Basic Energy Principles in Stiffness Analysis


 Stephanie Loren Neal
 1 years ago
 Views:
Transcription
1 Basic Energy Principles in Stiffness Analysis StressStrain 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 stressstrain 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 shortduration 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 zaxis; z = d v/dx = curvature strain about the zaxis; 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 premultiplying matrix is the transpose of the postmultiplying 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 forcedisplacement 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
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 informationME Final Exam. PROBLEM NO. 4 Part A (2 points max.) M (x) y. z (neutral axis) beam crosssec+on. 20 kip ft. 0.2 ft. 10 ft. 0.1 ft.
ME 323  Final Exam Name December 15, 2015 Instructor (circle) PROEM NO. 4 Part A (2 points max.) Krousgrill 11:30AM12:20PM Ghosh 2:303:20PM Gonzalez 12:301:20PM Zhao 4:305:20PM M (x) y 20 kip ft 0.2
More informationConsider 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 informationCHAPTER 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 EulerBernoulli assumptions One of its dimensions
More informationCHAPTER 6 BENDING Part 1
Ishik University / Sulaimani Civil Engineering Department Mechanics of Materials CE 211 CHAPTER 6 BENDING Part 11 CHAPTER 6 Bending Outlines of this chapter: 6.1. Chapter Objectives 6.2. Shear and
More informationINTRODUCTION TO STRAIN
SIMPLE STRAIN INTRODUCTION TO STRAIN In general terms, Strain is a geometric quantity that measures the deformation of a body. There are two types of strain: normal strain: characterizes dimensional changes,
More informationMechanics in Energy Resources Engineering  Chapter 5 Stresses in Beams (Basic topics)
Week 7, 14 March Mechanics in Energy Resources Engineering  Chapter 5 Stresses in Beams (Basic topics) KiBok Min, PhD Assistant Professor Energy Resources Engineering i Seoul National University Shear
More informationCHAPTER 4: BENDING OF BEAMS
(74) CHAPTER 4: BENDING OF BEAMS This chapter will be devoted to the analysis of prismatic members subjected to equal and opposite couples M and M' acting in the same longitudinal plane. Such members are
More informationε 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 informationChapter 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 informationKINGS COLLEGE OF ENGINEERING DEPARTMENT OF MECHANICAL ENGINEERING QUESTION BANK. Subject code/name: ME2254/STRENGTH OF MATERIALS Year/Sem:II / IV
KINGS COLLEGE OF ENGINEERING DEPARTMENT OF MECHANICAL ENGINEERING QUESTION BANK Subject code/name: ME2254/STRENGTH OF MATERIALS Year/Sem:II / IV UNIT I STRESS, STRAIN DEFORMATION OF SOLIDS PART A (2 MARKS)
More informationChapter 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 informationChapter Two: Mechanical Properties of materials
Chapter Two: Mechanical Properties of materials Time : 16 Hours An important consideration in the choice of a material is the way it behave when subjected to force. The mechanical properties of a material
More informationSTRENGTH OF MATERIALSI. Unit1. Simple stresses and strains
STRENGTH OF MATERIALSI Unit1 Simple stresses and strains 1. What is the Principle of surveying 2. Define Magnetic, True & Arbitrary Meridians. 3. Mention different types of chains 4. Differentiate between
More information1 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 informationFinite 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 informationSTRESS STRAIN AND DEFORMATION OF SOLIDS, STATES OF STRESS
1 UNIT I STRESS STRAIN AND DEFORMATION OF SOLIDS, STATES OF STRESS 1. Define: Stress When an external force acts on a body, it undergoes deformation. At the same time the body resists deformation. The
More informationAdvanced Structural Analysis EGF Section Properties and Bending
Advanced Structural Analysis EGF316 3. Section Properties and Bending 3.1 Loads in beams When we analyse beams, we need to consider various types of loads acting on them, for example, axial forces, shear
More informationUNIVERSITY OF SASKATCHEWAN ME MECHANICS OF MATERIALS I FINAL EXAM DECEMBER 13, 2008 Professor A. Dolovich
UNIVERSITY OF SASKATCHEWAN ME 313.3 MECHANICS OF MATERIALS I FINAL EXAM DECEMBER 13, 2008 Professor A. Dolovich A CLOSED BOOK EXAMINATION TIME: 3 HOURS For Marker s Use Only LAST NAME (printed): FIRST
More informationChapter 12. Static Equilibrium and Elasticity
Chapter 12 Static Equilibrium and Elasticity Static Equilibrium Equilibrium implies that the object moves with both constant velocity and constant angular velocity relative to an observer in an inertial
More informationPURE BENDING. If a simply supported beam carries two point loads of 10 kn as shown in the following figure, pure bending occurs at segment BC.
BENDING STRESS The effect of a bending moment applied to a crosssection of a beam is to induce a state of stress across that section. These stresses are known as bending stresses and they act normally
More informationStresses in Curved Beam
Stresses in Curved Beam Consider a curved beam subjected to bending moment M b as shown in the figure. The distribution of stress in curved flexural member is determined by using the following assumptions:
More informationSymmetric 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 informationExternal work and internal work
External work and internal work Consider a load gradually applied to a structure. Assume a linear relationship exists between the load and the deflection. This is the same assumption used in Hooke s aw
More informationModule 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 informationCIVIL DEPARTMENT MECHANICS OF STRUCTURES ASSIGNMENT NO 1. Brach: CE YEAR:
MECHANICS OF STRUCTURES ASSIGNMENT NO 1 SEMESTER: V 1) Find the least moment of Inertia about the centroidal axes XX and YY of an unequal angle section 125 mm 75 mm 10 mm as shown in figure 2) Determine
More informationGATE SOLUTIONS E N G I N E E R I N G
GATE SOLUTIONS C I V I L E N G I N E E R I N G From (1987018) Office : F16, (Lower Basement), Katwaria Sarai, New Delhi110016 Phone : 01165064 Mobile : 81309090, 9711853908 Email: info@iesmasterpublications.com,
More informationCE5510 Advanced Structural Concrete Design  Design & Detailing of Openings in RC Flexural Members
CE5510 Advanced Structural Concrete Design  Design & Detailing Openings in RC Flexural Members Assoc Pr Tan Kiang Hwee Department Civil Engineering National In this lecture DEPARTMENT OF CIVIL ENGINEERING
More informationEstimation of the Residual Stiffness of FireDamaged Concrete Members
Copyright 2011 Tech Science Press CMC, vol.22, no.3, pp.261273, 2011 Estimation of the Residual Stiffness of FireDamaged Concrete Members J.M. Zhu 1, X.C. Wang 1, D. Wei 2, Y.H. Liu 2 and B.Y. Xu 2 Abstract:
More informationMechanical Properties of Materials
Mechanical Properties of Materials Strains Material Model Stresses Learning objectives Understand the qualitative and quantitative description of mechanical properties of materials. Learn the logic of
More information14. *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 informationLecture 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 informationLab Exercise #5: Tension and Bending with Strain Gages
Lab Exercise #5: Tension and Bending with Strain Gages Prelab assignment: Yes No Goals: 1. To evaluate tension and bending stress models and Hooke s Law. a. σ = Mc/I and σ = P/A 2. To determine material
More informationQUESTION BANK DEPARTMENT: CIVIL SEMESTER: III SUBJECT CODE: CE2201 SUBJECT NAME: MECHANICS OF SOLIDS UNIT 1 STRESS AND STRAIN PART A
DEPARTMENT: CIVIL SUBJECT CODE: CE2201 QUESTION BANK SEMESTER: III SUBJECT NAME: MECHANICS OF SOLIDS UNIT 1 STRESS AND STRAIN PART A (2 Marks) 1. Define longitudinal strain and lateral strain. 2. State
More informationME 243. Mechanics of Solids
ME 243 Mechanics of Solids Lecture 2: Stress and Strain Ahmad Shahedi Shakil Lecturer, Dept. of Mechanical Engg, BUET Email: sshakil@me.buet.ac.bd, shakil6791@gmail.com Website: teacher.buet.ac.bd/sshakil
More informationUNIT 1 STRESS STRAIN AND DEFORMATION OF SOLIDS, STATES OF STRESS 1. Define stress. When an external force acts on a body, it undergoes deformation.
UNIT 1 STRESS STRAIN AND DEFORMATION OF SOLIDS, STATES OF STRESS 1. Define stress. When an external force acts on a body, it undergoes deformation. At the same time the body resists deformation. The magnitude
More informationD : SOLID MECHANICS. Q. 1 Q. 9 carry one mark each. Q.1 Find the force (in kn) in the member BH of the truss shown.
D : SOLID MECHANICS Q. 1 Q. 9 carry one mark each. Q.1 Find the force (in kn) in the member BH of the truss shown. Q.2 Consider the forces of magnitude F acting on the sides of the regular hexagon having
More informationWhere and are the factored end moments of the column and >.
11 LIMITATION OF THE SLENDERNESS RATIO( ) 1Nonsway (braced) frames: The ACI Code, Section 6.2.5 recommends the following limitations between short and long columns in braced (nonsway) frames: 1. The
More informationSTRESS, STRAIN AND DEFORMATION OF SOLIDS
VELAMMAL COLLEGE OF ENGINEERING AND TECHNOLOGY, MADURAI 625009 DEPARTMENT OF CIVIL ENGINEERING CE8301 STRENGTH OF MATERIALS I 
More informationMechanics of Materials II. Chapter III. A review of the fundamental formulation of stress, strain, and deflection
Mechanics of Materials II Chapter III A review of the fundamental formulation of stress, strain, and deflection Outline Introduction Assumtions and limitations Axial loading Torsion of circular shafts
More informationPost Graduate Diploma in Mechanical Engineering Computational mechanics using finite element method
9210220 Post Graduate Diploma in Mechanical Engineering Computational mechanics using finite element method You should have the following for this examination one answer book scientific calculator No
More informationInstitute 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 informationMarch 24, Chapter 4. Deflection and Stiffness. Dr. Mohammad Suliman Abuhaiba, PE
Chapter 4 Deflection and Stiffness 1 2 Chapter Outline Spring Rates Tension, Compression, and Torsion Deflection Due to Bending Beam Deflection Methods Beam Deflections by Superposition Strain Energy Castigliano
More informationInstitute 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 informationLecture 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 informationFinite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras. Lecture  06
Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras Lecture  06 In the last lecture, we have seen a boundary value problem, using the formal
More informationFLEXIBILITY 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 informationSTATICALLY INDETERMINATE STRUCTURES
STATICALLY INDETERMINATE STRUCTURES INTRODUCTION Generally the trusses are supported on (i) a hinged support and (ii) a roller support. The reaction components of a hinged support are two (in horizontal
More informationCIVL 8/7117 Chapter 12  Structural Dynamics 1/75. To discuss the dynamics of a singledegreeof freedom springmass
CIV 8/77 Chapter  /75 Introduction To discuss the dynamics of a singledegreeof freedom springmass system. To derive the finite element equations for the timedependent stress analysis of the onedimensional
More informationChapter Objectives. Copyright 2011 Pearson Education South Asia Pte Ltd
Chapter Objectives To generalize the procedure by formulating equations that can be plotted so that they describe the internal shear and moment throughout a member. To use the relations between distributed
More informationtwenty one concrete construction: shear & deflection ARCHITECTURAL STRUCTURES: FORM, BEHAVIOR, AND DESIGN DR. ANNE NICHOLS SUMMER 2014 lecture
ARCHITECTURAL STRUCTURES: FORM, BEHAVIOR, AND DESIGN DR. ANNE NICHOLS SUMMER 2014 lecture twenty one concrete construction: Copyright Kirk Martini shear & deflection Concrete Shear 1 Shear in Concrete
More information7 TRANSVERSE SHEAR transverse shear stress longitudinal shear stresses
7 TRANSVERSE SHEAR Before we develop a relationship that describes the shearstress distribution over the cross section of a beam, we will make some preliminary remarks regarding the way shear acts within
More informationStructural Analysis I Chapter 4  Torsion TORSION
ORSION orsional stress results from the action of torsional or twisting moments acting about the longitudinal axis of a shaft. he effect of the application of a torsional moment, combined with appropriate
More informationMECE 3321 MECHANICS OF SOLIDS CHAPTER 3
MECE 3321 MECHANICS OF SOLIDS CHAPTER 3 Samantha Ramirez TENSION AND COMPRESSION TESTS Tension and compression tests are used primarily to determine the relationship between σ avg and ε avg in any material.
More informationApplication of Finite Element Method to Create Animated Simulation of Beam Analysis for the Course of Mechanics of Materials
International Conference on Engineering Education and Research "Progress Through Partnership" 4 VSBTUO, Ostrava, ISSN 15635 Application of Finite Element Method to Create Animated Simulation of Beam
More informationENG2000 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 informationMATERIAL PROPERTIES. Material Properties Must Be Evaluated By Laboratory or Field Tests 1.1 INTRODUCTION 1.2 ANISOTROPIC MATERIALS
. MARIAL PROPRIS Material Properties Must Be valuated By Laboratory or Field ests. INRODUCION he fundamental equations of structural mechanics can be placed in three categories[]. First, the stressstrain
More informationChapter 3. Load and Stress Analysis
Chapter 3 Load and Stress Analysis 2 Shear Force and Bending Moments in Beams Internal shear force V & bending moment M must ensure equilibrium Fig. 3 2 Sign Conventions for Bending and Shear Fig. 3 3
More informationComb 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 informationEsben Byskov. Elementary Continuum. Mechanics for Everyone. With Applications to Structural Mechanics. Springer
Esben Byskov Elementary Continuum Mechanics for Everyone With Applications to Structural Mechanics Springer Contents Preface v Contents ix Introduction What Is Continuum Mechanics? "I Need Continuum Mechanics
More information3. 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 informationAircraft Stress Analysis and Structural Design Summary
Aircraft Stress Analysis and Structural Design Summary 1. Trusses 1.1 Determinacy in Truss Structures 1.1.1 Introduction to determinacy A truss structure is a structure consisting of members, connected
More informationChapter 5 Elastic Strain, Deflection, and Stability 1. Elastic StressStrain Relationship
Chapter 5 Elastic Strain, Deflection, and Stability Elastic StressStrain Relationship A stress in the xdirection causes a strain in the xdirection by σ x also causes a strain in the ydirection & zdirection
More information[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**********************************************************************
Department of Civil and Environmental Engineering School of Mining and Petroleum Engineering 333 Markin/CNRL Natural Resources Engineering Facility www.engineering.ualberta.ca/civil Tel: 780.492.4235
More informationMechanics of Materials Primer
Mechanics of Materials rimer Notation: A = area (net = with holes, bearing = in contact, etc...) b = total width of material at a horizontal section d = diameter of a hole D = symbol for diameter E = modulus
More informationME 2570 MECHANICS OF MATERIALS
ME 2570 MECHANICS OF MATERIALS Chapter III. Mechanical Properties of Materials 1 Tension and Compression Test The strength of a material depends on its ability to sustain a load without undue deformation
More information2012 MECHANICS OF SOLIDS
R10 SET  1 II B.Tech II Semester, Regular Examinations, April 2012 MECHANICS OF SOLIDS (Com. to ME, AME, MM) Time: 3 hours Max. Marks: 75 Answer any FIVE Questions All Questions carry Equal Marks ~~~~~~~~~~~~~~~~~~~~~~
More informationA HIGHERORDER BEAM THEORY FOR COMPOSITE BOX BEAMS
A HIGHERORDER BEAM THEORY FOR COMPOSITE BOX BEAMS A. Kroker, W. Becker TU Darmstadt, Department of Mechanical Engineering, Chair of Structural Mechanics Hochschulstr. 1, D64289 Darmstadt, Germany kroker@mechanik.tudarmstadt.de,
More informationA METHOD OF LOAD INCREMENTS FOR THE DETERMINATION OF SECONDORDER LIMIT LOAD AND COLLAPSE SAFETY OF REINFORCED CONCRETE FRAMED STRUCTURES
A METHOD OF LOAD INCREMENTS FOR THE DETERMINATION OF SECONDORDER LIMIT LOAD AND COLLAPSE SAFETY OF REINFORCED CONCRETE FRAMED STRUCTURES Konuralp Girgin (Ph.D. Thesis, Institute of Science and Technology,
More informationJeff Brown Hope College, Department of Engineering, 27 Graves Pl., Holland, Michigan, USA UNESCO EOLSS
MECHANICS OF MATERIALS Jeff Brown Hope College, Department of Engineering, 27 Graves Pl., Holland, Michigan, USA Keywords: Solid mechanics, stress, strain, yield strength Contents 1. Introduction 2. Stress
More informationBEAM 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 crosssectional area of a beam. Supports that apply a moment
More informationMECHANICS OF MATERIALS
STATICS AND MECHANICS OF MATERIALS Ferdinand P. Beer E. Russell Johnston, Jr, John T. DeWolf David E Mazurek \Cawect Mc / iur/» Craw SugomcT Hilt Introduction 1 1.1 What is Mechanics? 2 1.2 Fundamental
More informationMechanics of Structure
S.Y. Diploma : Sem. III [CE/CS/CR/CV] Mechanics of Structure Time: Hrs.] Prelim Question Paper Solution [Marks : 70 Q.1(a) Attempt any SIX of the following. [1] Q.1(a) Define moment of Inertia. State MI
More informationCHAPTER 4. ANALYSIS AND DESIGN OF COLUMNS
4.1. INTRODUCTION CHAPTER 4. ANALYSIS AND DESIGN OF COLUMNS A column is a vertical structural member transmitting axial compression loads with or without moments. The cross sectional dimensions of a column
More informationLecture 7: The Beam Element Equations.
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
More informationRigid Pavement Mechanics. Curling Stresses
Rigid Pavement Mechanics Curling Stresses Major Distress Conditions Cracking Bottomup transverse cracks Topdown transverse cracks Longitudinal cracks Corner breaks Punchouts (CRCP) 2 Major Distress Conditions
More informationStress 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 crosssectional areas of A ) and A ) over the lengths l ) and l ), respectively.
More informationName :. Roll No. :... Invigilator s Signature :.. CS/B.TECH (CENEW)/SEM3/CE301/ SOLID MECHANICS
Name :. Roll No. :..... Invigilator s Signature :.. 2011 SOLID MECHANICS Time Allotted : 3 Hours Full Marks : 70 The figures in the margin indicate full marks. Candidates are required to give their answers
More informationQUESTION BANK SEMESTER: III SUBJECT NAME: MECHANICS OF SOLIDS
QUESTION BANK SEMESTER: III SUBJECT NAME: MECHANICS OF SOLIDS UNIT 1 STRESS AND STRAIN PART A (2 Marks) 1. Define longitudinal strain and lateral strain. 2. State Hooke s law. 3. Define modular ratio,
More informationChapter 5 CENTRIC TENSION OR COMPRESSION ( AXIAL LOADING )
Chapter 5 CENTRIC TENSION OR COMPRESSION ( AXIAL LOADING ) 5.1 DEFINITION A construction member is subjected to centric (axial) tension or compression if in any cross section the single distinct stress
More information4. BEAMS: CURVED, COMPOSITE, UNSYMMETRICAL
4. BEMS: CURVED, COMPOSITE, UNSYMMETRICL Discussions of beams in bending are usually limited to beams with at least one longitudinal plane of symmetry with the load applied in the plane of symmetry or
More informationModule III  Macromechanics of Lamina. Lecture 23. MacroMechanics of Lamina
Module III  Macromechanics of Lamina Lecture 23 MacroMechanics of Lamina For better understanding of the macromechanics of lamina, the knowledge of the material properties in essential. Therefore, the
More informationStructural Dynamics Lecture Eleven: Dynamic Response of MDOF Systems: (Chapter 11) By: H. Ahmadian
Structural Dynamics Lecture Eleven: Dynamic Response of MDOF Systems: (Chapter 11) By: H. Ahmadian ahmadian@iust.ac.ir Dynamic Response of MDOF Systems: ModeSuperposition Method ModeSuperposition Method:
More informationThis Technical Note describes how the program checks column capacity or designs reinforced concrete columns when the ACI code is selected.
COMPUTERS AND STRUCTURES, INC., BERKELEY, CALIFORNIA DECEMBER 2001 CONCRETE FRAME DESIGN ACI31899 Technical Note This Technical Note describes how the program checks column capacity or designs reinforced
More informationThe aims of this experiment were to obtain values for Young s modulus and Poisson s ratio for
The Cornu Method Nikki Truss 09369481 Abstract: The aims of this experiment were to obtain values for Young s modulus and Poisson s ratio for Perspex using the Cornu Method. A value of was found for Young
More informationCIV100 Mechanics. Module 5: Internal Forces and Design. by: Jinyue Zhang. By the end of this Module you should be able to:
CIV100 Mechanics Module 5: Internal Forces and Design by: Jinyue Zhang Module Objective By the end of this Module you should be able to: Find internal forces of any structural members Understand how Shear
More informationCHAPTER THREE SYMMETRIC BENDING OF CIRCLE PLATES
CHAPTER THREE SYMMETRIC BENDING OF CIRCLE PLATES * Governing equations in beam and plate bending ** Solution by superposition 1.1 From Beam Bending to Plate Bending 1.2 Governing Equations For Symmetric
More informationFinite 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 informationPDDC 1 st Semester Civil Engineering Department Assignments of Mechanics of Solids [ ] Introduction, Fundamentals of Statics
Page1 PDDC 1 st Semester Civil Engineering Department Assignments of Mechanics of Solids [2910601] Introduction, Fundamentals of Statics 1. Differentiate between Scalar and Vector quantity. Write S.I.
More informationMECHANICS OF SOLIDS. (For B.E. Mechanical Engineering Students) As per New Revised Syllabus of APJ Abdul Kalam Technological University
MECHANICS OF SOLIDS (For B.E. Mechanical Engineering Students) As per New Revised Syllabus of APJ Abdul Kalam Technological University Dr. S.Ramachandran, M.E., Ph.D., Mr. V.J. George, M.E., Mr. S. Kumaran,
More informationTheory at a Glance (for IES, GATE, PSU)
1. Stress and Strain Theory at a Glance (for IES, GATE, PSU) 1.1 Stress () When a material is subjected to an external force, a resisting force is set up within the component. The internal resistance force
More informationDecember 10, PROBLEM NO points max.
PROBLEM NO. 1 25 points max. PROBLEM NO. 2 25 points max. B 3A A C D A H k P L 2L Given: Consider the structure above that is made up of rod segments BC and DH, a spring of stiffness k and rigid connectors
More information6.4 A cylindrical specimen of a titanium alloy having an elastic modulus of 107 GPa ( psi) and
6.4 A cylindrical specimen of a titanium alloy having an elastic modulus of 107 GPa (15.5 10 6 psi) and an original diameter of 3.8 mm (0.15 in.) will experience only elastic deformation when a tensile
More informationReview 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 informationBE Semester I ( ) Question Bank (MECHANICS OF SOLIDS)
BE Semester I ( ) Question Bank (MECHANICS OF SOLIDS) All questions carry equal marks(10 marks) Q.1 (a) Write the SI units of following quantities and also mention whether it is scalar or vector: (i)
More informationPart 1 is to be completed without notes, beam tables or a calculator. DO NOT turn Part 2 over until you have completed and turned in Part 1.
NAME CM 3505 Fall 06 Test 2 Part 1 is to be completed without notes, beam tables or a calculator. Part 2 is to be completed after turning in Part 1. DO NOT turn Part 2 over until you have completed and
More informationStructural Displacements
Structural Displacements Beam Displacement 1 Structural Displacements P Truss Displacements 2 The deflections of civil engineer ing structures under the action of usual design loads are known to be small
More informationEE C245 ME C218 Introduction to MEMS Design
EE C245 ME C218 Introduction to MEMS Design Fall 2007 Prof. Clark T.C. Nguyen Dept. of Electrical Engineering & Computer Sciences University of California at Berkeley Berkeley, CA 94720 Lecture 16: Energy
More information4.3 Moment Magnification
CHAPTER 4: Reinforced Concrete Columns 4.3 Moment Magnification Description An ordinary or first order frame analysis does not include either the effects of the lateral sidesway deflections of the column
More information