LATERAL STABILITY OF DEEP BEAMS WITH SHEAR-BEAM SUPPORT
|
|
- Cameron Richards
- 6 years ago
- Views:
Transcription
1 U. FOREST SERVICE RESEARCH PAPER FPL 43 OCTOBER U. S. DEPARTMENT OF AGRICULTURE FOREST SERVICE FOREST PRODUCTS LABORATORY MADISON, WIS. LATERAL STABILITY OF DEEP BEAMS WITH SHEAR-BEAM SUPPORT
2 The FOREST SERVICE of the U. S. DEPARTMENT OF AGRICULTURE is dedicated to the principle of tiple use management of the Nation s forest resources for sustained yields of wood, water, forage, wildlife, and recreation. Through forestry research, cooperation with the States and private forest owner s, and management of the National Forests and National Grasslands, it directed by Congress to provide increasingly greater service to a growing
3 SUMMARY The stability of roof and floor systems whose proportions allow lateral buckling of the supporting beams is analyzed, with particular attention to the stabilizing influence of the stiffness of the attached deck. For the class of problems considered, the stabilizing effect is shown to be mathematically analogous to that of an axial pull acting on the beams along the line of deck attachment. The model employed for the deck is a "shear-beam" in which the effects of shear are primary and those of bending are negligible. Such a beam is a dual of the usual Euler-Bernoulli beam. The main result is a pair of linear differential equations governing lateral displacement and twist in which the influence of vertical deflection is disregarded. In particular cases, a power-series solution was assumed and eigenvalue pairs were determined with the aid of a computer. Numerical results are presented in the form of curves for four cases where the vertical loading and deck attachment are at the centroidal axis of the beam. i
4 CONTENTS Page INTRODUCTION DEFINITIONS OF SYMBOLS USED MATHEMATICAL MODEL OF DECK--THE "SHEAR-BEAM" THE SUPPORTING BEAM Equilibium analysis Load-Deflection Relations GENERAL DIFFERENTIAL EQUATIONS STABILITY CRITERIA FOR PARTICULAR CASES Case 1.--Pure Bending and Simple Support Case 2.--Uniform Load and Simple Support End Loaded Cantilever Case 4.--Uniformly Loaded Cantilever DISCUSSION AND RECOMMENDATIONS LITERATURE CITED APPENDIX--DETAILS OF SERIES SOLUTION FOR CASE ii
5 Lateral Stability of Deep Beams with Shear-Beam Support By JOHN J. ZAHN, Engineer Forest Products Laboratory, 1 Forest Service U.S. Department of Agriculture INTRODUCTION In many modern roof structures, a few deep laminated beams have replaced a system of many smaller rafters. As a result, the possibility of failure by lateral buckling has arisen calculating the buckling load, the only formulas presently available to the designer do not include the stabilizing effect of the decking attached to the beams, although it seems likely that this effect has a major influence on the stability of the system. It is clear that present design practice depends upon the decking to provide stability, although the margin of safety contained in these designs is difficult to estimate. Early papers (4, 5) 2,3 contain formulas for the critical loads of cantilevers and simple beams under various loadings. Timoshenko (7) summarized these and indicated how an approximate solutition for point loadings could be obtained by an energy method. Temple and Bickley (6) improved and clarified the energy approach. Flint (2) and Hooley and Madsen (3) reviewed these solutions, verified them experimentally, and 1 Maintained at Madison, Wis., in cooperation with the University of Wisconsin. 2 Underlined numbers in parentheses refer to Literature Cited at the end of this report. 3 Prandtl, L. "Kipperscheinungen.'' Ph.D. thesis at Nuremberg
6 urged their adoption in design codes. Flint (1) also used Timoshenko's energy method to investigate the effects of elasticity of supports and elastic restraint at an intermediate point on the span. The purpose of this Research Paper is to assist the designer of roof and floor systems by theoretically analyzing the effect of deck stiffness on the lateral stability of the supporting beams. Figure 1 shows the basic beam and deck system, which is periodic with period S and extends to infinity in the z direction; that is, any influence of end walls is neglected. The deep beams are assumed to be supported by sidewalls (not shown) either as cantilevers or as simple beams whose ends are restrained against axial rotation. Under these assumptions, the degree of lateral support which a deck system offer the supporting beams is determined by the shear stiffness of the deck. To analyze the effect of this stiffness on the lateral buckling load of the beams, the deck system is mathematically modeled as a "shear-beam," that is, a beam in the horizontal plane whose deflections are entirely due to shear and whose bending can be neglected. The stabilizing effect of the deck on the lateral stability of the supporting beams turns out to be mathematically analogous to the stabilizing effect of an axial pull on the beams along the line of deck attachment, and when both influences are present their effects are additive. The results are therefore presented in terms of a single dimensionless parameter η, which represents the sum of an axial pull and the shear stiffness of the attached deck. When η is negative, the buckling load is found to be reduced and this is interpreted as buckling under combined axial compression and vertical loading in the plane of greatest flexural rigidity. The buckled shape is a combination of twist and lateral displacement in the plane of least flexural rigidity. The theoretical equations are quite general, but numerical results are presented only for the case where the deck is FPL 43 2
7 attached at the centroidal axis of the beam and vertical loading is through the deck. This solution should next be perturbed by a small parameter representing the height of the deck attachment. DEFINITION OF SYMBOLS USED A n, B n, C n, D n Series coefficients. Refer to equations (A1) and 4 mn, B mn, C mn, D mn} (A6) in Appendix. c Distance from centroidal axis to line of deck attachment, Refer to figure 4. EI 2 F F x JG k Least flexural rigidity of deep beam. Shear force in deck. Axial force in deep beam. Torsional rigidity of deep beam. Shear of deck, in pounds. Refer to figure 3 and equation (2). L Mx, My, Mz Representative length in x direction. Internal moments in deep beam. Refer to figure 4, Refer to equations (17). P P q Distributed vertical load. Refer to figure 4. Representative vertical force on deep beam. Distributed load in z direction transmitted by deck attachment. u, v, w Displacements of centroidal axis of deep beam in x, y, and z directions. We neglect u and v. Vx, Vy Shear forces in deep beam. Refer to figure 4. Refer to equations (17). w D Displacement of deck in z direction. 3
8 Refer to equations (17). Coordinate axes. to figure 4. Rotation of cross section of deep beam due to twist. Refer to figure 4. Refer to equations (17) MATHEMATICAL MODEL OF DECK--THE SHEAR-BEAM Consider a roof system consisting of regularly spaced deep beams whose top edges are bridged by deck planks (fig. 1). Figure 1.--Basic beam and deck system used for analysis in this study. FPL 43 4
9 For convenience, the planks are considered to be infinitely long and any influence from their end support is overlooked. When the beams buckle laterally, the planks are displaced. Although the deck offers no resistance to a rigid-body displacement, it is assumed that differential displacement of planks is resisted by internailing between planks so that the deck has shear stiffness; that is, a deck is considered which has two kinds of nailing: one which joins adjacent deckboards to each other, and another which attaches the deck to the supporting beams is assumed to be such that the The attachment to the can transmit only a lateral force to the beams. This rather loose attachment renders our system slightly less stable than a real one. Since the system is periodic, our attention may be restricted to one period. Figure 2 isolates a single deckboard with a length of one period, such as the one shown shaded in figure 1. M Figure 2.--Deckboard considered as an element of a "shear-beam." In this diagram, F is the force in the nails between deckboards, q x is the force at the attachment to the beam, and x is the board width. Lateral forces between boards are not shown and deformations associated 5
10 with them will be neglected. Thus, the board in figure 2 is like an element of a "shear-beam," whose deflection is due entirely to shear. Figure 3 shows the relation between shear force and slope for such a beam, where w is the "shear-beam" displacement and k is a shear stiffness which has the dimension of force: (1) M Figure 3.--Deck modeled as a "shear-beam" of depth S. For convenience, the deck is further idealized by permitting x to approach zero. The individual attachment forces q x then become infinitesimal and a model is provided with continuous deck attachment that transmits a distributed force of q pounds per foot to the support beams. Then, (2) For equilibrium of vertical forces: (3) FPL 43 6
11 From (2) and (3) the load-deflection relationship of a shear-beam is (4) Notice that the application of a concentrated load to a "shear-beam" requires a jump discontinuity in the shear force F and, according to (1), in the slope as well. Since in our system the "shear-beam" is attached to an Euler-Bernoulli beam where the slope of the line of attachment must be continuous, there can be no concentrated force transmitted between the deck and the supporting beams except possibly at the ends of the span. Equilibrium Analysis THE SUPPORTING BEAM Figure 4 shows an element of the support beam in a deflected position, M Figure 3.--Free-body diagram of a support bean element.
12 Here p and q are distributed forces acting a distance c above the centroidal axis. The displacements in the y and z directions are denoted by v and w, respectively, and the angle of twist is ß, measured positive from y toward z. The vertical displacement v is much smaller than the lateral displacement w since the supporting beam is assumed to be deep; hence v is neglected. This effectively limits the length-to-depth ratio of the supporting beams. The equations of equilibrium are: (5) (6) (7) (8) (9) Load- Deflection Relations The following are classical results and will not be derived here. For details refer to (5, 7). The equation governing lateral deflection is (10) where EI2 is the flexural rigidity in the x-z plane. The equation governing twist is FPL 43 8
13 (11) where JG is the torsional rigidity. GENERAL DIFFERENTIAL EQUATIONS The "shear-beam" is attached to the support beam by matching displacements (12) Using (4) and (12), equation (6) becomes which integrates to (13) where sub-zero denotes evaluation at x = 0. Using (13), equation (8) becomes which integrates to (14) 9
14 Substituting (14) into (10) we get (15) Differentiating (11) we get Using (7) and (9) this becomes Inserting q from (4) and (12) we get, finally, (16) where the last term can be neglected in comparison with the left side. At this point we nondimensionalize the main equations. Let (17) FPL 43 10
15 (17) We will use primers to denote We repeat the main results in nondimensional form, retaining the original equation numbers and adding a "hat" (^). ^ (13) ^ (14) ^ (15) ^ (16) 11
16 It is unfortunate that the deck stiffness k occurs in two parameters τ. and η instead of only one, but the quantity will be zero if the point x = 0 is either: condition 1--A free boundary or, condition 2--A point of symmetry. For symmetrically loaded, simply supported beams we will place the origin at the center and argue from symmetry that and must vanish, thus establishing condition 2. For cantilevers, we will place the origin at the free end and appeal to condition 1. To prove condition 1, recall that there can be no concentrated force transmitted between the deck and the beam except possibly at the ends of the beam. Figure 5 shows how such a force could act at the free end of a cantilever. M Figure 5.--Concentrated force between deck and support beam at free end. FPL 43 12
17 From statics, we have that In nondimensional notation this means (18) which establishes condition 1. Thus the second term on the right in equation (15) ^ will be zero for the cases to be considered in this paper. Then k and Fx combine into the group k + F x and it is seen that the stabilizing influence of the deck is exactly analogous to the effect of an axial pull acting on the beam along the line of attachment. Thus, a single parameter η suffices in the cases about to be considered. STABILITY CRITERIA FOR PARTICULAR CASES Four cases are discussed, namely: 1. Pure bending and simple support 2. Uniform load and simple support 3. End loaded cantilever 4. Uniformly loaded cantilever For simplicity, the only cases considered are those in which c. = o; that is, the distributed loads are assumed to act at the centroidal axis of the beam, including that due to deck attachment. An approximate assessment of the influence of c can be obtained later by using the solution for the case c = o and adding a first-order correction term. 13
18 Case 1.--Pure Bending and Simple Support Figure 6.--Asimply supported beam in pure bending. M Attached deck is not shown. In figure 6 the length is 2L and the ends x = +L from rotating about the x axis. take P to be M L. Then are restrained Since there are no vertical loads, we and (19) (20) Thus (15) ^ and (16) ^ become (21) FPL 43 14
19 with boundary conditions (22) The general solution of (21) is: (23) where the C i are integration constants. Applying the first two conditions of (22) it is noted that C1 = C 3 = 0 From the remaining two conditions of (22) (24) from which a nonzero solution is possible only if the determinant of the coefficients is zero. This requires that or or (25) 15
20 which checks the classical result if η = 0, and reduces to the Euler column formula if θ = 0; that is, if M = 0, buckling can still occur if or (26) which is possible since F x can be negative (compressive). Note that L in equation (26) is the half-length. In the next three cases the differential equations do not have constant coefficients, but they can be solved numerically by assuming a power-series solution and programming a computer to set the determinant of the boundary-condition equations to zero by trial. In this way critical pairs (θ, η) can be found and presented graphically in lieu of a closed expression such as (25). For ready comparison with (25) θ 2 is plotted versus η when presenting these critical relationships. Since θ 2 is always positive, this will further serve to indicate that the sign of θ is as it is by symmetry. Case 2.--Uniform Load and Simple Support M Figure 7.--A simply supported beam uniformly loaded. the attached deck (not shown) and the distributed load p act at the centtroidal axis. FPL 43 16
21 Here the boundary conditions are (22), the same as in Case 1. Again L is the half-length. P is taken to be 1 pl, where p is the 2 distributed load, and since we have Of course Equation (20) still holds as in Case 1 and for the same reasons. the differential equations (15) ^ and (16) ^ become: Thus, and (27) Details of the series solution of equations (27) subject to boundary conditions (22) are given in the Appendix. The resulting critical condition is presented in figure 8. 17
22 M Figure 8.--Relation between load parameter θ and stabilizer parameter η (both dimensionless) for Case 2. Case 3.--End Loaded Cantilever M Figure 9.--A cantilever beam under a concentrated load at centroid of the end cross section. Attached deck is not shown. FPL 43 18
23 In figure 9, L is the length of the beam and P is the end load, Then The basic torque-twist relation (11) is (11) Since we are considering only the case where c = 0, and no torque Mx is applied at the free end, it is concluded that Obviously, M ^ ^ ^ yo = 0 so the differential equations (15) and (16) become (28) with the boundary conditions (29) 19
24 The eigenvalues of system (28, 29) were obtained by assuming a series solution in a manner entirely analogous to Case 2. The critical condition is presented in figure Figure 10.--Relation between load parameter θ and stabiiizer parameter η (both dimensionless) for Case-3. FPL 43 20
25 Case 4 Loaded Cantilever M Figure 11.--A cantilever beam uniformly loaded. Attached deck (not shown) and distributed load p act at the centroidal axis. In figure 11, L is the length of the beam and P is taken to be 1 2 pl. Again ^ Mzo = 0. Thus, and the differential equations (15) ^ and (16) ^ are (30) subject to the same boundary conditions (29) as in Case 3. The eigenvalues of system (29, 30), as obtained by a series solution, are presented in figure
26 Figure 12.--Relation between load parameter θ and stabiiizer parameter η (both dimensionless) for Case-4. DISCUSSION AND RECOMMENDATIONS Particular stability criteria are presented in equation (25) and in figures 8, 10, and 12. These can be used to predict the lateral-buckling load of flat roof and floor systems employing widely spaced deep beams with an attached deck. The "diaphragm action" of the deck, that is, the stabilizing influence obtained from its shear rigidity, is here quantitatively accounted or in the parameter? for the four cases considered. FPL GPO
27 The assumptions made in the derivation are that the beam is thin and deep--so that vertical deflections were neglected--and the deck is taken to be attached at the centroidal axis (c = 0). It is believed the influence of c will be small for relatively long beams, but this question deserves further study. should be checked experimentally. The theoretical results contained here 23
28 LITERATURE CITED (1) Flint, A. R The influence of restraints on the stability of beams. Structural Engineer, Vol. 29(9): (2) The stability and strength of slender beams. Engineering, Vol. 170: (3) Hooley, R. F., and Madsen, B Lateral stability of glued laminated beams. Jour. Amer. Soc. Engrs., Part I, Vol. 90(ST3) : (4) Mitchell, A. G. M Elastic stability of long beams under transverse forces. Phil. Mag., Vol. 48: (5) Prescott, J The buckling of deep beams. Phil. Mag., Series 6, 36: ; Vol. 39: , (6) Temple, G., and Bickley, W. G Rayleigh's principle and its applications to engineering. Dover, NewYork. (7) Timoshenko, S Theory of elastic stability. McGraw-Hill, New York. FPL 43 24
29 APPENDIX Details of Series Solution for Case 2 Given the differential equations (27.1) (27.2) and the boundary conditions (22) we assume a solution in the form (A1) with (A2) so that ao and bo are integration constants. Note (A1) is even in ξ so that only the last two boundary conditions (22) remain to be satisfied. They require that 25
30 (A3) System (A3) has a nontrivial solution only if the determinant of coefficients vanishes. Let H be this determinant. Then, (A4) is the critical condition. In (A4) the subscripted variables are all functions of η and θ. We shall want to specify η and solve for θ by trial. convenient to write Hence it is (A5) where the double-subscripted variables are functions of η. Then (A1) becomes FPL 43 26
31 (A6) where, by (A2) (A7) Substituting (A6) into the differential equations (27), we from (27.1) obtain, (A8) 27
32 and from (27.2) (A9) From (A8) and (A9) four recursion relations are obtained, as follows: Equating coefficients of in (A8), we obtain (A10) FPL 43 28
33 Equating coefficients of in (A8), we obtain (A11) Equating coefficients of in (A9), we obtain (A12) Equating coefficients of in (A9), we obtain (A13) These relations can be made to hold for all integers n, including n = 0, if we define that is, (A14) and agree that if any subscript is negative, the subscripted variable is zero. Critical condition (A4) now reads: 29
34 (A15) Now, given any value for θ, the recursion relations (A10) through (A13) and the initial values (A7) and (A14) will generate the arrays Amn, Bmn, Cmn, and Dmn.Summing these over n will yield one-dimensional arrays of coefficients of powers of θ in (A15). It is found that (A15) is a polynominal in powers of θ 2. (This could have been expected from the fact that the system is symmetrical about the x-z plane and therefore the sense of the critical load in the y direction--that is, the sign of θ--should be immaterial.) The critical value of θ 2 corresponding to the given η is then the lowest root of the polynominal (A15). An IBM 1620 computer was programmed to follow this procedure automatically and figure 8 was generated. A similar method was used in Cases 3 and 4 to produce figures 10 and 12. FPL GPO
LATERAL STABILITY OF BEAMS WITH ELASTIC END RESTRAINTS
LATERAL STABILITY OF BEAMS WITH ELASTIC END RESTRAINTS By John J. Zahn, 1 M. ASCE ABSTRACT: In the analysis of the lateral buckling of simply supported beams, the ends are assumed to be rigidly restrained
More informationChapter 1 General Introduction Instructor: Dr. Mürüde Çelikağ Office : CE Building Room CE230 and GE241
CIVL222 STRENGTH OF MATERIALS Chapter 1 General Introduction Instructor: Dr. Mürüde Çelikağ Office : CE Building Room CE230 and GE241 E-mail : murude.celikag@emu.edu.tr 1. INTRODUCTION There are three
More informationLecture 15 Strain and stress in beams
Spring, 2019 ME 323 Mechanics of Materials Lecture 15 Strain and stress in beams Reading assignment: 6.1 6.2 News: Instructor: Prof. Marcial Gonzalez Last modified: 1/6/19 9:42:38 PM Beam theory (@ ME
More informationSTRESSES WITHIN CURVED LAMINATED BEAMS OF DOUGLAS-FIR
UNITED STATES DEPARTMENT OF AGRICULTURE. FOREST SERVICE - FOREST PRODUCTS LABORATORY - MADISON, WIS. STRESSES WITHIN CURVED LAMINATED BEAMS OF DOUGLAS-FIR NOVEMBER 1963 FPL-020 STRESSES WITHIN CURVED LAMINATED
More informationTHEORETICAL DESIGN OF A NAILED OR BOLTED JOINT UNDER LATERAL LOAD 1. Summary
THEORETICAL DESIGN OF A NAILED OR BOLTED JOINT UNDER LATERAL LOAD 1 BY EDWARD W. KUENZI, 2 Engineer Forest Products Laboratory,3 Forest Service U. S. Department of Agriculture Summary This report presents
More informationPh.D. Preliminary Examination Analysis
UNIVERSITY OF CALIFORNIA, BERKELEY Spring Semester 2014 Dept. of Civil and Environmental Engineering Structural Engineering, Mechanics and Materials Name:......................................... Ph.D.
More informationStructural Dynamics Lecture Eleven: Dynamic Response of MDOF Systems: (Chapter 11) By: H. Ahmadian
Structural Dynamics Lecture Eleven: Dynamic Response of MDOF Systems: (Chapter 11) By: H. Ahmadian ahmadian@iust.ac.ir Dynamic Response of MDOF Systems: Mode-Superposition Method Mode-Superposition Method:
More informationWilliam J. McCutcheon U.S. Department of Agriculture, Forest Service Forest Products Laboratory Madison, Wisconsin 53705
This article appeared in Civil Engineering for Practicing and Design Engineers 2: 207-233; 1983. McCutcheon, William J. Deflections and stresses in circular tapered beams and poles. Civil Eng. Pract. Des,
More informationLecture 11: The Stiffness Method. Introduction
Introduction Although the mathematical formulation of the flexibility and stiffness methods are similar, the physical concepts involved are different. We found that in the flexibility method, the unknowns
More informationMODULE C: COMPRESSION MEMBERS
MODULE C: COMPRESSION MEMBERS This module of CIE 428 covers the following subjects Column theory Column design per AISC Effective length Torsional and flexural-torsional buckling Built-up members READING:
More information3. BEAMS: STRAIN, STRESS, DEFLECTIONS
3. BEAMS: STRAIN, STRESS, DEFLECTIONS The beam, or flexural member, is frequently encountered in structures and machines, and its elementary stress analysis constitutes one of the more interesting facets
More informationPresented By: EAS 6939 Aerospace Structural Composites
A Beam Theory for Laminated Composites and Application to Torsion Problems Dr. BhavaniV. Sankar Presented By: Sameer Luthra EAS 6939 Aerospace Structural Composites 1 Introduction Composite beams have
More informationConsider an elastic spring as shown in the Fig.2.4. When the spring is slowly
.3 Strain Energy Consider an elastic spring as shown in the Fig..4. When the spring is slowly pulled, it deflects by a small amount u 1. When the load is removed from the spring, it goes back to the original
More informationPLAT DAN CANGKANG (TKS 4219)
PLAT DAN CANGKANG (TKS 4219) SESI I: PLATES Dr.Eng. Achfas Zacoeb Dept. of Civil Engineering Brawijaya University INTRODUCTION Plates are straight, plane, two-dimensional structural components of which
More informationGeneral 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 informationMarch 24, Chapter 4. Deflection and Stiffness. Dr. Mohammad Suliman Abuhaiba, PE
Chapter 4 Deflection and Stiffness 1 2 Chapter Outline Spring Rates Tension, Compression, and Torsion Deflection Due to Bending Beam Deflection Methods Beam Deflections by Superposition Strain Energy Castigliano
More informationIf the number of unknown reaction components are equal to the number of equations, the structure is known as statically determinate.
1 of 6 EQUILIBRIUM OF A RIGID BODY AND ANALYSIS OF ETRUCTURAS II 9.1 reactions in supports and joints of a two-dimensional structure and statically indeterminate reactions: Statically indeterminate structures
More informationQuintic 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 informationCHAPTER -6- BENDING Part -1-
Ishik University / Sulaimani Civil Engineering Department Mechanics of Materials CE 211 CHAPTER -6- BENDING Part -1-1 CHAPTER -6- Bending Outlines of this chapter: 6.1. Chapter Objectives 6.2. Shear and
More information6. Bending CHAPTER OBJECTIVES
CHAPTER OBJECTIVES Determine stress in members caused by bending Discuss how to establish shear and moment diagrams for a beam or shaft Determine largest shear and moment in a member, and specify where
More informationMECHANICS OF MATERIALS. Prepared by Engr. John Paul Timola
MECHANICS OF MATERIALS Prepared by Engr. John Paul Timola Mechanics of materials branch of mechanics that studies the internal effects of stress and strain in a solid body. stress is associated with the
More informationKINGS COLLEGE OF ENGINEERING DEPARTMENT OF MECHANICAL ENGINEERING QUESTION BANK. Subject code/name: ME2254/STRENGTH OF MATERIALS Year/Sem:II / IV
KINGS COLLEGE OF ENGINEERING DEPARTMENT OF MECHANICAL ENGINEERING QUESTION BANK Subject code/name: ME2254/STRENGTH OF MATERIALS Year/Sem:II / IV UNIT I STRESS, STRAIN DEFORMATION OF SOLIDS PART A (2 MARKS)
More informationMachine Direction Strength Theory of Corrugated Fiberboard
Thomas J. Urbanik 1 Machine Direction Strength Theory of Corrugated Fiberboard REFERENCE: Urbanik.T.J., Machine Direction Strength Theory of Corrugated Fiberboard, Journal of Composites Technology & Research,
More informationFlexural-Torsional Buckling of General Cold-Formed Steel Columns with Unequal Unbraced Lengths
Proceedings of the Annual Stability Conference Structural Stability Research Council San Antonio, Texas, March 21-24, 2017 Flexural-Torsional Buckling of General Cold-Formed Steel Columns with Unequal
More informationLecture 4: PRELIMINARY CONCEPTS OF STRUCTURAL ANALYSIS. Introduction
Introduction In this class we will focus on the structural analysis of framed structures. We will learn about the flexibility method first, and then learn how to use the primary analytical tools associated
More informationLecture Slides. Chapter 4. Deflection and Stiffness. The McGraw-Hill Companies 2012
Lecture Slides Chapter 4 Deflection and Stiffness The McGraw-Hill Companies 2012 Chapter Outline Force vs Deflection Elasticity property of a material that enables it to regain its original configuration
More informationModule 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method
Module 2 Analysis of Statically Indeterminate Structures by the Matrix Force Method Lesson 8 The Force Method of Analysis: Beams Instructional Objectives After reading this chapter the student will be
More informationChapter 4 Deflection and Stiffness
Chapter 4 Deflection and Stiffness Asst. Prof. Dr. Supakit Rooppakhun Chapter Outline Deflection and Stiffness 4-1 Spring Rates 4-2 Tension, Compression, and Torsion 4-3 Deflection Due to Bending 4-4 Beam
More informationShear force and bending moment of beams 2.1 Beams 2.2 Classification of beams 1. Cantilever Beam Built-in encastre' Cantilever
CHAPTER TWO Shear force and bending moment of beams 2.1 Beams A beam is a structural member resting on supports to carry vertical loads. Beams are generally placed horizontally; the amount and extent of
More informationStructural Steelwork Eurocodes Development of A Trans-national Approach
Structural Steelwork Eurocodes Development of A Trans-national Approach Course: Eurocode Module 7 : Worked Examples Lecture 0 : Simple braced frame Contents: 1. Simple Braced Frame 1.1 Characteristic Loads
More informationAccordingly, the nominal section strength [resistance] for initiation of yielding is calculated by using Equation C-C3.1.
C3 Flexural Members C3.1 Bending The nominal flexural strength [moment resistance], Mn, shall be the smallest of the values calculated for the limit states of yielding, lateral-torsional buckling and distortional
More informationDEFLECTION OF BEAMS WlTH SPECIAL REFERENCE TO SHEAR DEFORMATIONS
DEFLECTION OF BEAMS WlTH SPECIAL REFERENCE TO SHEAR DEFORMATIONS THE INFLUENCE OF THE FORM OF A WOODEN BEAM ON ITS STIFFNESS AND STRENGTH-I (REPRINT FROM NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS REPORT
More informationMechanical Design in Optical Engineering
OPTI Buckling Buckling and Stability: As we learned in the previous lectures, structures may fail in a variety of ways, depending on the materials, load and support conditions. We had two primary concerns:
More information7.4 The Elementary Beam Theory
7.4 The Elementary Beam Theory In this section, problems involving long and slender beams are addressed. s with pressure vessels, the geometry of the beam, and the specific type of loading which will be
More informationMechanics of Materials II. Chapter III. A review of the fundamental formulation of stress, strain, and deflection
Mechanics of Materials II Chapter III A review of the fundamental formulation of stress, strain, and deflection Outline Introduction Assumtions and limitations Axial loading Torsion of circular shafts
More informationFLEXIBILITY METHOD FOR INDETERMINATE FRAMES
UNIT - I FLEXIBILITY METHOD FOR INDETERMINATE FRAMES 1. What is meant by indeterminate structures? Structures that do not satisfy the conditions of equilibrium are called indeterminate structure. These
More informationDESIGN OF BEAMS AND SHAFTS
DESIGN OF EAMS AND SHAFTS! asis for eam Design! Stress Variations Throughout a Prismatic eam! Design of pristmatic beams! Steel beams! Wooden beams! Design of Shaft! ombined bending! Torsion 1 asis for
More informationD : SOLID MECHANICS. Q. 1 Q. 9 carry one mark each. Q.1 Find the force (in kn) in the member BH of the truss shown.
D : SOLID MECHANICS Q. 1 Q. 9 carry one mark each. Q.1 Find the force (in kn) in the member BH of the truss shown. Q.2 Consider the forces of magnitude F acting on the sides of the regular hexagon having
More informationMINIMUM WEIGHT STRUCTURAL SANDWICH
U.S. DEPARTMENT OF AGRICULTURE FOREST SERVICE FOREST PRODUCTS LABORATORY MADISON, WIS. In Cooperation with the University of Wisconsin U.S.D.A. FOREST SERVICE RESEARCH NOTE Revised NOVEMBER 1970 MINIMUM
More informationPart D: Frames and Plates
Part D: Frames and Plates Plane Frames and Thin Plates A Beam with General Boundary Conditions The Stiffness Method Thin Plates Initial Imperfections The Ritz and Finite Element Approaches A Beam with
More informationMECHANICS OF STRUCTURES SCI 1105 COURSE MATERIAL UNIT - I
MECHANICS OF STRUCTURES SCI 1105 COURSE MATERIAL UNIT - I Engineering Mechanics Branch of science which deals with the behavior of a body with the state of rest or motion, subjected to the action of forces.
More informationUnit 15 Shearing and Torsion (and Bending) of Shell Beams
Unit 15 Shearing and Torsion (and Bending) of Shell Beams Readings: Rivello Ch. 9, section 8.7 (again), section 7.6 T & G 126, 127 Paul A. Lagace, Ph.D. Professor of Aeronautics & Astronautics and Engineering
More informationDES140: Designing for Lateral-Torsional Stability in Wood Members
DES140: Designing for Lateral-Torsional Stability in Wood embers Welcome to the Lateral Torsional Stability ecourse. 1 Outline Lateral-Torsional Buckling Basic Concept Design ethod Examples In this ecourse,
More informationCHAPTER 5. Beam Theory
CHPTER 5. Beam Theory SangJoon Shin School of Mechanical and erospace Engineering Seoul National University ctive eroelasticity and Rotorcraft Lab. 5. The Euler-Bernoulli assumptions One of its dimensions
More informationFIXED BEAMS IN BENDING
FIXED BEAMS IN BENDING INTRODUCTION Fixed or built-in beams are commonly used in building construction because they possess high rigidity in comparison to simply supported beams. When a simply supported
More informationSRI CHANDRASEKHARENDRA SARASWATHI VISWA MAHAVIDHYALAYA
SRI CHANDRASEKHARENDRA SARASWATHI VISWA MAHAVIDHYALAYA (Declared as Deemed-to-be University under Section 3 of the UGC Act, 1956, Vide notification No.F.9.9/92-U-3 dated 26 th May 1993 of the Govt. of
More informationTransactions on Modelling and Simulation vol 18, 1997 WIT Press, ISSN X
An integral equation formulation of the coupled vibrations of uniform Timoshenko beams Masa. Tanaka & A. N. Bercin Department of Mechanical Systems Engineering, Shinshu University 500 Wakasato, Nagano
More informationDesign of Beams (Unit - 8)
Design of Beams (Unit - 8) Contents Introduction Beam types Lateral stability of beams Factors affecting lateral stability Behaviour of simple and built - up beams in bending (Without vertical stiffeners)
More informationA HIGHER-ORDER BEAM THEORY FOR COMPOSITE BOX BEAMS
A HIGHER-ORDER BEAM THEORY FOR COMPOSITE BOX BEAMS A. Kroker, W. Becker TU Darmstadt, Department of Mechanical Engineering, Chair of Structural Mechanics Hochschulstr. 1, D-64289 Darmstadt, Germany kroker@mechanik.tu-darmstadt.de,
More informationMechanics of Materials Primer
Mechanics of Materials rimer Notation: A = area (net = with holes, bearing = in contact, etc...) b = total width of material at a horizontal section d = diameter of a hole D = symbol for diameter E = modulus
More informationD : SOLID MECHANICS. Q. 1 Q. 9 carry one mark each.
GTE 2016 Q. 1 Q. 9 carry one mark each. D : SOLID MECHNICS Q.1 single degree of freedom vibrating system has mass of 5 kg, stiffness of 500 N/m and damping coefficient of 100 N-s/m. To make the system
More informationA study of the critical condition of a battened column and a frame by classical methods
University of South Florida Scholar Commons Graduate Theses and Dissertations Graduate School 003 A study of the critical condition of a battened column and a frame by classical methods Jamal A.H Bekdache
More informationQUESTION BANK SEMESTER: III SUBJECT NAME: MECHANICS OF SOLIDS
QUESTION BANK SEMESTER: III SUBJECT NAME: MECHANICS OF SOLIDS UNIT 1- STRESS AND STRAIN PART A (2 Marks) 1. Define longitudinal strain and lateral strain. 2. State Hooke s law. 3. Define modular ratio,
More informationDISTORTION ANALYSIS OF TILL -WALLED BOX GIRDERS
Nigerian Journal of Technology, Vol. 25, No. 2, September 2006 Osadebe and Mbajiogu 36 DISTORTION ANALYSIS OF TILL -WALLED BOX GIRDERS N. N. OSADEBE, M. Sc., Ph. D., MNSE Department of Civil Engineering
More informationChapter Objectives. Design a beam to resist both bendingand shear loads
Chapter Objectives Design a beam to resist both bendingand shear loads A Bridge Deck under Bending Action Castellated Beams Post-tensioned Concrete Beam Lateral Distortion of a Beam Due to Lateral Load
More informationThe Homotopy Perturbation Method for free vibration analysis of beam on elastic foundation
Shiraz University of Technology From the SelectedWorks of Habibolla Latifizadeh 2011 The Homotopy Perturbation Method for free vibration analysis of beam on elastic foundation Habibolla Latifizadeh, Shiraz
More informationLinear elastic analysis of thin laminated beams with uniform and symmetric cross-section
Applied and Computational Mechanics 2 (2008) 397 408 Linear elastic analysis of thin laminated beams with uniform and symmetric cross-section M. Zajíček a, a Faculty of Applied Sciences, UWB in Pilsen,
More informationCHAPTER 5. T a = 0.03 (180) 0.75 = 1.47 sec 5.12 Steel moment frame. h n = = 260 ft. T a = (260) 0.80 = 2.39 sec. Question No.
CHAPTER 5 Question Brief Explanation No. 5.1 From Fig. IBC 1613.5(3) and (4) enlarged region 1 (ASCE 7 Fig. -3 and -4) S S = 1.5g, and S 1 = 0.6g. The g term is already factored in the equations, thus
More informationModule 3. Analysis of Statically Indeterminate Structures by the Displacement Method
odule 3 Analysis of Statically Indeterminate Structures by the Displacement ethod Lesson 14 The Slope-Deflection ethod: An Introduction Introduction As pointed out earlier, there are two distinct methods
More informationTheory of structure I 2006/2013. Chapter one DETERMINACY & INDETERMINACY OF STRUCTURES
Chapter one DETERMINACY & INDETERMINACY OF STRUCTURES Introduction A structure refers to a system of connected parts used to support a load. Important examples related to civil engineering include buildings,
More informationComb resonator design (2)
Lecture 6: Comb resonator design () -Intro Intro. to Mechanics of Materials School of Electrical l Engineering i and Computer Science, Seoul National University Nano/Micro Systems & Controls Laboratory
More informationChapter 7: Bending and Shear in Simple Beams
Chapter 7: Bending and Shear in Simple Beams Introduction A beam is a long, slender structural member that resists loads that are generally applied transverse (perpendicular) to its longitudinal axis.
More informationA New Method of Analysis of Continuous Skew Girder Bridges
A New Method of Analysis of Continuous Skew Girder Bridges Dr. Salem Awad Ramoda Hadhramout University of Science & Technology Mukalla P.O.Box : 50511 Republic of Yemen ABSTRACT One three girder two span
More information1859. Forced transverse vibration analysis of a Rayleigh double-beam system with a Pasternak middle layer subjected to compressive axial load
1859. Forced transverse vibration analysis of a Rayleigh double-beam system with a Pasternak middle layer subjected to compressive axial load Nader Mohammadi 1, Mehrdad Nasirshoaibi 2 Department of Mechanical
More information2. Determine the deflection at C of the beam given in fig below. Use principal of virtual work. W L/2 B A L C
CE-1259, Strength of Materials UNIT I STRESS, STRAIN DEFORMATION OF SOLIDS Part -A 1. Define strain energy density. 2. State Maxwell s reciprocal theorem. 3. Define proof resilience. 4. State Castigliano
More informationLaboratory 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[8] Bending and Shear Loading of Beams
[8] Bending and Shear Loading of Beams Page 1 of 28 [8] Bending and Shear Loading of Beams [8.1] Bending of Beams (will not be covered in class) [8.2] Bending Strain and Stress [8.3] Shear in Straight
More information9.1 Introduction to bifurcation of equilibrium and structural
Module 9 Stability and Buckling Readings: BC Ch 14 earning Objectives Understand the basic concept of structural instability and bifurcation of equilibrium. Derive the basic buckling load of beams subject
More informationWhere and are the factored end moments of the column and >.
11 LIMITATION OF THE SLENDERNESS RATIO----( ) 1-Nonsway (braced) frames: The ACI Code, Section 6.2.5 recommends the following limitations between short and long columns in braced (nonsway) frames: 1. The
More informationFINAL EXAMINATION. (CE130-2 Mechanics of Materials)
UNIVERSITY OF CLIFORNI, ERKELEY FLL SEMESTER 001 FINL EXMINTION (CE130- Mechanics of Materials) Problem 1: (15 points) pinned -bar structure is shown in Figure 1. There is an external force, W = 5000N,
More informationLab Exercise #5: Tension and Bending with Strain Gages
Lab Exercise #5: Tension and Bending with Strain Gages Pre-lab assignment: Yes No Goals: 1. To evaluate tension and bending stress models and Hooke s Law. a. σ = Mc/I and σ = P/A 2. To determine material
More informationCH. 4 BEAMS & COLUMNS
CH. 4 BEAMS & COLUMNS BEAMS Beams Basic theory of bending: internal resisting moment at any point in a beam must equal the bending moments produced by the external loads on the beam Rx = Cc + Tt - If the
More informationCompression Members Columns II
Compression Members Columns II 1. Introduction. Main aspects related to the derivation of typical columns buckling lengths for. Analysis of imperfections, leading to the derivation of the Ayrton-Perry
More informationQUESTION BANK DEPARTMENT: CIVIL SEMESTER: III SUBJECT CODE: CE2201 SUBJECT NAME: MECHANICS OF SOLIDS UNIT 1- STRESS AND STRAIN PART A
DEPARTMENT: CIVIL SUBJECT CODE: CE2201 QUESTION BANK SEMESTER: III SUBJECT NAME: MECHANICS OF SOLIDS UNIT 1- STRESS AND STRAIN PART A (2 Marks) 1. Define longitudinal strain and lateral strain. 2. State
More informationTORSION 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 informationCOURSE TITLE : APPLIED MECHANICS & STRENGTH OF MATERIALS COURSE CODE : 4017 COURSE CATEGORY : A PERIODS/WEEK : 6 PERIODS/ SEMESTER : 108 CREDITS : 5
COURSE TITLE : APPLIED MECHANICS & STRENGTH OF MATERIALS COURSE CODE : 4017 COURSE CATEGORY : A PERIODS/WEEK : 6 PERIODS/ SEMESTER : 108 CREDITS : 5 TIME SCHEDULE MODULE TOPICS PERIODS 1 Simple stresses
More informationtwenty one concrete construction: shear & deflection ARCHITECTURAL STRUCTURES: FORM, BEHAVIOR, AND DESIGN DR. ANNE NICHOLS SUMMER 2014 lecture
ARCHITECTURAL STRUCTURES: FORM, BEHAVIOR, AND DESIGN DR. ANNE NICHOLS SUMMER 2014 lecture twenty one concrete construction: Copyright Kirk Martini shear & deflection Concrete Shear 1 Shear in Concrete
More informationDesign of Steel Structures Prof. S.R.Satish Kumar and Prof. A.R.Santha Kumar
5.4 Beams As stated previousl, the effect of local buckling should invariabl be taken into account in thin walled members, using methods described alread. Laterall stable beams are beams, which do not
More informationNON LINEAR BUCKLING OF COLUMNS Dr. Mereen Hassan Fahmi Technical College of Erbil
Abstract: NON LINEAR BUCKLING OF COLUMNS Dr. Mereen Hassan Fahmi Technical College of Erbil The geometric non-linear total potential energy equation is developed and extended to study the behavior of buckling
More informationFinite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras. Module - 01 Lecture - 13
Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras (Refer Slide Time: 00:25) Module - 01 Lecture - 13 In the last class, we have seen how
More informationMechanics of materials is one of the first application-based engineering
In This Chapter Chapter 1 Predicting Behavior with Mechanics of Materials Defining mechanics of materials Introducing stresses and strains Using mechanics of materials to aid in design Mechanics of materials
More informationDeflection of Flexural Members - Macaulay s Method 3rd Year Structural Engineering
Deflection of Flexural Members - Macaulay s Method 3rd Year Structural Engineering 008/9 Dr. Colin Caprani 1 Contents 1. Introduction... 3 1.1 General... 3 1. Background... 4 1.3 Discontinuity Functions...
More informationChapter 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 informationFirst-Order Solutions for the Buckling Loads of Euler-Bernoulli Weakened Columns
First-Order Solutions for the Buckling Loads of Euler-Bernoulli Weakened Columns J. A. Loya ; G. Vadillo 2 ; and J. Fernández-Sáez 3 Abstract: In this work, closed-form expressions for the buckling loads
More informationLaminated Beams of Isotropic or Orthotropic Materials Subjected to Temperature Change
United States Department of Agriculture Forest Service Forest Products Laboratory Research Paper FPL 375 June 1980 Laminated Beams of Isotropic or Orthotropic Materials Subjected to Temperature Change
More informationDeflection of Flexural Members - Macaulay s Method 3rd Year Structural Engineering
Deflection of Flexural Members - Macaulay s Method 3rd Year Structural Engineering 009/10 Dr. Colin Caprani 1 Contents 1. Introduction... 4 1.1 General... 4 1. Background... 5 1.3 Discontinuity Functions...
More informationBy Dr. Mohammed Ramidh
Engineering Materials Design Lecture.6 the design of beams By Dr. Mohammed Ramidh 6.1 INTRODUCTION Finding the shear forces and bending moments is an essential step in the design of any beam. we usually
More informationSymmetric Bending of Beams
Symmetric Bending of Beams beam is any long structural member on which loads act perpendicular to the longitudinal axis. Learning objectives Understand the theory, its limitations and its applications
More informationM.S Comprehensive Examination Analysis
UNIVERSITY OF CALIFORNIA, BERKELEY Spring Semester 2014 Dept. of Civil and Environmental Engineering Structural Engineering, Mechanics and Materials Name:......................................... M.S Comprehensive
More informationModule 3. Analysis of Statically Indeterminate Structures by the Displacement Method
odule 3 Analysis of Statically Indeterminate Structures by the Displacement ethod Lesson 16 The Slope-Deflection ethod: rames Without Sidesway Instructional Objectives After reading this chapter the student
More information7.6 Stress in symmetrical elastic beam transmitting both shear force and bending moment
7.6 Stress in symmetrical elastic beam transmitting both shear force and bending moment à It is more difficult to obtain an exact solution to this problem since the presence of the shear force means that
More informationAdvanced Structural Analysis EGF Section Properties and Bending
Advanced Structural Analysis EGF316 3. Section Properties and Bending 3.1 Loads in beams When we analyse beams, we need to consider various types of loads acting on them, for example, axial forces, shear
More informationNational Exams May 2015
National Exams May 2015 04-BS-6: Mechanics of Materials 3 hours duration Notes: If doubt exists as to the interpretation of any question, the candidate is urged to submit with the answer paper a clear
More informationProperties of Southern Pine in Relation to Strength Grading of Dimension Lumber
U. S. FOREST SERVICE RESEARCH PAPER FPL-64 JULY U.S. DEPARTMENT OF AGRICULTURE FOREST SERVICE FOREST PRODUCTS LABORATORY MADISON, WISCONSIN Properties of Southern Pine in Relation to Strength Grading of
More informationBUCKLING COEFFICIENTS FOR SIMPLY SUPPORTED, FLAT, RECTANGULAR SANDWICH PANELS UNDER BIAXIAL COMPRESSION
U. S. FOREST SERVICE RESEARCH PAPER FPL 135 APRIL 1970 BUCKLING COEFFICIENTS FOR SIMPLY SUPPORTED, FLAT, RECTANGULAR SANDWICH PANELS UNDER BIAXIAL COMPRESSION FOREST PRODUCTS LABORATORY, FOREST SERVICE
More informationA STUDY OF THE STRENGTH OF SHORT AND INTERMEDIATE WOOD COLUMNS BY EXPERIMENTAL AND ANALYTICAL METHODS
UNITED STATES DEPARTMENT OF AGRICULTURE. FOREST SERVICE. FOREST PRODUCTS LABORATORY. MADISON, WIS A STUDY OF THE STRENGTH OF SHORT AND INTERMEDIATE WOOD COLUMNS BY EXPERIMENTAL AND ANALYTICAL METHODS January
More informationSeismic design of bridges
NATIONAL TECHNICAL UNIVERSITY OF ATHENS LABORATORY FOR EARTHQUAKE ENGINEERING Seismic design of bridges Lecture 3 Ioannis N. Psycharis Capacity design Purpose To design structures of ductile behaviour
More informationSabah Shawkat Cabinet of Structural Engineering Walls carrying vertical loads should be designed as columns. Basically walls are designed in
Sabah Shawkat Cabinet of Structural Engineering 17 3.6 Shear walls Walls carrying vertical loads should be designed as columns. Basically walls are designed in the same manner as columns, but there are
More information7 TRANSVERSE SHEAR transverse shear stress longitudinal shear stresses
7 TRANSVERSE SHEAR Before we develop a relationship that describes the shear-stress distribution over the cross section of a beam, we will make some preliminary remarks regarding the way shear acts within
More informationEMA 3702 Mechanics & Materials Science (Mechanics of Materials) Chapter 6 Shearing Stress in Beams & Thin-Walled Members
EMA 3702 Mechanics & Materials Science (Mechanics of Materials) Chapter 6 Shearing Stress in Beams & Thin-Walled Members Beams Bending & Shearing EMA 3702 Mechanics & Materials Science Zhe Cheng (2018)
More information