of Thin Walled Sections and.. Kollbruner Section 5.3 Bending without Twist thin walled => (cross section shape arbitrary and thickness can vary) axial

More information

UNIT II SLOPE DEFLECION AND MOMENT DISTRIBUTION METHOD

UNIT II SLOPE DEFLECION AND MOMENT DISTRIBUTION METHOD SIDDHARTH GROUP OF INSTITUTIONS :: PUTTUR Siddharth Nagar, Narayanavanam Road 517583 QUESTION BANK (DESCRIPTIVE) Subject with Code : SA-II (13A01505) Year & Sem: III-B.Tech & I-Sem Course & Branch: B.Tech

More information

1 Bending of beams Mindlin theory

1 Bending of beams Mindlin theory 1 BENDNG OF BEAMS MNDLN THEORY 1 1 Bending of beams Mindlin theory Cross-section kinematics assumptions Distributed load acts in the xz plane, which is also a plane of symmetry of a body Ω v(x = 0 m Vertical

More information

12. Stresses and Strains

12. Stresses and Strains 12. Stresses and Strains Finite Element Method Differential Equation Weak Formulation Approximating Functions Weighted Residuals FEM - Formulation Classification of Problems Scalar Vector 1-D T(x) u(x)

More information

Chapter 2: Deflections of Structures

Chapter 2: Deflections of Structures Chapter 2: Deflections of Structures Fig. 4.1. (Fig. 2.1.) ASTU, Dept. of C Eng., Prepared by: Melkamu E. Page 1 (2.1) (4.1) (2.2) Fig.4.2 Fig.2.2 ASTU, Dept. of C Eng., Prepared by: Melkamu E. Page 2

More information

Rigid Pavement Mechanics. Curling Stresses

Rigid Pavement Mechanics. Curling Stresses Rigid Pavement Mechanics Curling Stresses Major Distress Conditions Cracking Bottom-up transverse cracks Top-down transverse cracks Longitudinal cracks Corner breaks Punchouts (CRCP) 2 Major Distress Conditions

More information

Comb resonator design (2)

Comb resonator design (2) Lecture 6: Comb resonator design () -Intro Intro. to Mechanics of Materials School of Electrical l Engineering i and Computer Science, Seoul National University Nano/Micro Systems & Controls Laboratory

More information

Chapter 7 FORCES IN BEAMS AND CABLES

Chapter 7 FORCES IN BEAMS AND CABLES hapter 7 FORES IN BEAMS AN ABLES onsider a straight two-force member AB subjected at A and B to equal and opposite forces F and -F directed along AB. utting the member AB at and drawing the free-body B

More information

Basic Energy Principles in Stiffness Analysis

Basic Energy Principles in Stiffness Analysis Basic Energy Principles in Stiffness Analysis Stress-Strain Relations The application of any theory requires knowledge of the physical properties of the material(s) comprising the structure. We are limiting

More information

Portal Frame Calculations Lateral Loads

Portal Frame Calculations Lateral Loads Portal Frame Calculations Lateral Loads Consider the following multi-story frame: The portal method makes several assumptions about the internal forces of the columns and beams in a rigid frame: 1) Inflection

More information

Lecture Slides. Chapter 4. Deflection and Stiffness. The McGraw-Hill Companies 2012

Lecture Slides. Chapter 4. Deflection and Stiffness. The McGraw-Hill Companies 2012 Lecture Slides Chapter 4 Deflection and Stiffness The McGraw-Hill Companies 2012 Chapter Outline Force vs Deflection Elasticity property of a material that enables it to regain its original configuration

More information

Mechanics PhD Preliminary Spring 2017

Mechanics PhD Preliminary Spring 2017 Mechanics PhD Preliminary Spring 2017 1. (10 points) Consider a body Ω that is assembled by gluing together two separate bodies along a flat interface. The normal vector to the interface is given by n

More information

L2. Bending of beams: Normal stresses, PNA: , CCSM: chap 9.1-2

L2. Bending of beams: Normal stresses, PNA: , CCSM: chap 9.1-2 L2. Bending of beams: ormal stresses, P: 233-239, CCSM: chap 9.1-2 àcoordinate system üship coordinate system übeam coordinate system üstress resultats - Section forces üstress - strain tensors (recall

More information

7. Hierarchical modeling examples

7. Hierarchical modeling examples 7. Hierarchical modeling examples The objective of this chapter is to apply the hierarchical modeling approach discussed in Chapter 1 to three selected problems using the mathematical models studied in

More information

Strength of Materials Prof. Dr. Suraj Prakash Harsha Mechanical and Industrial Engineering Department Indian Institute of Technology, Roorkee

Strength of Materials Prof. Dr. Suraj Prakash Harsha Mechanical and Industrial Engineering Department Indian Institute of Technology, Roorkee Strength of Materials Prof. Dr. Suraj Prakash Harsha Mechanical and Industrial Engineering Department Indian Institute of Technology, Roorkee Lecture - 28 Hi, this is Dr. S. P. Harsha from Mechanical and

More information

Elasto-plastic concrete beam analysis by 1-dimensional Finite Element Method

Elasto-plastic concrete beam analysis by 1-dimensional Finite Element Method AALBORG UNIVERSITY MASTER S THESIS Elasto-plastic concrete beam analysis by 1-dimensional Finite Element Method Authors: Niels F. Overgaard Martin B. Andreasen Supervisors: Johan Clausen Lars V. Andersen

More information

Chapter 7: Internal Forces

Chapter 7: Internal Forces Chapter 7: Internal Forces Chapter Objectives To show how to use the method of sections for determining the internal loadings in a member. To generalize this procedure by formulating equations that can

More information

Lecture 11: The Stiffness Method. Introduction

Lecture 11: The Stiffness Method. Introduction Introduction Although the mathematical formulation of the flexibility and stiffness methods are similar, the physical concepts involved are different. We found that in the flexibility method, the unknowns

More information

Lecture notes Models of Mechanics

Lecture notes Models of Mechanics Lecture notes Models of Mechanics Anders Klarbring Division of Mechanics, Linköping University, Sweden Lecture 7: Small deformation theories Klarbring (Mechanics, LiU) Lecture notes Linköping 2012 1 /

More information

THE USE OF DYNAMIC RELAXATION TO SOLVE THE DIFFERENTIAL EQUATION DESCRIBING THE SHAPE OF THE TALLEST POSSIBLE BUILDING

THE USE OF DYNAMIC RELAXATION TO SOLVE THE DIFFERENTIAL EQUATION DESCRIBING THE SHAPE OF THE TALLEST POSSIBLE BUILDING VII International Conference on Textile Composites and Inflatable Structures STRUCTURAL MEMBRANES 2015 E. Oñate, K.-U.Bletzinger and B. Kröplin (Eds) THE USE OF DYNAMIC RELAXATION TO SOLVE THE DIFFERENTIAL

More information

Lecture 4: PRELIMINARY CONCEPTS OF STRUCTURAL ANALYSIS. Introduction

Lecture 4: PRELIMINARY CONCEPTS OF STRUCTURAL ANALYSIS. Introduction Introduction In this class we will focus on the structural analysis of framed structures. We will learn about the flexibility method first, and then learn how to use the primary analytical tools associated

More information

UNSYMMETRICAL BENDING

UNSYMMETRICAL BENDING UNSYMMETRICAL BENDING The general bending stress equation for elastic, homogeneous beams is given as (II.1) where Mx and My are the bending moments about the x and y centroidal axes, respectively. Ix and

More information

Chapter 7: Bending and Shear in Simple Beams

Chapter 7: Bending and Shear in Simple Beams Chapter 7: Bending and Shear in Simple Beams Introduction A beam is a long, slender structural member that resists loads that are generally applied transverse (perpendicular) to its longitudinal axis.

More information

14. *14.8 CASTIGLIANO S THEOREM

14. *14.8 CASTIGLIANO S THEOREM *14.8 CASTIGLIANO S THEOREM Consider a body of arbitrary shape subjected to a series of n forces P 1, P 2, P n. Since external work done by forces is equal to internal strain energy stored in body, by

More information

Using the finite element method of structural analysis, determine displacements at nodes 1 and 2.

Using the finite element method of structural analysis, determine displacements at nodes 1 and 2. Question 1 A pin-jointed plane frame, shown in Figure Q1, is fixed to rigid supports at nodes and 4 to prevent their nodal displacements. The frame is loaded at nodes 1 and by a horizontal and a vertical

More information

Due Tuesday, September 21 st, 12:00 midnight

Due Tuesday, September 21 st, 12:00 midnight Due Tuesday, September 21 st, 12:00 midnight The first problem discusses a plane truss with inclined supports. You will need to modify the MatLab software from homework 1. The next 4 problems consider

More information

Problem d d d B C E D. 0.8d. Additional lecturebook examples 29 ME 323

Problem d d d B C E D. 0.8d. Additional lecturebook examples 29 ME 323 Problem 9.1 Two beam segments, AC and CD, are connected together at C by a frictionless pin. Segment CD is cantilevered from a rigid support at D, and segment AC has a roller support at A. a) Determine

More information

Module 3. Analysis of Statically Indeterminate Structures by the Displacement Method

Module 3. Analysis of Statically Indeterminate Structures by the Displacement Method odule 3 Analysis of Statically Indeterminate Structures by the Displacement ethod Lesson 14 The Slope-Deflection ethod: An Introduction Introduction As pointed out earlier, there are two distinct methods

More information

Numerical and Analytical Approach of Thermo-Mechanical Stresses in FGM Beams

Numerical and Analytical Approach of Thermo-Mechanical Stresses in FGM Beams Numerical and Analytical Approach of Thermo-Mechanical Stresses in FGM Beams Fatemeh Farhatnia, Gholam-Ali Sharifi and Saeid Rasouli Abstract In this paper, thermo-mechanical stress distribution has been

More information

1 Static Plastic Behaviour of Beams

1 Static Plastic Behaviour of Beams 1 Static Plastic Behaviour of Beams 1.1 Introduction Many ductile materials which are used in engineering practice have a considerable reserve capacity beyond the initial yield condition. The uniaxial

More information