Chapter 7: Forces in Beams and Cables


 Alexia Lester
 1 years ago
 Views:
Transcription
1 Chapter 7: Forces in Beams and Cables 최해진
2 Contents Introduction Internal Forces in embers Sample Problem 7.1 Various Types of Beam Loading and Support Shear and Bending oment in a Beam Sample Problem 7. Sample Problem 7.3 Relations Among Load, Shear, and Bending oment Sample Problem 7.4 Sample Problem 7.6 Cables With Concentrated Loads Cables With Distributed Loads Parabolic Cable Sample Problem 7.8 Catenary 7
3 Introduction Preceding chapters dealt with: a) determining eternal forces acting on a structure and b) determining forces which hold together the various members of a structure. The current chapter is concerned with determining the internal forces (i.e., tension/compression, shear, and bending) which hold together the various parts of a given member. Focus is on two important types of engineering structures: a) Beams  usually long, straight, prismatic members designed to support loads applied at various points along the member. b) Cables  fleible members capable of withstanding only tension, designed to support concentrated or distributed loads. 73
4 Internal Forces in embers Straight twoforce member AB is in equilibrium under application of F and F. Internal forces equivalent to F and F are required for equilibrium of freebodies AC and CB. ultiforce member ABCD is in equilibrium under application of cable and member contact forces. Internal forces equivalent to a forcecouple system are necessary for equilibrium of freebodies JD and ABCJ. An internal forcecouple system is required for equilibrium of twoforce members which are not straight. 74
5 Sample Problem 7.1 SOLUTION: Compute reactions and forces at connections for each member. Cut member ACF at J. The internal forces at J are represented by equivalent forcecouple system which is determined by considering equilibrium of either part. Determine the internal forces (a) in member ACF at point J and (b) in member BCD at K. Cut member BCD at K. Determine forcecouple system equivalent to internal forces at K by applying equilibrium conditions to either part. 75
6 Sample Problem 7.1 SOLUTION: Compute reactions and connection forces. Consider entire frame as a freebody: å å F y å E : ( 4 N)( 3.6 m) + ( 4.8m)  F F 18 N :  y 4 N + 18 N + E E y 6 F : E N 76
7 Sample Problem 7.1 Consider member BCD as freebody: å B : ( 4 N)( 3.6m) + C (.4m)  y å C : ( 4 N)( 1.m) + B (.4m)  y å F : B + C  C y B y 36 N 1 N Consider member ABE as freebody: å : B (.4 m) A B å F : B A  å F : A + B + 6 N y From member BCD, å  y y A A y F : B + C  18 N C 77
8 Sample Problem 7.1 Cut member ACF at J. The internal forces at J are represented by equivalent forcecouple system. Consider freebody AJ: å J : ( 18 N)( 1. m) N m å F F  å F y : ( 18 N) cos 41.7 : ( 18 N) sin 41.7 F 1344 N V + V 1197 N 78
9 Sample Problem 7.1 Cut member BCD at K. Determine a forcecouple system equivalent to internal forces at K. Consider freebody BK: å K : ( 1 N)( 1.5m) å N m F : F å F y : 1 N V V 1 N 79
10 Various Types of Beam Loading and Support Beam  structural member designed to support loads applied at various points along its length. Beam can be subjected to concentrated loads or distributed loads or combination of both. Beam design is twostep process: 1) determine shearing forces and bending moments produced by applied loads ) select crosssection best suited to resist shearing forces and bending moments 71
11 Various Types of Beam Loading and Support Beams are classified according to way in which they are supported. Reactions at beam supports are determinate if they involve only three unknowns. Otherwise, they are statically indeterminate. 711
12 Shear and Bending oment in a Beam Wish to determine bending moment and shearing force at any point in a beam subjected to concentrated and distributed loads. Determine reactions at supports by treating whole beam as freebody. Cut beam at C and draw freebody diagrams for AC and CB. By definition, positive sense for internal forcecouple systems are as shown. From equilibrium considerations, determine and V or and V. 71
13 Shear and Bending oment Diagrams Variation of shear and bending moment along beam may be plotted. Determine reactions at supports. Cut beam at C and consider member AC, V + P + P Cut beam at E and consider member EB, V  P + P L  ( ) For a beam subjected to concentrated loads, shear is constant between loading points and moment varies linearly. 713
14 Sample Problem 7. SOLUTION: Taking entire beam as a freebody, calculate reactions at B and D. Draw the shear and bending moment diagrams for the beam and loading shown. Find equivalent internal forcecouple systems for freebodies formed by cutting beam on either side of load application points. Plot results. 714
15 Sample Problem 7. SOLUTION: Taking entire beam as a freebody, calculate reactions at B and D. Find equivalent internal forcecouple systems at sections on either side of load application points. å Fy :  kn V 1 V1  kn å ( kn)( m) 1 : Similarly, + 1 V V V V kn 6kN 14kN 14 kn kn m + 8kN m + 8kN m 715
16 Sample Problem 7. Plot results. Note that shear is of constant value between concentrated loads and bending moment varies linearly. 716
17 Sample Problem 7.3 SOLUTION: Taking entire beam as freebody, calculate reactions at A and B. Determine equivalent internal forcecouple systems at sections cut within segments AC, CD, and DB. Draw the shear and bending moment diagrams for the beam AB. The distributed load of 4 N/m etends over.3 m of the beam, from A to C, and the 4N load is applied at E. Plot results. 717
18 Sample Problem 7.3 SOLUTION: Taking entire beam as a freebody, calculate reactions at A and B. : å B y å A (.8m) ( 1 N)(.3m)  ( 4 N)(.55m) B : B y 365N ( 1N)(.65m) + ( 4 N)(.5m)  A(.8m) å A 135 N F : Note: The 4 N load at E may be replaced by a 4 N force and 16 Nm couple at D. B 718
19 Sample Problem 7.3 Evaluate equivalent internal forcecouple systems at sections cut within segments AC, CD, and DB. From A to C: å F : V y å ( ) + 1 : From C to D: : å F : V y å ( .15) + V V 13 N ( ) N m 719
20 Sample Problem 7.3 Evaluate equivalent internal forcecouple systems at sections cut within segments AC, CD, and DB. From D to B: å F y : 1351m  4 V å : ( .15) ( .45) V 77 N ( ) N cm 7
21 Sample Problem 7.3 Plot results. From A to C: V From C to D: V 13 N From D to B: V ( ) N m 77 N ( ) N m 71
22 Relations Among Load, Shear, and Bending oment Relations between load and shear: V V  dv d D ( V + DV ) lim D  wd DV D V  ò D wd C C w  ( area under load curve) Relations between shear and bending moment: ( + D ) d d D D  VD + wd D lim lim D D D  ò D V d C C ( V  1 wd) V ( area under shear curve) 7
23 Relations Among Load, Shear, and Bending oment Reactions at supports, Shear curve, V VA ò wd V V A  w oment curve,  ò ma A æ L wç è ò wl 8 w wl  w ö  d ø Vd æ ç è w at RA RB æ L wç è ( ) L  d d V ö  ø wl ö ø 73
24 Sample Problem 7.4 SOLUTION: Draw the shear and bendingmoment diagrams for the beam and loading shown. Taking entire beam as a freebody, determine reactions at supports. Between concentrated load application points, dv d w and shear is constant. With uniform loading between D and E, the shear variation is linear. Between concentrated load application points, d d V constant. The change in moment between load application points is equal to area under shear curve between points. With a linear shear variation between D and E, the bending moment diagram is a parabola. 74
25 Sample Problem 7.4 SOLUTION: Taking entire beam as a freebody, determine reactions at supports. å A : D 7. m  kn 1.8 m  1 kn ( ) ( )( ) ( )( 4. m)  ( 1 kn)( 8.4 m) å F y :  kn 1 kn + 6 kn 1 kn A y D 6 kn A y Between concentrated load application points, dv d w and shear is constant. With uniform loading between D and E, the shear variation is linear. 18 kn 75
26 Sample Problem 7.4 Between concentrated load application points, d d V constant. The change in moment between load application points is equal to area under the shear curve between points. B C D E A B C D kn m + 9 kn m 48 kn m With a linear shear variation between D and E, the bending moment diagram is a parabola. B C D E 76
27 Sample Problem 7.6 SOLUTION: Sketch the shear and bendingmoment diagrams for the cantilever beam and loading shown. The change in shear between A and B is equal to the negative of area under load curve between points. The linear load curve results in a parabolic shear curve. With zero load, change in shear between B and C is zero. The change in moment between A and B is equal to area under shear curve between points. The parabolic shear curve results in a cubic moment curve. The change in moment between B and C is equal to area under shear curve between points. The constant shear curve results in a linear moment curve. 77
28 Sample Problem 7.6 SOLUTION: The change in shear between A and B is equal to negative of area under load curve between points. The linear load curve results in a parabolic shear curve. dv at A, VA, w w d V V  1 w a  1 w a B A V B dv at B, w d With zero load, change in shear between B and C is zero. 78
29 Sample Problem 7.6 The change in moment between A and B is equal to area under shear curve between the points. The parabolic shear curve results in a cubic moment curve. d at A, A, V d B C   A B w w a a B w ( L  a)  1 w a( 3L  a) C a The change in moment between B and C is equal to area under shear curve between points. The constant shear curve results in a linear moment curve. 79
30 Cables With Concentrated Loads Cables are applied as structural elements in suspension bridges, transmission lines, aerial tramways, guy wires for high towers, etc. For analysis, assume: a) concentrated vertical loads on given vertical lines, b) weight of cable is negligible, c) cable is fleible, i.e., resistance to bending is small, d) portions of cable between successive loads may be treated as two force members Wish to determine shape of cable, i.e., vertical distance from support A to each load point. 73
31 Cables With Concentrated Loads Consider entire cable as freebody. Slopes of cable at A and B are not known  two reaction components required at each support. Four unknowns are involved and three equations of equilibrium are not sufficient to determine the reactions. Additional equation is obtained by considering equilibrium of portion of cable AD and assuming that coordinates of point D on the cable are known. The additional equation is å. For other points on cable, å C yields y å F, å F yieldt, T T T cos q A constant D y y 731
32 Cables With Distributed Loads For cable carrying a distributed load: a) cable hangs in shape of a curve b) internal force is a tension force directed along tangent to curve. Consider freebody for portion of cable etending from lowest point C to given point D. Forces are horizontal force T at C and tangential force T at D. From force triangle: T cosq T T T + W T sinq W tanq Horizontal component of T is uniform over cable. Vertical component of T is equal to magnitude of W measured from lowest point. Tension is minimum at lowest point and maimum at A and B. W T 73
33 Parabolic Cable Consider a cable supporting a uniform, horizontally distributed load, e.g., support cables for a suspension bridge. With loading on cable from lowest point C to a point D given by W w, internal tension force magnitude and direction are w T T + w tanq T Summing moments about D, å D : w T y or w y T The cable forms a parabolic curve. 733
34 Sample Problem 7.8 SOLUTION: Determine reaction force components at A from solution of two equations formed from taking entire cable as freebody and summing moments about E, and from taking cable portion ABC as a freebody and summing moments about C. The cable AE supports three vertical loads from the points indicated. If point C is 1.5 m below the left support, determine (a) the elevation of points B and D, and (b) the maimum slope and maimum tension in the cable. Calculate elevation of B by considering AB as a freebody and summing moments B. Similarly, calculate elevation of D using ABCD as a freebody. Evaluate maimum slope and maimum tension which occur in DE. 734
35 Sample Problem 7.8 SOLUTION: Determine two reaction force components at A from solution of two equations formed from taking entire cable as a freebody and summing moments about E, å E : and from taking cable portion ABC as a freebody and summing moments about C. å C 1.5A :  9A y + 18 Solving simultaneously, A 18 kn A y 5 kn 735
36 Sample Problem 7.8 Calculate elevation of B by considering AB as a freebody and summing moments B. å B : y  B ( 18) ( 5)( 6) y B m Similarly, calculate elevation of D using ABCD as a freebody. å  : ( 18)  ( 5) (13.5) + (6)(7.5) + (1)(4.5) y D y D 1.75 m 736
37 Sample Problem 7.8 Evaluate maimum slope and maimum tension which occur in DE tan q q kn Tma cosq T ma 4.8 kn 737
38 Catenary Consider a cable uniformly loaded along the cable itself, e.g., cables hanging under their own weight. With loading on the cable from lowest point C to a point D given by W ws, the internal tension force magnitude is T T T + w s w c + s c To relate horizontal distance to cable length s, T ds d ds cosq cosq T q + s c s ò ds q + s c c sinh 1 s c and s c w sinh c 738
39 Catenary To relate and y cable coordinates, dy y c d y  c tanq ò cosh sinh c c W T d d c s c d cosh c sinh  c which is the equation of a catenary. c d 739
Shear Forces and Bending Moments
Chapter 4 Shear Forces and Bending Moments 4.1 Introduction Consider a beam subjected to transverse loads as shown in figure, the deflections occur in the plane same as the loading plane, is called the
More informationBeam Structures and Internal Forces
RCH 331 Note Set 7 Su2011abn Beam Structures and Internal Forces Notation: a = algebraic quantity, as is b, c, d = name for area b = intercept of a straight line d = calculus symbol for differentiation
More informationMechanics of Materials. Chapter 4 Shear and Moment In Beams
Mechanics of Materials Chapter 4 Shear and Moment In Beams 4.1 Introduction The term beam refers to a slender bar that carries transverse loading; that is, the applied force are perpendicular to the bar.
More information8.2 Shear and BendingMoment Diagrams: Equation Form
8.2 Shear and endingoment Diagrams: Equation Form 8.2 Shear and endingoment Diagrams: Equation Form Eample 1, page 1 of 6 1. Epress the shear and bending moment as functions of, the distance from the
More informationRecitation #5. Understanding Shear Force and Bending Moment Diagrams
Recitation #5 Understanding Shear Force and Bending Moment Diagrams Shear force and bending moment are examples of interanl forces that are induced in a structure when loads are applied to that structure.
More informationProblem 1: Computation of Reactions. Problem 2: Computation of Reactions. Problem 3: Computation of Reactions
Problem 1: Computation of Reactions Problem 2: Computation of Reactions Problem 3: Computation of Reactions Problem 4: Computation of forces and moments Problem 5: Bending Moment and Shear force Problem
More informationAn introduction to the relationship between bending moments and shear forces using the Push Me Pull Me models on Expedition Workshed
Worksheet 6 Bending moments & shear force An introduction to the relationship between bending moments and shear forces using the Push Me Pull Me models on Expedition Workshed National HE STEM Programme
More informationENGR1100 Introduction to Engineering Analysis. Lecture 13
ENGR1100 Introduction to Engineering Analysis Lecture 13 EQUILIBRIUM OF A RIGID BODY & FREEBODY DIAGRAMS Today s Objectives: Students will be able to: a) Identify support reactions, and, b) Draw a freebody
More informationBEAMS: SHEAR AND MOMENT DIAGRAMS (GRAPHICAL)
LECTURE Third Edition BES: SHER ND OENT DIGRS (GRPHICL). J. Clark School of Engineering Department of Civil and Environmental Engineering 3 Chapter 5.3 by Dr. Ibrahim. ssakkaf SPRING 003 ENES 0 echanics
More informationStatics of Structural Supports
Statics of Structural Supports TYPES OF FORCES External Forces actions of other bodies on the structure under consideration. Internal Forces forces and couples exerted on a member or portion of the structure
More informationStatics of Structural Supports
Statics of Structural Supports TYPES OF FORCES External Forces actions of other bodies on the structure under consideration. Internal Forces forces and couples exerted on a member or portion of the structure
More informationShear and Moment Diagrams. Shear and Moment Diagrams. Shear and Moment Diagrams. Shear and Moment Diagrams. Shear and Moment Diagrams
CI 3 Shear Force and Bending oment Diagrams /8 If the variation of and are written as functions of position,, and plotted, the resulting graphs are called the shear diagram and the moment diagram. Developing
More information2. Axial Force, Shear Force, Torque and Bending Moment Diagrams
2. Axial Force, Shear Force, Torque and Bending Moment Diagrams In this section, we learn how to summarize the internal actions (shear force and bending moment) that occur throughout an axial member, shaft,
More informationChapter 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
More information8.2 Continuous Beams (Part I)
8.2 Continuous Beams (Part I) This section covers the following topics. Analysis Incorporation of Moment due to Reactions Pressure Line due to Prestressing Force Introduction Beams are made continuous
More informationBending Stress and Strain
Bending Stress and Strain DEFLECTIONS OF BEAMS When a beam with a straight longitudinal ais is loaded by lateral forces, the ais is deformed into a curve, called the deflection curve of the beam. We will
More informationFy = P sin 50 + F cos = 0 Solving the two simultaneous equations for P and F,
ENGR0135  Statics and Mechanics of Materials 1 (161) Homework # Solution Set 1. Summing forces in the ydirection allows one to determine the magnitude of F : Fy 1000 sin 60 800 sin 37 F sin 45 0 F 543.8689
More informationDeflections. Question: What are Structural Deflections?
Question: What are Structural Deflections? Answer: The deformations or movements of a structure and its components, such as beams and trusses, from their original positions. It is as important for the
More informationChapter 9 Deflections of Beams
Chapter 9 Deflections of Beams 9.1 Introduction in this chapter, we describe methods for determining the equation of the deflection curve of beams and finding deflection and slope at specific points along
More informationStructural Axial, Shear and Bending Moments
Structural Axial, Shear and Bending Moments Positive Internal Forces Acting Recall from mechanics of materials that the internal forces P (generic axial), V (shear) and M (moment) represent resultants
More information4 Shear Forces and Bending Moments
4 Shear Forces and ending oments Shear Forces and ending oments 8 lb 16 lb roblem 4.31 alculate the shear force and bending moment at a cross section just to the left of the 16lb load acting on the simple
More informationHØGSKOLEN I GJØVIK Avdeling for teknologi, økonomi og ledelse. Løsningsforslag for kontinuasjonseksamen i Mekanikk 4/110
Løsningsforslag for kontinuasjonseksamen i 4/110 Oppgave 1 (T betyr tension, dvs. strekk, og C betyr compression, dvs. trykk.) Side 1 av 9 Leif Erik Storm Oppgave 2 Løsning (fra http://www.public.iastate.edu/~statics/examples/vmdiags/vmdiaga.html
More information4 Shear Forces and Bending Moments
4 Shear Forces and ending oments Shear Forces and ending oments 8 lb 16 lb roblem 4.31 alculate the shear force and bending moment at a cross section just to the left of the 16lb load acting on the simple
More informationMECHANICS OF SOLIDS  BEAMS TUTORIAL 2 SHEAR FORCE AND BENDING MOMENTS IN BEAMS
MECHANICS OF SOLIDS  BEAMS TUTORIAL 2 SHEAR FORCE AND BENDING MOMENTS IN BEAMS This is the second tutorial on bending of beams. You should judge your progress by completing the self assessment exercises.
More informationApproximate Analysis of Statically Indeterminate Structures
Approximate Analysis of Statically Indeterminate Structures Every successful structure must be capable of reaching stable equilibrium under its applied loads, regardless of structural behavior. Exact analysis
More informationENGINEERING SCIENCE H1 OUTCOME 1  TUTORIAL 3 BENDING MOMENTS EDEXCEL HNC/D ENGINEERING SCIENCE LEVEL 4 H1 FORMERLY UNIT 21718P
ENGINEERING SCIENCE H1 OUTCOME 1  TUTORIAL 3 BENDING MOMENTS EDEXCEL HNC/D ENGINEERING SCIENCE LEVEL 4 H1 FORMERLY UNIT 21718P This material is duplicated in the Mechanical Principles module H2 and those
More informationPin 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
More informationShear Forces and Bending Moments in Beams
Shear Forces and ending Moments in eams ending Stress: Moment of Inertia: σ = My I I x = y 2 d Parallel xis Theorem: I y = x 2 d I x = I xc + d 2 eam Classifications: I y = I yc + d 2 eams are also classified
More informationThe Hanging Chain. Kirk Gordon, Torey Seward. 6th of May, 2011
The Hanging Chain Kirk Gordon, Torey Seward 6th of May, 2011 1 Contents 1 Introduction 4 2 Deriving the Catenary Curve from a Chain of Uniform Mass Density 4 2.1 Setting up the Equation.......................
More informationTutorial 02 Statics (Chapter )
Tutorial  Statics (Chapter 3. 5.3) MECH Spring 9 eb6 th Tutor: LEO Contents Reviews ree Bod Diagram (BD) Truss structure Practices Review orce vector, Moment, Equilibrium of orces, D/3D Vectors (position,
More informationMODULE 3 ANALYSIS OF STATICALLY INDETERMINATE STRUCTURES BY THE DISPLACEMENT METHOD. Version 2 CE IIT, Kharagpur
ODULE 3 ANALYI O TATICALLY INDETERINATE TRUCTURE BY THE DIPLACEENT ETHOD LEON 19 THE OENT DITRIBUTION ETHOD: TATICALLY INDETERINATE BEA WITH UPPORT ETTLEENT Instructional Objectives After reading this
More information4. Analysis of Plane Trusses and Frames
4. Analysis of Plane Trusses and Frames December 2009 () Analysis of Plane Trusses and Frames 12/09 1 / 28 Introduction An engineering structure is any connected system of members built to support or transfer
More informationStatics of Bending: Shear and Bending Moment Diagrams
Statics of Bending: Shear and Bending Moment Diagrams David Roylance Department of Materials Science and Engineering Massachusetts Institute of Technology Cambridge, MA 02139 November 15, 2000 Introduction
More informationP4 Stress and Strain Dr. A.B. Zavatsky MT07 Lecture 3 Statically Indeterminate Structures
4 Stress and Strain Dr... Zavatsky MT07 ecture 3 Statically Indeterminate Structures Statically determinate structures. Statically indeterminate structures (equations of equilibrium, compatibility, and
More informationShear Forces and Bending Moments
Shear Forces and ending oments lanar (D) Structures: ll loads act in the same plane and all deflections occurs in the same plane (y plane) ssociated with the shear forces and bending moments are normal
More information4.2 Free Body Diagrams
CE297FA09Ch4 Page 1 Friday, September 18, 2009 12:11 AM Chapter 4: Equilibrium of Rigid Bodies A (rigid) body is said to in equilibrium if the vector sum of ALL forces and all their moments taken about
More informationInterconnected Rigid Bodies with Multiforce Members Rigid Noncollapsible
Frames and Machines Interconnected Rigid Bodies with Multiforce Members Rigid Noncollapsible structure constitutes a rigid unit by itself when removed from its supports first find all forces external
More information2.2 Losses in Prestress (Part II)
2.2 Losses in Prestress (Part II) This section covers the following topics Friction Anchorage Slip Force Variation Diagram 2.2.1 Friction The friction generated at the interface of concrete and steel during
More informationMECHANICS OF MATERIALS
2009 The McGrawHill Companies, Inc. All rights reserved. Fifth SI Edition CHAPTER 4 MECHANICS OF MATERIALS Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf David F. Mazurek Pure Bending Lecture
More informationENGINEERING MECHANICS STATIC
EX 16 Using the method of joints, determine the force in each member of the truss shown. State whether each member in tension or in compression. Sol Freebody diagram of the pin at B X = 0 500 BC sin
More informationModule 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
More informationBending Stress in Beams
93673600 Bending Stress in Beams Derive a relationship for bending stress in a beam: Basic Assumptions:. Deflections are very small with respect to the depth of the beam. Plane sections before bending
More informationCE 201 (STATICS) DR. SHAMSHAD AHMAD CIVIL ENGINEERING ENGINEERING MECHANICSSTATICS
COURSE: CE 201 (STATICS) LECTURE NO.: 28 to 30 FACULTY: DR. SHAMSHAD AHMAD DEPARTMENT: CIVIL ENGINEERING UNIVERSITY: KING FAHD UNIVERSITY OF PETROLEUM & MINERALS, DHAHRAN, SAUDI ARABIA TEXT BOOK: ENGINEERING
More informationMethod of Sections for Truss Analysis
Method of Sections for Truss Analysis Joint Configurations (special cases to recognize for faster solutions) Case 1) Two Bodies Connected F AB has to be equal and opposite to F BC Case 2) Three Bodies
More informationCHAPTER 13. Definite Integrals. Since integration can be used in a practical sense in many applications it is often
7 CHAPTER Definite Integrals Since integration can be used in a practical sense in many applications it is often useful to have integrals evaluated for different values of the variable of integration.
More informationAnalysis of Plane Frames
Plane frames are twodimensional structures constructed with straight elements connected together by rigid and/or hinged connections. rames are subjected to loads and reactions that lie in the plane of
More informationMECE 3400 Summer 2016 Homework #1 Forces and Moments as Vectors
ASSIGNED 1) Knowing that α = 40, determine the resultant of the three forces shown: 2) Two cables, AC and BC, are tied together at C and pulled by a force P, as shown. Knowing that P = 500 N, α = 60, and
More informationR A = R B = = 3.6 kn. ΣF y = 3.6 V = 0 V = 3.6 kn. A similar calculation for any section through the beam at 3.7 < x < 7.
ENDNG STRESSES & SHER STRESSES N EMS (SSGNMENT SOLUTONS) Question 1: 89 mm 3 mm Parallam beam has a length of 7.4 m and supports a concentrated load of 7.2 kn, as illustrated below. Draw shear force and
More informationAnnouncements. Moment of a Force
Announcements Test observations Units Significant figures Position vectors Moment of a Force Today s Objectives Understand and define Moment Determine moments of a force in 2D and 3D cases Moment of
More informationSTRESS AND DEFORMATION ANALYSIS OF LINEAR ELASTIC BARS IN TENSION
Chapter 11 STRESS AND DEFORMATION ANALYSIS OF LINEAR ELASTIC BARS IN TENSION Figure 11.1: In Chapter10, the equilibrium, kinematic and constitutive equations for a general threedimensional solid deformable
More informationRevision Coordinate Geometry. Question 1: The gradient of the line joining the points V (k, 8) and W (2, k +1) is is.
Question 1: 2 The gradient of the line joining the points V (k, 8) and W (2, k +1) is is. 3 a) Find the value of k. b) Find the length of VW. Question 2: A straight line passes through A( 2, 3) and B(
More informationCHAPTER 3 SHEARING FORCE AND BENDING MOMENT DIAGRAMS. Summary
CHAPTER 3 SHEARING FORCE AND BENDING MOMENT DIAGRAMS Summary At any section in a beam carrying transverse loads the shearing force is defined as the algebraic sum of the forces taken on either side of
More informationSIMPLE TRUSSES, THE METHOD OF JOINTS, & ZEROFORCE MEMBERS
SIMPLE TRUSSES, THE METHOD OF JOINTS, & ZEROFORCE MEMBERS Today s Objectives: Students will be able to: a) Define a simple truss. b) Determine the forces in members of a simple truss. c) Identify zeroforce
More information8.2 Elastic Strain Energy
Section 8. 8. Elastic Strain Energy The strain energy stored in an elastic material upon deformation is calculated below for a number of different geometries and loading conditions. These expressions for
More informationIndeterminate Analysis Force Method 1
Indeterminate Analysis Force Method 1 The force (flexibility) method expresses the relationships between displacements and forces that exist in a structure. Primary objective of the force method is to
More informationSOLUTION 6 6. Determine the force in each member of the truss, and state if the members are in tension or compression.
6 6. etermine the force in each member of the truss, and state if the members are in tension or compression. 600 N 4 m Method of Joints: We will begin by analyzing the equilibrium of joint, and then proceed
More informationChapter 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
More informationStrength of Materials Prof: S.K.Bhattacharya Dept of Civil Engineering, IIT, Kharagpur Lecture no 23 Bending of Beams II
Strength of Materials Prof: S.K.Bhattacharya Dept of Civil Engineering, IIT, Kharagpur Lecture no 23 Bending of Beams II Welcome to the second lesson of the fifth module which is on Bending of Beams part
More information1 CHAPTER 18 THE CATENARY. 18.1 Introduction
1 CHAPER 18 HE CAENARY 18.1 Introduction If a flexible chain or rope is loosely hung between two fixed points, it hangs in a curve that looks a little like a parabola, but in fact is not quite a parabola;
More informationTwoForce Members, ThreeForce Members, Distributed Loads
TwoForce Members, ThreeForce Members, Distributed Loads TwoForce Members  Examples ME 202 2 TwoForce Members Only two forces act on the body. The line of action (LOA) of forces at both A and B must
More informationThe Free Body Diagram. The Concurrent System
T A B The Free Body Diagram The Concurrent System Free Body Diagrams Essential step in solving Equilibrium problems Complex Structural systems reduced into concise FORCE systems WHAT IS A FREE BODY DIAGRAM?
More informationcos 2u  t xy sin 2u (Q.E.D.)
09 Solutions 46060 6/8/10 3:13 PM Page 619 010 Pearson Education, Inc., Upper Saddle River, NJ. ll rights reserved. This material is protected under all copyright laws as they currently 9 1. Prove that
More informationChapter 8. Shear Force and Bending Moment Diagrams for Uniformly Distributed Loads.
hapter 8 Shear Force and ending Moment Diagrams for Uniformly Distributed Loads. 8.1 Introduction In Unit 4 we saw how to calculate moments for uniformly distributed loads. You might find it worthwhile
More informationLaboratory 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
More informationEML 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
More information5 Solutions /23/09 5:11 PM Page 377
5 Solutions 44918 1/23/09 5:11 PM Page 377 5 71. The rod assembly is used to support the 250lb cylinder. Determine the components of reaction at the ballandsocket joint, the smooth journal bearing E,
More informationCH 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
More informationSign Conventions. Applied and Reaction Force Sign Conventions Are Static Sign Conventions
Sign Conventions Sign conventions are a standardized set of widely accepted rules that provide a consistent method of setting up, solving, and communicating solutions to engineering mechanics problemsstatics,
More informationAnswers to Selected EvenNumbered Problems
Answers to Selected EvenNumbered Problems NOTE TO INSTRUCTORS CONSIDERING ADOPTION: Additional content (e.g., FBDs, shear and moment diagrams, etc.) is in the process of being added to this document.
More informationModule 6. Approximate Methods for Indeterminate Structural Analysis. Version 2 CE IIT, Kharagpur
odule 6 Approximate ethods for Indeterminate Structural Analsis Lesson 36 Building Frames Instructional Objectives: After reading this chapter the student will be able to 1. Analse building frames b approximate
More informationModule 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method. Version 2 CE IIT, Kharagpur
Module Analysis of Statically Indeterminate Structures by the Matrix Force Method esson 11 The Force Method of Analysis: Frames Instructional Objectives After reading this chapter the student will be able
More information1. (a) Find an equation of the line joining A (7, 4) and B (2, 0), giving your answer in the form ax + by + c = 0, where a, b and c are integers.
1. (a) Find an equation of the line joining A (7, 4) and B (2, 0), giving your answer in the form a + by + c = 0, where a, b and c are integers. (b) Find the length of AB, leaving your answer in surd form.
More informationEQUILIBRIUM AND ELASTICITY
Chapter 12: EQUILIBRIUM AND ELASTICITY 1 A net torque applied to a rigid object always tends to produce: A linear acceleration B rotational equilibrium C angular acceleration D rotational inertia E none
More informationPLANE TRUSSES. Definitions
Definitions PLANE TRUSSES A truss is one of the major types of engineering structures which provides a practical and economical solution for many engineering constructions, especially in the design of
More informationThe Mathematics of Beam Deflection
The athematics of eam Deflection Scenario s a structural engineer you are part of a team working on the design of a prestigious new hotel comple in a developing city in the iddle East. It has been decided
More informationGeneral tests Algebra
General tests Algebra Question () : Choose the correct answer :  If = then = a)0 b) 6 c)5 d)4  The shape which represents Y is a function of is :   V A B C D o   y   V o    V o    V o  if
More information6.3 Trusses: Method of Sections
6.3 Trusses: Method of Sections 6.3 Trusses: Method of Sections xample 1, page 1 of 2 1. etermine the force in members,, and, and state whether the force is tension or compression. 5 m 4 kn 4 kn 5 m 1
More informationv v ax v a x a v a v = = = Since F = ma, it follows that a = F/m. The mass of the arrow is unchanged, and ( )
Week 3 homework IMPORTANT NOTE ABOUT WEBASSIGN: In the WebAssign versions of these problems, various details have been changed, so that the answers will come out differently. The method to find the solution
More informationStresses 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
More informationMechanics of Materials. Chapter 6 Deflection of Beams
Mechanics of Materials Chapter 6 Deflection of Beams 6.1 Introduction Because the design of beams is frequently governed by rigidity rather than strength. For example, building codes specify limits on
More informationChapter 5: Indeterminate Structures Force Method
Chapter 5: Indeterminate Structures Force Method 1. Introduction Statically indeterminate structures are the ones where the independent reaction components, and/or internal forces cannot be obtained by
More informationPhysics Exam 1 Review  Chapter 1,2
Physics 1401  Exam 1 Review  Chapter 1,2 13. Which of the following is NOT one of the fundamental units in the SI system? A) newton B) meter C) kilogram D) second E) All of the above are fundamental
More information11. THE STABILITY OF SLOPES
111 11. THE STABILITY OF SLOPES 11.1 INTRODUCTION The quantitative determination of the stability of slopes is necessary in a number of engineering activities, such as: (a) (b) (c) (d) the design of earth
More informationChapter (3) SLOPE DEFLECTION METHOD
Chapter (3) SOPE DEFECTION ETHOD 3.1 Introduction: The methods of three moment equation, and consistent deformation method are represent the FORCE ETHOD of structural analysis, The slope deflection method
More informationAnnouncements. Dry Friction
Announcements Dry Friction Today s Objectives Understand the characteristics of dry friction Draw a FBD including friction Solve problems involving friction Class Activities Applications Characteristics
More informationStructural Analysis: Space Truss
Structural Analysis: Space Truss Space Truss  6 bars joined at their ends to form the edges of a tetrahedron as the basic noncollapsible unit  3 additional concurrent bars whose ends are attached to
More informationUnit 4: Science and Materials in Construction and the Built Environment. Chapter 14. Understand how Forces act on Structures
Chapter 14 Understand how Forces act on Structures 14.1 Introduction The analysis of structures considered here will be based on a number of fundamental concepts which follow from simple Newtonian mechanics;
More informationThe Locus of the Focus of a Rolling Parabola
The Locus of the Focus of a Rolling Parabola Anurag Agarwal & James Marengo October 6, 2009 1 Introduction The catenary can easily be mistaken for a parabola. Even Galileo made this error. In his Discorsi
More informationwww.mathsbox.org.uk Displacement (x) Velocity (v) Acceleration (a) x = f(t) differentiate v = dx Acceleration Velocity (v) Displacement x
Mechanics 2 : Revision Notes 1. Kinematics and variable acceleration Displacement (x) Velocity (v) Acceleration (a) x = f(t) differentiate v = dx differentiate a = dv = d2 x dt dt dt 2 Acceleration Velocity
More informationP4 Stress and Strain Dr. A.B. Zavatsky MT07 Lecture 4 Stresses on Inclined Sections
4 Stress and Strain Dr. A.B. Zavatsky MT07 Lecture 4 Stresses on Inclined Sections Shear stress and shear strain. Equality of shear stresses on perpendicular planes. Hooke s law in shear. Normal and shear
More informationSimple Harmonic Motion Concepts
Simple Harmonic Motion Concepts INTRODUCTION Have you ever wondered why a grandfather clock keeps accurate time? The motion of the pendulum is a particular kind of repetitive or periodic motion called
More informationCHAPTER 6 Space Trusses
CHAPTER 6 Space Trusses INTRODUCTION A space truss consists of members joined together at their ends to form a stable threedimensional structures A stable simple space truss can be built from the basic
More informationMechanics Lecture Notes. 1 Notes for lectures 12 and 13: Motion in a circle
Mechanics Lecture Notes Notes for lectures 2 and 3: Motion in a circle. Introduction The important result in this lecture concerns the force required to keep a particle moving on a circular path: if the
More informationChapter 1: Statics. A) Newtonian Mechanics B) Relativistic Mechanics
Chapter 1: Statics 1. The subject of mechanics deals with what happens to a body when is / are applied to it. A) magnetic field B) heat C ) forces D) neutrons E) lasers 2. still remains the basis of most
More informationUnit 7: Right Triangles and Trigonometry Lesson 7.1 Use Inequalities in a Triangle Lesson 5.5 from textbook
Unit 7: Right Triangles and Trigonometry Lesson 7.1 Use Inequalities in a Triangle Lesson 5.5 from textbook Objectives Use the triangle measurements to decide which side is longest and which angle is largest.
More informationStatics 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
More informationINTRODUCTION TO BEAMS
CHAPTER Structural Steel Design LRFD Method INTRODUCTION TO BEAMS Third Edition A. J. Clark School of Engineering Department of Civil and Environmental Engineering Part II Structural Steel Design and Analysis
More informationDESIGN AND NONLINEAR ANALYSIS OF A PARABOLIC LEAF SPRING
47 DESIGN AND NONLINEAR ANALYSIS OF A PARABOLIC LEAF SPRING Muhammad Ashiqur Rahman*, Muhammad Tareq Siddiqui and Muhammad Arefin Kowser Department of Mechanical Engineering, Bangladesh University of
More information4.4 Concavity and Curve Sketching
Concavity and Curve Sketching Section Notes Page We can use the second derivative to tell us if a graph is concave up or concave down To see if something is concave down or concave up we need to look at
More informationChapter 2: Concurrent force systems. Department of Mechanical Engineering
Chapter : Concurrent force sstems Objectives To understand the basic characteristics of forces To understand the classification of force sstems To understand some force principles To know how to obtain
More information