Analysis of Plane Frames

Size: px
Start display at page:

Download "Analysis of Plane Frames"

Transcription

1 Plane frames are two-dimensional structures constructed with straight elements connected together by rigid and/or hinged connections. rames are subjected to loads and reactions that lie in the plane of the structure. Analysis of Plane rames Under the action of external loads, the elements of a plane frame are subjected to axial forces similar to truss members as well as bending moments and shears one would see in a beam. ence the analysis of plane frame members can be conveniently conducted by treating the frame as a composite structure with beam elements that can be subject to axial loads. These elements are usually rigidly connected or semi-rigidly connected depending on the amount of rotational restraint designed into the connection. D rame Moment connections throughout

2 Equivalent Joint oads The calculations of displacements in larger more extensive structures by the means of the matrix methods derived later requires that the structure be subject to loads applied only at the joints. Thus in general, loads are categorized into those applied at joints, and those that are not. oads that are not applied to joints must be replaced with statically equivalent loads. Consider the statically indeterminate frame with a distributed load between joints B and C:

3 The frame on the previous page is statically equivalent to the following two structure: The frame to the left is statically equivalent to the original frame in the sense that the reactions would be the same. owever the internal bending moment in the cross beam would not be the same. The frame would not be kinematically consistent with the original frame. owever, the reactions at the foundations and the joint actions would be equivalent. Keep in mind we are trying to find these unknown actions (forces and moments).

4 Example 8.1 Redundants are forces and moments Zero and nonzero displacements in the released structure due to external loads are shown 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 axial deformations. The flexural rigidity EI is constant. The axial rigidity EA is also constant. Joint B is a rigid connection and we will endeavor to preserve equilibrium at Joint B and throughout the structure. The structure is statically indeterminate to the third degree. A released structure is obtained by cutting the frame at joint B and the released actions Q 1, Q and Q are redundants. ind the magnitude and direction of these redundants.

5 The displacements in the released structure caused by P and corresponding to Q 1, Q and Q are depicted in the previous figures. These displacements in the released structure caused by the external load are designated D Q1, D Q and D Q respectively. The displacement D Q1 consists of the sum of two translations which are found by analyzing the released structure as a set of two cantilever beams AB and. irst analyze the cantilever beam AB. The load P will cause a downward translation at B and a clockwise rotation at B. There is no axial displacement thus D Q 1 AB The displacements D Q and D Q in the released structure AB consist of a vertical displacement and a rotations, i.e., D Q AB 5P 48EI D Q AB P 8EI But these displacements in the released structure AB have corresponding displacements in the released structure.

6 owever, since there is no load on member in the released structure, there will be no displacement at end B and Even though the displacements at B in are zero, the total displacement of joint B would be the summation of the two D Q components from AB and. Thus the D Q matrix is D Q D D D Q1 Q Q 5P 48EI P 8EI 5 6 We need to assemble the flexibility matrix. Consider the released structure with Q Q 1 1 Q P 48EI

7 The displacements at end B of member BC are The displacements corresponding to unit values of Q 1, Q and Q are shown as flexibility coefficients 11, 1 and 1. If both axial and flexural deformations are considered, the displacements at end B of member AB are AE AB AB AB EI EI The flexibility coefficients take on the following values AE 11 EI 1 1 EI

8 The same analysis must be made with Q Q 1 1 Q or frame sections AB and AB : EI 1 AB AB AB EI AE : 1 Which leads to flexibility coefficients 1 EI AE EI

9 Once again from frame sections AB and (Q = 1, Q 1 = Q = ) AB : EI 1 AB AB AB EI : EI 1 EI leading to flexibility coefficients 1 EI EI EI EI

10 Assembly of the flexibility matrix leads to EA EI EI EI EA EI EI EI EI EI Now let P 1 K 1 ft 144 inches E I, in 4 ksi A 1 in

11 When these numerical values and D Q EA EI EA 1 144, 144,.48 in / kip in / kip The axial compliance (flexibility) of each component is quite small relative to the flexural compliance (flexibility). We will ignore the axial compliance of the beam and the column when assembling the flexibility matrix. The inverse of compliance is stiffness. This is equivalent to stating that the axial stiffness of the beam and column is so large relative to flexural stiffness of the beam and column that axial displacements are negligible.

12 Omitting axial deformations leads to the following flexibility matrix When the flexibility matrix is inverted we obtain

13 With -1 and D Q we can compute the unknown redundants utilizing the matrix equation i.e., 1 Q D Q D Q Q kips 4.69 kips kip inches Note that the displacements associated with the redundants in the original structural, represented by the matrix {D Q } are zero because joint B is a rigid connection. One can rationalize the rotation D Q is zero from this assumption.

14 Now examine the structure by omitting axial deformations. The flexibility matrix inding the inverse of this matrix (homework assignment) and substitution into 1 Q D Q D Q leads to Q.9167 kips kips 88.6 kip inches which is less then % different from the computations where axial deformations are included. This frequently happens in the analysis of typical frames and bolsters the assumption that D Q1 and D Q (axial deformations associated with axial force redundants) are zero. This allows the seasoned engineer to make judgments about considering only bending in frame analyses.

15 Example 8.

16 Example 8.

Structural Axial, Shear and Bending Moments

Structural Axial, Shear and Bending Moments Structural Axial, Shear and Bending Moments Positive Internal Forces Acting Recall from mechanics of materials that the internal forces P (generic axial), V (shear) and M (moment) represent resultants

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

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

Module 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method. Version 2 CE IIT, Kharagpur Module Analysis of Statically Indeterminate Structures by the Matrix Force Method esson 11 The Force Method of Analysis: Frames Instructional Objectives After reading this chapter the student will be able

More information

Structural Analysis - II Prof. P. Banerjee Department of Civil Engineering Indian Institute of Technology, Bombay. Lecture - 02

Structural Analysis - II Prof. P. Banerjee Department of Civil Engineering Indian Institute of Technology, Bombay. Lecture - 02 Structural Analysis - II Prof. P. Banerjee Department of Civil Engineering Indian Institute of Technology, Bombay Lecture - 02 Good morning. Today is the second lecture in the series of lectures on structural

More information

Chapter 5: Indeterminate Structures Slope-Deflection Method

Chapter 5: Indeterminate Structures Slope-Deflection Method Chapter 5: Indeterminate Structures Slope-Deflection Method 1. Introduction Slope-deflection method is the second of the two classical methods presented in this course. This method considers the deflection

More information

Module 4. Analysis of Statically Indeterminate Structures by the Direct Stiffness Method. Version 2 CE IIT, Kharagpur

Module 4. Analysis of Statically Indeterminate Structures by the Direct Stiffness Method. Version 2 CE IIT, Kharagpur Module Analysis of Statically Indeterminate Structures by the Direct Stiffness Method Version CE IIT, haragpur esson 7 The Direct Stiffness Method: Beams Version CE IIT, haragpur Instructional Objectives

More information

Approximate Analysis of Statically Indeterminate Structures

Approximate Analysis of Statically Indeterminate Structures Approximate Analysis of Statically Indeterminate Structures Every successful structure must be capable of reaching stable equilibrium under its applied loads, regardless of structural behavior. Exact analysis

More information

Deflections. Question: What are Structural Deflections?

Deflections. Question: What are Structural Deflections? Question: What are Structural Deflections? Answer: The deformations or movements of a structure and its components, such as beams and trusses, from their original positions. It is as important for the

More information

CHAPTER 3. INTRODUCTION TO MATRIX METHODS FOR STRUCTURAL ANALYSIS

CHAPTER 3. INTRODUCTION TO MATRIX METHODS FOR STRUCTURAL ANALYSIS 1 CHAPTER 3. INTRODUCTION TO MATRIX METHODS FOR STRUCTURAL ANALYSIS Written by: Sophia Hassiotis, January, 2003 Last revision: February, 2015 Modern methods of structural analysis overcome some of the

More information

Statics of Structural Supports

Statics of Structural Supports Statics of Structural Supports TYPES OF FORCES External Forces actions of other bodies on the structure under consideration. Internal Forces forces and couples exerted on a member or portion of the structure

More information

Advanced Structural Analysis. Prof. Devdas Menon. Department of Civil Engineering. Indian Institute of Technology, Madras. Module - 5.3.

Advanced Structural Analysis. Prof. Devdas Menon. Department of Civil Engineering. Indian Institute of Technology, Madras. Module - 5.3. Advanced Structural Analysis Prof. Devdas Menon Department of Civil Engineering Indian Institute of Technology, Madras Module - 5.3 Lecture - 29 Matrix Analysis of Beams and Grids Good morning. This is

More information

Chapter 5: Indeterminate Structures Force Method

Chapter 5: Indeterminate Structures Force Method Chapter 5: Indeterminate Structures Force Method 1. Introduction Statically indeterminate structures are the ones where the independent reaction components, and/or internal forces cannot be obtained by

More information

Unit 21 Influence Coefficients

Unit 21 Influence Coefficients Unit 21 Influence Coefficients Readings: Rivello 6.6, 6.13 (again), 10.5 Paul A. Lagace, Ph.D. Professor of Aeronautics & Astronautics and Engineering Systems Have considered the vibrational behavior of

More information

Mechanics of Materials. Chapter 4 Shear and Moment In Beams

Mechanics of Materials. Chapter 4 Shear and Moment In Beams Mechanics of Materials Chapter 4 Shear and Moment In Beams 4.1 Introduction The term beam refers to a slender bar that carries transverse loading; that is, the applied force are perpendicular to the bar.

More information

4.2 Free Body Diagrams

4.2 Free Body Diagrams CE297-FA09-Ch4 Page 1 Friday, September 18, 2009 12:11 AM Chapter 4: Equilibrium of Rigid Bodies A (rigid) body is said to in equilibrium if the vector sum of ALL forces and all their moments taken about

More information

Design Example 1 Reinforced Concrete Wall

Design Example 1 Reinforced Concrete Wall Design Example 1 Reinforced Concrete Wall OVERVIEW The structure in this design example is an eight-story office with load-bearing reinforced concrete walls as its seismic-force-resisting system. This

More information

Truss Structures. See also pages in the supplemental notes. Truss: Mimic Beam Behavior. Truss Definitions and Details

Truss Structures. See also pages in the supplemental notes. Truss: Mimic Beam Behavior. Truss Definitions and Details Truss Structures Truss: Mimic Beam Behavior Truss Definitions and Details 1 2 Framing of a Roof Supported Truss Bridge Truss Details 3 4 See also pages 12-15 in the supplemental notes. 1 Common Roof Trusses

More information

Plane Trusses. Section 7: Flexibility Method - Trusses. A plane truss is defined as a twodimensional

Plane Trusses. Section 7: Flexibility Method - Trusses. A plane truss is defined as a twodimensional lane Trusses A plane truss is defined as a twodiensional fraework of straight prisatic ebers connected at their ends by frictionless hinged joints, and subjected to loads and reactions that act only at

More information

Advanced Structural Analysis Prof. Devdas Menon Department of Civil Engineering Indian Institute of Technology, Madras. Module

Advanced Structural Analysis Prof. Devdas Menon Department of Civil Engineering Indian Institute of Technology, Madras. Module Advanced Structural Analysis Prof. Devdas Menon Department of Civil Engineering Indian Institute of Technology, Madras Module - 2.2 Lecture - 08 Review of Basic Structural Analysis-2 Good morning to you.

More information

Shear Forces and Bending Moments

Shear Forces and Bending Moments Chapter 4 Shear Forces and Bending Moments 4.1 Introduction Consider a beam subjected to transverse loads as shown in figure, the deflections occur in the plane same as the loading plane, is called the

More information

Shear Force and Moment Diagrams

Shear Force and Moment Diagrams C h a p t e r 9 Shear Force and Moment Diagrams In this chapter, you will learn the following to World Class standards: Making a Shear Force Diagram Simple Shear Force Diagram Practice Problems More Complex

More information

The elements used in commercial codes can be classified in two basic categories:

The elements used in commercial codes can be classified in two basic categories: CHAPTER 3 Truss Element 3.1 Introduction The single most important concept in understanding FEA, is the basic understanding of various finite elements that we employ in an analysis. Elements are used for

More information

STRESS AND DEFORMATION ANALYSIS OF LINEAR ELASTIC BARS IN TENSION

STRESS AND DEFORMATION ANALYSIS OF LINEAR ELASTIC BARS IN TENSION Chapter 11 STRESS AND DEFORMATION ANALYSIS OF LINEAR ELASTIC BARS IN TENSION Figure 11.1: In Chapter10, the equilibrium, kinematic and constitutive equations for a general three-dimensional solid deformable

More information

Statically Indeterminate Structure. : More unknowns than equations: Statically Indeterminate

Statically Indeterminate Structure. : More unknowns than equations: Statically Indeterminate Statically Indeterminate Structure : More unknowns than equations: Statically Indeterminate 1 Plane Truss :: Determinacy No. of unknown reactions = 3 No. of equilibrium equations = 3 : Statically Determinate

More information

Finite Element Formulation for Beams - Handout 2 -

Finite Element Formulation for Beams - Handout 2 - Finite Element Formulation for Beams - Handout 2 - Dr Fehmi Cirak (fc286@) Completed Version Review of Euler-Bernoulli Beam Physical beam model midline Beam domain in three-dimensions Midline, also called

More information

Method of Sections for Truss Analysis

Method of Sections for Truss Analysis Method of Sections for Truss Analysis Joint Configurations (special cases to recognize for faster solutions) Case 1) Two Bodies Connected F AB has to be equal and opposite to F BC Case 2) Three Bodies

More information

8.2 Elastic Strain Energy

8.2 Elastic Strain Energy Section 8. 8. Elastic Strain Energy The strain energy stored in an elastic material upon deformation is calculated below for a number of different geometries and loading conditions. These expressions for

More information

MODULE E: BEAM-COLUMNS

MODULE E: BEAM-COLUMNS MODULE E: BEAM-COLUMNS This module of CIE 428 covers the following subjects P-M interaction formulas Moment amplification Web local buckling Braced and unbraced frames Members in braced frames Members

More information

Shear and Moment Diagrams. Shear and Moment Diagrams. Shear and Moment Diagrams. Shear and Moment Diagrams. Shear and Moment Diagrams

Shear and Moment Diagrams. Shear and Moment Diagrams. Shear and Moment Diagrams. Shear and Moment Diagrams. Shear and Moment Diagrams CI 3 Shear Force and Bending oment Diagrams /8 If the variation of and are written as functions of position,, and plotted, the resulting graphs are called the shear diagram and the moment diagram. Developing

More information

P4 Stress and Strain Dr. A.B. Zavatsky MT07 Lecture 3 Statically Indeterminate Structures

P4 Stress and Strain Dr. A.B. Zavatsky MT07 Lecture 3 Statically Indeterminate Structures 4 Stress and Strain Dr... Zavatsky MT07 ecture 3 Statically Indeterminate Structures Statically determinate structures. Statically indeterminate structures (equations of equilibrium, compatibility, and

More information

Copyright 2011 Casa Software Ltd. www.casaxps.com. Centre of Mass

Copyright 2011 Casa Software Ltd. www.casaxps.com. Centre of Mass Centre of Mass A central theme in mathematical modelling is that of reducing complex problems to simpler, and hopefully, equivalent problems for which mathematical analysis is possible. The concept of

More information

CHAP 4 FINITE ELEMENT ANALYSIS OF BEAMS AND FRAMES INTRODUCTION

CHAP 4 FINITE ELEMENT ANALYSIS OF BEAMS AND FRAMES INTRODUCTION CHAP FINITE EEMENT ANAYSIS OF BEAMS AND FRAMES INTRODUCTION We learned Direct Stiffness Method in Chapter imited to simple elements such as D bars we will learn Energ Method to build beam finite element

More information

Linear Dependence Tests

Linear Dependence Tests Linear Dependence Tests The book omits a few key tests for checking the linear dependence of vectors. These short notes discuss these tests, as well as the reasoning behind them. Our first test checks

More information

Type of Force 1 Axial (tension / compression) Shear. 3 Bending 4 Torsion 5 Images 6 Symbol (+ -)

Type of Force 1 Axial (tension / compression) Shear. 3 Bending 4 Torsion 5 Images 6 Symbol (+ -) Cause: external force P Force vs. Stress Effect: internal stress f 05 Force vs. Stress Copyright G G Schierle, 2001-05 press Esc to end, for next, for previous slide 1 Type of Force 1 Axial (tension /

More information

Optimum proportions for the design of suspension bridge

Optimum proportions for the design of suspension bridge Journal of Civil Engineering (IEB), 34 (1) (26) 1-14 Optimum proportions for the design of suspension bridge Tanvir Manzur and Alamgir Habib Department of Civil Engineering Bangladesh University of Engineering

More information

CLASSICAL STRUCTURAL ANALYSIS

CLASSICAL STRUCTURAL ANALYSIS Table of Contents CASSCA STRUCTURA ANAYSS... Conjugate beam method... External work and internal work... 3 Method of virtual force (unit load method)... 5 Castigliano s second theorem... Method of consistent

More information

Chapter 11 Equilibrium

Chapter 11 Equilibrium 11.1 The First Condition of Equilibrium The first condition of equilibrium deals with the forces that cause possible translations of a body. The simplest way to define the translational equilibrium of

More information

FOUNDATION DESIGN. Instructional Materials Complementing FEMA 451, Design Examples

FOUNDATION DESIGN. Instructional Materials Complementing FEMA 451, Design Examples FOUNDATION DESIGN Proportioning elements for: Transfer of seismic forces Strength and stiffness Shallow and deep foundations Elastic and plastic analysis Foundation Design 14-1 Load Path and Transfer to

More information

Solving Simultaneous Equations and Matrices

Solving Simultaneous Equations and Matrices Solving Simultaneous Equations and Matrices The following represents a systematic investigation for the steps used to solve two simultaneous linear equations in two unknowns. The motivation for considering

More information

ENGINEERING MECHANICS STATIC

ENGINEERING MECHANICS STATIC EX 16 Using the method of joints, determine the force in each member of the truss shown. State whether each member in tension or in compression. Sol Free-body diagram of the pin at B X = 0 500- BC sin

More information

METHODS FOR ACHIEVEMENT UNIFORM STRESSES DISTRIBUTION UNDER THE FOUNDATION

METHODS FOR ACHIEVEMENT UNIFORM STRESSES DISTRIBUTION UNDER THE FOUNDATION International Journal of Civil Engineering and Technology (IJCIET) Volume 7, Issue 2, March-April 2016, pp. 45-66, Article ID: IJCIET_07_02_004 Available online at http://www.iaeme.com/ijciet/issues.asp?jtype=ijciet&vtype=7&itype=2

More information

SECTION 5 ANALYSIS OF CONTINUOUS SPANS DEVELOPED BY THE PTI EDC-130 EDUCATION COMMITTEE LEAD AUTHOR: BRYAN ALLRED

SECTION 5 ANALYSIS OF CONTINUOUS SPANS DEVELOPED BY THE PTI EDC-130 EDUCATION COMMITTEE LEAD AUTHOR: BRYAN ALLRED SECTION 5 ANALYSIS OF CONTINUOUS SPANS DEVELOPED BY THE PTI EDC-130 EDUCATION COMMITTEE LEAD AUTHOR: BRYAN ALLRED NOTE: MOMENT DIAGRAM CONVENTION In PT design, it is preferable to draw moment diagrams

More information

PLANE TRUSSES. Definitions

PLANE TRUSSES. Definitions Definitions PLANE TRUSSES A truss is one of the major types of engineering structures which provides a practical and economical solution for many engineering constructions, especially in the design of

More information

DESIGN OF BEAM-COLUMNS - I

DESIGN OF BEAM-COLUMNS - I 13 DESIGN OF BEA-COLUNS - I INTRODUCTION Columns in practice rarely experience concentric axial compression alone. Since columns are usually parts of a frame, they experience both bending moment and axial

More information

STRUCTURAL ANALYSIS II (A60131)

STRUCTURAL ANALYSIS II (A60131) LECTURE NOTES ON STRUCTURAL ANALYSIS II (A60131) III B. Tech - II Semester (JNTUH-R13) Dr. Akshay S. K. Naidu Professor, Civil Engineering Department CIVIL ENGINEERING INSTITUTE OF AERONAUTICAL ENGINEERING

More information

The Basics of FEA Procedure

The Basics of FEA Procedure CHAPTER 2 The Basics of FEA Procedure 2.1 Introduction This chapter discusses the spring element, especially for the purpose of introducing various concepts involved in use of the FEA technique. A spring

More information

2. Axial Force, Shear Force, Torque and Bending Moment Diagrams

2. Axial Force, Shear Force, Torque and Bending Moment Diagrams 2. Axial Force, Shear Force, Torque and Bending Moment Diagrams In this section, we learn how to summarize the internal actions (shear force and bending moment) that occur throughout an axial member, shaft,

More information

Mechanics lecture 7 Moment of a force, torque, equilibrium of a body

Mechanics lecture 7 Moment of a force, torque, equilibrium of a body G.1 EE1.el3 (EEE1023): Electronics III Mechanics lecture 7 Moment of a force, torque, equilibrium of a body Dr Philip Jackson http://www.ee.surrey.ac.uk/teaching/courses/ee1.el3/ G.2 Moments, torque and

More information

DISTRIBUTION OF LOADSON PILE GROUPS

DISTRIBUTION OF LOADSON PILE GROUPS C H A P T E R 7 DISTRIBUTION OF LOADSON PILE GROUPS Section I. DESIGN LOADS 7-1. Basic design. The load carried by an individual pile or group of piles in a foundation depends upon the structure concerned

More information

Analysis of Stresses and Strains

Analysis of Stresses and Strains Chapter 7 Analysis of Stresses and Strains 7.1 Introduction axial load = P / A torsional load in circular shaft = T / I p bending moment and shear force in beam = M y / I = V Q / I b in this chapter, we

More information

4B-2. 2. The stiffness of the floor and roof diaphragms. 3. The relative flexural and shear stiffness of the shear walls and of connections.

4B-2. 2. The stiffness of the floor and roof diaphragms. 3. The relative flexural and shear stiffness of the shear walls and of connections. Shear Walls Buildings that use shear walls as the lateral force-resisting system can be designed to provide a safe, serviceable, and economical solution for wind and earthquake resistance. Shear walls

More information

SEISMIC DESIGN. Various building codes consider the following categories for the analysis and design for earthquake loading:

SEISMIC DESIGN. Various building codes consider the following categories for the analysis and design for earthquake loading: SEISMIC DESIGN Various building codes consider the following categories for the analysis and design for earthquake loading: 1. Seismic Performance Category (SPC), varies from A to E, depending on how the

More information

Stress and Deformation Analysis. Representing Stresses on a Stress Element. Representing Stresses on a Stress Element con t

Stress and Deformation Analysis. Representing Stresses on a Stress Element. Representing Stresses on a Stress Element con t Stress and Deformation Analysis Material in this lecture was taken from chapter 3 of Representing Stresses on a Stress Element One main goals of stress analysis is to determine the point within a load-carrying

More information

Question 2: How do you solve a matrix equation using the matrix inverse?

Question 2: How do you solve a matrix equation using the matrix inverse? Question : How do you solve a matrix equation using the matrix inverse? In the previous question, we wrote systems of equations as a matrix equation AX B. In this format, the matrix A contains the coefficients

More information

6.3 Trusses: Method of Sections

6.3 Trusses: Method of Sections 6.3 Trusses: Method of Sections 6.3 Trusses: Method of Sections xample 1, page 1 of 2 1. etermine the force in members,, and, and state whether the force is tension or compression. 5 m 4 kn 4 kn 5 m 1

More information

Finite Element Simulation of Simple Bending Problem and Code Development in C++

Finite Element Simulation of Simple Bending Problem and Code Development in C++ EUROPEAN ACADEMIC RESEARCH, VOL. I, ISSUE 6/ SEPEMBER 013 ISSN 86-48, www.euacademic.org IMPACT FACTOR: 0.485 (GIF) Finite Element Simulation of Simple Bending Problem and Code Development in C++ ABDUL

More information

Finite Element Formulation for Plates - Handout 3 -

Finite Element Formulation for Plates - Handout 3 - Finite Element Formulation for Plates - Handout 3 - Dr Fehmi Cirak (fc286@) Completed Version Definitions A plate is a three dimensional solid body with one of the plate dimensions much smaller than the

More information

Finite Element Method (ENGC 6321) Syllabus. Second Semester 2013-2014

Finite Element Method (ENGC 6321) Syllabus. Second Semester 2013-2014 Finite Element Method Finite Element Method (ENGC 6321) Syllabus Second Semester 2013-2014 Objectives Understand the basic theory of the FEM Know the behaviour and usage of each type of elements covered

More information

Rigid and Braced Frames

Rigid and Braced Frames Rigid Frames Rigid and raced Frames Rigid frames are identified b the lack of pinned joints within the frame. The joints are rigid and resist rotation. The ma be supported b pins or fied supports. The

More information

Statics and Mechanics of Materials

Statics and Mechanics of Materials Statics and Mechanics of Materials Chapter 4-1 Internal force, normal and shearing Stress Outlines Internal Forces - cutting plane Result of mutual attraction (or repulsion) between molecules on both

More information

New approaches in Eurocode 3 efficient global structural design

New approaches in Eurocode 3 efficient global structural design New approaches in Eurocode 3 efficient global structural design Part 1: 3D model based analysis using general beam-column FEM Ferenc Papp* and József Szalai ** * Associate Professor, Department of Structural

More information

Newton s Third Law. object 1 on object 2 is equal in magnitude and opposite in direction to the force exerted by object 2 on object 1

Newton s Third Law. object 1 on object 2 is equal in magnitude and opposite in direction to the force exerted by object 2 on object 1 Newton s Third Law! If two objects interact, the force exerted by object 1 on object 2 is equal in magnitude and opposite in direction to the force exerted by object 2 on object 1!! Note on notation: is

More information

Problem 1: Computation of Reactions. Problem 2: Computation of Reactions. Problem 3: Computation of Reactions

Problem 1: Computation of Reactions. Problem 2: Computation of Reactions. Problem 3: Computation of Reactions Problem 1: Computation of Reactions Problem 2: Computation of Reactions Problem 3: Computation of Reactions Problem 4: Computation of forces and moments Problem 5: Bending Moment and Shear force Problem

More information

Worked Examples of mathematics used in Civil Engineering

Worked Examples of mathematics used in Civil Engineering Worked Examples of mathematics used in Civil Engineering Worked Example 1: Stage 1 Engineering Surveying (CIV_1010) Tutorial - Transition curves and vertical curves. Worked Example 1 draws from CCEA Advanced

More information

EDEXCEL NATIONAL CERTIFICATE/DIPLOMA MECHANICAL PRINCIPLES AND APPLICATIONS NQF LEVEL 3 OUTCOME 1 - LOADING SYSTEMS

EDEXCEL NATIONAL CERTIFICATE/DIPLOMA MECHANICAL PRINCIPLES AND APPLICATIONS NQF LEVEL 3 OUTCOME 1 - LOADING SYSTEMS EDEXCEL NATIONAL CERTIFICATE/DIPLOMA MECHANICAL PRINCIPLES AND APPLICATIONS NQF LEVEL 3 OUTCOME 1 - LOADING SYSTEMS TUTORIAL 1 NON-CONCURRENT COPLANAR FORCE SYSTEMS 1. Be able to determine the effects

More information

Element 5. Statics Truss Problem. 2.1 Statics

Element 5. Statics Truss Problem. 2.1 Statics Statics Truss Problem. Statics We are going to start our discussion of Finite Element Analysis (FEA) with something very familiar. We are going to look at a simple statically determinate truss. Trusses

More information

Design Parameters for Steel Special Moment Frame Connections

Design Parameters for Steel Special Moment Frame Connections SEAOC 2011 CONVENTION PROCEEDINGS Design Parameters for Steel Special Moment Frame Connections Scott M. Adan, Ph.D., S.E., SECB, Chair SEAONC Structural Steel Subcommittee Principal Adan Engineering Oakland,

More information

Nonlinear analysis and form-finding in GSA Training Course

Nonlinear analysis and form-finding in GSA Training Course Nonlinear analysis and form-finding in GSA Training Course Non-linear analysis and form-finding in GSA 1 of 47 Oasys Ltd Non-linear analysis and form-finding in GSA 2 of 47 Using the GSA GsRelax Solver

More information

Recitation #5. Understanding Shear Force and Bending Moment Diagrams

Recitation #5. Understanding Shear Force and Bending Moment Diagrams Recitation #5 Understanding Shear Force and Bending Moment Diagrams Shear force and bending moment are examples of interanl forces that are induced in a structure when loads are applied to that structure.

More information

Structural Analysis: Space Truss

Structural Analysis: Space Truss Structural Analysis: Space Truss Space Truss - 6 bars joined at their ends to form the edges of a tetrahedron as the basic non-collapsible unit - 3 additional concurrent bars whose ends are attached to

More information

When the fluid velocity is zero, called the hydrostatic condition, the pressure variation is due only to the weight of the fluid.

When the fluid velocity is zero, called the hydrostatic condition, the pressure variation is due only to the weight of the fluid. Fluid Statics When the fluid velocity is zero, called the hydrostatic condition, the pressure variation is due only to the weight of the fluid. Consider a small wedge of fluid at rest of size Δx, Δz, Δs

More information

Chapter 18 Static Equilibrium

Chapter 18 Static Equilibrium Chapter 8 Static Equilibrium 8. Introduction Static Equilibrium... 8. Lever Law... Example 8. Lever Law... 4 8.3 Generalized Lever Law... 5 8.4 Worked Examples... 7 Example 8. Suspended Rod... 7 Example

More information

INTRODUCTION TO BEAMS

INTRODUCTION TO BEAMS CHAPTER Structural Steel Design LRFD Method INTRODUCTION TO BEAMS Third Edition A. J. Clark School of Engineering Department of Civil and Environmental Engineering Part II Structural Steel Design and Analysis

More information

Solving Systems of Linear Equations. Substitution

Solving Systems of Linear Equations. Substitution Solving Systems of Linear Equations There are two basic methods we will use to solve systems of linear equations: Substitution Elimination We will describe each for a system of two equations in two unknowns,

More information

BASIC CONCEPTS AND CONVENTIONAL METHODS OF STUCTURAL ANALYSIS (LECTURE NOTES)

BASIC CONCEPTS AND CONVENTIONAL METHODS OF STUCTURAL ANALYSIS (LECTURE NOTES) BASIC CONCEPTS AND CONVENTIONA METHODS OF STUCTURA ANAYSIS (ECTURE NOTES) DR. MOHAN KAANI (Retired Professor of Structural Engineering) DEPARTMENT OF CIVI ENGINEERING INDIAN INSTITUTE OF TECHNOOGY (BOMBAY)

More information

Design MEMO 54a Reinforcement design for RVK 41

Design MEMO 54a Reinforcement design for RVK 41 Page of 5 CONTENTS PART BASIC ASSUMTIONS... 2 GENERAL... 2 STANDARDS... 2 QUALITIES... 3 DIMENSIONS... 3 LOADS... 3 PART 2 REINFORCEMENT... 4 EQUILIBRIUM... 4 Page 2 of 5 PART BASIC ASSUMTIONS GENERAL

More information

Stability Of Structures: Basic Concepts

Stability Of Structures: Basic Concepts 23 Stability Of Structures: Basic Concepts ASEN 3112 Lecture 23 Slide 1 Objective This Lecture (1) presents basic concepts & terminology on structural stability (2) describes conceptual procedures for

More information

Perspectives on the Evolution of Structures

Perspectives on the Evolution of Structures Perspectives on the Evolution of Structures The Eiffel Tower Structural Study Assignment In this assignment, you have the following tasks: 1. Calculate the force in the legs of the Eiffel tower (a) at

More information

Design of Fully Restrained Moment Connections per AISC LRFD 3rd Edition (2001)

Design of Fully Restrained Moment Connections per AISC LRFD 3rd Edition (2001) PDHonline Course S154 (4 PDH) Design of Fully Restrained Moment Connections per AISC LRFD 3rd Edition (2001) Instructor: Jose-Miguel Albaine, M.S., P.E. 2012 PDH Online PDH Center 5272 Meadow Estates Drive

More information

Distribution of Forces in Lateral Load Resisting Systems

Distribution of Forces in Lateral Load Resisting Systems Distribution of Forces in Lateral Load Resisting Systems Part 2. Horizontal Distribution and Torsion IITGN Short Course Gregory MacRae Many slides from 2009 Myanmar Slides of Profs Jain and Rai 1 Reinforced

More information

Unit 6 Plane Stress and Plane Strain

Unit 6 Plane Stress and Plane Strain Unit 6 Plane Stress and Plane Strain Readings: T & G 8, 9, 10, 11, 12, 14, 15, 16 Paul A. Lagace, Ph.D. Professor of Aeronautics & Astronautics and Engineering Systems There are many structural configurations

More information

MCE380: Measurements and Instrumentation Lab. Chapter 9: Force, Torque and Strain Measurements

MCE380: Measurements and Instrumentation Lab. Chapter 9: Force, Torque and Strain Measurements MCE380: Measurements and Instrumentation Lab Chapter 9: Force, Torque and Strain Measurements Topics: Elastic Elements for Force Measurement Dynamometers and Brakes Resistance Strain Gages Holman, Ch.

More information

Introduction to Engineering Analysis - ENGR1100 Course Description and Syllabus Monday / Thursday Sections. Fall '15.

Introduction to Engineering Analysis - ENGR1100 Course Description and Syllabus Monday / Thursday Sections. Fall '15. Introduction to Engineering Analysis - ENGR1100 Course Description and Syllabus Monday / Thursday Sections Fall 2015 All course materials are available on the RPI Learning Management System (LMS) website.

More information

Method of Joints. Method of Joints. Method of Joints. Method of Joints. Method of Joints. Method of Joints. CIVL 3121 Trusses - Method of Joints 1/5

Method of Joints. Method of Joints. Method of Joints. Method of Joints. Method of Joints. Method of Joints. CIVL 3121 Trusses - Method of Joints 1/5 IVL 3121 Trusses - 1/5 If a truss is in equilibrium, then each of its joints must be in equilibrium. The method of joints consists of satisfying the equilibrium equations for forces acting on each joint.

More information

4 Shear Forces and Bending Moments

4 Shear Forces and Bending Moments 4 Shear Forces and ending oments Shear Forces and ending oments 8 lb 16 lb roblem 4.3-1 alculate the shear force and bending moment at a cross section just to the left of the 16-lb load acting on the simple

More information

Solving Linear Systems, Continued and The Inverse of a Matrix

Solving Linear Systems, Continued and The Inverse of a Matrix , Continued and The of a Matrix Calculus III Summer 2013, Session II Monday, July 15, 2013 Agenda 1. The rank of a matrix 2. The inverse of a square matrix Gaussian Gaussian solves a linear system by reducing

More information

Modeling Beams on Elastic Foundations Using Plate Elements in Finite Element Method

Modeling Beams on Elastic Foundations Using Plate Elements in Finite Element Method Modeling Beams on Elastic Foundations Using Plate Elements in Finite Element Method Yun-gang Zhan School of Naval Architecture and Ocean Engineering, Jiangsu University of Science and Technology, Zhenjiang,

More information

Homework 2 Part B Perspectives on the Evolution of Structures (Analysis 1, Eiffel ++) Analysis 1 and The Eiffel Tower structural study assignment

Homework 2 Part B Perspectives on the Evolution of Structures (Analysis 1, Eiffel ++) Analysis 1 and The Eiffel Tower structural study assignment Homework 2 Part B Perspectives on the Evolution of Structures (Analysis 1, Eiffel ++) Analysis 1 and The Eiffel Tower structural study assignment Eiffel Tower Structural Study Problems; In this assignment,

More information

The Mathematics of Beam Deflection

The Mathematics of Beam Deflection The athematics of eam Deflection Scenario s a structural engineer you are part of a team working on the design of a prestigious new hotel comple in a developing city in the iddle East. It has been decided

More information

Bending Stress in Beams

Bending Stress in Beams 936-73-600 Bending Stress in Beams Derive a relationship for bending stress in a beam: Basic Assumptions:. Deflections are very small with respect to the depth of the beam. Plane sections before bending

More information

Design of reinforced concrete columns. Type of columns. Failure of reinforced concrete columns. Short column. Long column

Design of reinforced concrete columns. Type of columns. Failure of reinforced concrete columns. Short column. Long column Design of reinforced concrete columns Type of columns Failure of reinforced concrete columns Short column Column fails in concrete crushed and bursting. Outward pressure break horizontal ties and bend

More information

Analysis of Statically Determinate Trusses

Analysis of Statically Determinate Trusses Analysis of Statically Determinate Trusses THEORY OF STRUCTURES Asst. Prof. Dr. Cenk Üstündağ Common Types of Trusses A truss is one of the major types of engineering structures which provides a practical

More information

Beam Deflections: 4th Order Method and Additional Topics

Beam Deflections: 4th Order Method and Additional Topics 11 eam Deflections: 4th Order Method and dditional Topics 11 1 ecture 11: EM DEFECTIONS: 4TH ORDER METHOD ND DDITION TOICS TE OF CONTENTS age 11.1. Fourth Order Method Description 11 3 11.1.1. Example

More information

Torque and Rotational Equilibrium

Torque and Rotational Equilibrium Torque and Rotational Equilibrium Name Section Torque is the rotational analog of force. If you want something to move (translation), you apply a force; if you want something to rotate, you apply a torque.

More information

Pancake-type collapse energy absorption mechanisms and their influence on the final outcome (complete version)

Pancake-type collapse energy absorption mechanisms and their influence on the final outcome (complete version) Report, Structural Analysis and Steel Structures Institute, Hamburg University of Technology, Hamburg, June, 2013 Pancake-type collapse energy absorption mechanisms and their influence on the final outcome

More information

6. Vectors. 1 2009-2016 Scott Surgent (surgent@asu.edu)

6. Vectors. 1 2009-2016 Scott Surgent (surgent@asu.edu) 6. Vectors For purposes of applications in calculus and physics, a vector has both a direction and a magnitude (length), and is usually represented as an arrow. The start of the arrow is the vector s foot,

More information

MECHANICS OF MATERIALS

MECHANICS OF MATERIALS T dition CHTR MCHNICS OF MTRIS Ferdinand. Beer. Russell Johnston, Jr. John T. DeWolf ecture Notes: J. Walt Oler Texas Tech University Stress and Strain xial oading - Contents Stress & Strain: xial oading

More information

Linear Algebra Notes for Marsden and Tromba Vector Calculus

Linear Algebra Notes for Marsden and Tromba Vector Calculus Linear Algebra Notes for Marsden and Tromba Vector Calculus n-dimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of

More information

Trusses Theory and in LEGO TJ Avery, c. 2001 (updated May 2009)

Trusses Theory and in LEGO TJ Avery, c. 2001 (updated May 2009) Page 1 of 10 Trusses Theory and in LEGO TJ Avery, c. 2001 (updated May 2009) This article explains what a truss is and how it functions structurally. Also presented are examples of how to build a truss

More information

Structural Displacements. Structural Displacements. Beam Displacement. Truss Displacements 2

Structural Displacements. Structural Displacements. Beam Displacement. Truss Displacements 2 Structural Displacements Structural Displacements P Beam Displacement 1 Truss Displacements The deflections of civil engineering structures under the action of usual design loads are known to be small

More information