1 MECHANICS OF MATERIALS Prepared by Engr. John Paul Timola
2 Mechanics of materials branch of mechanics that studies the internal effects of stress and strain in a solid body. stress is associated with the strength of the material from which the body is made, while strain is a measure of the deformation of the body
3 vital importance for the design of any machine or structure many of the formulas and rules of design cited in engineering codes are based upon the principles of this subject. dates back to the beginning of the seventeenth century, when Galileo Galilei performed experiments to study the effects of loads on rods and beams made of various materials.
4 LOADS Surface loads act on a small area of contact reported by concentrated forces Distributed loading act over a larger surface area of the body. When coplanar, there is a resultant force equal to the area under the distributed loading diagram resultant acts through the geometric center or centroid of this area.
5 Body force developed when one body exerts a force on another body without direct physical contact between the bodies. examples include the effects caused by the earth s gravitation or its electromagnetic field. normally represented by a single concentrated force acting on the body. In the case of gravitation, this force is called the weight W of the body and acts through the body s center of gravity.
6 Support Reactions
7 Equations of Equilibrium Equilibrium of a body requires both a balance of forces, to prevent the body from translating or having accelerated motion along a straight or curved path, and a balance of moments, to prevent the body from rotating. F 0 M O 0
8 F 0 F 0 F 0 x y z M 0 M 0 M 0 x y z
9 Internal Resultant Loadings statics is primarily used to determine the resultant loadings that act within a body using the method of sections. Normal force, N. acts perpendicular to the area. developed whenever the external loads tend to push or pull on the two segments of the body. Shear force, V. lies in the plane of the area, the body to slide over one another.
10 Internal Resultant Loadings Torsional moment or torque, T. developed when the external loads tend to twist one segment of the body with respect to the other about an axis perpendicular to the area. Bending moment, M. caused by the external loads that tend to bend the body about an axis lying within the plane of the area.
11 Important Points 1. The study of the relationship between the external loads applied to a body is. 2. Internal loads within the body cause stress and. 3. produce a resultant force having a magnitude equal to the area under the load diagram, and having a location that passes through the centroid of this area. 4. The mathematical equations in order to prevent a body from translating with accelerated motion and from rotating are. 5. The method used to determine the internal resultant loadings acting on the surface of a sectioned body is. 6. When attached to a member, it prevents translation of the member in that direction, and it produces a couple moment to prevents rotation. 7. Three types of external forces that can be applied to a body.
12 Example 1 Determine the resultant internal loadings acting on the cross section at C of the cantilevered beam.
13 Example 2 The 500-kg engine is suspended from the crane boom. Determine the resultant internal loadings acting on the cross section of the boom at point E.
14 Example 3 Determine the resultant internal loadings acting on the cross section at C of the beam.
15 STRESS quotient that describes the intensity of the internal force acting on a specific plane (area) passing through a point
16 Normal Stress, σ (sigma) intensity of the force acting normal to A z lim A 0 F z A If the normal force or stress pulls on A as shown, it is tensile stress, whereas if it pushes on A, it is compressive stress.
17 Shear Stress, (tau) intensity of the force acting tangent to A zx lim A 0 F x A zy lim A 0 F y A
18 Units In the International Standard or SI system, the magnitudes of both normal and shear stress are specified in the basic units of newtons per square meter (N/m 2 ) or a Pascal (1 Pa = 1 N/m 2 ), In the Foot-Pound Second system of units, usually express stress in pounds per square inch (psi) or kilopounds per square inch (ksi), where 1 kilopound (kip) = 1000 lb.
19 AVERAGE NORMAL STRESS IN AN AXIALLY LOADED BAR Provided the material of the bar is both homogeneous and isotropic, then when the load P is applied to the bar through the centroid of its cross-sectional area, the bar will deform uniformly throughout the central region of its length.
20 Average Normal Stress Distribution If we pass a section through the bar, and separate it into two parts, then equilibrium requires the resultant normal force N at the section to be equal to P.
21 Because the material undergoes a uniform deformation, it is necessary that the cross section be subjected to a constant normal stress distribution. Each small area A on the cross section is subjected to a force N = σ A, and the sum of these forces acting over the entire cross-sectional area must be equivalent to the internal resultant force P at the section.
22 N A where σ = average normal stress at any point on the cross-sectional area N = internal resultant normal force, which acts through the centroid of the cross-sectional area. A = cross-sectional area of the bar where s is determined
23 Maximum Average Normal Stress If the bar may be subjected to several external axial loads, or a change in its cross-sectional area may occur, the normal stress within the bar may be different from one section to the next. The maximum average normal stress is to be determined, where the ratio N/A is a maximum.
24 Important Points 1. When a body subjected to external loads is sectioned, there is a distribution of force acting over the sectioned area which holds each segment of the body in equilibrium called. 2. Stress is the limiting value of per unit area, as the area approaches zero. 3. The of the stress components at a point depends upon the type of loading acting on the body, and the orientation of the element at the point. 4. When a prismatic bar is made of homogeneous and material, and is subjected to an axial force acting through the centroid of the cross-sectional area, then the center region of the bar will deform. 5. True or False. Stress can determine if the material is continuous and cohesive.
25 Example 1 The bar has a constant width of 35 mm and a thickness of 10 mm. Determine the maximum average normal stress in the bar when it is subjected to the loading.
26 Example 2 The 80-kg lamp is supported by two rods AB and BC. If AB has a diameter of 10 mm and BC has a diameter of 8 mm, determine the average normal stress in each rod.
27 Activity The cylinder is made of steel having a specific weight of 490 lb/ft 3. Determine the average compressive stress acting at points A and B.
28 AVERAGE SHEAR STRESS Shear stress is the component that acts in the plane of the sectioned area.
29 How it develops Consider the effect of applying a force F to the bar. If F is large enough, it can cause the material of the bar to deform and fail along the planes identified by AB and CD. A free-body diagram of the unsupported center segment of the bar indicates that the shear force V = F/2 must be applied at each section to hold the segment in equilibrium.
30 The average shear stress distributed over each sectioned area that develops this shear force is defined by V A avg = average shear stress at the section V = internal resultant shear force on the section determined from the equations of equilibrium A = area of the section
31 Procedure for Analysis 1. Section the member at the point where the average shear stress is to be determined. 2. Draw the necessary free-body diagram, and calculate the internal shear force V acting at the section that is necessary to hold the part in equilibrium.
32 Example 1 Determine the average shear stress in the 20-mmdiameter pin at A and the 30-mm-diameter pin at B that support the beam.
33 Example 2 If the wood joint has a thickness of 150 mm, determine the average shear stress along shear planes a a and b b of the connected member. For each plane, represent the state of stress on an element of the material.
34 Activity Determine the largest internal shear force resisted by the bolt with a diameter of 180 mm then solve for the average shear stress.
35 Example 3 The inclined member is subjected to a compressive force of 600 lb. Determine the average compressive stress along the smooth areas of contact at AB and BC, and the average shear stress along the horizontal plane DB.
36 ALLOWABLE STRESS DESIGN To ensure the safety of a structural or mechanical member, it is necessary to restrict the applied load to one that is less than the load the member can fully support.
37 Reasons 1. The intended measurements of a structure or machine may not be exact, due to errors in fabrication or in the assembly of its component parts. 2. Unknown vibrations, impact, or accidental loadings can occur that may not be accounted for in the design. 3. Atmospheric corrosion, decay, or weathering tend to cause materials to deteriorate during service. 4. Some materials, such as wood, concrete, or fiber-reinforced composites, can show high variability in mechanical properties.
38 Factor of Safety One method of specifying the allowable load for a member is to use a number Ratio of the failure load F fail to the allowable load F allow F.S. F F fail allow
39 Factor of Safety If the load applied to the member is linearly related to the stress developed within the member, F.S. fail fail allow allow
40 Required Area at Section N A or A= allow V allow
41 Procedure for Analysis First find the section over which the critical stress is acting Section the member through the area and draw a free-body diagram of a segment of the member. Determine the internal resultant force at the section using the equations of equilibrium. Solve the required area needed to sustain the calculated load or factored load at the section provided either the allowable stress or the load and resistance factors are known
42 Example 1 Determine the largest load P that can be applied to the bars of the lap joint. The bolt has a diameter of 10 mm and an allowable shear stress of 80 MPa. Each plate has an allowable tensile stress of 50 MPa, an allowable bearing stress of 80 MPa, and an allowable shear stress of 30 MPa.
43 Activity If each of the three nails has a diameter of 4 mm and can withstand an average shear stress of 60 MPa, determine the maximum allowable force P that can be applied to the board.
44 Example 2 The control arm is subjected to the loading. Determine to the nearest 1/4 inch the required diameters of the steel pins at A and C if the factor of safety for shear is F.S. = 1.5 and the failure shear stress is 12 ksi.
45 Example 3 The suspender rod is supported at its end by a fixed-connected circular disk. If the rod passes through a 40- mm-diameter hole, determine the minimum required diameter of the rod and the minimum thickness of the disk needed to support the 20-kN load. The allowable normal stress for the rod is σ allow = 60 MPa, and the allowable shear stress for the disk is τ allow = 35 MPa.
MECE 3321 MECHANICS O SOLIDS CHAPTER 1 Samantha Ramirez, MSE WHAT IS MECHANICS O MATERIALS? Rigid Bodies Statics Dynamics Mechanics Deformable Bodies Solids/Mech. Of Materials luids 1 WHAT IS MECHANICS
E X M P L E 1.1 Determine the resultant internal loadings acting on the cross section at of the beam shown in Fig. 1 a. 70 N/m m 6 m Fig. 1 Support Reactions. This problem can be solved in the most direct
STRESS! Stress Evisdom! verage Normal Stress in an xially Loaded ar! verage Shear Stress! llowable Stress! Design of Simple onnections 1 Equilibrium of a Deformable ody ody Force w F R x w(s). D s y Support
1 Chapter 3 Load and Stress Analysis 2 Chapter Outline Equilibrium & Free-Body Diagrams Shear Force and Bending Moments in Beams Singularity Functions Stress Cartesian Stress Components Mohr s Circle for
ME 108 - Statics Equilibrium of a Particle Chapter 3 Applications For a spool of given weight, what are the forces in cables AB and AC? Applications For a given weight of the lights, what are the forces
IDE 110 S08 Test 1 Name: 1. Determine the internal axial forces in segments (1), (2) and (3). (a) N 1 = kn (b) N 2 = kn (c) N 3 = kn 2. Rigid bar ABC supports a weight of W = 50 kn. Bar ABC is pinned at
Chapter 12 Static Equilibrium and Elasticity Static Equilibrium Equilibrium implies that the object moves with both constant velocity and constant angular velocity relative to an observer in an inertial
Strength of Material Shear Strain Dr. Attaullah Shah Shear Strain TRIAXIAL DEFORMATION Poisson's Ratio Relationship Between E, G, and ν BIAXIAL DEFORMATION Bulk Modulus of Elasticity or Modulus of Volume
(130) CHAPTER 6: Shearing Stresses in Beams When a beam is in pure bending, the only stress resultants are the bending moments and the only stresses are the normal stresses acting on the cross sections.
Third E CHAPTER 2 Stress MECHANICS OF MATERIALS Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf Lecture Notes: J. Walt Oler Texas Tech University and Strain Axial Loading Contents Stress & Strain:
Engineering Mechanics Department of Mechanical Engineering Dr. G. Saravana Kumar Indian Institute of Technology, Guwahati Module 3 Lecture 6 Internal Forces Today, we will see analysis of structures part
Fifth SI Edition CHTER 1 MECHNICS OF MTERILS Ferdinand. Beer E. Russell Johnston, Jr. John T. DeWolf David F. Mazurek Introduction Concept of Stress Lecture Notes: J. Walt Oler Teas Tech University Contents
Introduction Fundamentals of statics Applications of fundamentals of statics Friction Centroid & Moment of inertia Simple Stresses & Strain Stresses in Beam Torsion Principle Stresses DEPARTMENT OF CIVIL
UNIT 1 STRESS STRAIN AND DEFORMATION OF SOLIDS, STATES OF STRESS 1. Define stress. When an external force acts on a body, it undergoes deformation. At the same time the body resists deformation. The magnitude
3.1 CONDITIONS FOR RIGID-BODY EQUILIBRIUM Consider rigid body fixed in the x, y and z reference and is either at rest or moves with reference at constant velocity Two types of forces that act on it, the
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.
What Every Engineer Should Know About Structures Part C - Axial Strength of Materials by Professor Patrick L. Glon, P.E. This is a continuation of a series of courses in the area of study of physics called
4. SHAFTS A shaft is an element used to transmit power and torque, and it can support reverse bending (fatigue). Most shafts have circular cross sections, either solid or tubular. The difference between
ARC241 Structural Analysis I Lecture 10: MM1.3 MM1.13 MM1.4) Analysis and Design MM1.5) Axial Loading; Normal Stress MM1.6) Shearing Stress MM1.7) Bearing Stress in Connections MM1.9) Method of Problem
APPLIED ACHITECTURAL STRUCTURES: STRUCTURAL ANALYSIS AND SYSTEMS DR. ANNE NICHOLS SPRING 2017 lecture two structural analysis (statics & mechanics) Analysis 1 Structural Requirements strength serviceability
MECE 3321 MECHANICS OF SOLIDS CHAPTER 3 Samantha Ramirez TENSION AND COMPRESSION TESTS Tension and compression tests are used primarily to determine the relationship between σ avg and ε avg in any material.
Objectives Calculate torque given the lever arm (perpendicular distance) and the force. Calculate torque in newton meters and in pound feet. Interpret positive and negative signs in the context of torque.
Elasticity Homework Problems 2014 Section 1. The Strain Tensor. 1. A pure shear deformation is shown. The volume is unchanged. What is the strain tensor. 2. Given a steel bar compressed with a deformation
Static Equilibrium Static Equilibrium Definition: When forces acting on an object which is at rest are balanced, then the object is in a state of static equilibrium. - No translations - No rotations In
C h a p t e r 6 Equilibrium in Two Dimensions In this chapter, you will learn the following to World Class standards: 1. The Ladder Against the Wall 2. The Street Light 3. The Floor Beam 6-1 The Ladder
Failure from static loading Topics Quiz /1/07 Failures from static loading Reading Chapter 5 Homework HW 3 due /1 HW 4 due /8 What is Failure? Failure any change in a machine part which makes it unable
CHAPTER 5 Statically Determinate Plane Trusses TYPES OF ROOF TRUSS TYPES OF ROOF TRUSS ROOF TRUSS SETUP ROOF TRUSS SETUP OBJECTIVES To determine the STABILITY and DETERMINACY of plane trusses To analyse
UNIVERSITY PHYSICS I Professor Meade Brooks, Collin College Chapter 12: STATIC EQUILIBRIUM AND ELASTICITY Two stilt walkers in standing position. All forces acting on each stilt walker balance out; neither
EQUATIONS OF EQUILIBRIUM & TWO- AND THREE-FORCE MEMBERS Today s Objectives: Students will be able to: a) Apply equations of equilibrium to solve for unknowns, and, b) Recognize two-force members. APPLICATIONS
CHAPTER 5 Statically Determinate Plane Trusses TYPES OF ROOF TRUSS 1 TYPES OF ROOF TRUSS ROOF TRUSS SETUP 2 ROOF TRUSS SETUP OBJECTIVES To determine the STABILITY and DETERMINACY of plane trusses To analyse
MECHANICS OF MATERIALS Analysis of Beams for Bending By NUR FARHAYU ARIFFIN Faculty of Civil Engineering & Earth Resources Chapter Description Expected Outcomes Define the elastic deformation of an axially
Beams Beams are structural members that offer resistance to bending due to applied load 1 Beams Long prismatic members Non-prismatic sections also possible Each cross-section dimension Length of member
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
2011 earson Education, Inc., Upper Saddle River, NJ. ll rights reserved. This material is protected under all copyright laws as they currently 8 1. 3 1. concrete cylinder having a a diameter of of 6.00
D e s i g n o f R i v e t e d J o i n t s, C o t t e r & K n u c k l e J o i n t s 1. Design of various types of riveted joints under different static loading conditions, eccentrically loaded riveted joints.
Chapter 5 CENTRIC TENSION OR COMPRESSION ( AXIAL LOADING ) 5.1 DEFINITION A construction member is subjected to centric (axial) tension or compression if in any cross section the single distinct stress
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:
330:48 (g) achine Design Nageswara Rao Posinasetti P N Rao 5. Repeated Loading Objectives Identify the various kinds of loading encountered on a part and learn to combine them as appropriate. Determine
PHY241 - General Physics I Dr. Carlson, Fall 2013 Prep Exam 3 PREP Chapters 6, 7, 8 Name TRUE/FALSE. Write 'T' if the statement is true and 'F' if the statement is false. 1) Astronauts in orbiting satellites
ARCH 314 Structures I Test Primer Questions Dr.-Ing. Peter von Buelow Properties of Sections 1. Select all that apply to the characteristics of the Center of Gravity: A) 1. The point about which the body
8 3. If the coefficient of static friction at is m s = 0.4 and the collar at is smooth so it only exerts a horizontal force on the pipe, determine the minimum distance x so that the bracket can support
MECH 401 Mechanical Design Applications Dr. M. O Malley Master Notes Spring 008 Dr. D. M. McStravick Rice University Updates HW 1 due Thursday (1-17-08) Last time Introduction Units Reliability engineering
VALLIAMMAI ENGINEERING COLLEGE SRM NAGAR, KATTANKULATHUR 603203 DEPARTMENT OF MECHANICAL ENGINEERING BRANCH: MECHANICAL YEAR / SEMESTER: I / II UNIT 1 PART- A 1. State Newton's three laws of motion? 2.
Torque and Static Equilibrium Rigid Bodies Rigid body: An extended object in which the distance between any two points in the object is constant in time. Examples: sphere, disk Effect of external forces
MECHANICS OF MATERIALS Jeff Brown Hope College, Department of Engineering, 27 Graves Pl., Holland, Michigan, USA Keywords: Solid mechanics, stress, strain, yield strength Contents 1. Introduction 2. Stress
CHAPTER THE EFFECTS OF FORCES ON MATERIALS EXERCISE 1, Page 50 1. A rectangular bar having a cross-sectional area of 80 mm has a tensile force of 0 kn applied to it. Determine the stress in the bar. Stress
Chapters 8-9 Rotational Kinematics and Dynamics Rotational motion Rotational motion refers to the motion of an object or system that spins about an axis. The axis of rotation is the line about which the
CH. 5 TRUSSES BASIC PRINCIPLES Typical depth-to-span ratios range from 1:10 to 1:20 - Flat trusses require less overall depth than pitched trusses Spans: 40-200 Spacing: 10 to 40 on center - Residential
API 11E - Specification for Pumping Units 5 Beam Pump Structure Requirements 5.1 General Requirements for beam pump structures are specified in the following sections. Only loads imposed on the structure
Name: Date: Solid Mechanics Homework nswers Please show all of your work, including which equations you are using, and circle your final answer. Be sure to include the units in your answers. 1. The yield
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
CONNECTION DESIGN Connections must be designed at the strength limit state Average of the factored force effect at the connection and the force effect in the member at the same point At least 75% of the
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
I. Overview The objective of this report is to review the design of the trebuchet model from an engineering standpoint. This study analyzes the dimensions and material selection using the COMSOL multiphisics
ISBN : 978-81-909042-4-7 - www.airwalkpublications.com ANNA UNIVERSITY - R2013 GE6253 ENGINEERING MECHANICS UNIT I: BASICS AND STATICS OF PARTICLES 12 Introduction Units and Dimensions Laws of Mechanics
CHAPER ORSION ORSION orsion refers to the twisting of a structural member when it is loaded by moments/torques that produce rotation about the longitudinal axis of the member he problem of transmitting
MME131: Lecture 13 Mechanical properties 1 Elastic behaviour of materials A. K. M. B. Rashid Professor, Department of MME BUET, Dhaka Today s Topics Deformation of material under the action of a mechanical
Date: February-12-16 Time: 2:00:28 PM TEST REPORT Question file: P12-2006 Copyright: Test Date: 21/10/2010 Test Name: EquilibriumPractice Test Form: 0 Test Version: 0 Test Points: 138.00 Test File: EquilibriumPractice
Chapter 3 Mass quantity of matter composing a body represented by m Kinetic Concepts for Analyzing Human Motion units are kg Inertia tendency to resist change in state of motion proportional to mass has
Mechanics II 1. By applying a force F on a block, a person pulls a block along a rough surface at constant velocity v (see Figure below; directions, but not necessarily magnitudes, are indicated). Which
RODS: STTICLLY INDETERMINTE MEMERS Statically Indeterminate ackground In all of the problems discussed so far, it was possible to determine the forces and stresses in the members by utilizing the equations
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
10.3 Static Equilibrium and Torque SECTION OUTCOMES Use vector analysis in two dimensions for systems involving static equilibrium and torques. Apply static torques to structures such as seesaws and bridges.
1 1 wooden block of mass 0.60 kg is on a rough horizontal surface. force of 12 N is applied to the block and it accelerates at 4.0 m s 2. wooden block 4.0 m s 2 12 N hat is the magnitude of the frictional
CIVL222 STRENGTH OF MATERIALS Chapter 6 Torsion Definition Torque is a moment that tends to twist a member about its longitudinal axis. Slender members subjected to a twisting load are said to be in torsion.
SOLUTION (17.3) Known: A simply supported steel shaft is connected to an electric motor with a flexible coupling. Find: Determine the value of the critical speed of rotation for the shaft. Schematic and
CHTER MECHNICS OF MTERILS 10 Ferdinand. Beer E. Russell Johnston, Jr. Columns John T. DeWolf cture Notes: J. Walt Oler Texas Tech University 006 The McGraw-Hill Companies, Inc. ll rights reserved. Columns
University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Robert Katz Publications Research Papers in Physics and Astronomy 1-1958 Physics, Chapter 3: The Equilibrium of a Particle
Beam Bending Stresses and Shear Stress Notation: A = name or area Aweb = area o the web o a wide lange section b = width o a rectangle = total width o material at a horizontal section c = largest distance
Name ME 270 Fall 2005 Final Exam PROBLEM NO. 1 Given: A distributed load is applied to the top link which is, in turn, supported by link AC. Find: a) Draw a free body diagram of link BCDE and one of link
Introduction to Engineering Materials ENGR2 Chapter 6: Mechanical Properties of Metals Dr. Coates 6.2 Concepts of Stress and Strain tension compression shear torsion Tension Tests The specimen is deformed
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
BOOK OF COURSE WORKS ON STRENGTH OF MATERIALS FOR THE ND YEAR STUDENTS OF THE UACEG Assoc.Prof. Dr. Svetlana Lilkova-Markova, Chief. Assist. Prof. Dimitar Lolov Sofia, 011 STRENGTH OF MATERIALS GENERAL
his chapter is devoted to the study of torsion and of the stresses and deformations it causes. In the jet engine shown here, the central shaft links the components of the engine to develop the thrust that
4 5000 4000 (increased d ) (increased f (increased A s or f y ) c or b) Flexure: Behavior and Nominal Strength of Beam Sections Moment (kip-in.) 3000 2000 1000 0 0 (basic) (A s 0.5A s ) 0.0005 0.001 0.0015
558 A Textbook of Machine Design C H A P T E R 15 Levers 1. Introduction.. Application of Levers in Engineering Practice.. Design of a Lever. 4. Hand Lever. 5. Foot Lever. 6. Cranked Lever. 7. Lever for