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


 Noel Mason
 2 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 informationIf the number of unknown reaction components are equal to the number of equations, the structure is known as statically determinate.
1 of 6 EQUILIBRIUM OF A RIGID BODY AND ANALYSIS OF ETRUCTURAS II 9.1 reactions in supports and joints of a twodimensional structure and statically indeterminate reactions: Statically indeterminate structures
More informationGeometric Stiffness Effects in 2D and 3D Frames
Geometric Stiffness Effects in D and 3D Frames CEE 41. Matrix Structural Analsis Department of Civil and Environmental Engineering Duke Universit Henri Gavin Fall, 1 In situations in which deformations
More 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 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 informationMethods of Analysis. Force or Flexibility Method
INTRODUCTION: The structural analysis is a mathematical process by which the response of a structure to specified loads is determined. This response is measured by determining the internal forces or stresses
More information2 marks Questions and Answers
1. Define the term strain energy. A: Strain Energy of the elastic body is defined as the internal work done by the external load in deforming or straining the body. 2. Define the terms: Resilience and
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 informationChapter 4 Deflection and Stiffness
Chapter 4 Deflection and Stiffness Asst. Prof. Dr. Supakit Rooppakhun Chapter Outline Deflection and Stiffness 41 Spring Rates 42 Tension, Compression, and Torsion 43 Deflection Due to Bending 44 Beam
More informationMethod of Consistent Deformation
Method of onsistent eformation Structural nalysis y R.. Hibbeler Theory of StructuresII M Shahid Mehmood epartment of ivil Engineering Swedish ollege of Engineering and Technology, Wah antt FRMES Method
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 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 information3 2 6 Solve the initial value problem u ( t) 3. a If A has eigenvalues λ =, λ = 1 and corresponding eigenvectors 1
Math Problem a If A has eigenvalues λ =, λ = 1 and corresponding eigenvectors 1 3 6 Solve the initial value problem u ( t) = Au( t) with u (0) =. 3 1 u 1 =, u 1 3 = b True or false and why 1. if A is
More 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 informationFINAL EXAMINATION. (CE1302 Mechanics of Materials)
UNIVERSITY OF CLIFORNI, ERKELEY FLL SEMESTER 001 FINL EXMINTION (CE130 Mechanics of Materials) Problem 1: (15 points) pinned bar structure is shown in Figure 1. There is an external force, W = 5000N,
More information6. 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 informationLecture 4: PRELIMINARY CONCEPTS OF STRUCTURAL ANALYSIS. Introduction
Introduction In this class we will focus on the structural analysis of framed structures. We will learn about the flexibility method first, and then learn how to use the primary analytical tools associated
More informationSupplement: Statically Indeterminate Trusses and Frames
: Statically Indeterminate Trusses and Frames Approximate Analysis  In this supplement, we consider an approximate method of solving statically indeterminate trusses and frames subjected to lateral loads
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 informationFinite Element Method in Geotechnical Engineering
Finite Element Method in Geotechnical Engineering Short Course on + Dynamics Boulder, Colorado January 58, 2004 Stein Sture Professor of Civil Engineering University of Colorado at Boulder Contents Steps
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 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 informationSlender Structures Load carrying principles
Slender Structures Load carrying principles Basic cases: Extension, Shear, Torsion, Cable Bending (Euler) v0171 Hans Welleman 1 Content (preliminary schedule) Basic cases Extension, shear, torsion, cable
More informationStructural Analysis. For. Civil Engineering.
Structural Analysis For Civil Engineering By www.thegateacademy.com ` Syllabus for Structural Analysis Syllabus Statically Determinate and Indeterminate Structures by Force/ Energy Methods; Method of Superposition;
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 informationM5 Simple Beam Theory (continued)
M5 Simple Beam Theory (continued) Reading: Crandall, Dahl and Lardner 7.7.6 In the previous lecture we had reached the point of obtaining 5 equations, 5 unknowns by application of equations of elasticity
More informationSTRUCTURAL SURFACES & FLOOR GRILLAGES
STRUCTURAL SURFACES & FLOOR GRILLAGES INTRODUCTION Integral car bodies are 3D structures largely composed of approximately subassemblies SSS Planar structural subassemblies can be grouped into two categories
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 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 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 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 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 informationMechanics PhD Preliminary Spring 2017
Mechanics PhD Preliminary Spring 2017 1. (10 points) Consider a body Ω that is assembled by gluing together two separate bodies along a flat interface. The normal vector to the interface is given by n
More 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 informationChapter 2 Basis for Indeterminate Structures
Chapter  Basis for the Analysis of Indeterminate Structures.1 Introduction... 3.1.1 Background... 3.1. Basis of Structural Analysis... 4. Small Displacements... 6..1 Introduction... 6.. Derivation...
More informationTable of Contents. Preface...xvii. Part 1. Level
Preface...xvii Part 1. Level 1... 1 Chapter 1. The Basics of Linear Elastic Behavior... 3 1.1. Cohesion forces... 4 1.2. The notion of stress... 6 1.2.1. Definition... 6 1.2.2. Graphical representation...
More informationSEMM Mechanics PhD Preliminary Exam Spring Consider a twodimensional rigid motion, whose displacement field is given by
SEMM Mechanics PhD Preliminary Exam Spring 2014 1. Consider a twodimensional rigid motion, whose displacement field is given by u(x) = [cos(β)x 1 + sin(β)x 2 X 1 ]e 1 + [ sin(β)x 1 + cos(β)x 2 X 2 ]e
More informationME325 EXAM I (Sample)
ME35 EXAM I (Sample) NAME: NOTE: COSED BOOK, COSED NOTES. ONY A SINGE 8.5x" ORMUA SHEET IS AOWED. ADDITIONA INORMATION IS AVAIABE ON THE AST PAGE O THIS EXAM. DO YOUR WORK ON THE EXAM ONY (NO SCRATCH PAPER
More informationLecture M1 Slender (one dimensional) Structures Reading: Crandall, Dahl and Lardner 3.1, 7.2
Lecture M1 Slender (one dimensional) Structures Reading: Crandall, Dahl and Lardner 3.1, 7.2 This semester we are going to utilize the principles we learnt last semester (i.e the 3 great principles and
More informationShafts: Torsion of Circular Shafts Reading: Crandall, Dahl and Lardner 6.2, 6.3
M9 Shafts: Torsion of Circular Shafts Reading: Crandall, Dahl and Lardner 6., 6.3 A shaft is a structural member which is long and slender and subject to a torque (moment) acting about its long axis. We
More informationLecture Slides. Chapter 4. Deflection and Stiffness. The McGrawHill Companies 2012
Lecture Slides Chapter 4 Deflection and Stiffness The McGrawHill Companies 2012 Chapter Outline Force vs Deflection Elasticity property of a material that enables it to regain its original configuration
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 informationD : SOLID MECHANICS. Q. 1 Q. 9 carry one mark each.
GTE 2016 Q. 1 Q. 9 carry one mark each. D : SOLID MECHNICS Q.1 single degree of freedom vibrating system has mass of 5 kg, stiffness of 500 N/m and damping coefficient of 100 Ns/m. To make the system
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 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 informationUNIT IV FLEXIBILTY AND STIFFNESS METHOD
SIDDHARTH GROUP OF INSTITUTIONS :: PUTTUR Siddharth Nagar, Narayanavanam Road 517583 QUESTION BANK (DESCRIPTIVE) Subject with Code : SAII (13A01505) Year & Sem: IIIB.Tech & ISem Course & Branch: B.Tech
More informationCHAPTER 5 Statically Determinate Plane Trusses
CHAPTER 5 Statically Determinate Plane Trusses TYPES OF ROOF TRUSS TYPES OF ROOF TRUSS ROOF TRUSS SETUP ROOF TRUSS SETUP OBJECTIVES To determine the STABILITY and DETERMINACY of plane trusses To analyse
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 informationCHAPTER 5 Statically Determinate Plane Trusses TYPES OF ROOF TRUSS
CHAPTER 5 Statically Determinate Plane Trusses TYPES OF ROOF TRUSS 1 TYPES OF ROOF TRUSS ROOF TRUSS SETUP 2 ROOF TRUSS SETUP OBJECTIVES To determine the STABILITY and DETERMINACY of plane trusses To analyse
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 informationExample 3.7 Consider the undeformed configuration of a solid as shown in Figure 3.60.
162 3. The linear 3D elasticity mathematical model The 3D elasticity model is of great importance, since it is our highest order hierarchical model assuming linear elastic behavior. Therefore, it provides
More informationUnit 13 Review of Simple Beam Theory
MIT  16.0 Fall, 00 Unit 13 Review of Simple Beam Theory Readings: Review Unified Engineering notes on Beam Theory BMP 3.8, 3.9, 3.10 T & G 1015 Paul A. Lagace, Ph.D. Professor of Aeronautics & Astronautics
More informationSupport Idealizations
IVL 3121 nalysis of Statically Determinant Structures 1/12 nalysis of Statically Determinate Structures nalysis of Statically Determinate Structures The most common type of structure an engineer will analyze
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 informationModule 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method
Module 2 Analysis of Statically Indeterminate Structures by the Matrix Force Method Lesson 11 The Force Method of Analysis: Frames Instructional Objectives After reading this chapter the student will be
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 informationM.S Comprehensive Examination Analysis
UNIVERSITY OF CALIFORNIA, BERKELEY Spring Semester 2014 Dept. of Civil and Environmental Engineering Structural Engineering, Mechanics and Materials Name:......................................... M.S Comprehensive
More 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 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 informationUsing the finite element method of structural analysis, determine displacements at nodes 1 and 2.
Question 1 A pinjointed plane frame, shown in Figure Q1, is fixed to rigid supports at nodes and 4 to prevent their nodal displacements. The frame is loaded at nodes 1 and by a horizontal and a vertical
More information2. Determine the deflection at C of the beam given in fig below. Use principal of virtual work. W L/2 B A L C
CE1259, Strength of Materials UNIT I STRESS, STRAIN DEFORMATION OF SOLIDS Part A 1. Define strain energy density. 2. State Maxwell s reciprocal theorem. 3. Define proof resilience. 4. State Castigliano
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 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 information3.4 Analysis for lateral loads
3.4 Analysis for lateral loads 3.4.1 Braced frames In this section, simple hand methods for the analysis of statically determinate or certain lowredundant braced structures is reviewed. Member Force Analysis
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 informationLevel 7 Postgraduate Diploma in Engineering Computational mechanics using finite element method
9210203 Level 7 Postgraduate Diploma in Engineering Computational mechanics using finite element method You should have the following for this examination one answer book No additional data is attached
More informationModule 3. Analysis of Statically Indeterminate Structures by the Displacement Method
odule 3 Analysis of Statically Indeterminate Structures by the Displacement ethod Lesson 16 The SlopeDeflection ethod: rames Without Sidesway Instructional Objectives After reading this chapter the student
More informationCHAPTER 14 BUCKLING ANALYSIS OF 1D AND 2D STRUCTURES
CHAPTER 14 BUCKLING ANALYSIS OF 1D AND 2D STRUCTURES 14.1 GENERAL REMARKS In structures where dominant loading is usually static, the most common cause of the collapse is a buckling failure. Buckling may
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 informationPh.D. Preliminary Examination Analysis
UNIVERSITY OF CALIFORNIA, BERKELEY Spring Semester 2014 Dept. of Civil and Environmental Engineering Structural Engineering, Mechanics and Materials Name:......................................... Ph.D.
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 informationComputational Stiffness Method
Computational Stiffness Method Hand calculations are central in the classical stiffness method. In that approach, the stiffness matrix is established columnbycolumn by setting the degrees of freedom
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 informationMechanics of Materials
Mechanics of Materials 2. Introduction Dr. Rami Zakaria References: 1. Engineering Mechanics: Statics, R.C. Hibbeler, 12 th ed, Pearson 2. Mechanics of Materials: R.C. Hibbeler, 9 th ed, Pearson 3. Mechanics
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 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 informationPh.D. Preliminary Examination Analysis
UNIVERSITY OF CALIFORNIA, BERKELEY Spring Semester 2017 Dept. of Civil and Environmental Engineering Structural Engineering, Mechanics and Materials Name:......................................... Ph.D.
More informationTorsion/Axial Illustration: 1 (3/30/00)
Torsion/Axial Illustration: 1 (3/30/00) Table of Contents Intro / General Strategy Axial: Different Materia The Displacement Method 1 2 Calculate the Stresses General Strategy The same structure is loaded
More informationStructural Dynamics Lecture 4. Outline of Lecture 4. MultiDegreeofFreedom Systems. Formulation of Equations of Motions. Undamped Eigenvibrations.
Outline of MultiDegreeofFreedom Systems Formulation of Equations of Motions. Newton s 2 nd Law Applied to Free Masses. D Alembert s Principle. Basic Equations of Motion for Forced Vibrations of Linear
More informationEquilibrium of a Particle
ME 108  Statics Equilibrium of a Particle Chapter 3 Applications For a spool of given weight, what are the forces in cables AB and AC? Applications For a given weight of the lights, what are the forces
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 informationModule 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method
Module 2 Analysis of Statically Indeterminate Structures by the Matrix Force Method Lesson 10 The Force Method of Analysis: Trusses Instructional Objectives After reading this chapter the student will
More informationModule 3. Analysis of Statically Indeterminate Structures by the Displacement Method
odule 3 Analysis of Statically Indeterminate Structures by the Displacement ethod Lesson 21 The oment Distribution ethod: rames with Sidesway Instructional Objectives After reading this chapter the student
More informationModule 1. Energy Methods in Structural Analysis
Module 1 Energy Methods in Structural Analysis esson 5 Virtual Work Instructional Objecties After studying this lesson, the student will be able to: 1. Define Virtual Work.. Differentiate between external
More informationTruss Structures: The Direct Stiffness Method
. Truss Structures: The Companies, CHAPTER Truss Structures: The Direct Stiffness Method. INTRODUCTION The simple line elements discussed in Chapter introduced the concepts of nodes, nodal displacements,
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 informationEML4507 Finite Element Analysis and Design EXAM 1
21715 Name (underline last name): EML4507 Finite Element Analysis and Design EXAM 1 In this exam you may not use any materials except a pencil or a pen, an 8.5x11 formula sheet, and a calculator. Whenever
More information6/6/2008. Qualitative Influence Lines for Statically Indeterminate Structures: MullerBreslau s Principle
Qualitative Influence Lines for Statically Indeterminate Structures: MullerBreslau s Principle The influence line for a force (or moment) response function is given by the deflected shape of the released
More informationUNIT I ENERGY PRINCIPLES
UNIT I ENERGY PRINCIPLES Strain energy and strain energy density strain energy in traction, shear in flexure and torsion Castigliano s theorem Principle of virtual work application of energy theorems
More informationChapter 11. Displacement Method of Analysis Slope Deflection Method
Chapter 11 Displacement ethod of Analysis Slope Deflection ethod Displacement ethod of Analysis Two main methods of analyzing indeterminate structure Force method The method of consistent deformations
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 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 informationk 21 k 22 k 23 k 24 k 31 k 32 k 33 k 34 k 41 k 42 k 43 k 44
CE 6 ab Beam Analysis by the Direct Stiffness Method Beam Element Stiffness Matrix in ocal Coordinates Consider an inclined bending member of moment of inertia I and modulus of elasticity E subjected shear
More informationLecture 11: The Stiffness Method. Introduction
Introduction Although the mathematical formulation of the flexibility and stiffness methods are similar, the physical concepts involved are different. We found that in the flexibility method, the unknowns
More information202 Index. failure, 26 field equation, 122 force, 1
Index acceleration, 12, 161 admissible function, 155 admissible stress, 32 Airy's stress function, 122, 124 d'alembert's principle, 165, 167, 177 amplitude, 171 analogy, 76 anisotropic material, 20 aperiodic
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 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 information