EMA 3702 Mechanics & Materials Science (Mechanics of Materials) Chapter 3 Torsion


 Helena Perry
 4 years ago
 Views:
Transcription
1 EMA 3702 Mechanics & Materials Science (Mechanics of Materials) Chapter 3 Torsion
2 Introduction Stress and strain in components subjected to torque T Circular Crosssection shape Material Shaft design Noncircular Irregular shape Elastic Elastoplastic Solid Hollow
3 Introduction Component subject to twisting couples, or torques of T & T T is a vector and has two ways of representations Curved arrow Couple vector (right hand rule) Example: Transmission of torque in shafts
4 Stresses in a Shaft under Torsion (1) A shaft subjected to equal and opposite torques (or moments of force) of T and T at A and B In a normal crosssection at C, for an elementary area da, the element shearing force df and local stress τ satisfy: df = da Define as (local) lever arm, i.e., the perpendicular distance from the elemental area to the axis (center), total torque T is: df T
5 Stresses in a Shaft under Torsion (2) From previous df = da df T Therefore, (da) T Complications for torsion: Distributions of and the resulting, i.e., how they change over the crosssection plane is statically indeterminate. Unlike normal stress or simple shearing, distribution of for torsion is NOT uniform!
6 Axial Shearing Stress in Shaft under Torsion Consider a small element as illustrated. Based on previous considerations for shearing stress, if xz 0, axial shearing stress zx = xz 0 zx xz Implication: under torsion, shearing stress exists along longitude planes, and neighboring elements have tendency to slide against each other along axial direction! z y x
7 Axisymmetric Property of a Circular Shaft For circular shaft under torsion Crosssections remain planar Crosssection remain undistorted
8 Shearing Strain in Circular Shaft under Torsion Within each crosssection, NO change Along axial direction, there is strain (deformation) due to axial shear zx Define the following terms L Length along shaft axis Radial distance from the shaft axis Angle of twist for a crosssection Shearing strain (change in angle from 90 o ) Far away from location of loading & for small strain and angle L L
9 Shearing Strain in Circular Shaft under Torsion L For a given L, when = c, i.e., radius of the shaft max c L Additionally, for a given L max c c max
10 Shearing Stresses in Circular Shaft under Torsion Hooke s Law for shear G From previous page, torsion shearing strain L Therefore, G shearing stress L For given L and ϕ, when = c = radius of the shaft, shearing stress reaches maximum: The ratio between τ and τ max c max max G max Gc L Use when shaft twist angle, length, AND materials G are known Distribution of shearing stress is linear w.r.t. radius from the axis ρ
11 Shearing Stresses & Torque in Circular Shaft under Torsion c max Recall relationship of T and local shearing stress τ For a circular shaft with fixed radius c under torque T T Recall the definition for moment of inertia: We have Or (da) max T max da da 2 ( ) max da c c 2 max T c Tc and T da Use when shaft geometry AND applied torque known
12 Solid & Hollow Circular Shaft under Torsion For a solid cylinder T max 1 c 2 4 Tc For a hollow cylinder ( c 2 c 2 T min max ) c c 1 2
13 EMA 3702 Mechanics & Materials Science Zhe Cheng (2018) 3 Torsion Example Problem for Torsion 3.1 Shaft ABCD subject to torques as illustrated. Knowing section BC is hollow with ID = 90 mm and OD = 120 mm. Sections AB and CD are solid. Calculate (a) The maximum and minimum shearing stress in section BC; (b) The required minimal diameter for AB and CD if shearing stress should not exceed 65 MPa.
14 Example Problem for Torsion 3.1 Section AB, Net internal torque T AB = 6 knm Section BC, Net internal torque T BC = (6 +14) knm = 20 knm EMA 3702 Mechanics & Materials Science Zhe Cheng (2018) 3 Torsion
15 Example Problem for Torsion 3.1 For section BC, T BC = 20 knm section BC is hollow with ID = 90 mm and OD = 120 mm. Max stress in BC occurs at the outer surface BC BCc T N m0.06m outer max 1 BC Min stress in BC occurs at the inner surface min T c 2010 N m0.045m BC 3 BC inner BC EMA 3702 Mechanics & Materials Science Zhe Cheng (2018) 3 Torsion m m MPa MPa
16 Example Problem for Torsion 3.1 Section AB, T AB = 6 knm Section BC, T BC = 20 knm Section CD, T CD = 6 knm For both AB and CD, shearing stress should not exceed 65 MPa max T c c 2 Minimum radius for AB or CD: T MPa T c 1 2 max 1/3 1 2 Minimum diameter = 7.78 cm EMA 3702 Mechanics & Materials Science Zhe Cheng (2018) 3 Torsion 3 N m 6 N / m 2 1/ cm
17 Class Example A hollow cylinder shaft is 1 m long and has inner and outer diameter of 20 and 40 mm. (a) What is the largest torque that can be applied if shearing stress should not exceed 100 MPa? (b) What is the minimum shearing stress when maximum reaching 100 MPa? From geometry, moment of inertia 1 1 outer inner 2 2 shearing stress should not exceed 100 MPa : Largest torque that can be applied: c c m T 100MPa c out Min shearing stress when max is 100 MPa max outer EMA 3702 Mechanics & Materials Science Zhe Cheng (2018) 3 Torsion m m cinner min max c 6 Tc outer 100 MPa N / m N m MPa 50MPa 0.02
18 Normal Stress in Circular Shaft under Torsion For element a at shaft surface max Tc For half of a at 45 o, shear force along BC and BD are: F BC F BD max A 0 To balance, force on CD surface F CD must be normal force F CD Since A CD 2A0 Normal stress due to torsion F F A BC CD CD / cos 45 2 A max Tc max 0 F CD For torsion, significant normal stress still exists and may cause failure!
19 Failure of Material under Torsion For circular shaft under torsion Ductile materials are weaker in shear and fail with 90 o fracture (i.e., fracture surface perpendicular to axis) Brittle materials are weaker in tension and fail with 45 o fracture (i.e., fracture surface at 45 o from axis) Ductile torsion failure Brittle torsion failure Photo by eff Thomas, 11/1997
20 Angle of Twist within Elastic Range For torsion, based on geometry max c L Within elastic limit, Hooke s law max G max Max sharing stress due to torque max Tc max Tc G L T G TL G Angle of twist increases with Increasing T and L Decreasing and G
21 Angle of Twist for Multiple CrossSection Shafts Overall twist of A w.r.t B i Ti Li G in which T i, i, and L i, G i are obtained and/or analyzed for each section Angle of Twist for Variable CrossSection Shafts For element dx d Tdx G i i Overall Tdx L 0 G
22 Class Exercise A cylindrical shaft is 0.5 m long and has diameter of 20 mm fixed at one end. If a torque of 76 Nm is applied to its free end, and the measured angle of twist at the loading end is 1.8 o, please estimate the shear modulus knowing moment of inertia is Angle of twist Shear modulus G 1 c 2 TL G G TL 76N m0.5m GPa m EMA 3702 Mechanics & Materials Science Zhe Cheng (2018) 3 Torsion 4
23 Example of Statically Indeterminate Shaft (1) A cylindrical shaft AB with two sections is fixed between two rigid end support at A and B. Section AC is solid with A C B diameter of 1.0 in; 10 in 10 in Section CB is hollow with outside diameter of 1.0 in and inner diameter of 0.8 in. Both sections have length of 10 in. If an external torque of 100 lbft is applied at center C, please determine the reaction torque at A and B, assuming the deformation is within proportional limit.
24 Example of Statically Indeterminate Shaft (2) From statics, balance of torque T T T 100lb in A B One equation two variables Statically indeterminate Consider geometry/deformation: For section AC, angle of twist AC For section CB, angle of twist CB AC CB Therefore, TALAC TBLCB AC CB G G AC CB Two equations & two variables, problem could be solved 100 AC CB T A AC T B CB
25 Example of Statically Indeterminate Shaft (3) Therefore, T A T B 100 T A T B TA T B Solving these two T A T B 62. 9lb 37. 1lb ft ft
26 Design of Transmission Shafts (1) Two parameters in transmission shaft design: Power P Speed of rotation For power: P T is angular velocity, = 2, and has unit of radians/s is frequency in unit Hz or s 1 Therefore, power P 2fT has unit of Nm/s = W As a result, for given P and f, the resulting torque will be T P 2f
27 Design of Transmission Shafts (2) Relationship between power output P, angular speed (or frequency f), and torque T P 2f On the other hand, max shearing stress for a given shaft geometry and applied torque max Tc Example of a solid circular shaft, moment of inertia: T c c 3 2 c 2 c T c 2T T max max max_ allowable 1/3
28 Class Exercise What size of solid shaft should be used for motor of 592 hp (1 hp = 6600 lbin/s) operating at 3600 rpm (f = 60 Hz) if the shearing stress is not to exceed 6600 psi in the shaft? Power for the shaft P 2fT P lb in s Torque for the shaft T 2f s Maximum shearing stress Tc max Tc 2T lb / in c 4 c 3 2 Minimum shaft radius: c 2T max_ allowable EMA 3702 Mechanics & Materials Science Zhe Cheng (2018) 3 Torsion 1/ lb in lb / in 1 1 1/ lb in 1.00 in
29 Stress Concentrations in Circular Shafts Similar to normal stress, shearing stress may concentrate (i.e., show higher value) at certain locations Concentration factor K max_ actual K Tc / or Tc max_ actual K Local max stress increases with Sharper transition or joining Larger ratio of radius
30 Plastic Deformation in Circular Shafts (1) For a given shaft construction (geometry and material), when torque is low and within linear elastic region, linear stress distribution w.r.t. radius from axis T As torque increases further, shearing stress would reach yield strength Y, it will go into nonlinear (e.g., ideal elasoplastic) distribution of w.r.t. radial from axis in certain region T
31 Plastic Deformation in Circular Shafts (2) For angle of twist When torque/shearing stress is low and within linear elastic region, increases linearly with torque T TL G T As torque increases further, shearing stress reaches yield strength Y, it will go into nonlinear distribution of w.r.t. T
32 Torsion of Noncircular Members A rectangular or square shaft does not contain axisymmetry The crosssections generally do NOT remain flat or planar under torsion Shearing stress at the edge (corner) of the shaft is zero Maximum shearing stress occur in the middle of flat face
33 ThinWalled Hollow Shafts For arbitrary shaped thin walled hollow tube subjected to torque T, the shearing stress can be approximated as τ = T 2ta a
34 EMA 3702 Mechanics & Materials Science Zhe Cheng (2018) 3 Torsion Homework 3.0 Read chapter 3 textbook sections 3.1 to 3.5 and give an honor statement confirm reading.
35 EMA 3702 Mechanics & Materials Science Zhe Cheng (2018) 3 Torsion Homework 3.1 Electric motor applies a torque of 2.8 kn m on shaft EF at E. Knowing each shaft section is solid. Based on the additional torques as depicted, please determine the maximum shearing stress in (a) shaft EF (radius c EF = 25 mm), (b) shaft FG (radius c FG =23 mm), and c) shaft GH (radius c GH =21 mm) E F G H E T F = 1.6 kn m T G = 0.8 kn m T H = 0.4 kn m
36 Homework 3.2 EMA 3702 Mechanics & Materials Science Zhe Cheng (2018) 3 Torsion A component with both solid rod EF and hollow tube with flange GH welded onto a flat end plate of GFI, and the rod EF and tube GH share the same longitude axis, as illustrated. The hollow tube GH is fixed at rigid flange of AA BB at H using bolts. The solid rod EF has a diameter d EF = 3.81 cm and is made of steel with allowable shearing stress of 82.7 MPa. Tube GH is made of copper alloy with OD d 2 =7.62 cm and ID d 1 =6.35 cm with allowable shearing stress of 48.3 MPa. Determine the largest torque T that can be applied at E. G T H E A A B B F I
37 Homework 3.3 EMA 3702 Mechanics & Materials Science Zhe Cheng (2018) 3 Torsion Two solid shafts GH and I are connected by gears K and L as illustrated. The radius for gear K is R K = 4 in, while radius for gear L is R L = 2.5 in. Both two shafts of GH and I are made of same steel with allowable shearing stress of 8 ksi. The radius for the shafts are c GH = 0.8 in and c I = in. Please determine the largest torque that can be applied at H. K L G R K = 4.0 in R L = 2.5 in I H
38 EMA 3702 Mechanics & Materials Science Zhe Cheng (2018) 3 Torsion Homework 3.4 Motor applies torque T = 500 N m on the HGFE shaft when it is rotating at a constant speed. Knowing shear modulus G = 27 GPa and the torques exerted on pulleys B and C are shown. Please calculate the angle of twist between (a) F and G, and (b) between F and H. Knowing shaft radius c GF = 2.2 cm; c HG = 2.4 cm; c EF = 2.0 cm; T = 500 N m 0.9 m 1.2 m H G F E Motor T G = 300 N m T F = 200 N m
39 EMA 3702 Mechanics & Materials Science Zhe Cheng (2018) 3 Torsion Homework 3.5 Solid aluminum rod (G Al =26 GPa) is bonded to solid copper alloy rod (G Al =39 GPa). The radius for both rods are 10 mm. Calculate the angle of twist with respect to base D at E point and at F point, respectively. D copper E aluminum F 0.2 m 0.4 m T F = 50 N m
40 EMA 3702 Mechanics & Materials Science Zhe Cheng (2018) 3 Torsion Homework 3.6 A 6 foot long composite shaft consists of 0.2 inch thick copper shell (G Cu = psi) bonded to 1.2 inch (G Steel = psi) diameter iron core. If the shaft is subject to 5000 lb in torque. Please calculate the maximum shearing stress in the steel core and the angle of twist of one end versus the other. T Cu shell Fe core 6 ft
41 EMA 3702 Mechanics & Materials Science Zhe Cheng (2018) 3 Torsion Homework 3.7 Determine maximum shearing stress in a solid shaft of 10 mm diameter as it transmits 2.4 kw at a frequency of (a) 30 Hz and (b) 60 Hz.
42 EMA 3702 Mechanics & Materials Science Zhe Cheng (2018) 3 Torsion Homework 3.8 Two solid shafts EF and I and gears G (R G = 3 inch) and H (R H = 5 inch) are used to transmit 16 hp (1 hp = 6600 lbin) from motor at E operating at 1200 rpm (f = 20 Hz) to machine tool at. Knowing the maximum allowable shearing stress is 7.5 ksi. Please calculate the required radius for shaft EF and shaft I, respectively. H I R H = 5.0 in R G = 3.0 in E F G
MECHANICS OF MATERIALS
2009 The McGrawHill Companies, Inc. All rights reserved. Fifth SI Edition CHAPTER 3 MECHANICS OF MATERIALS Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf David F. Mazurek Torsion Lecture Notes:
More information[7] Torsion. [7.1] Torsion. [7.2] Statically Indeterminate Torsion. [7] Torsion Page 1 of 21
[7] Torsion Page 1 of 21 [7] Torsion [7.1] Torsion [7.2] Statically Indeterminate Torsion [7] Torsion Page 2 of 21 [7.1] Torsion SHEAR STRAIN DUE TO TORSION 1) A shaft with a circular cross section is
More information(48) CHAPTER 3: TORSION
(48) CHAPTER 3: TORSION Introduction: In this chapter structural members and machine parts that are in torsion will be considered. More specifically, you will analyze the stresses and strains in members
More informationMECHANICS OF MATERIALS
GE SI CHAPTER 3 MECHANICS OF MATERIALS Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf David F. Mazurek Torsion Lecture Notes: J. Walt Oler Texas Tech University Torsional Loads on Circular Shafts
More informationStress Analysis Lecture 3 ME 276 Spring Dr./ Ahmed Mohamed Nagib Elmekawy
Stress Analysis Lecture 3 ME 276 Spring 20172018 Dr./ Ahmed Mohamed Nagib Elmekawy Axial Stress 2 Beam under the action of two tensile forces 3 Beam under the action of two tensile forces 4 Shear Stress
More informationMECE 3321: MECHANICS OF SOLIDS CHAPTER 5
MECE 3321: MECHANICS OF SOLIDS CHAPTER 5 SAMANTHA RAMIREZ TORSION Torque A moment that tends to twist a member about its longitudinal axis 1 TORSIONAL DEFORMATION OF A CIRCULAR SHAFT Assumption If the
More informationThe problem of transmitting a torque or rotary motion from one plane to another is frequently encountered in machine design.
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
More informationChapter 5: Torsion. 1. Torsional Deformation of a Circular Shaft 2. The Torsion Formula 3. Power Transmission 4. Angle of Twist CHAPTER OBJECTIVES
CHAPTER OBJECTIVES Chapter 5: Torsion Discuss effects of applying torsional loading to a long straight member (shaft or tube) Determine stress distribution within the member under torsional load Determine
More informationWORCESTER POLYTECHNIC INSTITUTE
WORCESTER POLYTECHNIC INSTITUTE MECHANICAL ENGINEERING DEPARTMENT STRESS ANALYSIS ES2502, C 2012 Lecture 17: 10 February 2012 General information Instructor: Cosme Furlong HL151 (508) 8315126 cfurlong@wpi.edu
More informationStructural Analysis I Chapter 4  Torsion TORSION
ORSION orsional stress results from the action of torsional or twisting moments acting about the longitudinal axis of a shaft. he effect of the application of a torsional moment, combined with appropriate
More informationEMA 3702 Mechanics & Materials Science (Mechanics of Materials) Chapter 2 Stress & Strain  Axial Loading
MA 3702 Mechanics & Materials Science (Mechanics of Materials) Chapter 2 Stress & Strain  Axial Loading MA 3702 Mechanics & Materials Science Zhe Cheng (2018) 2 Stress & Strain  Axial Loading Statics
More informationCIVL222 STRENGTH OF MATERIALS. Chapter 6. Torsion
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.
More information[5] Stress and Strain
[5] Stress and Strain Page 1 of 34 [5] Stress and Strain [5.1] Internal Stress of Solids [5.2] Design of Simple Connections (will not be covered in class) [5.3] Deformation and Strain [5.4] Hooke s Law
More informationChapter 5 Torsion STRUCTURAL MECHANICS: CE203. Notes are based on Mechanics of Materials: by R. C. Hibbeler, 7th Edition, Pearson
STRUCTURAL MECHANICS: CE203 Chapter 5 Torsion Notes are based on Mechanics of Materials: by R. C. Hibbeler, 7th Edition, Pearson Dr B. Achour & Dr Eng. K. Elkashif Civil Engineering Department, University
More informationTuesday, February 11, Chapter 3. Load and Stress Analysis. Dr. Mohammad Suliman Abuhaiba, PE
1 Chapter 3 Load and Stress Analysis 2 Chapter Outline Equilibrium & FreeBody Diagrams Shear Force and Bending Moments in Beams Singularity Functions Stress Cartesian Stress Components Mohr s Circle for
More information3.5 STRESS AND STRAIN IN PURE SHEAR. The next element is in a state of pure shear.
3.5 STRESS AND STRAIN IN PURE SHEAR The next element is in a state of pure shear. Fig. 320 Stresses acting on a stress element cut from a bar in torsion (pure shear) Stresses on inclined planes Fig. 321
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 informationTorsion of Shafts Learning objectives
Torsion of Shafts Shafts are structural members with length significantly greater than the largest crosssectional dimension used in transmitting torque from one plane to another. Learning objectives Understand
More informationMAAE 2202 A. Come to the PASS workshop with your mock exam complete. During the workshop you can work with other students to review your work.
It is most beneficial to you to write this mock final exam UNDER EXAM CONDITIONS. This means: Complete the exam in 3 hours. Work on your own. Keep your textbook closed. Attempt every question. After the
More informationTorsion Stresses in Tubes and Rods
Torsion Stresses in Tubes and Rods This initial analysis is valid only for a restricted range of problem for which the assumptions are: Rod is initially straight. Rod twists without bending. Material is
More informationThis 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
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
More informationThe example of shafts; a) Rotating Machinery; Propeller shaft, Drive shaft b) Structural Systems; Landing gear strut, Flap drive mechanism
TORSION OBJECTIVES: This chapter starts with torsion theory in the circular cross section followed by the behaviour of torsion member. The calculation of the stress stress and the angle of twist will be
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 informationTorsion. Torsion is a moment that twists/deforms a member about its longitudinal axis
Mehanis of Solids I Torsion Torsional loads on Cirular Shafts Torsion is a moment that twists/deforms a member about its longitudinal axis 1 Shearing Stresses due to Torque o Net of the internal shearing
More informationM. Vable Mechanics of Materials: Chapter 5. Torsion of Shafts
Torsion of Shafts Shafts are structural members with length significantly greater than the largest crosssectional dimension used in transmitting torque from one plane to another. Learning objectives Understand
More information2. Rigid bar ABC supports a weight of W = 50 kn. Bar ABC is pinned at A and supported at B by rod (1). What is the axial force in rod (1)?
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
More informationNAME: Given Formulae: Law of Cosines: Law of Sines:
NME: Given Formulae: Law of Cosines: EXM 3 PST PROBLEMS (LESSONS 21 TO 28) 100 points Thursday, November 16, 2017, 7pm to 9:30, Room 200 You are allowed to use a calculator and drawing equipment, only.
More informationSolution: The strain in the bar is: ANS: E =6.37 GPa Poison s ration for the material is:
Problem 10.4 A prismatic bar with length L 6m and a circular cross section with diameter D 0.0 m is subjected to 0kN compressive forces at its ends. The length and diameter of the deformed bar are measured
More informationMECHANICS OF MATERIALS
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:
More informationMECHANICS OF MATERIALS
CHAPER MECHANICS OF MAERIALS Ferdinand P. Beer E. Russell Johnston, Jr. John. DeWolf orsion Leture Notes: J. Walt Oler exas eh University 006 he MGrawHill Companies, In. All rights reserved. Contents
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 informationChapter Objectives. Copyright 2011 Pearson Education South Asia Pte Ltd
Chapter Objectives To determine the torsional deformation of a perfectly elastic circular shaft. To determine the support reactions when these reactions cannot be determined solely from the moment equilibrium
More informationMECE 3321 MECHANICS OF SOLIDS CHAPTER 1
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
More informationCHAPTER 2 Failure/Fracture Criterion
(11) CHAPTER 2 Failure/Fracture Criterion (12) Failure (Yield) Criteria for Ductile Materials under Plane Stress Designer engineer: 1 Analysis of loading (for simple geometry using what you learn here
More informationSamantha Ramirez, MSE. Stress. The intensity of the internal force acting on a specific plane (area) passing through a point. F 2
Samantha Ramirez, MSE Stress The intensity of the internal force acting on a specific plane (area) passing through a point. Δ ΔA Δ z Δ 1 2 ΔA Δ x Δ y ΔA is an infinitesimal size area with a uniform force
More informationThe University of Melbourne Engineering Mechanics
The University of Melbourne 436291 Engineering Mechanics Tutorial Four Poisson s Ratio and Axial Loading Part A (Introductory) 1. (Problem 922 from Hibbeler  Statics and Mechanics of Materials) A short
More informationPDDC 1 st Semester Civil Engineering Department Assignments of Mechanics of Solids [ ] Introduction, Fundamentals of Statics
Page1 PDDC 1 st Semester Civil Engineering Department Assignments of Mechanics of Solids [2910601] Introduction, Fundamentals of Statics 1. Differentiate between Scalar and Vector quantity. Write S.I.
More informationChapter 3. Load and Stress Analysis. Lecture Slides
Lecture Slides Chapter 3 Load and Stress Analysis 2015 by McGraw Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner.
More informationUNITI STRESS, STRAIN. 1. A Member A B C D is subjected to loading as shown in fig determine the total elongation. Take E= 2 x10 5 N/mm 2
UNITI STRESS, STRAIN 1. A Member A B C D is subjected to loading as shown in fig determine the total elongation. Take E= 2 x10 5 N/mm 2 Young s modulus E= 2 x10 5 N/mm 2 Area1=900mm 2 Area2=400mm 2 Area3=625mm
More informationAluminum shell. Brass core. 40 in
PROBLEM #1 (22 points) A solid brass core is connected to a hollow rod made of aluminum. Both are attached at each end to a rigid plate as shown in Fig. 1. The moduli of aluminum and brass are EA=11,000
More informationDownloaded from Downloaded from / 1
PURWANCHAL UNIVERSITY III SEMESTER FINAL EXAMINATION2002 LEVEL : B. E. (Civil) SUBJECT: BEG256CI, Strength of Material Full Marks: 80 TIME: 03:00 hrs Pass marks: 32 Candidates are required to give their
More informationEMA 3702 Mechanics & Materials Science (Mechanics of Materials) Chapter 4 Pure Bending
EA 3702 echanics & aterials Science (echanics of aterials) Chapter 4 Pure Bending Pure Bending Ch 2 Aial Loading & Parallel Loading: uniform normal stress and shearing stress distribution Ch 3 Torsion:
More informationR13. II B. Tech I Semester Regular Examinations, Jan MECHANICS OF SOLIDS (Com. to ME, AME, AE, MTE) PARTA
SET  1 II B. Tech I Semester Regular Examinations, Jan  2015 MECHANICS OF SOLIDS (Com. to ME, AME, AE, MTE) Time: 3 hours Max. Marks: 70 Note: 1. Question Paper consists of two parts (PartA and PartB)
More informationTorsion of shafts with circular symmetry
orsion of shafts with circular symmetry Introduction Consider a uniform bar which is subject to a torque, eg through the action of two forces F separated by distance d, hence Fd orsion is the resultant
More informationMechanical Design in Optical Engineering
Torsion Torsion: Torsion refers to the twisting of a structural member that is loaded by couples (torque) that produce rotation about the member s longitudinal axis. In other words, the member is loaded
More informationMechanical Properties of Materials
Mechanical Properties of Materials Strains Material Model Stresses Learning objectives Understand the qualitative and quantitative description of mechanical properties of materials. Learn the logic of
More informationMECHANICS OF MATERIALS
CHATR Stress MCHANICS OF MATRIALS and Strain Axial Loading Stress & Strain: Axial Loading Suitability of a structure or machine may depend on the deformations in the structure as well as the stresses induced
More informationPES Institute of Technology
PES Institute of Technology Bangalore south campus, Bangalore5460100 Department of Mechanical Engineering Faculty name : Madhu M Date: 29/06/2012 SEM : 3 rd A SEC Subject : MECHANICS OF MATERIALS Subject
More informationIDE 110 Mechanics of Materials Spring 2006 Final Examination FOR GRADING ONLY
Spring 2006 Final Examination STUDENT S NAME (please print) STUDENT S SIGNATURE STUDENT NUMBER IDE 110 CLASS SECTION INSTRUCTOR S NAME Do not turn this page until instructed to start. Write your name on
More informationFree Body Diagram: Solution: The maximum load which can be safely supported by EACH of the support members is: ANS: A =0.217 in 2
Problem 10.9 The angle β of the system in Problem 10.8 is 60. The bars are made of a material that will safely support a tensile normal stress of 8 ksi. Based on this criterion, if you want to design the
More informationSub. Code:
Important Instructions to examiners: ) The answers should be examined by key words and not as wordtoword as given in the model answer scheme. ) The model answer and the answer written by candidate may
More informationTORSION By Prof. Ahmed Amer
ORSION By Prof. Ahmed Amer orque wisting moments or torques are fores ating through distane so as to promote rotation. Example Using a wrenh to tighten a nut in a bolt. If the bolt, wrenh and fore are
More information4. SHAFTS. A shaft is an element used to transmit power and torque, and it can support
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
More informationStrength of Materials Prof S. K. Bhattacharya Department of Civil Engineering Indian Institute of Technology, Kharagpur Lecture  18 Torsion  I
Strength of Materials Prof S. K. Bhattacharya Department of Civil Engineering Indian Institute of Technology, Kharagpur Lecture  18 Torsion  I Welcome to the first lesson of Module 4 which is on Torsion
More informationPROBLEM 3.3 ( )(45 10 ) T 5.17 kn m. A c c. 2 J c, (2)( ) 2 ( ) mm ( )
0 mm PROEM..4 m 45 mm (a) Determine the torque that causes a maximum shearing stress of 45 MPa in the hollow cylindrical steel shaft shown. Determine the maximum shearing stress caused by the same torque
More informationINTRODUCTION TO STRAIN
SIMPLE STRAIN INTRODUCTION TO STRAIN In general terms, Strain is a geometric quantity that measures the deformation of a body. There are two types of strain: normal strain: characterizes dimensional changes,
More informationBME 207 Introduction to Biomechanics Spring 2017
April 7, 2017 UNIVERSITY OF RHODE ISAND Department of Electrical, Computer and Biomedical Engineering BE 207 Introduction to Biomechanics Spring 2017 Homework 7 Problem 14.3 in the textbook. In addition
More informationHigh Tech High Top Hat Technicians. An Introduction to Solid Mechanics. Is that supposed to bend there?
High Tech High Top Hat Technicians An Introduction to Solid Mechanics Or Is that supposed to bend there? Why don't we fall through the floor? The power of any Spring is in the same proportion with the
More informationMECHANICS OF MATERIALS
Third CHTR Stress MCHNICS OF MTRIS Ferdinand. Beer. Russell Johnston, Jr. John T. DeWolf ecture Notes: J. Walt Oler Texas Tech University and Strain xial oading Contents Stress & Strain: xial oading Normal
More informationMembers Subjected to Torsional Loads
Members Subjected to Torsional Loads Torsion of circular shafts Definition of Torsion: Consider a shaft rigidly clamped at one end and twisted at the other end by a torque T = F.d applied in a plane perpendicular
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 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 Ns/m. To make the system
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 informationIf the solution does not follow a logical thought process, it will be assumed in error.
Please indicate your group number (If applicable) Circle Your Instructor s Name and Section: MWF 8:309:20 AM Prof. Kai Ming Li MWF 2:303:20 PM Prof. Fabio Semperlotti MWF 9:3010:20 AM Prof. Jim Jones
More informationPERIYAR CENTENARY POLYTECHNIC COLLEGE PERIYAR NAGAR  VALLAM THANJAVUR. DEPARTMENT OF MECHANICAL ENGINEERING QUESTION BANK
PERIYAR CENTENARY POLYTECHNIC COLLEGE PERIYAR NAGAR  VALLAM  613 403  THANJAVUR. DEPARTMENT OF MECHANICAL ENGINEERING QUESTION BANK Sub : Strength of Materials Year / Sem: II / III Sub Code : MEB 310
More informationNORMAL STRESS. The simplest form of stress is normal stress/direct stress, which is the stress perpendicular to the surface on which it acts.
NORMAL STRESS The simplest form of stress is normal stress/direct stress, which is the stress perpendicular to the surface on which it acts. σ = force/area = P/A where σ = the normal stress P = the centric
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 informationStrength of Materials (15CV 32)
Strength of Materials (15CV 32) Module 1 : Simple Stresses and Strains Dr. H. Ananthan, Professor, VVIET,MYSURU 8/21/2017 Introduction, Definition and concept and of stress and strain. Hooke s law, StressStrain
More informationEMA 3702 Mechanics & Materials Science (Mechanics of Materials) Chapter 10 Columns
EMA 370 Mechanics & Materials Science (Mechanics of Materials) Chapter 10 Columns Columns Introduction Columns are vertical prismatic members subjected to compressive forces Goals: 1. Study the stability
More informationTorsion.
Torsion mi@seu.edu.cn Contents Introduction to Torsion( 扭转简介 ) Examples of Torsion Shafts( 扭转轴示例 ) Sign Convention of Torque( 扭矩符号规则 ) Torque Diagram( 扭矩图 ) Power & Torque( 功率与扭矩 ) Internal Torque & Stress
More informationChapter 4b Axially Loaded Members
CIVL 222 STRENGTH OF MATERIALS Chapter 4b Axially Loaded Members AXIAL LOADED MEMBERS Today s Objectives: Students will be able to: a) Determine the elastic deformation of axially loaded member b) Apply
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 informationMechanical Engineering Ph.D. Preliminary Qualifying Examination Solid Mechanics February 25, 2002
student personal identification (ID) number on each sheet. Do not write your name on any sheet. #1. A homogeneous, isotropic, linear elastic bar has rectangular cross sectional area A, modulus of elasticity
More informationSolution: The moment of inertia for the crosssection is: ANS: ANS: Problem 15.6 The material of the beam in Problem
Problem 15.4 The beam consists of material with modulus of elasticity E 14x10 6 psi and is subjected to couples M 150, 000 in lb at its ends. (a) What is the resulting radius of curvature of the neutral
More informationStress Transformation Equations: u = +135 (Fig. a) s x = 80 MPa s y = 0 t xy = 45 MPa. we obtain, cos u + t xy sin 2u. s x = s x + s y.
014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently 9 7. Determine the normal stress and shear stress acting
More informationSolid Mechanics Chapter 1: Tension, Compression and Shear
Solid Mechanics Chapter 1: Tension, Compression and Shear Dr. Imran Latif Department of Civil and Environmental Engineering College of Engineering University of Nizwa (UoN) 1 Why do we study Mechanics
More informationEMA 3702 Mechanics & Materials Science (Mechanics of Materials) Chapter 4 Pure Bending Homework Answers
EA 3702 echanics & aterials Science (echanics of aterials) Chapter 4 Pure Bending Homework Answers 100 mm Homework 4.1 For pure bending moment of 5 kn m on hollow beam with uniform wall thickness of 10
More informationSRI CHANDRASEKHARENDRA SARASWATHI VISWA MAHAVIDHYALAYA
SRI CHANDRASEKHARENDRA SARASWATHI VISWA MAHAVIDHYALAYA (Declared as Deemedtobe University under Section 3 of the UGC Act, 1956, Vide notification No.F.9.9/92U3 dated 26 th May 1993 of the Govt. of
More informationMechanics of Materials
Mechanics of Materials 2. Introduction Dr. Rami Zakaria References: 1. Engineering Mechanics: Statics, R.C. Hibbeler, 12 th ed, Pearson 2. Mechanics of Materials: R.C. Hibbeler, 9 th ed, Pearson 3. Mechanics
More informationPROBLEM #1.1 (4 + 4 points, no partial credit)
PROBLEM #1.1 ( + points, no partial credit A thermal switch consists of a copper bar which under elevation of temperature closes a gap and closes an electrical circuit. The copper bar possesses a length
More informationUNIT 1 STRESS STRAIN AND DEFORMATION OF SOLIDS, STATES OF STRESS 1. Define stress. When an external force acts on a body, it undergoes deformation.
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
More information5. What is the moment of inertia about the x  x axis of the rectangular beam shown?
1 of 5 Continuing Education Course #274 What Every Engineer Should Know About Structures Part D  Bending Strength Of Materials NOTE: The following question was revised on 15 August 2018 1. The moment
More informationMECHANICS OF SOLIDS TORSION  TUTORIAL 1. You should judge your progress by completing the self assessment exercises.
MECHANICS OF SOIS TORSION  TUTORIA 1 You should judge your progress by completing the self assessment exercises. On completion of this tutorial you should be able to do the following. erive the torsion
More information18.Define the term modulus of resilience. May/June Define Principal Stress. 20. Define Hydrostatic Pressure.
CE6306 STREGNTH OF MATERIALS Question Bank UnitI STRESS, STRAIN, DEFORMATION OF SOLIDS PARTA 1. Define Poison s Ratio May/June 2009 2. What is thermal stress? May/June 2009 3. Estimate the load carried
More informationMECHANICS OF MATERIALS
Third CHTR Stress MCHNICS OF MTRIS Ferdinand. Beer. Russell Johnston, Jr. John T. DeWolf ecture Notes: J. Walt Oler Texas Tech University and Strain xial oading Contents Stress & Strain: xial oading Normal
More informationPurpose of this Guide: To thoroughly prepare students for the exact types of problems that will be on Exam 3.
ES230 STRENGTH OF MTERILS Exam 3 Study Guide Exam 3: Wednesday, March 8 th inclass Updated 3/3/17 Purpose of this Guide: To thoroughly prepare students for the exact types of problems that will be on
More informationStructural Metals Lab 1.2. Torsion Testing of Structural Metals. Standards ASTM E143: Shear Modulus at Room Temperature
Torsion Testing of Structural Metals Standards ASTM E143: Shear Modulus at Room Temperature Purpose To determine the shear modulus of structural metals Equipment TiniusOlsen LoTorq Torsion Machine (figure
More information2. Polar moment of inertia As stated above, the polar second moment of area, J is defined as. Sample copy
GATE PATHSHALA  91. Polar moment of inertia As stated above, the polar second moment of area, is defined as z π r dr 0 R r π R π D For a solid shaft π (6) QP 0 π d Solid shaft π d Hollow shaft, " ( do
More informationME311 Machine Design
ME311 Machine Design Lecture : Materials; Stress & Strain; Power Transmission W Dornfeld 13Sep018 airfield University School of Engineering StressStrain Curve for Ductile Material Ultimate Tensile racture
More informationFailure from static loading
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
More informationEMA 3702 Mechanics & Materials Science (Mechanics of Materials) Chapter 6 Shearing Stress in Beams & ThinWalled Members
EMA 3702 Mechanics & Materials Science (Mechanics of Materials) Chapter 6 Shearing Stress in Beams & ThinWalled Members Beams Bending & Shearing EMA 3702 Mechanics & Materials Science Zhe Cheng (2018)
More informationMECE 3321 MECHANICS OF SOLIDS CHAPTER 3
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.
More informationCHAPTER 4: BENDING OF BEAMS
(74) CHAPTER 4: BENDING OF BEAMS This chapter will be devoted to the analysis of prismatic members subjected to equal and opposite couples M and M' acting in the same longitudinal plane. Such members are
More informationSpherical Pressure Vessels
Spherical Pressure Vessels Pressure vessels are closed structures containing liquids or gases under essure. Examples include tanks, pipes, essurized cabins, etc. Shell structures : When essure vessels
More informationLab Exercise #3: Torsion
Lab Exercise #3: Prelab assignment: Yes No Goals: 1. To evaluate the equations of angular displacement, shear stress, and shear strain for a shaft undergoing torsional stress. Principles: testing of round
More informationSOLUTION (17.3) Known: A simply supported steel shaft is connected to an electric motor with a flexible coupling.
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
More informationσ = Eα(T T C PROBLEM #1.1 (4 + 4 points, no partial credit)
PROBLEM #1.1 (4 + 4 points, no partial credit A thermal switch consists of a copper bar which under elevation of temperature closes a gap and closes an electrical circuit. The copper bar possesses a length
More informationI certify that I have not given unauthorized aid nor have I received aid in the completion of this exam.
NAME: ME 270 Fall 2012 Examination No. 3  Makeup Please review the following statement: Group No.: I certify that I have not given unauthorized aid nor have I received aid in the completion of this exam.
More informationSolution: T, A1, A2, A3, L1, L2, L3, E1, E2, E3, P are known Five equations in five unknowns, F1, F2, F3, ua and va
ME 323 Examination # 1 February 18, 2016 Name (Print) (Last) (First) Instructor PROBLEM #1 (20 points) A structure is constructed from members 1, 2 and 3, with these members made up of the same material
More informationME Final Exam. PROBLEM NO. 4 Part A (2 points max.) M (x) y. z (neutral axis) beam crosssec+on. 20 kip ft. 0.2 ft. 10 ft. 0.1 ft.
ME 323  Final Exam Name December 15, 2015 Instructor (circle) PROEM NO. 4 Part A (2 points max.) Krousgrill 11:30AM12:20PM Ghosh 2:303:20PM Gonzalez 12:301:20PM Zhao 4:305:20PM M (x) y 20 kip ft 0.2
More information