Basic Energy Principles in Stiffness Analysis

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Basic Energy Principles in Stiffness Analysis"

Transcription

1 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 our attention to linear elastic structural response. Further assuming that the material is homogenous and isotropic, we only need to know two of the following three material constants: E = Elastic (or Young s) modulus G = Shear modulus = Poisson ratio Normally, the shear modulus is expressed in terms of the elastic modulus and Poisson ratio as E G ( ) The most widely used civil engineering structural materials, steel and concrete, have uniaxial stress-strain diagrams of the types shown in Fig.. Mild steels yield Fig. : Typical Stress () Strain (e) Curves for (a) Steel and (b) Concrete with a pronounced permanent elongation at a stress ym (Fig.a). High strength steels yield gradually, which requires an arbitrary definition of its yield strength yh, offset criterion. Yield strengths for steel vary from less than 5 MPa to more than 7 MPa. For practical purposes, steel behaves as an ideal material in both tension and compression below the yield or buckling stress. The elastic modulus and Poisson ratio for steel are always close to, MPa and.3, respectively. Concrete is less predictable, but under short-duration compressive stress not greater than u /3 u /, its behavior is reasonably linear such as the commonly used.% 3 4

2 (Fig. b in which typical values for u are: 3 MPa u 5 MPa). An elastic modulus of E =, MPa and Poisson ratio of =.5 are typical for concrete. In using concrete for analysis, the ACI code specifies using the gross cross area properties to perform analyses to determine the force distributions in frame structures, i.e., ignore the reinforcing steel and tension cracking in calculating the force distributions. 5 Work and Energy The principle of conservation of energy is fundamentally important in structural analysis. This principle, expressed as energy or work balance, is applicable to both rigid and deformable structures. Rigid structures only require multiplying the external forces by the respective displacements. Deformable structures also require the summation of the internal stresses acting through the 6 respective deformations. Internal work is called strain energy and must be accounted for in the energy balance. The work dw of a force F acting through a change in displacement d in the direction of F is dw Fd () Over, the total work is W Fd () imiting attention to gradually applied forces, i.e., ignoring inertial forces caused by dynamic loads, and linear elastic response leads to W Fdk d k F F F k (3) 7 8

3 Expanding to a vector of forces and displacements leads to W F { } (4) The special case shown in the right figure: u W F x Fxu v where U= strain energy for the element. Equation (5) is a homogeneous, quadratic polynomial in terms of the local coordinate element displacements {u} or global coordinate element displacement {v}. Expanding (4) for a single element ({F} = [k] {u} or {F} = [K] {v}): W u [k]{u} v [K]{v} U (5) 9 Principle of Virtual Displacements to constructing stiffness equations. In prior chapters we established The principle of virtual the relationships of framework displacements can be stated as analysis directly utilizing the basic If a deformable structure is in conditions of equilibrium and equilibrium and remains in displacement continuity. Henceforth, we will use energy principles, equilibrium while it is subject to a virtual distortion, the external specifically the principle of virtual virtual work done by the external displacements since it permits forces acting on the structure is mathematical manipulations that equal to the internal virtual work are not possible with direct done by the stress resultants. procedures. We restrict our attention to virtual displacements Recall: virtual imaginary, not real, or in essence but not in fact since this principle is applicable 3

4 The principle of virtual displacements is expressed mathematically as W ext = W int (6) F F W ext W where W ext = F = external virtual work (shaded blue area in the figure) and W int = internal virtual work. 3 Equation (6) is based on the conservation of energy principle, i.e. the work done by the external forces going through a virtual displacement equals the work done by the internal forces due to the same virtual displacement. The external virtual work can be generalized to a system of forces as s ext i i (7) i W qdx ( )P 4 The internal virtual work (W int ) is a function of the structure type. Since this course focuses on frame members, only axial and bending deformations will be considered. Axial Deformation Consider the axial force system shown in Fig.. The differential internal virtual work (dw int ) is d( u) dw int Fdx x (8a) dx where u = virtual axial displacement and F x = real axial force. Recalling from your mechanics of materials class that axial strain e x = du/dx and the axial force F x = x A (axial stress times area), (8a) can be rewritten as dw int ex x Adx (8b) Fig. : Axial Deformation 5 Integrating (8b) over the length of 6 the element and substituting 4

5 Hooke s law ( x = Ee x ) leads to W e e dx int x x d( u) du dx (9) dx dx For the beam bending (flexure) case (Fig. 3), the internal virtual work is Wint z Mz dx z EIz dx d ( v) d v () EI dx dx dx where v = virtual transverse displacement; z = d(v)/dx = virtual rotation; M z = real moment about the z-axis; z = d v/dx = curvature strain about the z-axis; and M z = EI k z. Fig. 3: Bending Deformation 7 8 NOTE: A difficulty in applying the principle of virtual displacements is that functions must be assumed or developed for the real and virtual displacement functions in (9) and (). Development of these expressions will follow finite element mechanics, which is covered in a later section. 9 Analytical Solutions Using Principle of Virtual Displacements Consider the simple axial force structure shown in Fig. 4. The real x, u F x, u Fig. 4: Axial Deformation Structure displacement u: u = x/ u The real strain is e x = du/dx = u / Imposing a virtual displacement 5

6 u results in an external virtual work of W ext = u F x In order to calculate the internal virtual work d( u) du Wint dx dx dx expressions for u and u over the length of the axial deformation structure must be assumed. We will consistently assume the real displacement u: u = (x/) u We will consider various expressions for the virtual displacement to demonstrate the principle of virtual displacements. First, consider u = (x/) u The internal virtual work: u u int W dx u u Equating the external and internal virtual works gives u F x = u (/) u or u = F x / which is exact. Consider next: u = (x/) u The internal virtual work: u u int W xdx u u Which again gives the exact solution: u = F x / astly, consider: u = u sin(x/) 3 The internal virtual work: u x u Wint cos dx u u Which again gives the exact solution: u = F x / These three virtual displacement expressions all resulted in an exact solution since the real displacement solution was exact. If the chosen real displacements 4 6

7 correspond to stresses that identically satisfy the conditions of equilibrium, any form of admissible virtual displacement will suffice to produce the exact solution. Notice the adjective admissible in front of virtual displacement. Admissible means that the chosen function is physically continuous and satisfies all essential boundary conditions, i.e., is appropriately zero at all A = A (-x/) Consider next the nonprismatic axial deformation structure of Fig. 5. We will repeat the process considered for Fig. 4 with reference to the geometry of Fig. 5. Considering the first case: u = (x/) u 5 6 supports. F x x, u Fig. 5: Nonprismatic Axial Deformation Structure u x u int W A ( )dx E 3 W u u 4 int Equating the external and internal virtual works leads to 4Fx u 3 Considering the second virtual displacement expression: u = (x/) u leads to u x u int W x dx u u 3 7 Equating the external and internal virtual works leads to 3Fx u Considering the third virtual displacement expression: u = u sin(x/) leads to u x x u int W cos dx u u (.88) u u 8 7

8 Again, equating the external and internal virtual works leads to u.fx NOTE: None of the three solutions match. This is because neither the real or virtual displacements are exact. However, we produced three good approximate solutions. The exact solution for Fig. 5 is u.387fx The principle of virtual displacements has its greatest application in producing approximate solutions. The standard procedure is to adopt a virtual displacement of the same form as the real displacement. Adopting different forms for the real and virtual displacements can lead to unsymmetric stiffness matrices. 9 3 Special Transformations in Analysis Congruent Transformation A matrix triple product in which the pre-multiplying matrix is the transpose of the post-multiplying matrix, e.g. T T [C] [A] [B][A] or [D] [A][B][A] Significance of the transformation is that [C] and [D] will each be symmetric if [B] is symmetric, which is one of the reasons all our stiffness Contragradience Principal If one transformation is known, e.g., the local to global displacements, the force transformation will be transpose of the displacement transformation provided both sets of forces and displacements are conjugate and vice versa. Such a transformation is known as contragradient (or contragredient) under the stipulated conditions of conjugacy. Conjugate simply means that the force-displacement pair only produce work in the matrices were symmetric. 3 3 direction of the displacement. 8

9 For linear analysis, this is always the case when using orthogonal coordinate systems. A good example are the coordinate transformations for a truss member (7.) in which the transformation matrices are rectangular: {u a } = [T a ] {v a } T {F a} [T a] {Q a} cos sin [T a ] cos sin 33 9

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

Flexure: Behavior and Nominal Strength of Beam Sections

Flexure: Behavior and Nominal Strength of Beam Sections 4 5000 4000 (increased d ) (increased f (increased A s or f y ) c or b) Flexure: Behavior and Nominal Strength of Beam Sections Moment (kip-in.) 3000 2000 1000 0 0 (basic) (A s 0.5A s ) 0.0005 0.001 0.0015

More information

Civil Engineering Design (1) Design of Reinforced Concrete Columns 2006/7

Civil Engineering Design (1) Design of Reinforced Concrete Columns 2006/7 Civil Engineering Design (1) Design of Reinforced Concrete Columns 2006/7 Dr. Colin Caprani, Chartered Engineer 1 Contents 1. Introduction... 3 1.1 Background... 3 1.2 Failure Modes... 5 1.3 Design Aspects...

More information

CONSTITUTIVE RELATIONS FOR LINEAR ELASTIC SOLIDS

CONSTITUTIVE RELATIONS FOR LINEAR ELASTIC SOLIDS Chapter 9 CONSTITUTIV RLATIONS FOR LINAR LASTIC SOLIDS Figure 9.1: Hooke memorial window, St. Helen s, Bishopsgate, City of London 211 212 CHAPTR 9. CONSTITUTIV RLATIONS FOR LINAR LASTIC SOLIDS 9.1 Mechanical

More information

2. Rigid bar ABC supports a weight of W = 50 kn. Bar ABC is pinned at A and supported at B by rod (1). What is the axial force in rod (1)?

2. Rigid bar ABC supports a weight of W = 50 kn. Bar ABC is pinned at A and supported at B by rod (1). What is the axial force in rod (1)? IDE 110 S08 Test 1 Name: 1. Determine the internal axial forces in segments (1), (2) and (3). (a) N 1 = kn (b) N 2 = kn (c) N 3 = kn 2. Rigid bar ABC supports a weight of W = 50 kn. Bar ABC is pinned at

More information

International Journal of Advanced Engineering Technology E-ISSN

International Journal of Advanced Engineering Technology E-ISSN Research Article INTEGRATED FORCE METHOD FOR FIBER REINFORCED COMPOSITE PLATE BENDING PROBLEMS Doiphode G. S., Patodi S. C.* Address for Correspondence Assistant Professor, Applied Mechanics Department,

More information

Flexural-Torsional Buckling of General Cold-Formed Steel Columns with Unequal Unbraced Lengths

Flexural-Torsional Buckling of General Cold-Formed Steel Columns with Unequal Unbraced Lengths Proceedings of the Annual Stability Conference Structural Stability Research Council San Antonio, Texas, March 21-24, 2017 Flexural-Torsional Buckling of General Cold-Formed Steel Columns with Unequal

More information

SERVICEABILITY LIMIT STATE DESIGN

SERVICEABILITY LIMIT STATE DESIGN CHAPTER 11 SERVICEABILITY LIMIT STATE DESIGN Article 49. Cracking Limit State 49.1 General considerations In the case of verifications relating to Cracking Limit State, the effects of actions comprise

More information

UNIT I SIMPLE STRESSES AND STRAINS

UNIT I SIMPLE STRESSES AND STRAINS Subject with Code : SM-1(15A01303) Year & Sem: II-B.Tech & I-Sem SIDDHARTH GROUP OF INSTITUTIONS :: PUTTUR Siddharth Nagar, Narayanavanam Road 517583 QUESTION BANK (DESCRIPTIVE) UNIT I SIMPLE STRESSES

More information

Introduction to Engineering Materials ENGR2000. Dr. Coates

Introduction to Engineering Materials ENGR2000. Dr. Coates Introduction to Engineering Materials ENGR2 Chapter 6: Mechanical Properties of Metals Dr. Coates 6.2 Concepts of Stress and Strain tension compression shear torsion Tension Tests The specimen is deformed

More information

2. Mechanics of Materials: Strain. 3. Hookes's Law

2. Mechanics of Materials: Strain. 3. Hookes's Law Mechanics of Materials Course: WB3413, Dredging Processes 1 Fundamental Theory Required for Sand, Clay and Rock Cutting 1. Mechanics of Materials: Stress 1. Introduction 2. Plane Stress and Coordinate

More information

Question 1. Ignore bottom surface. Solution: Design variables: X = (R, H) Objective function: maximize volume, πr 2 H OR Minimize, f(x) = πr 2 H

Question 1. Ignore bottom surface. Solution: Design variables: X = (R, H) Objective function: maximize volume, πr 2 H OR Minimize, f(x) = πr 2 H Question 1 (Problem 2.3 of rora s Introduction to Optimum Design): Design a beer mug, shown in fig, to hold as much beer as possible. The height and radius of the mug should be not more than 20 cm. The

More information

Tuesday, February 11, Chapter 3. Load and Stress Analysis. Dr. Mohammad Suliman Abuhaiba, PE

Tuesday, February 11, Chapter 3. Load and Stress Analysis. Dr. Mohammad Suliman Abuhaiba, PE 1 Chapter 3 Load and Stress Analysis 2 Chapter Outline Equilibrium & Free-Body Diagrams Shear Force and Bending Moments in Beams Singularity Functions Stress Cartesian Stress Components Mohr s Circle for

More information

Parametric analysis and torsion design charts for axially restrained RC beams

Parametric analysis and torsion design charts for axially restrained RC beams Structural Engineering and Mechanics, Vol. 55, No. 1 (2015) 1-27 DOI: http://dx.doi.org/10.12989/sem.2015.55.1.001 1 Parametric analysis and torsion design charts for axially restrained RC beams Luís F.A.

More information

Enhancing Prediction Accuracy In Sift Theory

Enhancing Prediction Accuracy In Sift Theory 18 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS Enhancing Prediction Accuracy In Sift Theory J. Wang 1 *, W. K. Chiu 1 Defence Science and Technology Organisation, Fishermans Bend, Australia, Department

More information

Bending of Simply Supported Isotropic and Composite Laminate Plates

Bending of Simply Supported Isotropic and Composite Laminate Plates Bending of Simply Supported Isotropic and Composite Laminate Plates Ernesto Gutierrez-Miravete 1 Isotropic Plates Consider simply a supported rectangular plate of isotropic material (length a, width b,

More information

Introduction to Continuous Systems. Continuous Systems. Strings, Torsional Rods and Beams.

Introduction to Continuous Systems. Continuous Systems. Strings, Torsional Rods and Beams. Outline of Continuous Systems. Introduction to Continuous Systems. Continuous Systems. Strings, Torsional Rods and Beams. Vibrations of Flexible Strings. Torsional Vibration of Rods. Bernoulli-Euler Beams.

More information

Chapter 7. Highlights:

Chapter 7. Highlights: Chapter 7 Highlights: 1. Understand the basic concepts of engineering stress and strain, yield strength, tensile strength, Young's(elastic) modulus, ductility, toughness, resilience, true stress and true

More information

SANDWICH COMPOSITE BEAMS for STRUCTURAL APPLICATIONS

SANDWICH COMPOSITE BEAMS for STRUCTURAL APPLICATIONS SANDWICH COMPOSITE BEAMS for STRUCTURAL APPLICATIONS de Aguiar, José M., josemaguiar@gmail.com Faculdade de Tecnologia de São Paulo, FATEC-SP Centro Estadual de Educação Tecnológica Paula Souza. CEETEPS

More information

Lecture 8: Flexibility Method. Example

Lecture 8: Flexibility Method. Example ecture 8: lexibility Method Example The plane frame shown at the left has fixed supports at A and C. The frame is acted upon by the vertical load P as shown. In the analysis account for both flexural and

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

Chapter 6: Mechanical Properties of Metals. Dr. Feras Fraige

Chapter 6: Mechanical Properties of Metals. Dr. Feras Fraige Chapter 6: Mechanical Properties of Metals Dr. Feras Fraige Stress and Strain Tension Compression Shear Torsion Elastic deformation Plastic Deformation Yield Strength Tensile Strength Ductility Toughness

More information

Beam Bending Stresses and Shear Stress

Beam Bending Stresses and Shear Stress Beam Bending Stresses and Shear Stress Notation: A = name or area Aweb = area o the web o a wide lange section b = width o a rectangle = total width o material at a horizontal section c = largest distance

More information

PLATE GIRDERS II. Load. Web plate Welds A Longitudinal elevation. Fig. 1 A typical Plate Girder

PLATE GIRDERS II. Load. Web plate Welds A Longitudinal elevation. Fig. 1 A typical Plate Girder 16 PLATE GIRDERS II 1.0 INTRODUCTION This chapter describes the current practice for the design of plate girders adopting meaningful simplifications of the equations derived in the chapter on Plate Girders

More information

RECURSIVE DIFFERENTIATION METHOD FOR BOUNDARY VALUE PROBLEMS: APPLICATION TO ANALYSIS OF A BEAM-COLUMN ON AN ELASTIC FOUNDATION

RECURSIVE DIFFERENTIATION METHOD FOR BOUNDARY VALUE PROBLEMS: APPLICATION TO ANALYSIS OF A BEAM-COLUMN ON AN ELASTIC FOUNDATION Journal of Theoretical and Applied Mechanics, Sofia, 2014, vol. 44, No. 2, pp. 57 70 RECURSIVE DIFFERENTIATION METHOD FOR BOUNDARY VALUE PROBLEMS: APPLICATION TO ANALYSIS OF A BEAM-COLUMN ON AN ELASTIC

More information

Behavior and Modeling of Existing Reinforced Concrete Columns

Behavior and Modeling of Existing Reinforced Concrete Columns Behavior and Modeling of Existing Reinforced Concrete Columns Kenneth J. Elwood University of British Columbia with contributions from Jose Pincheira, Univ of Wisconsin John Wallace, UCLA Questions? What

More information

COURSE STE6289 Modern Materials and Computations (Moderne materialer og beregninger 7.5 stp.)

COURSE STE6289 Modern Materials and Computations (Moderne materialer og beregninger 7.5 stp.) Narvik University College (Høgskolen i Narvik) EXAMINATION TASK COURSE STE6289 Modern Materials and Computations (Moderne materialer og beregninger 7.5 stp.) CLASS: Master students in Engineering Design

More information

Stiffness Matrices, Spring and Bar Elements

Stiffness Matrices, Spring and Bar Elements CHAPTER Stiffness Matrices, Spring and Bar Elements. INTRODUCTION The primary characteristics of a finite element are embodied in the element stiffness matrix. For a structural finite element, the stiffness

More information

ENCE 455 Design of Steel Structures. III. Compression Members

ENCE 455 Design of Steel Structures. III. Compression Members ENCE 455 Design of Steel Structures III. Compression Members C. C. Fu, Ph.D., P.E. Civil and Environmental Engineering Department University of Maryland Compression Members Following subjects are covered:

More information

CORRELATING OFF-AXIS TENSION TESTS TO SHEAR MODULUS OF WOOD-BASED PANELS

CORRELATING OFF-AXIS TENSION TESTS TO SHEAR MODULUS OF WOOD-BASED PANELS CORRELATING OFF-AXIS TENSION TESTS TO SHEAR MODULUS OF WOOD-BASED PANELS By Edmond P. Saliklis 1 and Robert H. Falk ABSTRACT: The weakness of existing relationships correlating off-axis modulus of elasticity

More information

STRUCTURAL ANALYSIS CHAPTER 2. Introduction

STRUCTURAL ANALYSIS CHAPTER 2. Introduction CHAPTER 2 STRUCTURAL ANALYSIS Introduction The primary purpose of structural analysis is to establish the distribution of internal forces and moments over the whole part of a structure and to identify

More information

Large deflection analysis of planar solids based on the Finite Particle Method

Large deflection analysis of planar solids based on the Finite Particle Method yuying@uiuc.edu 10 th US National Congress on Computational Mechanics Large deflection analysis of planar solids based on the Finite Particle Method 1, 2 Presenter: Ying Yu Advisors: Prof. Glaucio H. Paulino

More information

Unit 15 Shearing and Torsion (and Bending) of Shell Beams

Unit 15 Shearing and Torsion (and Bending) of Shell Beams Unit 15 Shearing and Torsion (and Bending) of Shell Beams Readings: Rivello Ch. 9, section 8.7 (again), section 7.6 T & G 126, 127 Paul A. Lagace, Ph.D. Professor of Aeronautics & Astronautics and Engineering

More information

MECHANICS OF SOLIDS Credit Hours: 6

MECHANICS OF SOLIDS Credit Hours: 6 MECHANICS OF SOLIDS Credit Hours: 6 Teaching Scheme Theory Tutorials Practical Total Credit Hours/week 4 0 6 6 Marks 00 0 50 50 6 A. Objective of the Course: Objectives of introducing this subject at second

More information

DETERMINING THE STRESS PATTERN IN THE HH RAILROAD TIES DUE TO DYNAMIC LOADS 1

DETERMINING THE STRESS PATTERN IN THE HH RAILROAD TIES DUE TO DYNAMIC LOADS 1 PERIODICA POLYTECHNICA SER. CIV. ENG. VOL. 46, NO. 1, PP. 125 148 (2002) DETERMINING THE STRESS PATTERN IN THE HH RAILROAD TIES DUE TO DYNAMIC LOADS 1 Nándor LIEGNER Department of Highway and Railway Engineering

More information

The 5rd International Conference on. COMEC OCTOBER 2013, Brasov, Romania

The 5rd International Conference on. COMEC OCTOBER 2013, Brasov, Romania The 5rd International Conference on Computational Mechanics and Virtual Engineering COMEC 2013 24 25 OCTOBER 2013, Brasov, Romania THEORETICAL STUDIES AND EXPERIMENTAL RESEARCH FOR THE INCREASE OF THE

More information

ACCURATE FREE VIBRATION ANALYSIS OF POINT SUPPORTED MINDLIN PLATES BY THE SUPERPOSITION METHOD

ACCURATE FREE VIBRATION ANALYSIS OF POINT SUPPORTED MINDLIN PLATES BY THE SUPERPOSITION METHOD Journal of Sound and Vibration (1999) 219(2), 265 277 Article No. jsvi.1998.1874, available online at http://www.idealibrary.com.on ACCURATE FREE VIBRATION ANALYSIS OF POINT SUPPORTED MINDLIN PLATES BY

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

Structural Analysis III Compatibility of Displacements & Principle of Superposition

Structural Analysis III Compatibility of Displacements & Principle of Superposition Structural Analysis III Compatibility of Displacements & Principle of Superposition 2007/8 Dr. Colin Caprani, Chartered Engineer 1 1. Introduction 1.1 Background In the case of 2-dimensional structures

More information

MECH 344/X Machine Element Design

MECH 344/X Machine Element Design 1 MECH 344/X Machine Element Design Time: M 14:45-17:30 Lecture 2 Contents of today's lecture Introduction to Static Stresses Axial, Shear and Torsional Loading Bending in Straight and Curved Beams Transverse

More information

Damage detection of damaged beam by constrained displacement curvature

Damage detection of damaged beam by constrained displacement curvature Journal of Mechanical Science and Technology Journal of Mechanical Science and Technology 22 (2008) 1111~1120 www.springerlink.com/content/1738-494x Damage detection of damaged beam by constrained displacement

More information

Design of reinforced concrete sections according to EN and EN

Design of reinforced concrete sections according to EN and EN Design of reinforced concrete sections according to EN 1992-1-1 and EN 1992-2 Validation Examples Brno, 21.10.2010 IDEA RS s.r.o. South Moravian Innovation Centre, U Vodarny 2a, 616 00 BRNO tel.: +420-511

More information

Available online at ScienceDirect. Procedia Engineering 172 (2017 )

Available online at  ScienceDirect. Procedia Engineering 172 (2017 ) Available online at www.sciencedirect.com ScienceDirect Procedia Engineering 172 (2017 ) 1093 1101 Modern Building Materials, Structures and Techniques, MBMST 2016 Iterative Methods of Beam-Structure Analysis

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

MECHANICAL PROPERTIES OF SOLIDS

MECHANICAL PROPERTIES OF SOLIDS Chapter Nine MECHANICAL PROPERTIES OF SOLIDS MCQ I 9.1 Modulus of rigidity of ideal liquids is (a) infinity. (b) zero. (c) unity. (d) some finite small non-zero constant value. 9. The maximum load a wire

More information

structural analysis Excessive beam deflection can be seen as a mode of failure.

structural analysis Excessive beam deflection can be seen as a mode of failure. Structure Analysis I Chapter 8 Deflections Introduction Calculation of deflections is an important part of structural analysis Excessive beam deflection can be seen as a mode of failure. Extensive glass

More information

Chapter 13 ELASTIC PROPERTIES OF MATERIALS

Chapter 13 ELASTIC PROPERTIES OF MATERIALS Physics Including Human Applications 280 Chapter 13 ELASTIC PROPERTIES OF MATERIALS GOALS When you have mastered the contents of this chapter, you will be able to achieve the following goals: Definitions

More information

4. SHAFTS. A shaft is an element used to transmit power and torque, and it can support

4. SHAFTS. A shaft is an element used to transmit power and torque, and it can support 4. SHAFTS A shaft is an element used to transmit power and torque, and it can support reverse bending (fatigue). Most shafts have circular cross sections, either solid or tubular. The difference between

More information

Understand basic stress-strain response of engineering materials.

Understand basic stress-strain response of engineering materials. Module 3 Constitutive quations Learning Objectives Understand basic stress-strain response of engineering materials. Quantify the linear elastic stress-strain response in terms of tensorial quantities

More information

Name (Print) ME Mechanics of Materials Exam # 3 Date: December 9, 2013 Time: 7:00 9:00 PM Location: EE 129 & EE170

Name (Print) ME Mechanics of Materials Exam # 3 Date: December 9, 2013 Time: 7:00 9:00 PM Location: EE 129 & EE170 Name (Print) (Last) (First) Instructions: ME 323 - Mechanics of Materials Exam # 3 Date: December 9, 2013 Time: 7:00 9:00 PM Location: EE 129 & EE170 Circle your lecturer s name and your class meeting

More information

Nonlinear Analysis of Reinforced Concrete Shells Subjected to Impact Loads

Nonlinear Analysis of Reinforced Concrete Shells Subjected to Impact Loads Transactions of the 7 th International Conference on Structural Mechanics in Reactor Technology (SMiRT 7) Prague, Czech Republic, August 7, 00 Paper # J0- Nonlinear Analysis of Reinforced Concrete Shells

More information

The Frictional Regime

The Frictional Regime The Frictional Regime Processes in Structural Geology & Tectonics Ben van der Pluijm WW Norton+Authors, unless noted otherwise 1/25/2016 10:08 AM We Discuss The Frictional Regime Processes of Brittle Deformation

More information

Mechanics of Materials MENG 270 Fall 2003 Exam 3 Time allowed: 90min. Q.1(a) Q.1 (b) Q.2 Q.3 Q.4 Total

Mechanics of Materials MENG 270 Fall 2003 Exam 3 Time allowed: 90min. Q.1(a) Q.1 (b) Q.2 Q.3 Q.4 Total Mechanics of Materials MENG 70 Fall 00 Eam Time allowed: 90min Name. Computer No. Q.(a) Q. (b) Q. Q. Q.4 Total Problem No. (a) [5Points] An air vessel is 500 mm average diameter and 0 mm thickness, the

More information

999 TOWN & COUNTRY ROAD ORANGE, CALIFORNIA TITLE PUSHOVER ANALYSIS EXAMPLE BY R. MATTHEWS DATE 5/21/01

999 TOWN & COUNTRY ROAD ORANGE, CALIFORNIA TITLE PUSHOVER ANALYSIS EXAMPLE BY R. MATTHEWS DATE 5/21/01 DESCRIPTION Nonlinear static (pushover) analysis will be performed on a railroad bridge bent using several methods to determine its ultimate lateral deflection capability. 1. SAP2000 Nonlinear with axial-moment

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

LINEAR AND NONLINEAR SHELL THEORY. Contents

LINEAR AND NONLINEAR SHELL THEORY. Contents LINEAR AND NONLINEAR SHELL THEORY Contents Strain-displacement relations for nonlinear shell theory Approximate strain-displacement relations: Linear theory Small strain theory Small strains & moderate

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

THE BENDING STIFFNESSES OF CORRUGATED BOARD

THE BENDING STIFFNESSES OF CORRUGATED BOARD AMD-Vol. 145/MD-Vol. 36, Mechanics of Cellulosic Materials ASME 1992 THE BENDING STIFFNESSES OF CORRUGATED BOARD S. Luo and J. C. Suhling Department of Mechanical Engineering Auburn University Auburn,

More information

Bridge deck modelling and design process for bridges

Bridge deck modelling and design process for bridges EU-Russia Regulatory Dialogue Construction Sector Subgroup 1 Bridge deck modelling and design process for bridges Application to a composite twin-girder bridge according to Eurocode 4 Laurence Davaine

More information

Analysis of the geometrical dependence of auxetic behavior in reentrant structures by finite elements

Analysis of the geometrical dependence of auxetic behavior in reentrant structures by finite elements Acta Mech. Sin. (2016) 32(2):295 300 DOI 10.1007/s10409-015-0534-2 RESEARCH PAPER Analysis of the geometrical dependence of auxetic behavior in reentrant structures by finite elements V. H. Carneiro 1

More information

Modeling of the Bending Stiffness of a Bimaterial Beam by the Approximation of One-Dimensional of Laminated Theory

Modeling of the Bending Stiffness of a Bimaterial Beam by the Approximation of One-Dimensional of Laminated Theory . Flores-Domínguez Int. Journal of Engineering Research and Applications RESEARCH ARTICLE OPEN ACCESS odeling of the Bending Stiffness of a Bimaterial Beam by the Approimation of One-Dimensional of Laminated

More information

Section Downloads. Section Downloads. Handouts & Slides can be printed. Other documents cannot be printed Course binders are available for purchase

Section Downloads. Section Downloads. Handouts & Slides can be printed. Other documents cannot be printed Course binders are available for purchase Level II: Section 04 Simplified Method (optional) Section Downloads Section Downloads Handouts & Slides can be printed Version.0 Other documents cannot be printed Course binders are available for purchase

More information

Pressure Vessels Stresses Under Combined Loads Yield Criteria for Ductile Materials and Fracture Criteria for Brittle Materials

Pressure Vessels Stresses Under Combined Loads Yield Criteria for Ductile Materials and Fracture Criteria for Brittle Materials Pressure Vessels Stresses Under Combined Loads Yield Criteria for Ductile Materials and Fracture Criteria for Brittle Materials Pressure Vessels: In the previous lectures we have discussed elements subjected

More information

Torsion Stresses in Tubes and Rods

Torsion Stresses in Tubes and Rods Torsion Stresses in Tubes and Rods This initial analysis is valid only for a restricted range of problem for which the assumptions are: Rod is initially straight. Rod twists without bending. Material is

More information

UNIT-II MOVING LOADS AND INFLUENCE LINES

UNIT-II MOVING LOADS AND INFLUENCE LINES UNIT-II MOVING LOADS AND INFLUENCE LINES Influence lines for reactions in statically determinate structures influence lines for member forces in pin-jointed frames Influence lines for shear force and bending

More information

COORDINATE TRANSFORMATIONS

COORDINATE TRANSFORMATIONS COORDINAE RANSFORMAIONS Members of a structural system are typically oriented in differing directions, e.g., Fig. 17.1. In order to perform an analysis, the element stiffness equations need to be expressed

More information

Chapter 13 Elastic Properties of Materials

Chapter 13 Elastic Properties of Materials Chapter 13 Elastic Properties of Materials GOALS When you have mastered the contents of this chapter, you will be able to achieve the following goals: Definitions Define each of the following terms, and

More information

DISTRIBUTION OF STRESS IN GROUND-SUPPORTED SLABS

DISTRIBUTION OF STRESS IN GROUND-SUPPORTED SLABS Structural Concrete Software System TN207_sog_stresses_10 122005 DISTRIBUTION OF STRESS IN GROUND-SUPPORTED SLABS Bijan O Aalami 1 This Technical Note describes the distribution of stress in ground-supported

More information

3-D Bernoulli Beams within Akantu

3-D Bernoulli Beams within Akantu 3-D Bernoulli Beams within Akantu Semester Project Fall 2011 Fabian Barras Professor Jean-François Molinari Supervisors Seyedeh Mohadeseh Taheri Mousavi Guillaume Anciaux Nicolas Richart Computational

More information

6. NON-LINEAR PSEUDO-STATIC ANALYSIS OF ADOBE WALLS

6. NON-LINEAR PSEUDO-STATIC ANALYSIS OF ADOBE WALLS 6. NON-LINEAR PSEUDO-STATIC ANALYSIS OF ADOBE WALLS Blondet et al. [25] carried out a cyclic test on an adobe wall to reproduce its seismic response and damage pattern under in-plane loads. The displacement

More information

Flexural Analysis of Deep Aluminum Beam

Flexural Analysis of Deep Aluminum Beam Journal of Soft Computing in Civil Engineering -1 (018) 71-84 journal homepage: http://www.jsoftcivil.com/ Fleural Analysis of Deep Aluminum Beam P. Kapdis 1, U. Kalwane 1, U. Salunkhe 1 and A. Dahake

More information

LAMINATION THEORY FOR THE STRENGTH OF FIBER COMPOSITE MATERIALS

LAMINATION THEORY FOR THE STRENGTH OF FIBER COMPOSITE MATERIALS XXII. LAMINATION THEORY FOR THE STRENGTH OF FIBER COMPOSITE MATERIALS Introduction The lamination theory for the elastic stiffness of fiber composite materials is the backbone of the entire field, it holds

More information

Theory and Analysis of Structures

Theory and Analysis of Structures 7 Theory and nalysis of Structures J.Y. Richard iew National University of Singapore N.E. Shanmugam National University of Singapore 7. Fundamental Principles oundary Conditions oads and Reactions Principle

More information

Continuum mechanics V. Constitutive equations. 1. Constitutive equation: definition and basic axioms

Continuum mechanics V. Constitutive equations. 1. Constitutive equation: definition and basic axioms Continuum mechanics office Math 0.107 ales.janka@unifr.ch http://perso.unifr.ch/ales.janka/mechanics Mars 16, 2011, Université de Fribourg 1. Constitutive equation: definition and basic axioms Constitutive

More information

UNIVERSITÀ DEGLI STUDI DI PADOVA DIPARTIMENTO DI INGEGNERIA CIVILE, EDILE ED AMBIENTALE CORSO DI LAUREA MAGISTRALE IN INGEGNERIA CIVILE

UNIVERSITÀ DEGLI STUDI DI PADOVA DIPARTIMENTO DI INGEGNERIA CIVILE, EDILE ED AMBIENTALE CORSO DI LAUREA MAGISTRALE IN INGEGNERIA CIVILE UNIVERSITÀ DEGLI STUDI DI PADOVA DIPARTIMENTO DI INGEGNERIA CIVILE, EDILE ED AMBIENTALE CORSO DI LAUREA MAGISTRALE IN INGEGNERIA CIVILE Tesi di laurea Magistrale in Ingegneria Civile Curriculum Strutture

More information

Finite Element Modeling of an Aluminum Tricycle Frame

Finite Element Modeling of an Aluminum Tricycle Frame Finite Element Modeling of an Aluminum Tricycle Frame A. Rodríguez, B. Chiné*, and J. A. Ramírez Costa Rica Institute of Technology, School of Materials Science and Engineering, Cartago, Costa Rica *Corresponding

More information

Chapter 3: Stress and Equilibrium of Deformable Bodies

Chapter 3: Stress and Equilibrium of Deformable Bodies Ch3-Stress-Equilibrium Page 1 Chapter 3: Stress and Equilibrium of Deformable Bodies When structures / deformable bodies are acted upon by loads, they build up internal forces (stresses) within them to

More information

Finite Element Analysis of Composite Laminate By Using ABDH Matrix(Stiffness Matrix)

Finite Element Analysis of Composite Laminate By Using ABDH Matrix(Stiffness Matrix) Finite Element Analysis of Composite Laminate By Using ABDH Matrix(Stiffness Matrix) Nikhil J. Chaudhari 1 Post Graduate Student Department of Mechanical Engineering Veermata Jijabai Technological Institute

More information

Fracture Mechanics of Composites with Residual Thermal Stresses

Fracture Mechanics of Composites with Residual Thermal Stresses J. A. Nairn Material Science & Engineering, University of Utah, Salt Lake City, Utah 84 Fracture Mechanics of Composites with Residual Thermal Stresses The problem of calculating the energy release rate

More information

Mathematics FINITE ELEMENT ANALYSIS AS COMPUTATION. What the textbooks don't teach you about finite element analysis. Chapter 3

Mathematics FINITE ELEMENT ANALYSIS AS COMPUTATION. What the textbooks don't teach you about finite element analysis. Chapter 3 Mathematics FINITE ELEMENT ANALYSIS AS COMPUTATION What the textbooks don't teach you about finite element analysis Chapter 3 Completeness and continuity: How to choose shape functions? Gangan Prathap

More information

1.050 Content overview Engineering Mechanics I Content overview. Selection of boundary conditions: Euler buckling.

1.050 Content overview Engineering Mechanics I Content overview. Selection of boundary conditions: Euler buckling. .050 Content overview.050 Engineering Mechanics I Lecture 34 How things fail and how to avoid it Additional notes energy approach I. Dimensional analysis. On monsters, mice and mushrooms Lectures -3. Similarity

More information

Drucker-Prager yield criterion application to study the behavior of CFRP confined concrete under compression

Drucker-Prager yield criterion application to study the behavior of CFRP confined concrete under compression XXXVII IAHS World ongress on Housing October 6 9, 00, Santander, Spain Drucker-Prager yield criterion application to study the behavior of FRP confined concrete under compression Salvador Ivorra, Ramón

More information

Lab Exercise #3: Torsion

Lab Exercise #3: Torsion Lab Exercise #3: Pre-lab assignment: Yes No Goals: 1. To evaluate the equations of angular displacement, shear stress, and shear strain for a shaft undergoing torsional stress. Principles: testing of round

More information

Aalto University School of Engineering

Aalto University School of Engineering Aalto University School of Engineering Kul-4.4 Ship Structural Design (P) ecture 6 - Response of Web-frames, Girders and Grillages Kul-4.4 Ship Structures Response ecture 5: Tertiary Response: Bending

More information

Generic Strategies to Implement Material Grading in Finite Element Methods for Isotropic and Anisotropic Materials

Generic Strategies to Implement Material Grading in Finite Element Methods for Isotropic and Anisotropic Materials University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Engineering Mechanics Dissertations & Theses Mechanical & Materials Engineering, Department of Winter 12-9-2011 Generic

More information

MATERIAL ELASTIC ANISOTROPIC command

MATERIAL ELASTIC ANISOTROPIC command MATERIAL ELASTIC ANISOTROPIC command.. Synopsis The MATERIAL ELASTIC ANISOTROPIC command is used to specify the parameters associated with an anisotropic linear elastic material idealization. Syntax The

More information

MASTER'S THESIS. Robustness Analysis of Welding Simulations by Using Design of Experiments. Pirjo Koivuniemi. Luleå University of Technology

MASTER'S THESIS. Robustness Analysis of Welding Simulations by Using Design of Experiments. Pirjo Koivuniemi. Luleå University of Technology MASTER'S THESIS 28:1 CIV Robustness Analysis of Welding Simulations by Using Design of Experiments Pirjo Koivuniemi Luleå University of Technology MSc Programmes in Engineering Engineering Physics Department

More information

ADVANCED DESIGN OF GLASS STRUCTURES

ADVANCED DESIGN OF GLASS STRUCTURES ADVANCED DESIGN OF GLASS STRUCTURES Lecture L13 Design of compressed members Viorel Ungureanu / Martina Eliášová European Erasmus Mundus Master Course Sustainable Constructions under Natural Hazards and

More information

Prof. Dr. Zahid Ahmad Siddiqi BEAM COLUMNS

Prof. Dr. Zahid Ahmad Siddiqi BEAM COLUMNS BEA COLUNS Beam columns are structural members that are subjected to a combination of bending and axial stresses. The structural behaviour resembles simultaneousl to that of a beam and a column. ajorit

More information

A RATIONAL BUCKLING MODEL FOR THROUGH GIRDERS

A RATIONAL BUCKLING MODEL FOR THROUGH GIRDERS A RATIONAL BUCKLING MODEL FOR THROUGH GIRDERS (Hasan Santoso) A RATIONAL BUCKLING MODEL FOR THROUGH GIRDERS Hasan Santoso Lecturer, Civil Engineering Department, Petra Christian University ABSTRACT Buckling

More information

Ch 7 Summary - POLYNOMIAL FUNCTIONS

Ch 7 Summary - POLYNOMIAL FUNCTIONS Ch 7 Summary - POLYNOMIAL FUNCTIONS 1. An open-top box is to be made by cutting congruent squares of side length x from the corners of a 8.5- by 11-inch sheet of cardboard and bending up the sides. a)

More information

ASSESSMENT OF DYNAMICALLY LOADED CRACKS IN FILLETS

ASSESSMENT OF DYNAMICALLY LOADED CRACKS IN FILLETS ASSESSMENT OF DNAMICALL LOADED CRACKS IN FILLETS Uwe Zencker, Linan Qiao, Bernhard Droste Federal Institute for Materials Research and Testing (BAM) 12200 Berlin, Germany e-mail: zencker@web.de Abstract

More information

Pacific Earthquake Engineering Research Center

Pacific Earthquake Engineering Research Center Pacific Earthquake Engineering Research Center Analytical and Experimental Study of Fiber-Reinforced Elastomeric Isolators James M. Kelly Shakhzod M. Takhirov Department of Civil and Environmental Engineering

More information

ARC 341 Structural Analysis II. Lecture 10: MM1.3 MM1.13

ARC 341 Structural Analysis II. Lecture 10: MM1.3 MM1.13 ARC241 Structural Analysis I Lecture 10: MM1.3 MM1.13 MM1.4) Analysis and Design MM1.5) Axial Loading; Normal Stress MM1.6) Shearing Stress MM1.7) Bearing Stress in Connections MM1.9) Method of Problem

More information

PROPOSED SATSANG HALL TECHNICAL REPORT

PROPOSED SATSANG HALL TECHNICAL REPORT PROPOSED SATSANG HALL - VERTICAL STRIP V1 1 ------------------------------------------------------------------------------ ADAPT CORPORATION STRUCTURAL CONCRETE SOFTWARE SYSTEM 1733 Woodside Road, Suite

More information

THE EC3 CLASSIFICATION OF JOINTS AND ALTERNATIVE PROPOSALS

THE EC3 CLASSIFICATION OF JOINTS AND ALTERNATIVE PROPOSALS EUROSTEEL 2002, Coimbra, 19-20 September 2002, p.987-996 THE EC3 CLASSIFICATION OF JOINTS AND ALTERNATIVE PROPOSALS Fernando C. T. Gomes 1 ABSTRACT The Eurocode 3 proposes a classification of beam-to-column

More information

= 50 ksi. The maximum beam deflection Δ max is not = R B. = 30 kips. Notes for Strength of Materials, ET 200

= 50 ksi. The maximum beam deflection Δ max is not = R B. = 30 kips. Notes for Strength of Materials, ET 200 Notes for Strength of Materials, ET 00 Steel Six Easy Steps Steel beam design is about selecting the lightest steel beam that will support the load without exceeding the bending strength or shear strength

More information

COMPUTER AIDED DESIGN IN CASE OF THE LAMINATED COMPOSITE MATERIALS

COMPUTER AIDED DESIGN IN CASE OF THE LAMINATED COMPOSITE MATERIALS 6 th International Conference Computational Mechanics and Virtual Engineering COMEC 15 15-16 October 15, Braşov, Romania COMPUER AIDED DESIGN IN CASE OF HE LAMINAED COMPOSIE MAERIALS Camelia Cerbu ransilvania

More information

Unified Quiz M4 May 7, 2008 M - PORTION

Unified Quiz M4 May 7, 2008 M - PORTION 9:00-10: 00 (last four digits) 32-141 Unified Quiz M4 May 7, 2008 M - PORTION Put the last four digits of your MIT ID # on each page of the exam. Read all questions carefully. Do all work on that question

More information