FCP Short Course. Ductile and Brittle Fracture. Stephen D. Downing. Mechanical Science and Engineering

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "FCP Short Course. Ductile and Brittle Fracture. Stephen D. Downing. Mechanical Science and Engineering"

Transcription

1 FCP Short Course Ductile and Brittle Fracture Stephen D. Downing Mechanical Science and Engineering University of Illinois Board of Trustees, All Rights Reserved

2 Agenda Limit theorems Plane Stress Plane Strain Notched plates Notched bars Brittle Faracture FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 1 of 4

3 Beam in Bending M M M FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved of 4

4 Stress Distribution Elastic Elastic-Plastic Fully Plastic M FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 3 of 4

5 Plastic Moment y s Elastic: M e ys c I ys bh 6 y s Fully Plastic: M dy y o ys 4 M M o e h b ydy ys bh FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 4 of 4

6 Fully Plastic Moment F M o Beam can be loaded until M o is reached after which a plastic hinge is formed and the beam is free to rotate. FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 5 of 4

7 Beam With a Support L F L M o M o FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 6 of 4

8 Collapse F right end support prevents collapse M = M o F M = M o F left end support prevents collapse collapse M = M o M = M o FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 7 of 4

9 Lower Bound Theorem If an equilibrium distribution of stress can be found which balances the applied load and is everywhere below or at yield, the structure will not collapse or just be at the point of collapse. Guaranteed load capacity where system will not have large deformations FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 8 of 4

10 Upper Bound Theorem The structure must collapse if there is any compatible pattern of plastic deformation for which the rate at which external work is equal to or greater than the rate of internal dissipation. Guaranteed load capacity where the system will have large deformations. FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 9 of 4

11 Application to a Cracked Plate Assume a stress distribution that satisfied equilibrium P LB y b z x a W y = ys x = 0 z = 0 P LB = ys ( W a ) b xy = 0 FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 10 of 4

12 Upper Bound Solution t Rigid blocks allowed to slide Shear band of thickness t FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 11 of 4

13 Upper Bound Solution (continued) F s P UB du Slip plane area: A s W a b cos Force on shear plane: F s ys 3 W a b cos FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 1 of 4

14 Upper Bound Solution (continued) P UB Internal work: F s du duf du ys External work: s du 3 W a b cos du dw P du sin UB FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 13 of 4

15 Upper Bound Solution (continued) Upper Bound Theorem: du = dw P UB ys W a b du PUB dusin 3 cos ys W ab ys W ab 3 sin cos 3 sin Lowest upper bound for sin = 1 or = 45 P UB = 1.15 ys ( W a ) b P LB = ys ( W a ) b FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 14 of 4

16 Agenda Limit theorems Plane Stress Plane Strain Notched plates Notched bars Brittle Faracture FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 15 of 4

17 Plane Stress Plane Strain thick plate plane strain z = 0 z = 0 thin plate plane stress z = 0 z = 0 FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 16 of 4

18 Thick or Thin? Plane strain Plane stress FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 17 of 4

19 Transverse Strains Longitudinal Tensile Strain Transverse Compression Strain Thickness mm 0 mm 10 mm = = mm y z x 5 D 100 FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 18 of 4

20 Notch Stresses y z x t x z x z FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 19 of 4

21 Fracture Surfaces FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 0 of 4

22 Fracture Surfaces FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 1 of 4

23 Yielding plane stress z = 0 z = 0 x x y y 3 3 k plane strain z = 0 z = k x y FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved of 4

24 Flow Stress Ultimate strength Stress (MPa) Yield strength Strain Flow Stress: flow ys uts k flow 3 FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 3 of 4

25 Notches Edge Center Same net section, same K T FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 4 of 4

26 Plane Stress n = flow n = flow flow 0 0 FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 5 of 4

27 Edge Notch Plane Strain From the punch problem: n = k P =.96 flow By analogy: n =.96 flow.96 flow 1.81 flow.38 flow FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 6 of 4

28 Center Notch Plane Strain Slip Line Field x = 1.15 flow flow FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 7 of 4

29 Agenda Limit theorems Plane Stress Plane Strain Notched plates Notched bars Brittle Faracture FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 8 of 4

30 What does all this tell us? A B C D Lower Bound Theorem: P A = P B = P C = P D Upper Bound Theorem: P A = P B = P C < P D Slip Line Theory: P A = P B = P C < P D No effect of stress concentration on static strength! Some stress concentrations can increase the strength! FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 9 of 4

31 Failure Loads Edge Center Plane Stress P A net 1.15 flow P A net flow Plane Strain P A net.96 flow P A net 1.15 flow FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 30 of 4

32 Conclusion Net section area, state of stress and material strength control the failure load in a structure. FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 31 of 4

33 Stress or Strain Control? elastic material plastic zone Elastic material surrounding the plastic zone forces the displacements to be compatible, I.e. no gaps form in the structure. Boundary conditions acting on the plastic zone boundary are displacements. Strains are the first derivative of displacement FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 3 of 4

34 Define K and K S, e, Define: nominal stress, S nominal strain, e notch stress, notch strain, Stress concentration Strain concentration K K S e FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 33 of 4

35 K and K K T S Stress (MPa) K K S e K T e Strain FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 34 of 4

36 Stress and Strain Concentration Stress/Strain Concentration First yielding K K T K T K 1 Nominal Stress FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 35 of 4

37 Failure of a Notched Plate FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 36 of 4

38 Real Data Materials: 1018 Hot Rolled Steel 7075-T6 Aluminum /4 thick 1 1 FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 37 of 4

39 1018 Stress-Strain Curve true fracture strain, f = 0.86 true fracture stress, f = 770 MPa Strain FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 38 of 4

40 7075-T6 Stress-Strain Curve true fracture strain, f = 0.3 true fracture stress, f = 70 MPa Strain FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 39 of 4

41 7075-T6 Test Data 100 load, kn edge diamond slot hole displacement, mm FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 40 of 4

42 1018 Steel Test Data load, kn edge diamond slot hole displacement, mm FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 41 of 4

43 Agenda Limit theorems Plane Stress Plane Strain Notched plates Notched bars Brittle Faracture FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 4 of 4

44 Stress Concentration o Axial, z Tangential, Radial, r r FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 43 of 4

45 Bridgeman Analysis (1943) ys z ys r z r a r a r Elastic stress distribution Plastic stress distribution z r cons tan t FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 44 of 4

46 FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 45 of 4 Stresses a r a a ln 1 o z a 0 z z dr r P a 1 ln a 1 a P flow max P max = A net flow CF CF constraint factor

47 Constraint Factors a / CF P max = A net flow CF FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 46 of 4

48 Effect of Constraint Stress, z Increasing constraint Strain, z Higher strength and lower ductility FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 47 of 4

49 Failures from Stress Concentrations Net section stresses must be below the flow stress Notch strains must be below the fracture strain FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 48 of 4

50 Typical Stress Concentration K T ductile brittle K T brittle ductile Stress distribution Strain distribution A ductile material has the capacity for very large strains and it reaches the strength limit first A brittle material reaches the strain limit first FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 49 of 4

51 Strain Distribution Nominal strain will reach the yield strain when the strength limit is reached Strain distribution K T K T yield yield ductile transition K T f K yield T brittle yield f K f T yield K T yield FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 50 of 4

52 Notched Bars 1 8 R 1 16 R 1 R V notch FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 51 of 4

53 7075-T6 Test Data V displacement, mm FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 5 of 4

54 1018 Steel Test Data V displacement, mm FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 53 of 4

55 Conclusion Net section area, state of stress and material strength control the failure load in a structure only in ductile materials. In brittle materials, cracks will form before the maximum load capacity of the structure is reached. FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 54 of 4

56 Agenda Limit theorems Plane Stress Plane Strain Notched plates Notched bars Brittle Fracture FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 55 of 4

57 Brittle Fracture Failure of structures by the rapid growth of cracks (or crack-like defects) until loss of structural integrity. FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 56 of 4

58 Fracture FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 57 of 4

59 Stress Concentration applied Inglis Equation local a local 1 applied K t applied Crack dimensions a = 10-3 = 10-9 local Does this make sense? applied a Kt 000 FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 58 of 4

60 Griffith Approach l applied a Uncracked plate elastic energy 1 Ue w*l*t w*l*t E Introduce a crack which Reduces elastic energy by: a Ue a *t E Increases total surface energy by: w at s applied thickness, t For a crack to grow, the energy provided by new surfaces must equal that lost by elastic relief a U e FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 59 of 4

61 Griffith Approach applied Minimum criterion for stable crack growth: Strain energy goes into surface energy a U l a a a a a E t a s a 4 a t w a E s applied thickness, t s E a FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 60 of 4

62 Plastic Energy Term l a a w applied Previous derivation is for purely elastic material Most metals and polymers experience some plastic deformation Strain energy (U) goes into surface energy () & plastic energy (P) U P a a a Orowan introduced plastic deformation energy, γ p E ( s a p ) p s applied thickness, t Materials which exhibit plastic deformation absorb much more energy, removing it from the crack tip FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 61 of 4

63 Energy Release Rate, G l a a w applied applied thickness, t Irwin chose to define a term, G, that characterizes the energy per unit crack area required to extend the crack: G G 1 U t a a E Comparing to the previous expression, we see that: G s p Works well when plastic zone is small ( fraction of crack dimensions ) EG a Change in crack area FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 6 of 4

64 GExperimentally Measuring G a F F x x t Load ICa Ea U a + a Displacement atfracture Load cracked sample in elastic range Fix grips at given displacement Allow crack to grow length a Unload sample Compare energy under the curves G is the energy per unit crack area needed to extend a crack Experimentally measure combinations of stress and crack size at fracture to determine G IC FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 63 of 4

65 Stress Based Crack Analysis Westergaard, Irwin analyzed fracture of cracked components using elastic-based stress theory Three modes for crack loading Mode I opening Mode II in-plane shear Mode III out-of-plane shear From: Socie FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 64 of 4

66 Stress Intensity Factor, K af astress intensity factor flaw size operating stresses b Kspecimen geometry Critical parameter is now based on: Stress Flaw Size Specimen geometry included using correction factor crack shape specimen size and shape type of loading (i.e. tensile, bending, etc) FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 65 of 4

67 Stress Intensity Factor, K Stress Fields (near crack tip) x K 3 cos 1 sin sin r y K 3 cos 1 sin sin r xy K 3 cos sin cos r Note: as r 0, stress fields Plane stress - too thin to support stress through thickness z 0 z 0 Plane strain - so thick that constrains strain through thickness z 0 x y z FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 66 of 4

68 FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 67 of 4 Stress Intensity Factor, K Displacement Fields sin cos r G K u cos sin r G K v x u x x v y x v y u xy Displacement in x-direction Displacement in y-direction

69 K and G G K G E 1 E K plane stress plane strain FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 68 of 4

70 Linear Elastic Fracture Mechanics How can we use an elastic concept (K) when there is plastic deformation? Small scale yielding Blunts the crack tip At elastic-plastic boundary, y r r* K 3 1 sin sin y cos r* Note: zone of yielding will vary with plastic zone r p Consider stress at = 0 K y r* FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 69 of 4

71 Design Philosophy K IC a f a w fracture toughness flaw shape flaw size operating stresses FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 70 of 4

72 Critical Crack Sizes a critical 1 K IC f FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 71 of 4

73 Fracture Toughness From M F Ashby, Materials Selection in Mechanical Design, 1999, pg 431 FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 7 of 4

74 Thickness Effects fracture toughness, MPa m thickness, mm FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 73 of 4

75 FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 74 of 4 Plastic Zone Size plane stress plane strain ys p K 1 r ys p K 6 1 r

76 3D Plastic Zone plane stress plane strain FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 75 of 4

77 Fracture Surfaces FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 76 of 4

78 Size Requirements t,w a,a.5 K ys 50r ys K Ic t, mm 04- T T Ti-6Al-4V Ti-6Al-4V PH p FCP Fall Stephen Downing, University of Illinois at Urbana-Champaign, All Rights Reserved 77 of 4

Stress Concentration. Professor Darrell F. Socie Darrell Socie, All Rights Reserved

Stress Concentration. Professor Darrell F. Socie Darrell Socie, All Rights Reserved Stress Concentration Professor Darrell F. Socie 004-014 Darrell Socie, All Rights Reserved Outline 1. Stress Concentration. Notch Rules 3. Fatigue Notch Factor 4. Stress Intensity Factors for Notches 5.

More information

Mechanical Properties of Materials

Mechanical 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 information

Failure from static loading

Failure 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 information

Fracture mechanics fundamentals. Stress at a notch Stress at a crack Stress intensity factors Fracture mechanics based design

Fracture mechanics fundamentals. Stress at a notch Stress at a crack Stress intensity factors Fracture mechanics based design Fracture mechanics fundamentals Stress at a notch Stress at a crack Stress intensity factors Fracture mechanics based design Failure modes Failure can occur in a number of modes: - plastic deformation

More information

ME 2570 MECHANICS OF MATERIALS

ME 2570 MECHANICS OF MATERIALS ME 2570 MECHANICS OF MATERIALS Chapter III. Mechanical Properties of Materials 1 Tension and Compression Test The strength of a material depends on its ability to sustain a load without undue deformation

More information

Lecture #7: Basic Notions of Fracture Mechanics Ductile Fracture

Lecture #7: Basic Notions of Fracture Mechanics Ductile Fracture Lecture #7: Basic Notions of Fracture Mechanics Ductile Fracture by Dirk Mohr ETH Zurich, Department of Mechanical and Process Engineering, Chair of Computational Modeling of Materials in Manufacturing

More information

Multiaxial Fatigue. Professor Darrell F. Socie. Department of Mechanical Science and Engineering University of Illinois at Urbana-Champaign

Multiaxial Fatigue. Professor Darrell F. Socie. Department of Mechanical Science and Engineering University of Illinois at Urbana-Champaign Multiaxial Fatigue Professor Darrell F. Socie Department of Mechanical Science and Engineering University of Illinois at Urbana-Champaign 2001-2011 Darrell Socie, All Rights Reserved Contact Information

More information

4.MECHANICAL PROPERTIES OF MATERIALS

4.MECHANICAL PROPERTIES OF MATERIALS 4.MECHANICAL PROPERTIES OF MATERIALS The diagram representing the relation between stress and strain in a given material is an important characteristic of the material. To obtain the stress-strain diagram

More information

The objective of this experiment is to investigate the behavior of steel specimen under a tensile test and to determine it's properties.

The objective of this experiment is to investigate the behavior of steel specimen under a tensile test and to determine it's properties. Objective: The objective of this experiment is to investigate the behavior of steel specimen under a tensile test and to determine it's properties. Introduction: Mechanical testing plays an important role

More information

Mechanics of Earthquakes and Faulting

Mechanics of Earthquakes and Faulting Mechanics of Earthquakes and Faulting Lectures & 3, 9/31 Aug 017 www.geosc.psu.edu/courses/geosc508 Discussion of Handin, JGR, 1969 and Chapter 1 Scholz, 00. Stress analysis and Mohr Circles Coulomb Failure

More information

Introduction to Engineering Materials ENGR2000. Dr. Coates

Introduction to Engineering Materials ENGR2000. Dr. Coates 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

More information

Elastic-Plastic Fracture Mechanics. Professor S. Suresh

Elastic-Plastic Fracture Mechanics. Professor S. Suresh Elastic-Plastic Fracture Mechanics Professor S. Suresh Elastic Plastic Fracture Previously, we have analyzed problems in which the plastic zone was small compared to the specimen dimensions (small scale

More information

Mechanics of Materials Primer

Mechanics of Materials Primer Mechanics of Materials rimer Notation: A = area (net = with holes, bearing = in contact, etc...) b = total width of material at a horizontal section d = diameter of a hole D = symbol for diameter E = modulus

More information

Mechanics of Earthquakes and Faulting

Mechanics of Earthquakes and Faulting Mechanics of Earthquakes and Faulting www.geosc.psu.edu/courses/geosc508 Surface and body forces Tensors, Mohr circles. Theoretical strength of materials Defects Stress concentrations Griffith failure

More information

UNIT I SIMPLE STRESSES AND STRAINS

UNIT I SIMPLE STRESSES AND STRAINS Subject with Code : SM-1(15A01303) Year & Sem: II-B.Tech & I-Sem SIDDHARTH GROUP OF INSTITUTIONS :: PUTTUR Siddharth Nagar, Narayanavanam Road 517583 QUESTION BANK (DESCRIPTIVE) UNIT I SIMPLE STRESSES

More information

Advanced Strength of Materials Prof S. K. Maiti Mechanical Engineering Indian Institute of Technology, Bombay. Lecture 27

Advanced Strength of Materials Prof S. K. Maiti Mechanical Engineering Indian Institute of Technology, Bombay. Lecture 27 Advanced Strength of Materials Prof S. K. Maiti Mechanical Engineering Indian Institute of Technology, Bombay Lecture 27 Last time we considered Griffith theory of brittle fracture, where in it was considered

More information

Static Failure (pg 206)

Static Failure (pg 206) Static Failure (pg 06) All material followed Hookeʹs law which states that strain is proportional to stress applied, until it exceed the proportional limits. It will reach and exceed the elastic limit

More information

ME 243. Mechanics of Solids

ME 243. Mechanics of Solids ME 243 Mechanics of Solids Lecture 2: Stress and Strain Ahmad Shahedi Shakil Lecturer, Dept. of Mechanical Engg, BUET E-mail: sshakil@me.buet.ac.bd, shakil6791@gmail.com Website: teacher.buet.ac.bd/sshakil

More information

five Mechanics of Materials 1 ARCHITECTURAL STRUCTURES: FORM, BEHAVIOR, AND DESIGN DR. ANNE NICHOLS SUMMER 2017 lecture

five Mechanics of Materials 1 ARCHITECTURAL STRUCTURES: FORM, BEHAVIOR, AND DESIGN DR. ANNE NICHOLS SUMMER 2017 lecture ARCHITECTURAL STRUCTURES: FORM, BEHAVIOR, AND DESIGN DR. ANNE NICHOLS SUMMER 2017 lecture five mechanics www.carttalk.com of materials Mechanics of Materials 1 Mechanics of Materials MECHANICS MATERIALS

More information

five Mechanics of Materials 1 ARCHITECTURAL STRUCTURES: FORM, BEHAVIOR, AND DESIGN DR. ANNE NICHOLS SUMMER 2014 lecture

five Mechanics of Materials 1 ARCHITECTURAL STRUCTURES: FORM, BEHAVIOR, AND DESIGN DR. ANNE NICHOLS SUMMER 2014 lecture ARCHITECTURAL STRUCTURES: FORM, BEHAVIOR, AND DESIGN DR. ANNE NICHOLS SUMMER 2014 lecture five mechanics www.carttalk.com of materials Mechanics of Materials 1 Mechanics of Materials MECHANICS MATERIALS

More information

MECE 3321 MECHANICS OF SOLIDS CHAPTER 3

MECE 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 information

1 Static Plastic Behaviour of Beams

1 Static Plastic Behaviour of Beams 1 Static Plastic Behaviour of Beams 1.1 Introduction Many ductile materials which are used in engineering practice have a considerable reserve capacity beyond the initial yield condition. The uniaxial

More information

Mechanics of Materials

Mechanics of Materials Mechanics of Materials Notation: a = acceleration = area (net = with holes, bearing = in contact, etc...) SD = allowable stress design d = diameter of a hole = calculus symbol for differentiation e = change

More information

ME Final Exam. PROBLEM NO. 4 Part A (2 points max.) M (x) y. z (neutral axis) beam cross-sec+on. 20 kip ft. 0.2 ft. 10 ft. 0.1 ft.

ME Final Exam. PROBLEM NO. 4 Part A (2 points max.) M (x) y. z (neutral axis) beam cross-sec+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 11:30AM-12:20PM Ghosh 2:30-3:20PM Gonzalez 12:30-1:20PM Zhao 4:30-5:20PM M (x) y 20 kip ft 0.2

More information

Fatigue and Fracture

Fatigue and Fracture Fatigue and Fracture Multiaxial Fatigue Professor Darrell F. Socie Mechanical Science and Engineering University of Illinois 2004-2013 Darrell Socie, All Rights Reserved When is Multiaxial Fatigue Important?

More information

PDDC 1 st Semester Civil Engineering Department Assignments of Mechanics of Solids [ ] Introduction, Fundamentals of Statics

PDDC 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 information

Constitutive Equations (Linear Elasticity)

Constitutive Equations (Linear Elasticity) Constitutive quations (Linear lasticity) quations that characterize the physical properties of the material of a system are called constitutive equations. It is possible to find the applied stresses knowing

More information

Linear Elastic Fracture Mechanics

Linear Elastic Fracture Mechanics Measure what is measurable, and make measurable what is not so. - Galileo GALILEI Linear Elastic Fracture Mechanics Krishnaswamy Ravi-Chandar Lecture presented at the University of Pierre and Marie Curie

More information

Lap splice length and details of column longitudinal reinforcement at plastic hinge region

Lap splice length and details of column longitudinal reinforcement at plastic hinge region Lap length and details of column longitudinal reinforcement at plastic hinge region Hong-Gun Park 1) and Chul-Goo Kim 2) 1), 2 Department of Architecture and Architectural Engineering, Seoul National University,

More information

Strength of Materials (15CV 32)

Strength 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, Stress-Strain

More information

NORMAL 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. 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 information

D : SOLID MECHANICS. Q. 1 Q. 9 carry one mark each. Q.1 Find the force (in kn) in the member BH of the truss shown.

D : SOLID MECHANICS. Q. 1 Q. 9 carry one mark each. Q.1 Find the force (in kn) in the member BH of the truss shown. D : SOLID MECHANICS Q. 1 Q. 9 carry one mark each. Q.1 Find the force (in kn) in the member BH of the truss shown. Q.2 Consider the forces of magnitude F acting on the sides of the regular hexagon having

More information

MMJ1133 FATIGUE AND FRACTURE MECHANICS E ENGINEERING FRACTURE MECHANICS

MMJ1133 FATIGUE AND FRACTURE MECHANICS E ENGINEERING FRACTURE MECHANICS E ENGINEERING WWII: Liberty ships Reprinted w/ permission from R.W. Hertzberg, "Deformation and Fracture Mechanics of Engineering Materials", (4th ed.) Fig. 7.1(b), p. 6, John Wiley and Sons, Inc., 1996.

More information

EMA 3702 Mechanics & Materials Science (Mechanics of Materials) Chapter 2 Stress & Strain - Axial Loading

EMA 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 information

V Predicted Weldment Fatigue Behavior AM 11/03 1

V Predicted Weldment Fatigue Behavior AM 11/03 1 V Predicted Weldment Fatigue Behavior AM 11/03 1 Outline Heavy and Light Industry weldments The IP model Some predictions of the IP model AM 11/03 2 Heavy industry AM 11/03 3 Heavy industry AM 11/03 4

More information

MECHANICS OF MATERIALS

MECHANICS 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 information

CE5510 Advanced Structural Concrete Design - Design & Detailing of Openings in RC Flexural Members-

CE5510 Advanced Structural Concrete Design - Design & Detailing of Openings in RC Flexural Members- CE5510 Advanced Structural Concrete Design - Design & Detailing Openings in RC Flexural Members- Assoc Pr Tan Kiang Hwee Department Civil Engineering National In this lecture DEPARTMENT OF CIVIL ENGINEERING

More information

MATERIALS FOR CIVIL AND CONSTRUCTION ENGINEERS

MATERIALS FOR CIVIL AND CONSTRUCTION ENGINEERS MATERIALS FOR CIVIL AND CONSTRUCTION ENGINEERS 3 rd Edition Michael S. Mamlouk Arizona State University John P. Zaniewski West Virginia University Solution Manual FOREWORD This solution manual includes

More information

G1RT-CT A. BASIC CONCEPTS F. GUTIÉRREZ-SOLANA S. CICERO J.A. ALVAREZ R. LACALLE W P 6: TRAINING & EDUCATION

G1RT-CT A. BASIC CONCEPTS F. GUTIÉRREZ-SOLANA S. CICERO J.A. ALVAREZ R. LACALLE W P 6: TRAINING & EDUCATION A. BASIC CONCEPTS 6 INTRODUCTION The final fracture of structural components is associated with the presence of macro or microstructural defects that affect the stress state due to the loading conditions.

More information

Chapter 5 CENTRIC TENSION OR COMPRESSION ( AXIAL LOADING )

Chapter 5 CENTRIC TENSION OR COMPRESSION ( AXIAL LOADING ) 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

More information

Fracture Mechanics, Damage and Fatigue Linear Elastic Fracture Mechanics - Energetic Approach

Fracture Mechanics, Damage and Fatigue Linear Elastic Fracture Mechanics - Energetic Approach University of Liège Aerospace & Mechanical Engineering Fracture Mechanics, Damage and Fatigue Linear Elastic Fracture Mechanics - Energetic Approach Ludovic Noels Computational & Multiscale Mechanics of

More information

Theory at a Glance (for IES, GATE, PSU)

Theory at a Glance (for IES, GATE, PSU) 1. Stress and Strain Theory at a Glance (for IES, GATE, PSU) 1.1 Stress () When a material is subjected to an external force, a resisting force is set up within the component. The internal resistance force

More information

Chapter Two: Mechanical Properties of materials

Chapter Two: Mechanical Properties of materials Chapter Two: Mechanical Properties of materials Time : 16 Hours An important consideration in the choice of a material is the way it behave when subjected to force. The mechanical properties of a material

More information

Tuesday, February 11, Chapter 3. Load and Stress Analysis. Dr. Mohammad Suliman Abuhaiba, PE

Tuesday, February 11, Chapter 3. Load and Stress Analysis. Dr. Mohammad Suliman Abuhaiba, PE 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

More information

Chapter 3. Load and Stress Analysis

Chapter 3. Load and Stress Analysis Chapter 3 Load and Stress Analysis 2 Shear Force and Bending Moments in Beams Internal shear force V & bending moment M must ensure equilibrium Fig. 3 2 Sign Conventions for Bending and Shear Fig. 3 3

More information

Adhesive Joints Theory (and use of innovative joints) ERIK SERRANO STRUCTURAL MECHANICS, LUND UNIVERSITY

Adhesive Joints Theory (and use of innovative joints) ERIK SERRANO STRUCTURAL MECHANICS, LUND UNIVERSITY Adhesive Joints Theory (and use of innovative joints) ERIK SERRANO STRUCTURAL MECHANICS, LUND UNIVERSITY Wood and Timber Why I m intrigued From this to this! via this Fibre deviation close to knots and

More information

Mechanics 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 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 information

Lecture 4 Honeycombs Notes, 3.054

Lecture 4 Honeycombs Notes, 3.054 Honeycombs-In-plane behavior Lecture 4 Honeycombs Notes, 3.054 Prismatic cells Polymer, metal, ceramic honeycombs widely available Used for sandwich structure cores, energy absorption, carriers for catalysts

More information

THEME IS FIRST OCCURANCE OF YIELDING THE LIMIT?

THEME IS FIRST OCCURANCE OF YIELDING THE LIMIT? CIE309 : PLASTICITY THEME IS FIRST OCCURANCE OF YIELDING THE LIMIT? M M - N N + + σ = σ = + f f BENDING EXTENSION Ir J.W. Welleman page nr 0 kn Normal conditions during the life time WHAT HAPPENS DUE TO

More information

MECHANICS OF MATERIALS

MECHANICS 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 information

Laboratory 4 Bending Test of Materials

Laboratory 4 Bending Test of Materials Department of Materials and Metallurgical Engineering Bangladesh University of Engineering Technology, Dhaka MME 222 Materials Testing Sessional.50 Credits Laboratory 4 Bending Test of Materials. Objective

More information

INTRODUCTION TO STRAIN

INTRODUCTION 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 information

CHAPTER 6 MECHANICAL PROPERTIES OF METALS PROBLEM SOLUTIONS

CHAPTER 6 MECHANICAL PROPERTIES OF METALS PROBLEM SOLUTIONS CHAPTER 6 MECHANICAL PROPERTIES OF METALS PROBLEM SOLUTIONS Concepts of Stress and Strain 6.1 Using mechanics of materials principles (i.e., equations of mechanical equilibrium applied to a free-body diagram),

More information

Introduction to Fracture

Introduction to Fracture Introduction to Fracture Introduction Design of a component Yielding Strength Deflection Stiffness Buckling critical load Fatigue Stress and Strain based Vibration Resonance Impact High strain rates Fracture

More information

Stress-Strain Behavior

Stress-Strain Behavior Stress-Strain Behavior 6.3 A specimen of aluminum having a rectangular cross section 10 mm 1.7 mm (0.4 in. 0.5 in.) is pulled in tension with 35,500 N (8000 lb f ) force, producing only elastic deformation.

More information

Seismic Pushover Analysis Using AASHTO Guide Specifications for LRFD Seismic Bridge Design

Seismic Pushover Analysis Using AASHTO Guide Specifications for LRFD Seismic Bridge Design Seismic Pushover Analysis Using AASHTO Guide Specifications for LRFD Seismic Bridge Design Elmer E. Marx, Alaska Department of Transportation and Public Facilities Michael Keever, California Department

More information

A concrete cylinder having a a diameter of of in. mm and elasticity. Stress and Strain: Stress and Strain: 0.

A concrete cylinder having a a diameter of of in. mm and elasticity. Stress and Strain: Stress and Strain: 0. 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

More information

The University of Melbourne Engineering Mechanics

The University of Melbourne Engineering Mechanics The University of Melbourne 436-291 Engineering Mechanics Tutorial Four Poisson s Ratio and Axial Loading Part A (Introductory) 1. (Problem 9-22 from Hibbeler - Statics and Mechanics of Materials) A short

More information

[5] Stress and Strain

[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 information

Chapter 4. Test results and discussion. 4.1 Introduction to Experimental Results

Chapter 4. Test results and discussion. 4.1 Introduction to Experimental Results Chapter 4 Test results and discussion This chapter presents a discussion of the results obtained from eighteen beam specimens tested at the Structural Technology Laboratory of the Technical University

More information

Lecture 7, Foams, 3.054

Lecture 7, Foams, 3.054 Lecture 7, Foams, 3.054 Open-cell foams Stress-Strain curve: deformation and failure mechanisms Compression - 3 regimes - linear elastic - bending - stress plateau - cell collapse by buckling yielding

More information

Fracture Mechanics of Non-Shear Reinforced R/C Beams

Fracture Mechanics of Non-Shear Reinforced R/C Beams Irina Kerelezova Thomas Hansen M. P. Nielsen Fracture Mechanics of Non-Shear Reinforced R/C Beams DANMARKS TEKNISKE UNIVERSITET Report BYG DTU R-154 27 ISSN 161-2917 ISBN 97887-7877-226-5 Fracture Mechanics

More information

Tensile stress strain curves for different materials. Shows in figure below

Tensile stress strain curves for different materials. Shows in figure below Tensile stress strain curves for different materials. Shows in figure below Furthermore, the modulus of elasticity of several materials effected by increasing temperature, as is shown in Figure Asst. Lecturer

More information

MECHANICS OF MATERIALS

MECHANICS OF MATERIALS 2009 The McGraw-Hill 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

Presented by: Civil Engineering Academy

Presented by: Civil Engineering Academy Presented by: Civil Engineering Academy Structural Design and Material Properties of Steel Presented by: Civil Engineering Academy Advantages 1. High strength per unit length resulting in smaller dead

More information

STATICALLY INDETERMINATE STRUCTURES

STATICALLY INDETERMINATE STRUCTURES STATICALLY INDETERMINATE STRUCTURES INTRODUCTION Generally the trusses are supported on (i) a hinged support and (ii) a roller support. The reaction components of a hinged support are two (in horizontal

More information

2. 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)?

2. 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 information

Tentamen/Examination TMHL61

Tentamen/Examination TMHL61 Avd Hållfasthetslära, IKP, Linköpings Universitet Tentamen/Examination TMHL61 Tentamen i Skademekanik och livslängdsanalys TMHL61 lördagen den 14/10 2000, kl 8-12 Solid Mechanics, IKP, Linköping University

More information

Massachusetts Institute of Technology Department of Mechanical Engineering Cambridge, MA 02139

Massachusetts Institute of Technology Department of Mechanical Engineering Cambridge, MA 02139 Massachusetts Institute of Technology Department of Mechanical Engineering Cambridge, MA 02139 2.002 Mechanics and Materials II Spring 2004 Laboratory Module No. 6 Fracture Toughness Testing and Residual

More information

Chapter 7. Highlights:

Chapter 7. Highlights: Chapter 7 Highlights: 1. Understand the basic concepts of engineering stress and strain, yield strength, tensile strength, Young's(elastic) modulus, ductility, toughness, resilience, true stress and true

More information

High 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. 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 information

5. STRESS CONCENTRATIONS. and strains in shafts apply only to solid and hollow circular shafts while they are in the

5. STRESS CONCENTRATIONS. and strains in shafts apply only to solid and hollow circular shafts while they are in the 5. STRESS CONCENTRATIONS So far in this thesis, most of the formulas we have seen to calculate the stresses and strains in shafts apply only to solid and hollow circular shafts while they are in the elastic

More information

MECHANICS OF MATERIALS

MECHANICS 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 information

Strength of Material. Shear Strain. Dr. Attaullah Shah

Strength of Material. Shear Strain. Dr. Attaullah Shah 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

More information

SEMM Mechanics PhD Preliminary Exam Spring Consider a two-dimensional rigid motion, whose displacement field is given by

SEMM Mechanics PhD Preliminary Exam Spring Consider a two-dimensional rigid motion, whose displacement field is given by SEMM Mechanics PhD Preliminary Exam Spring 2014 1. Consider a two-dimensional rigid motion, whose displacement field is given by u(x) = [cos(β)x 1 + sin(β)x 2 X 1 ]e 1 + [ sin(β)x 1 + cos(β)x 2 X 2 ]e

More information

MECHANICS OF MATERIALS

MECHANICS 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 information

December 10, PROBLEM NO points max.

December 10, PROBLEM NO points max. PROBLEM NO. 1 25 points max. PROBLEM NO. 2 25 points max. B 3A A C D A H k P L 2L Given: Consider the structure above that is made up of rod segments BC and DH, a spring of stiffness k and rigid connectors

More information

Finite Element Modelling with Plastic Hinges

Finite Element Modelling with Plastic Hinges 01/02/2016 Marco Donà Finite Element Modelling with Plastic Hinges 1 Plastic hinge approach A plastic hinge represents a concentrated post-yield behaviour in one or more degrees of freedom. Hinges only

More information

Lecture #2: Split Hopkinson Bar Systems

Lecture #2: Split Hopkinson Bar Systems Lecture #2: Split Hopkinson Bar Systems by Dirk Mohr ETH Zurich, Department of Mechanical and Process Engineering, Chair of Computational Modeling of Materials in Manufacturing 2015 1 1 1 Uniaxial Compression

More information

Topics in Ship Structures

Topics in Ship Structures Topics in Ship Structures 8 Elastic-lastic Fracture Mechanics Reference : Fracture Mechanics by T.L. Anderson Lecture Note of Eindhoven University of Technology 17. 1 by Jang, Beom Seon Oen INteractive

More information

PES Institute of Technology

PES Institute of Technology PES Institute of Technology Bangalore south campus, Bangalore-5460100 Department of Mechanical Engineering Faculty name : Madhu M Date: 29/06/2012 SEM : 3 rd A SEC Subject : MECHANICS OF MATERIALS Subject

More information

Principal Stresses, Yielding Criteria, wall structures

Principal Stresses, Yielding Criteria, wall structures Principal Stresses, Yielding Criteria, St i thi Stresses in thin wall structures Introduction The most general state of stress at a point may be represented by 6 components, x, y, z τ xy, τ yz, τ zx normal

More information

University of Sheffield The development of finite elements for 3D structural analysis in fire

University of Sheffield The development of finite elements for 3D structural analysis in fire The development of finite elements for 3D structural analysis in fire Chaoming Yu, I. W. Burgess, Z. Huang, R. J. Plank Department of Civil and Structural Engineering StiFF 05/09/2006 3D composite structures

More information

CHAPTER 6: ULTIMATE LIMIT STATE

CHAPTER 6: ULTIMATE LIMIT STATE CHAPTER 6: ULTIMATE LIMIT STATE 6.1 GENERAL It shall be in accordance with JSCE Standard Specification (Design), 6.1. The collapse mechanism in statically indeterminate structures shall not be considered.

More information

Free Body Diagram: Solution: The maximum load which can be safely supported by EACH of the support members is: ANS: A =0.217 in 2

Free 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 information

Ratcheting and Rolling Contact Fatigue Crack Initiation Life of Rails under Service Loading. Wenyi YAN Monash University, Australia

Ratcheting and Rolling Contact Fatigue Crack Initiation Life of Rails under Service Loading. Wenyi YAN Monash University, Australia Ratcheting and Rolling Contact Fatigue Crack Initiation Life of Rails under Service Loading Wenyi YAN Monash University, Australia Chung Lun PUN Peter Mutton Qianhua Kan Guozheng Kang Contents Introduction

More information

Shafts: Torsion of Circular Shafts Reading: Crandall, Dahl and Lardner 6.2, 6.3

Shafts: Torsion of Circular Shafts Reading: Crandall, Dahl and Lardner 6.2, 6.3 M9 Shafts: Torsion of Circular Shafts Reading: Crandall, Dahl and Lardner 6., 6.3 A shaft is a structural member which is long and slender and subject to a torque (moment) acting about its long axis. We

More information

Influence of impact velocity on transition time for V-notched Charpy specimen*

Influence of impact velocity on transition time for V-notched Charpy specimen* [ 溶接学会論文集第 35 巻第 2 号 p. 80s-84s (2017)] Influence of impact velocity on transition time for V-notched Charpy specimen* by Yasuhito Takashima** and Fumiyoshi Minami** This study investigated the influence

More information

D : SOLID MECHANICS. Q. 1 Q. 9 carry one mark each.

D : SOLID MECHANICS. Q. 1 Q. 9 carry one mark each. GTE 2016 Q. 1 Q. 9 carry one mark each. D : SOLID MECHNICS Q.1 single degree of freedom vibrating system has mass of 5 kg, stiffness of 500 N/m and damping coefficient of 100 N-s/m. To make the system

More information

Name :. Roll No. :... Invigilator s Signature :.. CS/B.TECH (CE-NEW)/SEM-3/CE-301/ SOLID MECHANICS

Name :. Roll No. :... Invigilator s Signature :.. CS/B.TECH (CE-NEW)/SEM-3/CE-301/ SOLID MECHANICS Name :. Roll No. :..... Invigilator s Signature :.. 2011 SOLID MECHANICS Time Allotted : 3 Hours Full Marks : 70 The figures in the margin indicate full marks. Candidates are required to give their answers

More information

6.37 Determine the modulus of resilience for each of the following alloys:

6.37 Determine the modulus of resilience for each of the following alloys: 6.37 Determine the modulus of resilience for each of the following alloys: Yield Strength Material MPa psi Steel alloy 550 80,000 Brass alloy 350 50,750 Aluminum alloy 50 36,50 Titanium alloy 800 116,000

More information

N = Shear stress / Shear strain

N = Shear stress / Shear strain UNIT - I 1. What is meant by factor of safety? [A/M-15] It is the ratio between ultimate stress to the working stress. Factor of safety = Ultimate stress Permissible stress 2. Define Resilience. [A/M-15]

More information

GATE SOLUTIONS E N G I N E E R I N G

GATE SOLUTIONS E N G I N E E R I N G GATE SOLUTIONS C I V I L E N G I N E E R I N G From (1987-018) Office : F-16, (Lower Basement), Katwaria Sarai, New Delhi-110016 Phone : 011-65064 Mobile : 81309090, 9711853908 E-mail: info@iesmasterpublications.com,

More information

The science of elasticity

The science of elasticity The science of elasticity In 1676 Hooke realized that 1.Every kind of solid changes shape when a mechanical force acts on it. 2.It is this change of shape which enables the solid to supply the reaction

More information

MECHANICS OF MATERIALS

MECHANICS 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 information

Volume 2 Fatigue Theory Reference Manual

Volume 2 Fatigue Theory Reference Manual Volume Fatigue Theory Reference Manual Contents 1 Introduction to fatigue 1.1 Introduction... 1-1 1. Description of the applied loading... 1-1.3 Endurance curves... 1-3 1.4 Generalising fatigue data...

More information

[8] Bending and Shear Loading of Beams

[8] Bending and Shear Loading of Beams [8] Bending and Shear Loading of Beams Page 1 of 28 [8] Bending and Shear Loading of Beams [8.1] Bending of Beams (will not be covered in class) [8.2] Bending Strain and Stress [8.3] Shear in Straight

More information