DESIGN OF BEAM-COLUMNS - I

Save this PDF as:

Size: px
Start display at page:

Transcription

2 experience reverse curvature bending as shown in Fig. 1(e), causing variation of the nature (positive or negative) of the bending moment and curvature over the length of the column. (a) (b) (c) (d) z y x (f) Fig. 1 Beam-Columns in Frames resence of bending moments in the beam-columns reduces the axial force at which they fail. This topic presented in two parts, deals with the behaviour, and design of beam- Version II 13-2

3 columns. In art I initially, the behaviour and strength of short beam-column members under combined compression and bending moment are discussed. In such short beam column the failure is due to the strength of the material being reached (material failure). Subsequently, the behaviour and strength of practical, long beam-columns, as affected by stability and deformation, are discussed. In long columns the failure may be either due to material strength being reached at the ends of the column or instability of the overall column. In art II equations for the design of beam-columns subjected to combination of axial compression and biaxial bending are presented. A design example of a beamcolumn also is presented in art II. 2.0 SHORT BEA-COLUNS A short member (stub column), made of non-slender (plastic, compact or semi-compact) section under axial compression, fails by yielding (due to large deformation) at the squash load, d, given by [Fig. 2(a)] d = A g f y (1) where, f y section. is the yield strength of the material, and A g is the gross area of the cross If the stub column is made of slender cross section, the plate elements of the cross section undergo local buckling before reaching the yield stress. This causes reduction in the effective area of the cross section to a value below the gross area, A g, and the member fails at a load below d, given by Eqn.1. Similarly a short member made of plastic or compact section and subjected to only bending moment fails at the plastic moment capacity, p, given by [Fig. 2(b)] p = Z p. f y (2) where, Z p = plastic section modulus of the cross section, in the case of plastic and compact sections. f y f y f y f y Section f y (a) Only axial compression = + f y f y f y f y (b) Only bending (c) Combined compression moment and bending moment Fig. 2 Stresses in Short Beam-Columns A semi-compact section subjected to bending moment only fails by buckling of a plate element of the cross section before the plastification of the entire section as shown in Fig. 2 (b) but after the stress at the extreme fibre in compression reaches the yield stress. In a Version II 13-3

4 slender section, the plate elements buckle even before the extreme fibre stress in bending reaches the yield stress. Consequently, the semi-compact and slender sections fail under bending even before reaching the plastic moment, p, given by Eqn. 2. The discussions that follow generally assume that the cross section is either plastic or compact. In the case of slender and semi-compact sections the effect of earlier failure before complete section yielding has to be considered. The strength of such members may be analysed by following the procedure discussed in the chapters on cold-formed steel members. The stress distribution at failure over the (plastic or compact) cross-section of a beamcolumn under combined compression and bending moment is shown in Fig. 1(c). It can be modeled as superposition of only compressive stress over an area of the cross section close to the neutral axis of the cross section and the balance of the section subjected to compressive and tensile stress due to bending. Hence such beam-columns fail before reaching the squash load, d, given by equation 1 or the plastic moment, p, given by Eqn. 2. The typical failure envelope diagram of a stub beam-column made of I section and subjected to axial compression and bending moment is shown in Fig. 3, in a non-dimensional form. At smaller values of axial compression, only a small area of the cross section closer to the neutral axis is necessary to equilibrate the external compression,. Since the area closer to the neutral axis contributes very little to the plastic moment capacity, p, of the cross section, the reduction in the moment capacity,, is negligible when the axial compression is small. It is seen in the failure envelope that for smaller axial compression ( / d < 0.15) the reduction in the moment capacity is negligible (/ p ) Stub column loading curve / d 0 / d cl / d A cl δ p B Failure envelope Long columns loading curve Eqn. 3 O 0 / p max / p / p Fig. 3 Beam-Column Failure Envelope Eqn. 3 in a non-dimensional form gives the failure envelope under the major axis bending and the axial compression, as shown in Fig. 3. Version II 13-4

5 d p p () 3 Although there is a small reduction in the bending moment at lower values of axial compression as seen in Fig. 3, in Eqn. 3 this has been disregarded. The interaction equation (Eqn. 3) is linear up to the plastic moment capacity, p, of the member. The loading curve of a short beam-column by a compressive force at a constant eccentricity, is indicated by a straight line, such as OA (Fig. 3) having a constant slope. The slope is dictated by the eccentricity. When this loading path OA intersects the failure envelope curve at A, the beam-column strength is reached. The values of and corresponding to point A are the compression and moment capacity of short the beamcolumn under the given eccentricity. This may be calculated from Eqn. 3 by substituting.e for, where e is the eccentricity of the compressive force,. 3.0 LONG BEA-COLUNS Typically steel columns in practice are long and slender. Such slender columns when axially compressed tend to fail by buckling rather than yielding, as discussed in the chapter on Introduction to Column Buckling. Similarly, slender I sections subjected to bending moment about the major axis (z-axis) when not laterally supported, may fail by lateral-torsional buckling, as discussed earlier in the chapter on Unrestrained Beam Design. However, under minor axis (y-axis) bending, the plastic and compact sections will reach the plastic moment capacity, py, without undergoing premature lateral buckling or local buckling. Such long slender members subjected to combined axial compression and bending may experience different modes of instability or material failure. These are discussed in this section. Consider a slender beam-column subjected only to equal and opposite end moment, o, as shown in Fig. 4(a). The beam-column is bent into a single curvature with a maximum deflection δ 0, as shown by the dotted line in Fig. 4(a). If the axial compression is applied at the ends of the column now, additional bending moment is caused due to the axial load acting on the deformed shape. This additional bending moment causes additional deflection and so on, until the final maximum deflection δ is reached at the stage of equilibrium under combined axial force and bending moments. This is referred to as -δ effects. The final deflected shape and the final bending moment diagram, considering the -δ effect, are shown by dashed curves in Fig. 4(a). It is seen that due to -δ effect, the maximum moment in the beam-column, max, is larger than the externally applied end moments, o. The same beam-column, when subjected to equal end moments acting in the same direction, experiences double curvature bending in addition to axial compression as shown in Fig. 4(b). The deflected shape as well as the bending moment diagram of the Version II 13-5

6 beam column, not considering -δ effects, are shown in Fig. 4(b) by dotted curves and after considering the -δ effects is indicated by the dashed curve. It is seen that although δ is greater than δ o, in this case, the magnified moment considering the -δ effects need not be greater than the end moments o. Thus, it is seen that the -δ effects and the magnified moments depend upon the moment gradient over the length of the member. The discussion so far was about beam-columns in frames braced against sway δ 0 0 θ δ 0 max δ δ 0 0 (a) Single curvature δ 0 0 (b) Double curvature Linear Non-Linear (c) Sway Deformation Fig. 4 Deflection and oment agnification If a frame is not braced to prevent lateral sway, linear elastic analysis for lateral loads may indicate that column ends translate relative to one another by a distance 0, in addition to end rotations and reverse curvature deformation as indicated by dotted curve in Fig. 4(c). The axial forces,, acting on the frame with sway displacement, 0, increase the sway to, as shown by dashed curve and the column and beam moments also increase. This additional displacement can be obtained only by a non-linear analysis of the frame considering the equilibrium of the frame in the deformed configuration. This increase in sway deformation and bending moments, due to the load acting on the deformed structure, are referred to as - effects. Let us see the load-deformation behaviour of a beam-column, when the axial compression and end moments are applied one after another. If an axial compression,, is initially applied so that / cr is maintained constant while the moments at the two ends of the beam-column are proportionately increased in the elastic range of the material, the Version II 13-6

7 moment-end rotation diagram is linear as shown by Fig. 5(a). However, it is seen that the stiffness of the beam-column (the slope of the moment-rotation line) decreases with increase in the initially applied axial compression. When the / cr =, the end rotation, θ [Fig. 4(a)] increases to infinity even for =0, indicating instability under axial compression itself. On the other hand, if the ratios of the initially applied end moments, 0 / p, is maintained constant and the axial compression alone is increased, the compressive load versus lateral deformation behaviour of the beam-column in the elastic range is as per Fig. 5(b). In this case the load deformation behaviour is non-linear. It is seen that a perfect column subjected to axial compression without any end moments undergoes bifurcation type of buckling [OAB in Fig.5 (b)]. If some end moments are applied initially, the member undergoes initial deflection, δ o, the magnitude of which depends upon the magnitude of the end moments. Subsequently, as the axial compression is increased gradually, the lateral deflection increases even from the very beginning. Initially such an increase is seen to be at a slower rate, but nearer to the critical load it increases rapidly before failure occurs [Fig.5 (b)]. Usually the failure is triggered by yielding under the combined effect of axial compression and maximum moment. It is seen that the end moments modify the behaviour of an axially loaded column in a way, similar to the initial bow type of imperfections. 0 / cr A 0 / = cr 0.8 B 0.8 θ (a) At constant axial load, O (b) At constant end moments, 0 δ Fig. 5 Elastic Behaviour of Beam-Columns Conventional first order linear elastic analyses of frames do not reflect these additional bending moments in beam-columns due to -δ and - effects, since the equilibrium equations in these analyses methods are derived for the un-deformed structure. In the linear elastic analysis of a frame, the axial force and bending moments in a beam-column increase linearly, with the increase in the load on frame, as shown by straight line OA in Fig.3 drawn for short column. However, if we look at the equilibrium of the beamcolumn in the deformed configuration [Fig.4], (through a non-linear analysis), bending moments are magnified by - and -δ effects in a nonlinear fashion, as the load increases. Version II 13-7

10 axis bending strength is affected due to the lateral buckling and that pure bending strength can be less than the plastic moment capacity pz. The bending strength, when the axial compression is zero, is given by z, which is less than pz. The dashed curves represent the loading path corresponding to different levels of analysis. The dashed curves OA represent the linear analysis path and hence their intersections with the strength envelopes corrected for -δ and - effects (AS), give the member strengths, cl. The dashed curves OB represent the nonlinear analysis paths, considering -δ and - effects and hence intersections of these curves with the short column strength curve (BQ) give the member strengths, cl. The dashed curves OC represent the loading paths from nonlinear analysis considering - effects only and hence their intersections with the strength curves (CR) corrected for -δ effect give the member strengths, cl. 3.2 Effects of Slenderness Ratio and Axial Force on odes of Failure Beam-columns may fail by flexural yielding or torsional flexural buckling. The actual mode of failure would depend upon the magnitude of the axial load and eccentricity as well as the slenderness ratio. The sub-ultimate and failure behaviour of beam-columns, as affected by different parameters, are briefly reviewed in the following sections (Dowling et al., 1988). x px A / d < 0.25 D E F / d >0.5 F C F C Short Column Slender Column B O Fig. 7 Beam-column oment Rotation Behaviour θ Low axial load ratio (/ d < 0.33) Beam-columns having lower slenderness ratios (λ/r <50): When subjected to moments about the major axis at both ends of the beam, the moment-curvature relationship at the sub-ultimate stage may be linear or non-linear depending upon whether the axial force is applied first followed by bending moment or both of them are increased proportionately (Figs. 5). The deformation is only in the plane of bending moment. At the penultimate stage, yielding of the compression flange occurs first, which spreads through the section Version II 13-10

12 In the case of slender members under smaller axial load, there is very little reduction of moment capacity below p, since the lateral torsional buckling is not a problem in weak axis bending. The moment magnification is larger in the case of beam-columns bending about the weak axis. The failure of short/stocky members is either due to section strength being reached at the ends (under smaller axial load) or at the section of larger magnified moment (under larger axial load). The failure of even slender members is due to buckling about the weak axis only and no torsional deformation is experienced. The - failure envelope varies as shown in Fig.6 (a), due to the variations of slenderness ratio and axial force. 3.3 Beam-Column under Biaxial Bending The ultimate behaviour of beam-columns under biaxial bending is complicated by the effect of plastification, moment magnification and lateral torsional buckling. Typical failure envelope diagram is as shown in Fig.8. It is seen that the increase in the slenderness ratio of the member tends to reduce the strength of the member, except in the case of nearly pure bending about the weak axis (y-axis). Further, but for very small axial force ranges, increases in the axial compression tend to decrease the bending strength about both axes. / d 10 λ/r = 0 λ/r increases 10 y / py 10 z / pz Fig. 8 beam-columns under Biaxial Bending 4.0 SUARY Behaviour of beam-columns, which are members subjected to axial compression and bending simultaneously, was discussed in this chapter. The following were the main issues discussed in the chapter. The moment magnification due to -δ and - effects are important considerations affecting their strength, particularly in case of slender members subjected to larger axial compression. Under major axis bending lateral buckling is also an important consideration. Version II 13-12

13 Flexural yielding, flexural buckling, torsional flexural buckling are the different modes of failure, depending upon the slenderness ratio, axis of bending and extent of axial compression. In the next chapter, II of this topic, evaluation of strength of beam-columns will be dealt with and steps in designing beam-columns will be presented supported by an example. 5.0 REFERENCES 1. Dowling.J, Knowles,.R. and Owens, G.W., Structural Steel Design, Butterworths, London, Version II 13-13

End Restraint and Effective Lengths of Columns

CHAPTER Structural Steel Design LRFD Method Third Edition INTRODUCTION TO AXIALLY LOADED COMPRESSION MEMBERS A. J. Clark School of Engineering Department of Civil and Environmental Engineering Part II

III. Compression Members. Design of Steel Structures. Introduction. Compression Members (cont.)

ENCE 455 Design of Steel Structures III. Compression Members C. C. Fu, Ph.D., P.E. Civil and Environmental Engineering Department University it of Maryland Compression Members Following subjects are covered:

Design of Steel Structures Prof. S.R.Satish Kumar and Prof. A.R.Santha Kumar

6.3 Behaviour of steel beams Laterally stable steel beams can fail only by (a) Flexure (b) Shear or (c) Bearing, assuming the local buckling of slender components does not occur. These three conditions

A beam is a structural member that is subjected primarily to transverse loads and negligible

Chapter. Design of Beams Flexure and Shear.1 Section force-deformation response & Plastic Moment (M p ) A beam is a structural member that is subjected primarily to transverse loads and negligible axial

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

Objectives. Experimentally determine the yield strength, tensile strength, and modules of elasticity and ductility of given materials.

Lab 3 Tension Test Objectives Concepts Background Experimental Procedure Report Requirements Discussion Objectives Experimentally determine the yield strength, tensile strength, and modules of elasticity

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

Elastic Buckling Loads of Hinged Frames by the Newmark Method

International Journal of Applied Science and Technology Vol. 1 No. 3; June 011 Abstract Elastic Buckling Loads of Hinged Frames by the Newmark Method Ashraf Badir Department of Environmental and Civil

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

Design of Slender Columns

Design of Slender Columns When a stocky column (i.e. one that is not very slender, where slenderness is defined as the ratio between the column s height and width) is subjected to an excessive axial load

Statics and Mechanics of Materials

Statics and Mechanics of Materials Chapter 4 Stress, Strain and Deformation: Axial Loading Objectives: Learn and understand the concepts of internal forces, stresses, and strains Learn and understand the

CH 4: Deflection and Stiffness

CH 4: Deflection and Stiffness Stress analyses are done to ensure that machine elements will not fail due to stress levels exceeding the allowable values. However, since we are dealing with deformable

Concrete Frame Design Manual

Concrete Frame Design Manual Turkish TS 500-2000 with Turkish Seismic Code 2007 For SAP2000 ISO SAP093011M26 Rev. 0 Version 15 Berkeley, California, USA October 2011 COPYRIGHT Copyright Computers and Structures,

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

THREE DIMENSIONAL ACES MODELS FOR BRIDGES

THREE DIMENSIONAL ACES MODELS FOR BRIDGES Noel Wenham, Design Engineer, Wyche Consulting Joe Wyche, Director, Wyche Consulting SYNOPSIS Plane grillage models are widely used for the design of bridges,

Mechanics of Materials Stress-Strain Curve for Mild Steel

Stress-Strain Curve for Mild Steel 13-1 Definitions 13-2a Hooke s Law Shear Modulus: Stress: Strain: Poisson s Ratio: Normal stress or strain = " to the surface Shear stress = to the surface Definitions

Figure 12 1 Short columns fail due to material failure

12 Buckling Analysis 12.1 Introduction There are two major categories leading to the sudden failure of a mechanical component: material failure and structural instability, which is often called buckling.

Laboratory Weeks 9 10 Theory of Pure Elastic Bending

Laboratory Weeks 9 10 Theory of Pure Elastic Bending Objective To show the use of the Sagital method for finding the Radius of Curvature of a beam, to prove the theory of bending, and find the elastic

CONNECTION DESIGN DESIGN REQUIREMENTS

29 CONNECTION DESIGN DESIGN REQUIREMENTS 1.0 INTRODUCTION Steel sections are manufactured and shipped to some standard lengths, as governed by rolling, transportation and handling restrictions. However,

Bending Beam. Louisiana State University. Joshua Board

Bending Beam Louisiana State University Joshua Board Table of Contents: Table of Figures:... 4 Purpose... 5 Introduction... 5 Apparatus and Test Procedures... 11 Summary of Data... 14 Discussion of Results...

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

M x (a) (b) (c) Figure 2: Lateral Buckling The positions of the beam shown in Figures 2a and 2b should be considered as two possible equilibrium posit

Lateral Stability of a Slender Cantilever Beam With End Load Erik Thompson Consider the slender cantilever beam with an end load shown in Figure 1. The bending moment at any cross-section is in the x-direction.

FRAME IDEALISATION & ANALYSIS SUMMARY:

FRAME IDEALISATION & ANALYSIS SUMMARY: Introduce the various global frame modelling analysis approaches & basic concepts. Introduce the classification of frames as braced/unbraced, sway/non-sway. Introduce

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

A Comparative Study on Non-Linear Analysis of Frame with and without Structural Wall System

A Comparative Study on Non-Linear Analysis of Frame with and without Structural Wall System Dr.Binu Sukumar #1, A.Hemamathi *2, S.Kokila #3 C.Hanish #4 #1 Professor &Head, Department of Civil Engineering,

UNRESTRAINED BEAM DESIGN I

11 UNRESTRAINED BEA DESIGN I 1.0 INTRODUCTION Generally, a beam resists transverse loads by bending action. In a typical building frame, main beams are employed to span between adjacent columns; secondary

Dr. Seshu Adluri. Structural Steel Design Compression Members

Dr. Seshu Adluri Structural Steel Design Compression Members Columns in Buildings Columns in Buildings Column supports Compression members in trusses Compression members in trusses Compression members

Design of Steel Structures Prof. S.R.Satish Kumar and Prof. A.R.Santha Kumar. Fig. 7.21 some of the trusses that are used in steel bridges

7.7 Truss bridges Fig. 7.21 some of the trusses that are used in steel bridges Truss Girders, lattice girders or open web girders are efficient and economical structural systems, since the members experience

10 Space Truss and Space Frame Analysis

10 Space Truss and Space Frame Analysis 10.1 Introduction One dimensional models can be very accurate and very cost effective in the proper applications. For example, a hollow tube may require many thousands

Single angle members have a broad range of applications. Very often angle

Abstract Single angle members have a broad range of applications. Very often angle members are connected eccentrically. As a result, not only is an angle member subject to axial force, but it is also subject

CLASSIFICATION BOUNDARIES FOR STIFFNESS OF BEAM-TO- COLUMN JOINTS AND COLUMN BASES

Nordic Steel Construction Conference 2012 Hotel Bristol, Oslo, Norway 5-7 September 2012 CLASSIFICATION BOUNDARIES FOR STIFFNESS OF BEAM-TO- COLUMN JOINTS AND COLUMN BASES Ina Birkeland a,*, Arne Aalberg

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

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

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

ARCH 331 Structural Glossary S2014abn. Structural Glossary

Structural Glossary Allowable strength: Nominal strength divided by the safety factor. Allowable stress: Allowable strength divided by the appropriate section property, such as section modulus or cross

ETABS. Integrated Building Design Software. Concrete Frame Design Manual. Computers and Structures, Inc. Berkeley, California, USA

ETABS Integrated Building Design Software Concrete Frame Design Manual Computers and Structures, Inc. Berkeley, California, USA Version 8 January 2002 Copyright The computer program ETABS and all associated

PDHonline Course S121 (1 PDH) Leaner Columns. Instructor: D. Matthew Stuart, P.E., S.E., F.ASCE, F.SEI, SECB, MgtEng. PDH Online PDH Center

PDHonline Course S121 (1 PDH) Leaner Columns Instructor: D. Matthew Stuart, P.E., S.E., F.ASCE, F.SEI, SECB, MgtEng 2013 PDH Online PDH Center 5272 Meadow Estates Drive Fairfax, VA 22030-6658 Phone & Fax:

STEEL BUILDINGS IN EUROPE. Single-Storey Steel Buildings Part 4: Detailed Design of Portal Frames

STEEL BUILDIGS I EUROPE Single-Storey Steel Buildings Part 4: Detailed Design of Portal Frames Single-Storey Steel Buildings Part 4: Detailed Design of Portal Frames 4 - ii FOREWORD This publication is

POWER SCREWS (ACME THREAD) DESIGN There are at least three types of power screw threads: the square thread, the Acme thread, and the buttress thread. Of these, the square and buttress threads are the most

MATERIALS SELECTION FOR SPECIFIC USE

MATERIALS SELECTION FOR SPECIFIC USE-1 Sub-topics 1 Density What determines density and stiffness? Material properties chart Design problems LOADING 2 STRENGTH AND STIFFNESS Stress is applied to a material

CHAPTER 1 INTRODUCTION

CHAPTER 1 INTRODUCTION 1.1 Background of the research Beam is a main element in structural system. It is horizontal member that carries load through bending (flexure) action. Therefore, beam will deflect

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

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

Chapter 4: Summary and Conclusions

Chapter 4: Summary and Conclusions 4.1 Summary Three different models are presented and analyzed in this research for the purpose of studying the potential of using post-buckled or pre-bent elastic struts

Fire Engineering Research - Key Issues for the Future II Naples, Italy, 6 9 June Contents

Fire Engineering Research - Key Issues for the Future II Naples, Italy, 6 9 June 013 Contents (State of the art) PhD planning PhD: BEHAVIOUR OF COLD-FORMED STEEL BEAM-COLUMNS IN CASE OF FIRE Motivation

Lateral Buckling of Singly Symmetric Beams

Missouri University of Science and Technology Scholars' Mine International Specialty Conference on Cold- Formed Steel Structures (1992) - 11th International Specialty Conference on Cold-Formed Steel Structures

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

KINGS COLLEGE OF ENGINEERING DEPARTMENT OF MECHANICAL ENGINEERING QUESTION BANK. UNIT I STRESS STRAIN DEFORMATION OF SOLIDS PART- A (2 Marks)

KINGS COLLEGE OF ENGINEERING DEPARTMENT OF MECHANICAL ENGINEERING QUESTION BANK SUB CODE/NAME: CE1259 STRENGTH OF MATERIALS YEAR/SEM: II / IV 1. What is Hooke s Law? 2. What are the Elastic Constants?

Unit M4.7 The Column and Buckling

Unit M4.7 The Column and Buckling Readings: CDL 9.1-9.4 CDL 9.5, 9.6 16.003/004 -- Unified Engineering Department of Aeronautics and Astronautics Massachusetts Institute of Technology LEARNING OBJECTIVES

Buckling Analysis of Non-Prismatic Columns Using Slope-Deflection Method

Buckling Analysis of Non-Prismatic Columns Using Slope-Deflection Method H. Tajmir Riahi University of Isfahan, Iran A. Shojaei Barjoui, S. Bazazzadeh & S. M. A. Etezady Isfahan University of Technology,

MECHANICAL BEHAVIOR OF REINFORCED CONCRETE BEAM-COLUMN ASSEMBLAGES WITH ECCENTRICITY

13 th World Conference on Earthquake Engineering Vancouver, B.C., Canada August 1-6, 2004 Paper No. 4 MECHANICAL BEHAVIOR OF REINFORCED CONCRETE BEAM-COLUMN ASSEMBLAGES WITH ECCENTRICITY Tomohiko KAMIMURA

THE EFFECT OF FLEXURAL RIGIDITY ON THE MOMENT AND DEFLECTION OF STATICALLY INDETERMINATE REINFORCED CONCRETE ELEMENTS

2nd Int. PhD Symposium in Civil Engineering 1998 Budapest THE EFFECT OF FLEXURAL RIGIDITY ON THE OENT AND DEFLECTION OF STATICALLY INDETERINATE REINFORCED CONCRETE ELEENTS Katalin Rákóczy 1 and György

Structural Integrity Analysis

Structural Integrity Analysis 1. STRESS CONCENTRATION Igor Kokcharov 1.1 STRESSES AND CONCENTRATORS 1.1.1 Stress An applied external force F causes inner forces in the carrying structure. Inner forces

SAMPLE OF THE STUDY MATERIAL PART OF CHAPTER 1 STRESS AND STRAIN

SAMPLE OF THE STUDY MATERIAL PART OF CHAPTER 1 STRESS AND STRAIN 1.1 Stress & Strain Stress is the internal resistance offered by the body per unit area. Stress is represented as force per unit area. Typical

Civil Engineering. Strength of Materials. Comprehensive Theory with Solved Examples and Practice Questions. Publications

Civil Engineering Strength of Materials Comprehensive Theory with Solved Examples and Practice Questions Publications Publications MADE EASY Publications Corporate Office: 44-A/4, Kalu Sarai (Near Hauz

Since the Steel Joist Institute

SELECTING and SPECIFYING Wesley B. Myers, P.E. An insider s guide to selecting and specifying K-series, LH, DLH-series joists and joist girders Since the Steel Joist Institute adopted the first standard

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

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 Lesson 21 The oment- Distribution ethod: rames with Sidesway Instructional Objectives After reading this chapter the student

SEISMIC CODE EVALUATION. MEXICO Evaluation conducted by Jorge Gutiérrez

SEISMIC CODE EVALUATION MEXICO Evaluation conducted by Jorge Gutiérrez NAME OF DOCUMENT: Normas Técnicas Complementarias para Diseño por Sismo ( Complementary Technical Norms for Earthquake Resistant Design

Interaction between plate and column buckling

Delft, University of Technology Engineering office of Public works Rotterdam Interaction between plate and column buckling Master Thesis Name: Alex van Ham Student number: 1306138 Email: vanham.alex@gmail.com

Design of cross-girders and slabs in ladder deck bridges

130 Chris R Hendy Head of Bridge Design and Technology Highways & Transportation Atkins Jessica Sandberg Senior Engineer Highways & Transportation Atkins David Iles Steel Construction Institute Design

BEAM THEORIES The difference between Euler-Bernoulli and Timoschenko

BEAM THEORIES The difference between Euler-Bernoulli and Timoschenko Uemuet Goerguelue Two mathematical models, namely the shear-deformable (Timoshenko) model and the shearindeformable (Euler-Bernoulli)

THE STUDY OF LATERAL TORSIONAL BUCKLING BEHAVIOUR OF BEAM WITH TRAPEZOID WEB STEEL SECTION BY EXPERIMENTAL AND FINITE ELEMENT ANALYSIS

THE STUDY OF LATERAL TORSIONAL BUCKLING BEHAVIOUR OF BEAM WITH TRAPEZOID WEB STEEL SECTION BY EXPERIMENTAL AND FINITE ELEMENT ANALYSIS Fatimah Denan 1, Mohd Hanim Osman 2 & Sariffuddin Saad 3 1 School

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

Introduction to Mechanical Behavior of Biological Materials

Introduction to Mechanical Behavior of Biological Materials Ozkaya and Nordin Chapter 7, pages 127-151 Chapter 8, pages 173-194 Outline Modes of loading Internal forces and moments Stiffness of a structure

Design Analysis and Review of Stresses at a Point

Design Analysis and Review of Stresses at a Point Need for Design Analysis: To verify the design for safety of the structure and the users. To understand the results obtained in FEA, it is necessary to

Fundamentals of Structural Design Part of Steel Structures

undamentals of Structural Design Part of Steel Structures Civil Engineering for Bachelors 133STD Teacher: Zdeněk Sokol Office number: B619 1 Syllabus of lectures 1. Introduction, history of steel structures,

Pin jointed structures are often used because they are simple to design, relatively inexpensive to make, easy to construct, and easy to modify.

4. FORCES in PIN JOINTED STRUCTURES Pin jointed structures are often used because they are simple to design, relatively inexpensive to make, easy to construct, and easy to modify. They can be fixed structures

10. COMPRESSION AND BUCKLING

10. COMRESSION AND BUCKLING Whenever a structural member is designed, it is necessary that it satisfies specific strength, deflection and stability requirements. Typically strength (or in some cases fracture

Chapter 9 - Plastic Analysis

Chapter 9 lastic Analysis Chapter 9 - lastic Analysis 9.1 Introduction... 3 9.1.1 Background... 3 9.2 Basis of lastic Design... 4 9.2.1 aterial Behaviour... 4 9.2.2 Cross Section Behaviour... 6 9.2.3 lastic

Chapter 9 CONCRETE STRUCTURE DESIGN REQUIREMENTS

Chapter 9 CONCRETE STRUCTURE DESIGN REQUIREMENTS 9.1 GENERAL 9.1.1 Scope. The quality and testing of concrete and steel (reinforcing and anchoring) materials and the design and construction of concrete

Retrofitting of RCC Structure WIH Strengthening of Shear Wall with External Post Tensioning Cables

Retrofitting of RCC Structure WIH Strengthening of Shear Wall with External Post Tensioning Cables Yogesh Ghodke, G. R. Gandhe Department of Civil Engineering, Deogiri Institute of Engineering and Management

EML 5526 FEA Project 1 Alexander, Dylan. Project 1 Finite Element Analysis and Design of a Plane Truss

Problem Statement: Project 1 Finite Element Analysis and Design of a Plane Truss The plane truss in Figure 1 is analyzed using finite element analysis (FEA) for three load cases: A) Axial load: 10,000

The Behaviour of Drive-in Storage Structures

Missouri University of Science and Technology Scholars' Mine International Specialty Conference on Cold- Formed Steel Structures (2002) - 16th International Specialty Conference on Cold-Formed Steel Structures

NEESR SG: Behavior, Analysis and Design of Complex Wall Systems The laboratory testing presented here was conducted as part of a larger effort that employed laboratory testing and numerical simulation

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

CHAPTER 9 MULTI-DEGREE-OF-FREEDOM SYSTEMS Equations of Motion, Problem Statement, and Solution Methods

CHAPTER 9 MULTI-DEGREE-OF-FREEDOM SYSTEMS Equations of Motion, Problem Statement, and Solution Methods Two-story shear building A shear building is the building whose floor systems are rigid in flexure

Design of Steel Structures Prof. S.R.Satish Kumar and Prof. A.R.Santha Kumar

Problem 1 Design a hand operated overhead crane, which is provided in a shed, whose details are: Capacity of crane = 50 kn Longitudinal spacing of column = 6m Center to center distance of gantry girder

Steel and Cast Iron (Systematic Rehabilitation)

5. Steel and Cast Iron 5.1 Scope Rehabilitation measures for steel components and elements are described in this chapter. Information needed for systematic rehabilitation of steel buildings, as depicted

Stresses in Beam (Basic Topics)

Chapter 5 Stresses in Beam (Basic Topics) 5.1 Introduction Beam : loads acting transversely to the longitudinal axis the loads create shear forces and bending moments, stresses and strains due to V and

Numerical Optimization of Cold Formed Closed Built-up Steel Column

Numerical Optimization of Cold Formed Closed Built-up Steel Column Pradeep T 1, Arun N 1 P.G. Student, Department of Civil Engineering, K.S.Rangasamy College of Technology, Tiruchengode, Namakkal Tamil

5 Steel elements. 5.1 Structural design At present there are two British Standards devoted to the design of strucof tural steel elements:

5 Steel elements 5.1 Structural design At present there are two British Standards devoted to the design of strucof steelwork tural steel elements: BS 449 The use of structural steel in building. BS 5950

PRACTICAL METHODS FOR CRITICAL LOAD DETERMINATION AND STABILITY EVALUATION OF STEEL STRUCTURES PEDRO FERNANDEZ

PRACTICAL METHODS FOR CRITICAL LOAD DETERMINATION AND STABILITY EVALUATION OF STEEL STRUCTURES By PEDRO FERNANDEZ B.S., Instituto Tecnologico y de Estudios Superiores de Occidente, 1992 A Thesis submitted

CH 6: Fatigue Failure Resulting from Variable Loading Some machine elements are subjected to static loads and for such elements static failure theories are used to predict failure (yielding or fracture).

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

Chapter 3 THE STATIC ASPECT OF SOLICITATION

Chapter 3 THE STATIC ASPECT OF SOLICITATION 3.1. ACTIONS Construction elements interact between them and with the environment. The consequence of this interaction defines the system of actions that subject

STRENGTH OF MATERIALS - II

STRENGTH OF MATERIALS - II DEPARTMENT OF CIVIL ENGINEERING INSTITUTE OF AERONAUTICAL ENGINEERING UNIT 1 TORSION TORSION OF HOLLOW SHAFTS: From the torsion of solid shafts of circular x section,

Optimising plate girder design

Optimising plate girder design NSCC29 R. Abspoel 1 1 Division of structural engineering, Delft University of Technology, Delft, The Netherlands ABSTRACT: In the design of steel plate girders a high degree

INTERNATIONAL BUILDING CODE STRUCTURAL

INTERNATIONAL BUILDING CODE STRUCTURAL S5-06/07 1604.11 (New), 1605 (New) Proposed Change as Submitted: Proponent: William M. Connolly, State of New Jersey, Department of Community Affairs, Division of

1.1 To determine the effect the slenderness ratio has on the load carrying capacity of pin ended columns.

I. OBJECTIVES 1.1 To determine the effect the slenderness ratio has on the load carrying capacity of pin ended columns. 1. To observe short, intermediate and long column behavior under the application

Add-on Module STEEL EC3. Ultimate Limit State, Serviceability, Fire Resistance, and Stability Analyses According. Program Description

Version December 2014 Add-on Module STEEL EC3 Ultimate Limit State, Serviceability, Fire Resistance, and Stability Analyses According to Eurocode 3 Program Description All rights, including those of translations,

Designer s NOTEBOOK BLAST CONSIDERATIONS

Designer s NOTEBOOK BLAST CONSIDERATIONS For a surface blast, the most directly affected building elements are the façade and structural members on the lower four stories. Although the walls can be designed

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

σ = F / A o Chapter Outline Introduction Mechanical Properties of Metals How do metals respond to external loads?

Mechanical Properties of Metals How do metals respond to external loads? and Tension Compression Shear Torsion Elastic deformation Chapter Outline Introduction To understand and describe how materials

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

Non-linear behavior and seismic safety of reinforced concrete structures

Non-linear behavior and seismic safety of reinforced concrete structures K. Girgin & E. Oer Istanbul Technical University, Faculty of Civil Engineering, Istanbul, Turkey Keywords: Reinforced concrete structures,

Building a Better Grid

Building a Better Grid by Don White, Ph.D., and Domenic Coletti, P.E. Proposed improvements to 2D steel I-girder bridge analysis. Two-dimensional analysis methods are a popular choice when it comes to

MECHANICS OF SOLIDS COMPRESSION MEMBERS TUTORIAL 1 STRUTS. On completion of this tutorial you should be able to do the following.

MECHANICS OF SOLIDS COMPRESSION MEMBERS TUTORIAL 1 STRUTS You should judge your progress by completing the self assessment exercises. On completion of this tutorial you should be able to do the following.