k 21 k 22 k 23 k 24 k 31 k 32 k 33 k 34 k 41 k 42 k 43 k 44

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "k 21 k 22 k 23 k 24 k 31 k 32 k 33 k 34 k 41 k 42 k 43 k 44"

Transcription

1 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 force and bending moment at its ends We will consider only bending and not include axial force for this lab Define a local coordinate system x y where the x axis is attached to the long dimension of the member and runs from the i initial) end of the member to the j terminal) end of the member Note that the SAP local axis is analogous to the x axis We seek a relationship between the shear and bending moment at the member ends and the transverse displacement and rotation at the ends of the form:! " V i M i V j M j * * = * * * )* k k k k4 k k k k 4 k k k k 4 k 4 k 4 k 4 k 44! - - -" - -,- Δ i θ i Δ j θ j where the sixteen terms in the 4x4 matrix are the stiffness influence coefficients which make up the element stiffness matrix in local coordinates θ j V j x y y θ i Δ j j M j V i EI M i Δ i i φ x Vukazich CE 6 Beam Direct Stiffness ab []

2 We can find this relationship from the analysis of a fixed-fixed beam:! " V i M i V j M j ) = * EI EI 4EI EI EI EI EI 4EI,! " - Δ i θ i Δ j θ j ) we can write Eqn ) in shorthand form: Q = k where the 4x4 beam element stiffness matrix in local coordinates is [ k! ] = EI EI 4EI EI EI EI EI 4EI ) Note that, as was the case in the truss element, the [k ] matrix is symmetric k pq = k qp for p q) Note that all stiffness matrices are symmetric Example of Direct Stiffness Assembly of the Beam Structure System of Equations In order to illustrate the concept of the assembly of a system of equations for the entire beam structure, consider the beam that is made of two elements labeled and as shown: Vukazich CE 6 Beam Direct Stiffness ab []

3 y, y P x, x Note that the global degrees of freedom are labeled related to the joint numbering For a general joint with number n; n - n n For this example, the beam has joints and so there are a total of 6 global DOF for the structure and so, similar to the truss problem, the structure system of equations will be of the form: F = K ) where for this example; [K] = 6x6 global stiffness matrix; {F} = 6x vector of applied joint forces and support reactions); {Δ} = 6x vector of joint displacements Assembly of the Truss Structure Stiffness Matrix We will assemble the 6x6 structure stiffness matrix from the 4x4 element stiffness matrices Eqn 7) for elements and A connectivity table is constructed that maps each element DOF with its corresponding global DOF shaded) Element DOF Vukazich CE 6 Beam Direct Stiffness ab []

4 Δ i ) θ i ) Δ j ) 4 θ j ) Associated global DOF for element 4 Associated global DOF for element The structure global) stiffness matrix is assembled from the two element contributions [ k] = 4 k k k k4 k k k k 4 k k k k 4 4 k 4 k 4 k 4 k 44 [ k] = k k k k4 4 k k k k 4 5 k k k k 4 6 k 4 k 4 k 4 k 44 4) Note that the rows and columns of the element stiffness matrices are labeled with their corresponding global DOF in order to aid the assembly of the structure system of equations which yields the 6x6 structure global) stiffness matrix [ K] = k k k 4 k k k k k 4 k k k k k 4 k 5 k 6 k 4 k 4 k 4 k 4 k k 44 k k k 4 k k k k 4 k 4 k 4 k 4 k 44 5) Beam Structure Global) System of Equations Next the beam structure system of equations can be assembled Note that global DOF, 4 and 6 are unrestrained free) and DOF, and 5 are restrained supported) We can partition the structure system of equations by restrained and unrestrained DOF With the unrestrained partitions shaded below Vukazich CE 6 Beam Direct Stiffness ab [] 4

5 " V M P V 5 ) = * K K K K 4 K K K K 4 K K K K 4 K 5 K 6 K 4 K 4 K 4 K 44 K 45 K 46 K 5 K 54 K 55 K 56 K 6 K 64 K 65 K 66," - Δ 6) Note at the unrestrained DOF we know the forces or moments applied to the joints but the joint displacements and rotations are unknown At the restrained DOF we know that the displacements or rotations) are equal to zero at the supports but we do not know the support reactions We can solve the following system of equations for the unknown displacements at the unrestrained DOF " P ) = * K K 4 K 4 K 6 K 44 K 64 K 6 K 46 K 66," - Δ or equivalently! " K K 4 K 4 K 6 K 44 K 64 K 6 K 46 K 66 * ) * Δ, * * - = ) * * P, * - * 7) once Δ, θ4, and θ6 are found by solving the system of equations shown in Eqn 7, the support reactions can be found by performing the matrix multiplication! " V M V 5 * = * * )* K K 4 K K 4 K 5 K 54 K 56! - -" -,- Δ 8) and Eqn can be used to find the shear and bending moments at the ends of the individual beam members Vukazich CE 6 Beam Direct Stiffness ab [] 5

6 CE 6 Direct Stiffness Beam Analysis ab Problem y, y k x, x ft 6 ft For the beam shown the properties of the elements are: Member Section I E W8x 8 in 4 9 ksi W8x 8 in 4 9 ksi Using the coordinate system given in the figure: Find the 4x4 element stiffness matrices be guided by Eqn ) and write the values in the spaces below Use force units of kips and length units of inches for all calculations 4 [k] = 4 Vukazich CE 6 Beam Direct Stiffness ab [] 6

7 4 5 6 [k] = Assemble the 6x6 structure stiffness matrix be guided by Eqn 5) from the element contributions found in Step and write the values in the spaces below [K] = Vukazich CE 6 Beam Direct Stiffness ab [] 7

8 From the structure system of equations write the x system of equations Eqn 7) for the unrestrained DOF in the space below Δ 4 Verify that the solution to the x system of equations from Step is:! = 855 in θ! = 559 rad θ! = 54 rad 5 Find the support reactions V, M, and V 5 ) using Eqn 8 and your results from Step 4 6 Using statics, verify that the results from Step 5 satisfy equilibrium of the beam Vukazich CE 6 Beam Direct Stiffness ab [] 8

Method of Virtual Work Frame Deflection Example Steven Vukazich San Jose State University

Method of Virtual Work Frame Deflection Example Steven Vukazich San Jose State University Method of Virtual Work Frame Deflection xample Steven Vukazich San Jose State University Frame Deflection xample 9 k k D 4 ft θ " # The statically determinate frame from our previous internal force diagram

More information

Chapter 11. Displacement Method of Analysis Slope Deflection Method

Chapter 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 information

Lecture 8: Flexibility Method. Example

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

More information

P.E. Civil Exam Review:

P.E. Civil Exam Review: P.E. Civil Exam Review: Structural Analysis J.P. Mohsen Email: jpm@louisville.edu Structures Determinate Indeterminate STATICALLY DETERMINATE STATICALLY INDETERMINATE Stability and Determinacy of Trusses

More information

This procedure covers the determination of the moment of inertia about the neutral axis.

This procedure covers the determination of the moment of inertia about the neutral axis. 327 Sample Problems Problem 16.1 The moment of inertia about the neutral axis for the T-beam shown is most nearly (A) 36 in 4 (C) 236 in 4 (B) 136 in 4 (D) 736 in 4 This procedure covers the determination

More information

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

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

More information

Indeterminate Analysis Force Method 1

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

More information

Experimental Lab. Principles of Superposition

Experimental Lab. Principles of Superposition Experimental Lab Principles of Superposition Objective: The objective of this lab is to demonstrate and validate the principle of superposition using both an experimental lab and theory. For this lab you

More information

Lecture 11: The Stiffness Method. Introduction

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

More information

External Work. When a force F undergoes a displacement dx in the same direction i as the force, the work done is

External 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 information

Supplement: Statically Indeterminate Frames

Supplement: Statically Indeterminate Frames : Statically Indeterminate Frames Approximate Analysis - In this supplement, we consider another approximate method of solving statically indeterminate frames subjected to lateral loads known as the. Like

More information

Software Verification

Software Verification EXAMPLE 1-026 FRAME MOMENT AND SHEAR HINGES EXAMPLE DESCRIPTION This example uses a horizontal cantilever beam to test the moment and shear hinges in a static nonlinear analysis. The cantilever beam has

More information

General elastic beam with an elastic foundation

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

More information

5. What is the moment of inertia about the x - x axis of the rectangular beam shown?

5. What is the moment of inertia about the x - x axis of the rectangular beam shown? 1 of 5 Continuing Education Course #274 What Every Engineer Should Know About Structures Part D - Bending Strength Of Materials NOTE: The following question was revised on 15 August 2018 1. The moment

More information

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

Module 3. Analysis of Statically Indeterminate Structures by the Displacement Method odule 3 Analysis of Statically Indeterminate Structures by the Displacement ethod Lesson 21 The oment- Distribution ethod: rames with Sidesway Instructional Objectives After reading this chapter the student

More information

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

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

More information

Structural Analysis of Truss Structures using Stiffness Matrix. Dr. Nasrellah Hassan Ahmed

Structural Analysis of Truss Structures using Stiffness Matrix. Dr. Nasrellah Hassan Ahmed Structural Analysis of Truss Structures using Stiffness Matrix Dr. Nasrellah Hassan Ahmed FUNDAMENTAL RELATIONSHIPS FOR STRUCTURAL ANALYSIS In general, there are three types of relationships: Equilibrium

More information

Structural Dynamics. Spring mass system. The spring force is given by and F(t) is the driving force. Start by applying Newton s second law (F=ma).

Structural Dynamics. Spring mass system. The spring force is given by and F(t) is the driving force. Start by applying Newton s second law (F=ma). Structural Dynamics Spring mass system. The spring force is given by and F(t) is the driving force. Start by applying Newton s second law (F=ma). We will now look at free vibrations. Considering the free

More information

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

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

More information

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

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

More information

Introduction to Structural Member Properties

Introduction to Structural Member Properties Introduction to Structural Member Properties Structural Member Properties Moment of Inertia (I): a mathematical property of a cross-section (measured in inches 4 or in 4 ) that gives important information

More information

CIVL 8/7117 Chapter 12 - Structural Dynamics 1/75. To discuss the dynamics of a single-degree-of freedom springmass

CIVL 8/7117 Chapter 12 - Structural Dynamics 1/75. To discuss the dynamics of a single-degree-of freedom springmass CIV 8/77 Chapter - /75 Introduction To discuss the dynamics of a single-degree-of freedom springmass system. To derive the finite element equations for the time-dependent stress analysis of the one-dimensional

More information

EML4507 Finite Element Analysis and Design EXAM 1

EML4507 Finite Element Analysis and Design EXAM 1 2-17-15 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 information

DEFLECTION CALCULATIONS (from Nilson and Nawy)

DEFLECTION CALCULATIONS (from Nilson and Nawy) DEFLECTION CALCULATIONS (from Nilson and Nawy) The deflection of a uniformly loaded flat plate, flat slab, or two-way slab supported by beams on column lines can be calculated by an equivalent method that

More information

Level 7 Postgraduate Diploma in Engineering Computational mechanics using finite element method

Level 7 Postgraduate Diploma in Engineering Computational mechanics using finite element method 9210-203 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 information

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

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

More information

TORSION INCLUDING WARPING OF OPEN SECTIONS (I, C, Z, T AND L SHAPES)

TORSION INCLUDING WARPING OF OPEN SECTIONS (I, C, Z, T AND L SHAPES) Page1 TORSION INCLUDING WARPING OF OPEN SECTIONS (I, C, Z, T AND L SHAPES) Restrained warping for the torsion of thin-wall open sections is not included in most commonly used frame analysis programs. Almost

More information

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

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

More information

Methods of Analysis. Force or Flexibility Method

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

More information

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

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

More information

BEAM DEFLECTION THE ELASTIC CURVE

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

More information

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

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

More information

UNIT IV FLEXIBILTY AND STIFFNESS METHOD

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

More information

18. FAST NONLINEAR ANALYSIS. The Dynamic Analysis of a Structure with a Small Number of Nonlinear Elements is Almost as Fast as a Linear Analysis

18. FAST NONLINEAR ANALYSIS. The Dynamic Analysis of a Structure with a Small Number of Nonlinear Elements is Almost as Fast as a Linear Analysis 18. FAS NONLINEAR ANALYSIS he Dynamic Analysis of a Structure with a Small Number of Nonlinear Elements is Almost as Fast as a Linear Analysis 18.1 INRODUCION he response of real structures when subjected

More information

Ph.D. Preliminary Examination Analysis

Ph.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 information

Part 1 is to be completed without notes, beam tables or a calculator. DO NOT turn Part 2 over until you have completed and turned in Part 1.

Part 1 is to be completed without notes, beam tables or a calculator. DO NOT turn Part 2 over until you have completed and turned in Part 1. NAME CM 3505 Fall 06 Test 2 Part 1 is to be completed without notes, beam tables or a calculator. Part 2 is to be completed after turning in Part 1. DO NOT turn Part 2 over until you have completed and

More information

FEA CODE WITH MATLAB. Finite Element Analysis of an Arch ME 5657 FINITE ELEMENT METHOD. Submitted by: ALPAY BURAK DEMIRYUREK

FEA CODE WITH MATLAB. Finite Element Analysis of an Arch ME 5657 FINITE ELEMENT METHOD. Submitted by: ALPAY BURAK DEMIRYUREK FEA CODE WITH MATAB Finite Element Analysis of an Arch ME 5657 FINITE EEMENT METHOD Submitted by: APAY BURAK DEMIRYUREK This report summarizes the finite element analysis of an arch-beam with using matlab.

More information

Multi Linear Elastic and Plastic Link in SAP2000

Multi Linear Elastic and Plastic Link in SAP2000 26/01/2016 Marco Donà Multi Linear Elastic and Plastic Link in SAP2000 1 General principles Link object connects two joints, i and j, separated by length L, such that specialized structural behaviour may

More information

3 Relation between complete and natural degrees of freedom

3 Relation between complete and natural degrees of freedom Stiffness matrix for D tapered beams by ouie. Yaw, PhD, PE, SE Walla Walla University March 9, 9 Introduction This article presents information necessary for the construction of the stiffness matrix of

More information

FLEXIBILITY METHOD FOR INDETERMINATE FRAMES

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

More information

Structural Analysis III Compatibility of Displacements & Principle of Superposition

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

More information

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

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

More information

Stress Analysis Lecture 3 ME 276 Spring Dr./ Ahmed Mohamed Nagib Elmekawy

Stress Analysis Lecture 3 ME 276 Spring Dr./ Ahmed Mohamed Nagib Elmekawy Stress Analysis Lecture 3 ME 276 Spring 2017-2018 Dr./ Ahmed Mohamed Nagib Elmekawy Axial Stress 2 Beam under the action of two tensile forces 3 Beam under the action of two tensile forces 4 Shear Stress

More information

14. *14.8 CASTIGLIANO S THEOREM

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

More information

Mechanical Design in Optical Engineering

Mechanical 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 information

Due Monday, September 14 th, 12:00 midnight

Due 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 information

SLOPE-DEFLECTION METHOD

SLOPE-DEFLECTION METHOD SLOPE-DEFLECTION ETHOD The slope-deflection method uses displacements as unknowns and is referred to as a displacement method. In the slope-deflection method, the moments at the ends of the members are

More information

Sstan builds the global equation R = KG*r starting at the element level just like you did by hand

Sstan builds the global equation R = KG*r starting at the element level just like you did by hand CES - Stress Analysis Spring 999 Ex. #, the following -D truss is to be analyzed using Sstan (read the online stan intro first, and Ch- in Hoit) k k 0 ft E= 9000 ksi A= 0 in*in 0 ft Sstan builds the global

More information

4.5 The framework element stiffness matrix

4.5 The framework element stiffness matrix 45 The framework element stiffness matri Consider a 1 degree-of-freedom element that is straight prismatic and symmetric about both principal cross-sectional aes For such a section the shear center coincides

More information

Chapter 5 Structural Elements: The truss & beam elements

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

More information

Supplement: Statically Indeterminate Trusses and Frames

Supplement: 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 information

ANSWERS September 2014

ANSWERS September 2014 NSWERS September 2014 nswers to selected questions: Sheet # (1) (2) () (4) SCE-55 D SCE-86 D SCE-88 D C MCM-21 MCM-12 D MMC-80 C C D MCM-52 D D MCM-1 C D D MCM-51 D D MCM-57 D D MCM-60 D MLS-12 C D NS

More information

Lecture 15 Strain and stress in beams

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

More information

Deflections. Deflections. Deflections. Deflections. Deflections. Deflections. dx dm V. dx EI. dx EI dx M. dv w

Deflections. Deflections. Deflections. Deflections. Deflections. Deflections. dx dm V. dx EI. dx EI dx M. dv w CIVL 311 - Conjugate eam 1/5 Conjugate beam method The development of the conjugate beam method has been atributed to several strucutral engineers. any credit Heinrich üller-reslau (1851-195) with the

More information

Lecture 8: Assembly of beam elements.

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

More information

Laboratory 4 Topic: Buckling

Laboratory 4 Topic: Buckling Laboratory 4 Topic: Buckling Objectives: To record the load-deflection response of a clamped-clamped column. To identify, from the recorded response, the collapse load of the column. Introduction: Buckling

More information

Section Downloads. Section Downloads. Handouts & Slides can be printed. Course binders are available for purchase. Download & Print. Version 2.

Section Downloads. Section Downloads. Handouts & Slides can be printed. Course binders are available for purchase. Download & Print. Version 2. Level II: Section 03 Design Principles Section Downloads 2 Section Downloads Handouts & Slides can be printed Version 2.0 Course binders are available for purchase Not required Download & Print TTT II

More information

Chapter 8 Supplement: Deflection in Beams Double Integration Method

Chapter 8 Supplement: Deflection in Beams Double Integration Method Chapter 8 Supplement: Deflection in Beams Double Integration Method 8.5 Beam Deflection Double Integration Method In this supplement, we describe the methods for determining the equation of the deflection

More information

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

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

More information

Due Tuesday, September 21 st, 12:00 midnight

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

More information

ME Final Exam. PROBLEM NO. 4 Part A (2 points max.) M (x) y. z (neutral axis) beam cross-sec+on. 20 kip ft. 0.2 ft. 10 ft. 0.1 ft.

ME Final Exam. PROBLEM NO. 4 Part A (2 points max.) M (x) y. z (neutral axis) beam cross-sec+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 11:30AM-12:20PM Ghosh 2:30-3:20PM Gonzalez 12:30-1:20PM Zhao 4:30-5:20PM M (x) y 20 kip ft 0.2

More information

Towards The. Design of Super Columns. Prof. AbdulQader Najmi

Towards The. Design of Super Columns. Prof. AbdulQader Najmi Towards The Design of Super Columns Prof. AbdulQader Najmi Description: Tubular Column Square or Round Filled with Concrete Provided with U-Links welded to its Walls as shown in Figure 1 Compression Specimen

More information

Discretization Methods Exercise # 5

Discretization Methods Exercise # 5 Discretization Methods Exercise # 5 Static calculation of a planar truss structure: a a F Six steps: 1. Discretization 2. Element matrices 3. Transformation 4. Assembly 5. Boundary conditions 6. Solution

More information

Chapter 2: Deflections of Structures

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

More information

Geometric Stiffness Effects in 2D and 3D Frames

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

More information

Example: 5-panel parallel-chord truss. 8 ft. 5 k 5 k 5 k 5 k. F yield = 36 ksi F tension = 21 ksi F comp. = 10 ksi. 6 ft.

Example: 5-panel parallel-chord truss. 8 ft. 5 k 5 k 5 k 5 k. F yield = 36 ksi F tension = 21 ksi F comp. = 10 ksi. 6 ft. CE 331, Spring 2004 Beam Analogy for Designing Trusses 1 / 9 We need to make several decisions in designing trusses. First, we need to choose a truss. Then we need to determine the height of the truss

More information

D : SOLID MECHANICS. Q. 1 Q. 9 carry one mark each.

D : 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 N-s/m. To make the system

More information

Chapter 2: Rigid Bar Supported by Two Buckled Struts under Axial, Harmonic, Displacement Excitation..14

Chapter 2: Rigid Bar Supported by Two Buckled Struts under Axial, Harmonic, Displacement Excitation..14 Table of Contents Chapter 1: Research Objectives and Literature Review..1 1.1 Introduction...1 1.2 Literature Review......3 1.2.1 Describing Vibration......3 1.2.2 Vibration Isolation.....6 1.2.2.1 Overview.

More information

4.3 Moment Magnification

4.3 Moment Magnification CHAPTER 4: Reinforced Concrete Columns 4.3 Moment Magnification Description An ordinary or first order frame analysis does not include either the effects of the lateral sidesway deflections of the column

More information

10. Applications of 1-D Hermite elements

10. Applications of 1-D Hermite elements 10. Applications of 1-D Hermite elements... 1 10.1 Introduction... 1 10.2 General case fourth-order beam equation... 3 10.3 Integral form... 5 10.4 Element Arrays... 7 10.5 C1 Element models... 8 10.6

More information

ARCE 306: MATRIX STRUCTURAL ANALYSIS. Homework 1 (Due )

ARCE 306: MATRIX STRUCTURAL ANALYSIS. Homework 1 (Due ) Winter Quarter ARCE 6: MATRIX STRUCTURA ANAYSIS January 4, Problem Homework (Due -6-) k k/ft A B C D E ft 5 ft 5 ft 5 ft () Use the slope deflection method to find the bending moment diagram for the continuous

More information

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.

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. 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 information

STRUCTURAL SURFACES & FLOOR GRILLAGES

STRUCTURAL 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 information

Lecture 7: The Beam Element Equations.

Lecture 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 information

Presented By: EAS 6939 Aerospace Structural Composites

Presented 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 information

Symmetric Bending of Beams

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

More information

3.4 Analysis for lateral loads

3.4 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 low-redundant braced structures is reviewed. Member Force Analysis

More information

Portal Frame Calculations Lateral Loads

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

More information

Mechanics of Inflatable Fabric Beams

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

More information

2 marks Questions and Answers

2 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 information

Structural Dynamics Lecture 4. Outline of Lecture 4. Multi-Degree-of-Freedom Systems. Formulation of Equations of Motions. Undamped Eigenvibrations.

Structural Dynamics Lecture 4. Outline of Lecture 4. Multi-Degree-of-Freedom Systems. Formulation of Equations of Motions. Undamped Eigenvibrations. Outline of Multi-Degree-of-Freedom 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 information

University of California at Berkeley Structural Engineering Mechanics & Materials Department of Civil & Environmental Engineering Spring 2012 Student name : Doctoral Preliminary Examination in Dynamics

More information

CIV 207 Winter For practice

CIV 207 Winter For practice CIV 07 Winter 009 Assignment #10 Friday, March 0 th Complete the first three questions. Submit your work to Box #5 on the th floor of the MacDonald building by 1 noon on Tuesday March 31 st. No late submissions

More information

Finite Element Modelling with Plastic Hinges

Finite Element Modelling with Plastic Hinges 01/02/2016 Marco Donà Finite Element Modelling with Plastic Hinges 1 Plastic hinge approach A plastic hinge represents a concentrated post-yield behaviour in one or more degrees of freedom. Hinges only

More information

Mechanical Properties of Materials

Mechanical Properties of Materials Mechanical Properties of Materials Strains Material Model Stresses Learning objectives Understand the qualitative and quantitative description of mechanical properties of materials. Learn the logic of

More information

Module 3. Analysis of Statically Indeterminate Structures by the Displacement Method. Version 2 CE IIT, Kharagpur

Module 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 information

M.S Comprehensive Examination Analysis

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

More information

Seismic Pushover Analysis Using AASHTO Guide Specifications for LRFD Seismic Bridge Design

Seismic Pushover Analysis Using AASHTO Guide Specifications for LRFD Seismic Bridge Design Seismic Pushover Analysis Using AASHTO Guide Specifications for LRFD Seismic Bridge Design Elmer E. Marx, Alaska Department of Transportation and Public Facilities Michael Keever, California Department

More information

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

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

More information

techie-touch.blogspot.com DEPARTMENT OF CIVIL ENGINEERING ANNA UNIVERSITY QUESTION BANK CE 2302 STRUCTURAL ANALYSIS-I TWO MARK QUESTIONS UNIT I DEFLECTION OF DETERMINATE STRUCTURES 1. Write any two important

More information

Errors in FE Modelling (Section 5.10)

Errors in FE Modelling (Section 5.10) Errors in FE Modelling (Section 5.10) Modelling error : arises because physical reality is replaced by a mathematical model. Example: A beam that can resist both axial and transverse loads being modelled

More information

Method of Consistent Deformation

Method of Consistent Deformation Method of onsistent eformation Structural nalysis y R.. Hibbeler Theory of Structures-II M Shahid Mehmood epartment of ivil Engineering Swedish ollege of Engineering and Technology, Wah antt FRMES Method

More information

3.5 Reinforced Concrete Section Properties

3.5 Reinforced Concrete Section Properties CHAPER 3: Reinforced Concrete Slabs and Beams 3.5 Reinforced Concrete Section Properties Description his application calculates gross section moment of inertia neglecting reinforcement, moment of inertia

More information

COLUMNS: BUCKLING (DIFFERENT ENDS)

COLUMNS: BUCKLING (DIFFERENT ENDS) COLUMNS: BUCKLING (DIFFERENT ENDS) Buckling of Long Straight Columns Example 4 Slide No. 1 A simple pin-connected truss is loaded and supported as shown in Fig. 1. All members of the truss are WT10 43

More information

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

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

More information

Structural 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 Structural Dynamics Lecture Eleven: Dynamic Response of MDOF Systems: (Chapter 11) By: H. Ahmadian ahmadian@iust.ac.ir Dynamic Response of MDOF Systems: Mode-Superposition Method Mode-Superposition Method:

More information

1 of 12. Given: Law of Cosines: C. Law of Sines: Stress = E = G

1 of 12. Given: Law of Cosines: C. Law of Sines: Stress = E = G ES230 STRENGTH OF MATERIALS FINAL EXAM: WEDNESDAY, MAY 15 TH, 4PM TO 7PM, AEC200 Closed book. Calculator and writing supplies allowed. Protractor and compass required. 180 Minute Time Limit You must have

More information

UNIT II SLOPE DEFLECION AND MOMENT DISTRIBUTION METHOD

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

More information

Chapter 7: Internal Forces

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

More information

6. Bending CHAPTER OBJECTIVES

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

More information