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


 Noel Mason
 1 years ago
 Views:
Transcription
1 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 of Structural Analysis Strain Energy Strain energy is stored within an elastic solid when the solid is deformed under load. In the absence of energy losses, such as from friction, damping or yielding, the strain energy is equal to the work done on the solid by external loads. Strain energy is a type of potential energy. Consider the work done on an elastic solid by a single point force F. When the elastic solid carries the load, F, it deforms with strains (ɛ and γ) and the material is stressed (σ and τ). is a displacement in the same location and in the same direction as a point force, F. and F are colocated. The work done by the force F on the elastic solid is the area under the force vs. displacement curve. W = F d () This work is stored as strain energy U within the elastic solid. U = 2 V (σ xx ɛ xx + σ yy ɛ yy + σ zz ɛ zz + τ xy γ xy + τ xz γ xz + τ yz γ yz ) dv. (2) This is a very general expression for the strain energy, U, and is not very practical for structural elements like bars, beams, trusses, or frames.
2 2 CEE 42L. Matrix Structural Analysis uke University Fall 22 H.P. Gavin. Bars For a bar in tension or compression, we have internal axial force, N, only, N x N x x dl ε xx dl σ xx so σ yy =, σ zz =, τ xy =, τ xz =, and τ yz =, and U = 2 V σ xx ɛ xx dv, where σ xx = N/A and ɛ xx = N/. Substituting dv = A dx we get U = 2 L N(x) 2 E(x) A(x) dx, (3) and if N, E, and A are constant U = N 2 L 2 E A. Alternatively, we may express the strain as a function of the displacements along the bar u x (x), ɛ xx = u x (x)/ x, and σ xx = E u x (x)/ x. Again substituting dv = A dx, U = 2 L E(x) A(x) and if E, A and u x / x = (u 2 u )/L are constants, ( ) 2 ux (x) dx, (4) x U = 2 L (u 2 u ) 2
3 Strain Energy and Matrix Methods of Structural Analysis 3.2 Beams For a beam in bending we have internal bending moments, M, and internal shear forces, V. For slender beams the effects of shear deformation are usually neglected. M zz y v" dl M zz x σ xx dl As in the axially loaded bar, σ yy =, σ zz =, τ xy =, τ xz =, and τ yz =, and U = 2 V σ xx ɛ xx dv. For bending, σ xx = My/I and ɛ xx = My/EI. Substituting dv = da dx, where A y2 da = I, so U = 2 L U = 2 A L M(x) 2 y 2 da dx, E(x) I(x) 2 M(x) 2 E(x) I(x) dx. (5) Alternatively, we may express the moment in terms of the curvature of the beam, φ 2 u y / x 2, M(x) = E(x) I(x) 2 u y (x) x 2, from which σ xx = E ( 2 u y / x 2 ) y and ɛ xx = ( 2 u y / x 2 ) y, so that U = 2 where, again, A y2 da = I, so U = 2 L L A E(x) E(x) I(x) ( 2 ) 2 u y (x) y 2 da dx x 2 ( 2 ) 2 u y (x) dx. (6) x 2
4 4 CEE 42L. Matrix Structural Analysis uke University Fall 22 H.P. Gavin.3 Summary External work is done by a set of forces, F i, on a linear elastic solid, producing a set of displacements, i, in the same locations and directions. F Fi i j F j n Fn F F i Fi Fi i n j Fj F j j F n F n n The work done by these forces is W = 2 F + 2 F F The external forces are resisted by internal moments, M, and axial forces, N. The total strain energy stored within the solid is U = M 2 2 L E I dx + Nj 2 L j (7) 2 j E j A j where the first term is the integral over all lengths of all the beams and the second term is the sum over all the bars. If torsion and shear are included, then two additional terms are T 2 2 L G J dx and V 2 2 L G A/α dx. Alternatively, we can think of external forces producing curvatures ( 2 u y / x 2 ) by bending, and axial stretches ( u x / x). In this case U = ( 2 ) 2 u y E I dx + E j A j (u 2 L x 2 2j u j ) 2 (8) 2 j L j If torsion and shear are included, then two additional terms are ( ) 2 ( ) 2 uxθ uy G J dx, and G A/α dx, 2 L x 2 L x where u xθ is the torsional rotation about the xaxis, u xθ / x is the torsional shear strain, γ xθ, (on the face perpendicular to the xaxis and in the θdirection) and u y / x is the shear strain, γ xy, (on the face perpendicular to the xaxis and in the ydirection). Analyses using expressions of the form of equations (3), (5), or (7) are called force method or flexibility method analyses. Analyses using expressions of the form of equations (4), (6), or (8) are called displacement method or stiffness method analyses.
5 Strain Energy and Matrix Methods of Structural Analysis 5 2 Castigliano s Theorems 2. Castigliano s Theorem  Part I U = F d... strain energy F j U() j U + j j j F i = U i = U i A force, F i, on an elastic solid is equal to the derivative of the strain energy with respect to the displacement, i, in the direction and location of the force, F i. 2.2 Castigliano s Theorem  Part II U = df... complementary strain energy F+ j F j F j U* U*(F) F j j i = U F i = U F i A displacement, i, on an elastic solid is equal to the derivative of the complementary strain energy with respect to the force, F i, in the direction and location of the displacement, i. If the solid is linear elastic, then U = U.
6 6 CEE 42L. Matrix Structural Analysis uke University Fall 22 H.P. Gavin 3 Superposition Superposition is an extremely powerful idea that helps us solve problems that are statically indeterminate. To use the principle of superposition, the system must behave in a linear elastic fashion. The principle of superposition states: Any response of a system to multiple inputs can be represented as the sum of the responses to the inputs taken individually. By response we can mean a strain, a stress, a deflection, an internal force, a rotation, an internal moment, etc. By input we can mean an externally applied load, a temperature change, a support settlement, etc. 4 etailed Example of Castigliano s Theorem and Superposition An example of a statically indeterminate system with external loads w(x) and three redundant reaction forces, R B, R C, and R, is shown below. y w(x) EI A B L C x H In general, the displacements at the locations of the unknown reaction forces are known, and, in this example these displacements will be taken as zero: B =, C =, =. Invoking the principle of superposition, we may apply the external loads, (w(x)) and the unknown reactions (R B, R C, and R ) individually, and then sumup the responses to each individual load. Further, we may represent the response to a reaction force, (e.g., R B ) as the response to a unit force colocated with the reaction force, times the value of the reaction force. Note that all four systems to the right of the equal sign in the following figure are statically determinate. Expressions for Mo(x), m (x), m 2 (x), m 3 (x), N o (x), n (x), n 2 (x), and n 3 (x) may be found from static equilibrium alone.
7 Strain Energy and Matrix Methods of Structural Analysis 7 w(x) A B C M(x) R B R C N(x) R = + w(x) A B C M (x) o m (x) B N (x) o n (x) * R B + m (x) 2 C n (x) 2 * R C + m (x) 3 n (x) 3 * R In equation form, the principle of superposition says: M(x) = M o (x) + m (x)r B + m 2 (x)r C + m 3 (x)r (9) N = N o + n R B + n 2 R C + n 3 R () (Note that in this particular example, N o (x) =, n =, n 2 =, n 3 =, m (x) = for x > x B, and m 2 (x) = for x > x C.)
8 8 CEE 42L. Matrix Structural Analysis uke University Fall 22 H.P. Gavin The total strain energy, U, in systems with bending strain energy and axial strain energy is, U = L M(x) 2 dx + N 2 H () 2 EI 2 We are told that the displacements at points B, C, and are all zero and we need to assume the structure behaves linear elastically in order to invoke superposition in the first place. Therefore, from Castigliano s Second Theorem, i = U F i = U F i, we obtain three expressions for the facts that B =, C =, and =. B = = U R B C = = U R C = = U R Inserting equation () into the three expressions for zero displacement at the fixed reactions, noting that EI and are constants in this problem, and noting that the strain energy, U, depends on the reactions R, only through the internal forces, M and N, we obtain B = = L M(x) M(x) dx + H EI R B N N R B C = = L M(x) M(x) dx + H EI R C N N R C = = L M(x) M(x) dx + H EI R N N R Now, from the superposition equations (9) and (), M(x)/ R B = m (x), M(x)/ R C = m 2 (x), M(x)/ R = m 3 (x), N(x)/ R B = n, N(x)/ R C = n 2, and N(x)/ R = n 3. Inserting these expressions and the superposition equations (9) and () into the above equations for B, C, and, B = = L [M o + m R B + m 2 R C + m 3 R ] m dx + H EI [N o + n R B + n 2 R C + n 3 R ] n C = = L [M o + m R B + m 2 R C + m 3 R ] m 2 dx + H EI [N o + n R B + n 2 R C + n 3 R ] n 2 = = L [M o + m R B + m 2 R C + m 3 R ] m 3 dx + H EI [N o + n R B + n 2 R C + n 3 R ] n 3 These three expressions contain the three unknown reactions R B, R C, and R. Everything else in these equations (m (x), m 2 (x)... n 3 ) can be found without knowing the unknown
9 Strain Energy and Matrix Methods of Structural Analysis 9 reactions. By taking the unknown reactions out of the integrals (they are constants), we can write these three equations in matrix form. L m m EI L m 2m EI L m 3m EI dx + n n H dx + n 2n H dx + n 3n H L m m 2 EI L m 2m 2 EI L m 3m 2 EI dx + n n 2 H dx + n 2n 2 H dx + n 3n 2 H L m m 3 EI L m 2m 3 EI L m 3m 3 EI dx + n n 3 H dx + n 2n 3 H dx + n 3n 3 H R B R C R = L Mom EI L Mom 2 EI L Mom 3 EI (2) This 3by3 matrix is called a flexibility matrix, F. The values of the terms in the flexibility matrix depend only on the responses of the structure to unit loads placed at various points in the structure. The flexibility matrix is therefore a property of the structure alone, and does not depend upon the loads on the structure. The vector on the righthandside depends on the loads on the structure. Recall that this matrix looks a lot like the matrix from the threemoment equation. All flexibility matrices share several properties: dx + Non H dx + Non 2H dx + Non 3H All flexibility matrices are symmetric. No diagonal terms are negative. Flexibility matrices for structures which can not move or rotate without deforming are positive definite. This means that all of the eigenvalues of a flexibility matrix describing a fixed structure are positive. The unknowns in a flexibility matrix equation are forces (or moments). The number of equations (rows of the flexibility matrix) equals the number of unknown forces (or moments). There are some fascinating cases in which the behavior does depend upon the loads, but that is a story for another day!
10 CEE 42L. Matrix Structural Analysis uke University Fall 22 H.P. Gavin It is instructive to now examine the meaning of the terms in the matrix, F F = L m m EI dx + n n H = δ displacement at due to unit force at F A B C F 2 = L m 2 m EI dx + n 2n H = δ 2 displacement at 2 due to unit force at F 2 A B C F 2 = L m m 2 EI dx + n n 2 H = δ 2 displacement at due to unit force at 2 F 2 A B C F 3 F 3 = L m 3 m EI dx + n 3n H = δ 3 displacement at 3 due to unit force at A B C The fact that F 2 = F 2 is called Maxwell s Reciprocity Theorem.
11 Strain Energy and Matrix Methods of Structural Analysis 5 Introductory Example of the Stiffness Matrix Method In this simple example, elements are springs with stiffness k. A spring with stiffness k > connecting point i to point j, will have a force f = k(d j d i ) where d i is the displacement of point i and d j is the displacement of point j. (Tension is positive so d i points into the spring and d j points away from the spring.) The stiffness matrix for this structure can be found using equilibrium and forcedeflection relationships (f = kd) for the springs. #: F x = : f k d + k 2 (d 2 d ) = #2: F x = : f 2 k 2 (d 2 d ) k 4 d 2 + k 3 (d 3 d 2 ) = #3: F x = : f 3 k 3 (d 3 d 2 ) k 5 d 3 = In matrix form these three equations may be written: k + k 2 k 2 d k 2 k 2 + k 3 + k 4 k 3 d 2 k 3 k 3 + k 5 d 3 = The displacements are found by solving the stiffness matrix equation for d, d = K f. The matrix K is called a stiffness matrix. All stiffness matrices are symmetric. f f 2 f 3
12 2 CEE 42L. Matrix Structural Analysis uke University Fall 22 H.P. Gavin All diagonal terms of all stiffness matrices are positive. Stiffness matrices are diagonally dominant. This means that the diagonal terms are usually larger than the offdiagonal term. If the structure is not free to translate or rotate without deforming, then the stiffness matrix is positive definite. This mathematical property guarantees that the stiffness matrix is invertible, and a unique set of displacements, d, can be found by solving K d = f. The total potential energy, U, in this system of springs is U = 2 k d k 2(d 2 d ) k 3(d 3 d 2 ) k 4d k 5d 2 3. You should be able to confirm that this is equal to U = 2 dt K d Also, note that no matter what the values of the displacements, d, may be, the energy U is always positive. The statement 2 dt Kd > d is another way of saying that K is positivedefinite. The set of forces required to deflect coordinate i by a deflection of unit equals the ith column of the stiffness matrix. For example consider the case in which d =, d 2 =, and d 3 =, k + k 2 k 2 k 2 k 2 + k 3 + k 4 k 3 k 3 k 3 + k 5 = which is equal to the first column of the stiffness matrix. k + k 2 k 2,
13 Strain Energy and Matrix Methods of Structural Analysis 3 This fact may be used to derive the stiffness matrix: d =, d 2 =, d 3 = f = k + k 2, f 2 = k 2, f 3 =... st column d =, d 2 =, d 3 = f = k 2, f 2 = k 2 + k 3 + k 4, f 3 = k nd column d =, d 2 =, d 3 = f =, f 2 = k 3, f 3 = k 3 + k rd column
14 4 CEE 42L. Matrix Structural Analysis uke University Fall 22 H.P. Gavin 6 Basic Concepts of the Stiffness Matrix Method The previous example illustrates some of the basic concepts needed to apply the stiffness matrix method to structures made out of bars and beams. There are, however, a few additional complications. isplacements in structures can be vertical, horizontal, or rotational, and structural bars and beams have a more complicated forcedisplacement relationships than those of simple springs. In applying the matrix stiffness method of structural analysis, structures are described in terms of elements that connect nodes which can move in certain coordinate directions. 6. Elements In the stiffness matrix method, structures are modeled as assemblies of elements such as bars, beams, cables, shafts, plates, and walls. Elements connect the nodes of the structural model. Like the simple springs in the previous example, structural elements have clearly defined, albeit more complicated, forcedisplacement relationships. The stiffness properties of structural elements can be determined from equilibrium equations, Castigliano s Theorems, the principle of minimum potential energy, and/or the principle of virtual work. Structural elements can be mathematically assembled with one another (like making a structural system using a set of tinkertoys), into a system of equations for the entire structure. 6.2 Nodes The forcedisplacement relationship of a structural element is defined in terms of the forces and displacements at the nodes of the element. Nodes define the points where elements meet. The nodes in the model of a truss are at the joints between the truss bars. The nodes in the model of a beam or a frame are at the reaction locations, at locations at which elements connect to each other, and possibly at other intermediate locations. 6.3 Coordinates Coordinates describe the location and direction at which forces and displacements act on an element or on a structure. Trusses are loaded with vertical and horizontal forces at the joints. The joints of a 2 truss can move vertically and horizontally; so there are two coordinates per node in a 2 truss. Beams and frames carry vertical and horizontal loads as well as bending moments. The nodes of a 2 frame can move vertically, horizontally, and can rotate; so there are three coordinates per node in a 2 frame. Structural coordinates can be classified into two sets. isplacement coordinates have unknown displacements but know forces. Reaction coordinates have unknown forces but known displacements (usually zero).
15 Strain Energy and Matrix Methods of Structural Analysis Elements, Nodes and Coordinates Planar (2) truss bar elements have two nodes and four coordinates, two at each end. Space (3) truss bar elements have two nodes and six coordinates, three at each end. Planar (2) frame elements have two nodes and six coordinates, three at each end. Space (3) frame elements have two nodes and twelve coordinates, six at each end.
16 6 CEE 42L. Matrix Structural Analysis uke University Fall 22 H.P. Gavin 6.5 Structural Nodes and Coordinates Planar (2) truss nodes and coordinates Planar (2) frame nodes and coordinates
17 Strain Energy and Matrix Methods of Structural Analysis 7 7 Relate the Flexibility Matrix to the Stiffness Matrix Column j of the stiffness matrix: The set of forces at all coordinates required to produce a unit displacement at coordinate j Column j of the flexibility matrix: The set of displacements at all coordinates resulting from a unit force at coordinate j Stiffness matrix equation: Flexibility matrix equation: K K 2 K 3 K n K 2 K 22 K 23 K 2n K 3 K 32 K 33 K 3n K = F and K = F K n K n2 K n3 K nn F F 2 F 3 F n F 2 F 22 F 23 F 2n F 3 F 32 F 33 F 3n F n F n2 F n3 F nn d d 2 d 3. d n f f 2 f 3. f n = = f f 2 f 3. f n d d 2 d 3. d n A useful fact for 2by2 matrices... [ ] a b = c d ad bc [ d b c a ]... you should be able to prove this fact to yourself.
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 informationMathematical Properties of Stiffness Matrices
Mathematical Properties of Stiffness Matrices CEE 4L. Matrix Structural Analysis Department of Civil and Environmental Engineering Duke University Henri P. Gavin Fall, 0 These notes describe some of the
More informationLecture 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 informationPreliminaries: Beam Deflections Virtual Work
Preliminaries: Beam eflections Virtual Work There are several methods available to calculate deformations (displacements and rotations) in beams. They include: Formulating moment equations and then integrating
More informationIndeterminate 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 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 informationChapter 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 informationStructural 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 2dimensional structures
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 informationModule 3. Analysis of Statically Indeterminate Structures by the Displacement Method
odule 3 Analysis of Statically Indeterminate Structures by the Displacement ethod Lesson 14 The SlopeDeflection ethod: An Introduction Introduction As pointed out earlier, there are two distinct methods
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 informationUnit 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 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 informationMechanical Design in Optical Engineering
OPTI Buckling Buckling and Stability: As we learned in the previous lectures, structures may fail in a variety of ways, depending on the materials, load and support conditions. We had two primary concerns:
More informationStiffness 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 informationExternal Work. When a force F undergoes a displacement dx in the same direction i as the force, the work done is
Structure Analysis I Chapter 9 Deflection Energy Method External Work Energy Method When a force F undergoes a displacement dx in the same direction i as the force, the work done is du e = F dx If the
More informationModule 6. Approximate Methods for Indeterminate Structural Analysis. Version 2 CE IIT, Kharagpur
Module 6 Approximate Methods for Indeterminate Structural Analysis Lesson 35 Indeterminate Trusses and Industrial rames Instructional Objectives: After reading this chapter the student will be able to
More informationDiscontinuous Distributions in Mechanics of Materials
Discontinuous Distributions in Mechanics of Materials J.E. Akin, Rice University 1. Introduction The study of the mechanics of materials continues to change slowly. The student needs to learn about software
More informationInternational Journal of Advanced Engineering Technology EISSN
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 informationtwo structural analysis (statics & mechanics) APPLIED ACHITECTURAL STRUCTURES: DR. ANNE NICHOLS SPRING 2017 lecture STRUCTURAL ANALYSIS AND SYSTEMS
APPLIED ACHITECTURAL STRUCTURES: STRUCTURAL ANALYSIS AND SYSTEMS DR. ANNE NICHOLS SPRING 2017 lecture two structural analysis (statics & mechanics) Analysis 1 Structural Requirements strength serviceability
More informationPLAT DAN CANGKANG (TKS 4219)
PLAT DAN CANGKANG (TKS 4219) SESI I: PLATES Dr.Eng. Achfas Zacoeb Dept. of Civil Engineering Brawijaya University INTRODUCTION Plates are straight, plane, twodimensional structural components of which
More informationChapter 7 FORCES IN BEAMS AND CABLES
hapter 7 FORES IN BEAMS AN ABLES onsider a straight twoforce 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 freebody B
More informationThe bending moment diagrams for each span due to applied uniformly distributed and concentrated load are shown in Fig.12.4b.
From inspection, it is assumed that the support moments at is zero and support moment at, 15 kn.m (negative because it causes compression at bottom at ) needs to be evaluated. pplying three Hence, only
More informationLecture 3: The Concept of Stress, Generalized Stresses and Equilibrium
Lecture 3: The Concept of Stress, Generalized Stresses and Equilibrium 3.1 Stress Tensor We start with the presentation of simple concepts in one and two dimensions before introducing a general concept
More informationStrain Energy in Linear Elastic Solids
Strain Energ in Linear Eastic Soids CEE L. Uncertaint, Design, and Optimiation Department of Civi and Environmenta Engineering Duke Universit Henri P. Gavin Spring, 5 Consider a force, F i, appied gradua
More informationContent. Department of Mathematics University of Oslo
Chapter: 1 MEK4560 The Finite Element Method in Solid Mechanics II (January 25, 2008) (Epost:torgeiru@math.uio.no) Page 1 of 14 Content 1 Introduction to MEK4560 3 1.1 Minimum Potential energy..............................
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 informationGyroscopic matrixes of the straight beams and the discs
Titre : Matrice gyroscopique des poutres droites et des di[...] Date : 29/05/2013 Page : 1/12 Gyroscopic matrixes of the straight beams and the discs Summarized: This document presents the formulation
More informationModule 3. Analysis of Statically Indeterminate Structures by the Displacement Method. Version 2 CE IIT, Kharagpur
odule 3 Analysis of Statically Indeterminate Structures by the Displacement ethod Version CE IIT, Kharagpur Lesson The ultistory Frames with Sidesway Version CE IIT, Kharagpur Instructional Objectives
More information1.105 Solid Mechanics Laboratory
1.105 Solid Mechanics Laboratory General Information Fall 2003 Prof. Louis Bucciarelli Rm 5213 x34061 llbjr@mit.edu TA: Attasit Korchaiyapruk, Pong Rm 5330B x 35170 attasit@mit.edu Athena Locker: /mit/1.105/
More informationStress, Strain, Mohr s Circle
Stress, Strain, Mohr s Circle The fundamental quantities in solid mechanics are stresses and strains. In accordance with the continuum mechanics assumption, the molecular structure of materials is neglected
More informationStructural Element Stiffness, Mass, and Damping Matrices
Structural Element Stiffness, Mass, and Damping Matrices CEE 541. Structural Dynamics Department of Civil and Environmental Engineering Duke University Henri P. Gavin Fall 18 1 Preliminaries This document
More informationBeams. Beams are structural members that offer resistance to bending due to applied load
Beams Beams are structural members that offer resistance to bending due to applied load 1 Beams Long prismatic members Nonprismatic sections also possible Each crosssection dimension Length of member
More informationMaterials: engineering, science, processing and design, 2nd edition Copyright (c)2010 Michael Ashby, Hugh Shercliff, David Cebon.
Modes of Loading (1) tension (a) (2) compression (b) (3) bending (c) (4) torsion (d) and combinations of them (e) Figure 4.2 1 Standard Solution to Elastic Problems Three common modes of loading: (a) tie
More informationMECHANICS OF MATERIALS
Third E CHAPTER 2 Stress MECHANICS OF MATERIALS Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf Lecture Notes: J. Walt Oler Texas Tech University and Strain Axial Loading Contents Stress & Strain:
More informationDynamic Model of a Badminton Stroke
ISEA 28 CONFERENCE Dynamic Model of a Badminton Stroke M. Kwan* and J. Rasmussen Department of Mechanical Engineering, Aalborg University, 922 Aalborg East, Denmark Phone: +45 994 9317 / Fax: +45 9815
More informationRigid and Braced Frames
RH 331 Note Set 12.1 F2014abn Rigid and raced Frames Notation: E = modulus of elasticit or Young s modulus F = force component in the direction F = force component in the direction FD = free bod diagram
More informationBOOK OF COURSE WORKS ON STRENGTH OF MATERIALS FOR THE 2 ND YEAR STUDENTS OF THE UACEG
BOOK OF COURSE WORKS ON STRENGTH OF MATERIALS FOR THE ND YEAR STUDENTS OF THE UACEG Assoc.Prof. Dr. Svetlana LilkovaMarkova, Chief. Assist. Prof. Dimitar Lolov Sofia, 011 STRENGTH OF MATERIALS GENERAL
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 informationARC 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 informationBasic Energy Principles in Stiffness Analysis
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
More informationPortal Frame Calculations Lateral Loads
Portal Frame Calculations Lateral Loads Consider the following multistory frame: The portal method makes several assumptions about the internal forces of the columns and beams in a rigid frame: 1) Inflection
More informationNow we are going to use our free body analysis to look at Beam Bending (W3L1) Problems 17, F2002Q1, F2003Q1c
Now we are going to use our free body analysis to look at Beam Bending (WL1) Problems 17, F00Q1, F00Q1c One of the most useful applications of the free body analysis method is to be able to derive equations
More information2 Introduction to mechanics
21 Motivation Thermodynamic bodies are being characterized by two competing opposite phenomena, energy and entropy which some researchers in thermodynamics would classify as cause and chance or determinancy
More information7.4 The Elementary Beam Theory
7.4 The Elementary Beam Theory In this section, problems involving long and slender beams are addressed. s with pressure vessels, the geometry of the beam, and the specific type of loading which will be
More informationIntroduction 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. BernoulliEuler Beams.
More information2. 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 information11 Locate the centroid of the plane area shown. 12 Determine the location of centroid of the composite area shown.
Chapter 1 Review of Mechanics of Materials 11 Locate the centroid of the plane area shown 650 mm 1000 mm 650 x 1 Determine the location of centroid of the composite area shown. 00 150 mm radius 00 mm
More informationEMA 3702 Mechanics & Materials Science (Mechanics of Materials) Chapter 2 Stress & Strain  Axial Loading
MA 3702 Mechanics & Materials Science (Mechanics of Materials) Chapter 2 Stress & Strain  Axial Loading MA 3702 Mechanics & Materials Science Zhe Cheng (2018) 2 Stress & Strain  Axial Loading Statics
More informationLecture 27: Structural Dynamics  Beams.
Chapter #16: Structural Dynamics and Time Dependent Heat Transfer. Lectures #16 have discussed only steady systems. There has been no time dependence in any problems. We will investigate beam dynamics
More information7.5 Elastic Buckling Columns and Buckling
7.5 Elastic Buckling The initial theory of the buckling of columns was worked out by Euler in 1757, a nice example of a theory preceding the application, the application mainly being for the later invented
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 informationPLAXIS. 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 informationMETHOD OF LEAST WORK
METHOD OF EAST WORK 91 METHOD OF EAST WORK CHAPTER TWO The method of least work is used for the analysis of statically indeterminate beams, frames and trusses. Indirect use of the Castigliano s nd theorem
More informationOutline. Structural Matrices. Giacomo Boffi. Introductory Remarks. Structural Matrices. Evaluation of Structural Matrices
Outline in MDOF Systems Dipartimento di Ingegneria Civile e Ambientale, Politecnico di Milano May 8, 014 Additional Today we will study the properties of structural matrices, that is the operators that
More information1 Stress and Strain. Introduction
1 Stress and Strain Introduction This book is concerned with the mechanical behavior of materials. The term mechanical behavior refers to the response of materials to forces. Under load, a material may
More information3D problem: Fx Fy Fz. Forces act parallel to the members (2 5 ) / 29 (2 5 ) / 29
problem: x y z 0 t each joint a a a a 5a j i W k y z x x y z Equations:S x =S y =S z =0 at each joint () Unknowns: Total of : Member forces,,, () Reactions : x, y, z, x, y, z, x, y, z (9) y z x W orces
More informationDue Monday, September 14 th, 12:00 midnight
Due Monday, September 14 th, 1: midnight This homework is considering the analysis of plane and space (3D) trusses as discussed in class. A list of MatLab programs that were discussed in class is provided
More informationModeling Mechanical Systems
Modeling Mechanical Systems Mechanical systems can be either translational or rotational. Although the fundamental relationships for both types are derived from Newton s law, they are different enough
More informationMECH 401 Mechanical Design Applications
MECH 401 Mechanical Design Applications Dr. M. O Malley Master Notes Spring 008 Dr. D. M. McStravick Rice University Updates HW 1 due Thursday (11708) Last time Introduction Units Reliability engineering
More informationMoment Distribution The Real Explanation, And Why It Works
Moment Distribution The Real Explanation, And Why It Works Professor Louie L. Yaw c Draft date April 15, 003 To develop an explanation of moment distribution and why it works, we first need to develop
More informationσ = Eα(T T C PROBLEM #1.1 (4 + 4 points, no partial credit)
PROBLEM #1.1 (4 + 4 points, no partial credit A thermal switch consists of a copper bar which under elevation of temperature closes a gap and closes an electrical circuit. The copper bar possesses a length
More informationGeometrydependent MITC method for a 2node isobeam element
Structural Engineering and Mechanics, Vol. 9, No. (8) 33 Geometrydependent MITC method for a node isobeam element PhillSeung Lee Samsung Heavy Industries, Seocho, Seoul 37857, Korea HyuChun Noh
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 informationPLATE 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 informationUnit III Theory of columns. Dr.P.Venkateswara Rao, Associate Professor, Dept. of Civil Engg., SVCE, Sriperumbudir
Unit III Theory of columns 1 Unit III Theory of Columns References: Punmia B.C.,"Theory of Structures" (SMTS) Vol II, Laxmi Publishing Pvt Ltd, New Delhi 2004. Rattan.S.S., "Strength of Materials", Tata
More informationNonlinear static analysis PUSHOVER
Nonlinear static analysis PUSHOVER Adrian DOGARIU European Erasmus Mundus Master Course Sustainable Constructions under Natural Hazards and Catastrophic Events 520121120111CZERA MUNDUSEMMC Structural
More informationENCE 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 informationEQUILIBRIUM and ELASTICITY
PH 2211D Spring 2013 EQUILIBRIUM and ELASTICITY Lectures 3032 Chapter 12 (Halliday/Resnick/Walker, Fundamentals of Physics 9 th edition) 1 Chapter 12 Equilibrium and Elasticity In this chapter we will
More informationC:\Users\whit\Desktop\Active\304_2012_ver_2\_Notes\4_Torsion\1_torsion.docx 6
C:\Users\whit\Desktop\Active\304_2012_ver_2\_Notes\4_Torsion\1_torsion.doc 6 p. 1 of Torsion of circular bar The crosssections rotate without deformation. The deformation that does occur results from
More informationTheory 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 informationGame Physics. Game and Media Technology Master Program  Utrecht University. Dr. Nicolas Pronost
Game and Media Technology Master Program  Utrecht University Dr. Nicolas Pronost Soft body physics Soft bodies In reality, objects are not purely rigid for some it is a good approximation but if you hit
More informationTuesday, February 11, Chapter 3. Load and Stress Analysis. Dr. Mohammad Suliman Abuhaiba, PE
1 Chapter 3 Load and Stress Analysis 2 Chapter Outline Equilibrium & FreeBody Diagrams Shear Force and Bending Moments in Beams Singularity Functions Stress Cartesian Stress Components Mohr s Circle for
More informationFinite 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 informationUNITII MOVING LOADS AND INFLUENCE LINES
UNITII MOVING LOADS AND INFLUENCE LINES Influence lines for reactions in statically determinate structures influence lines for member forces in pinjointed frames Influence lines for shear force and bending
More informationStructural Analysis III The Moment Area Method Mohr s Theorems
Structural Analysis III The Moment Area Method Mohr s Theorems 009/10 Dr. Colin Caprani 1 Contents 1. Introduction... 4 1.1 Purpose... 4. Theory... 6.1 asis... 6. Mohr s First Theorem (Mohr I)... 8.3 Mohr
More informationCivil 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 informationRotational & RigidBody Mechanics. Lectures 3+4
Rotational & RigidBody Mechanics Lectures 3+4 Rotational Motion So far: point objects moving through a trajectory. Next: moving actual dimensional objects and rotating them. 2 Circular Motion  Definitions
More informationFinal Exam Solution Dynamics :45 12:15. Problem 1 Bateau
Final Exam Solution Dynamics 2 191157140 31012013 8:45 12:15 Problem 1 Bateau Bateau is a trapeze act by Cirque du Soleil in which artists perform aerial maneuvers on a boat shaped structure. The boat
More informationNumerical Limit Analysis of Rigid Plastic Structures
Numerical Limit Analysis of Rigid Plastic Structures Hector Andres Tinoco Navarro Institute of Technology and Innovation University of Southern Denmark Supervisors: Prof. Ph.D. Linh Cao Hoang University
More informationSLOPEDEFLECTION METHOD
SLOPEDEFLECTION ETHOD The slopedeflection method uses displacements as unknowns and is referred to as a displacement method. In the slopedeflection method, the moments at the ends of the members are
More informationStatics and Influence Functions From a Modern Perspective
Statics and Influence Functions From a Modern Perspective Friedel Hartmann Peter Jahn Statics and Influence Functions From a Modern Perspective 123 Friedel Hartmann Department of Civil Engineering University
More information4. 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 informationIntroduction to Finite Element Method
Introduction to Finite Element Method Dr. Rakesh K Kapania Aerospace and Ocean Engineering Department Virginia Polytechnic Institute and State University, Blacksburg, VA AOE 524, Vehicle Structures Summer,
More informationAdvanced Structural Analysis Prof. Devdas Menon Department of Civil Engineering Indian Institute of Technology, Madras
Advanced Structural Analysis Prof. Devdas Menon Department of Civil Engineering Indian Institute of Technology, Madras Module  5.2 Lecture  28 Matrix Analysis of Beams and Grids (Refer Slide Time: 00:23)
More informationStructural Matrices in MDOF Systems
in MDOF Systems http://intranet.dica.polimi.it/people/boffigiacomo Dipartimento di Ingegneria Civile Ambientale e Territoriale Politecnico di Milano April 9, 2016 Outline Additional Static Condensation
More informationMechanics of Solids. Mechanics Of Solids. Suraj kr. Ray Department of Civil Engineering
Mechanics Of Solids Suraj kr. Ray (surajjj2445@gmail.com) Department of Civil Engineering 1 Mechanics of Solids is a branch of applied mechanics that deals with the behaviour of solid bodies subjected
More informationPresented By: EAS 6939 Aerospace Structural Composites
A Beam Theory for Laminated Composites and Application to Torsion Problems Dr. BhavaniV. Sankar Presented By: Sameer Luthra EAS 6939 Aerospace Structural Composites 1 Introduction Composite beams have
More informationStress transformation and Mohr s circle for stresses
Stress transformation and Mohr s circle for stresses 1.1 General State of stress Consider a certain body, subjected to external force. The force F is acting on the surface over an area da of the surface.
More informationThermomechanical Effects
3 hermomechanical Effects 3 Chapter 3: HERMOMECHANICAL EFFECS ABLE OF CONENS Page 3. Introduction..................... 3 3 3.2 hermomechanical Behavior............... 3 3 3.2. hermomechanical Stiffness
More informationDeflections and Strains in Cracked Shafts due to Rotating Loads: A Numerical and Experimental Analysis
Rotating Machinery, 10(4): 283 291, 2004 Copyright c Taylor & Francis Inc. ISSN: 1023621X print / 15423034 online DOI: 10.1080/10236210490447728 Deflections and Strains in Cracked Shafts due to Rotating
More informationBasics of Finite Element Analysis. Strength of Materials, Solid Mechanics
Basics of Finite Element Analysis Strength of Materials, Solid Mechanics 1 Outline of Presentation Basic concepts in mathematics Analogies and applications Approximations to Actual Applications Improvisation
More informationFORMULATION OF THE INTERNAL STRESS EQUATIONS OF PINNED PORTAL FRAMES PUTTING AXIAL DEFORMATION INTO CONSIDERATION
FORMUATION OF THE INTERNA STRESS EQUATIONS OF PINNED PORTA FRAMES PUTTING AXIA DEFORMATION INTO CONSIDERATION Okonkwo V. O. B.Eng, M.Eng, MNSE, COREN.ecturer, Department of Civil Engineering, Nnamdi Azikiwe
More informationcos(θ)sin(θ) Alternative Exercise Correct Correct θ = 0 skiladæmi 10 Part A Part B Part C Due: 11:59pm on Wednesday, November 11, 2015
skiladæmi 10 Due: 11:59pm on Wednesday, November 11, 015 You will receive no credit for items you complete after the assignment is due Grading Policy Alternative Exercise 1115 A bar with cross sectional
More informationSixth Term Examination Papers 9475 MATHEMATICS 3
Sixth Term Examination Papers 9475 MATHEMATICS 3 Morning WEDNESDAY 26 JUNE 2013 Time: 3 hours Additional Materials: Answer Booklet Formulae Booklet INSTRUCTIONS TO CANDIDATES Please read this page carefully,
More informationA NEW SIMPLIFIED AND EFFICIENT TECHNIQUE FOR FRACTURE BEHAVIOR ANALYSIS OF CONCRETE STRUCTURES
Fracture Mechanics of Concrete Structures Proceedings FRAMCOS3 AEDFCATO Publishers, D79104 Freiburg, Germany A NEW SMPLFED AND EFFCENT TECHNQUE FOR FRACTURE BEHAVOR ANALYSS OF CONCRETE STRUCTURES K.
More informationAdvanced Structural Analysis Prof. Devdas Menon Department of Civil Engineering Indian Institute of Technology, Madras
Advanced Structural Analysis Prof. Devdas Menon Department of Civil Engineering Indian Institute of Technology, Madras Module  4.3 Lecture  24 Matrix Analysis of Structures with Axial Elements (Refer
More informationBeam 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 informationThe thin plate theory assumes the following deformational kinematics:
MEG6007 (Fall, 2017)  Solutions of Homework # 8 (ue on Tuesay, 29 November 2017) 8.0 Variational Derivation of Thin Plate Theory The thin plate theory assumes the following eformational kinematics: u
More informationPlastic Analysis and Design of Steel Structures
Plastic Analysis and Design of Steel Structures This page intentionally left blank Plastic Analysis and Design of Steel Structures M. Bill Wong Department of Civil Engineering Monash University, Australia
More information