Save this PDF as:

Size: px
Start display at page:

## Transcription

1 CHAPTER OBJECTIVES qanalyze the stress developed in thin-walled pressure vessels qreview the stress analysis developed in previous chapters regarding axial load, torsion, bending and shear qdiscuss the solution of problems where several of these internal loads occur simultaneously on a member s x-section 1 CHAPTER OUTLINE 1. Thin-Walled Pressure Vessels 2. State of Stress Caused by Combined Loadings 2 1

2 8.1 THIN-WALLED PRESSURE VESSELS qcylindrical or spherical vessels are commonly used in industry to serve as boilers or tanks qwhen under pressure, the material which they are made of, is subjected to a loading from all directions qhowever, we can simplify the analysis provided it has a thin wall q Thin wall refers to a vessel having an innerradius-thickness ratio of 10 or more (r/t 10) qwhen r/t = 10, results of a thin-wall analysis will predict a stress approximately 4% less than actual maximum stress in the vessel THIN-WALLED PRESSURE VESSELS qassumption taken before analysis is that the thickness of the pressure vessel is uniform or constant throughout qthe pressure in the vessel is understood to be the gauge pressure, since it measures the pressure above atmospheric pressure, which is assumed to exist both inside and outside the vessel s wall 4 2

3 8.1 THIN-WALLED PRESSURE VESSELS Cylindrical vessels qa gauge pressure p is developed within the vessel by a contained gas or fluid, and assumed to have negligible weight qdue to uniformity of loading, an element of the vessel is subjected to normal stresses σ 1 in the circumferential or hoop direction and σ 2 in the longitudinal or axial direction THIN-WALLED PRESSURE VESSELS Cylindrical vessels qwe use the method of sections and apply the equations of force equilibrium to get the magnitudes of the stress components qfor equilibrium in the x direction, we require σ 1 = pr t Equation

4 8.1 THIN-WALLED PRESSURE VESSELS Cylindrical vessels qas shown, σ 2 acts uniformly throughout the wall, and p acts on the section of gas or fluid. Thus for equilibrium in the y direction, we require σ 2 = pr 2t Equation THIN-WALLED PRESSURE VESSELS Cylindrical vessels qfor Eqns 8-1 and 8-2, σ 1, σ 2 = normal stress in the hoop and longitudinal directions, respectively. Each is assumed to be constant throughout the wall of the cylinder, and each subjects the material to tension p = internal gauge pressure developed by the contained gas or fluid r = inner radius of the cylinder t = thickness of the wall (r/t 10) 8 4

5 8.1 THIN-WALLED PRESSURE VESSELS Cylindrical vessels qcomparing Eqns 8-1 and 8-2, note that the hoop or circumferential stress is twice as large as the longitudinal or axial stress qwhen engineers fabricate cylindrical pressure vessels from rolled-form plates, the longitudinal joints must be designed to carry twice as much stress as the circumferential joints THIN-WALLED PRESSURE VESSELS Spherical vessels qthe analysis for a spherical pressure vessel can be done in a similar manner qlike the cylinder, equilibrium in the y direction requires σ 2 = pr 2t Equation

6 8.1 THIN-WALLED PRESSURE VESSELS Spherical vessels qnote that Eqn 8-3 is similar to Eqn 8-2. Thus, this stress is the same regardless of the orientation of the hemispheric free-body diagram An element of material taken from either a cylindrical or spherical pressure vessel is subjected to biaxial stress; normal stress existing in two directions THIN-WALLED PRESSURE VESSELS Spherical vessels qthe material is also subjected to radial stress, σ 3. It has a value equal to pressure p at the interior wall and decreases to zero at exterior surface of the vessel However, we ignore the radial stress component for thin-walled vessels, since the limiting assumption of r/t = 10, results in σ 3 and σ 2 being 5 and 10 times higher than maximum radial stress (σ 3 ) max = p 12 6

7 EXAMPLE 8.1 Cylindrical pressure vessel has an inner diameter of 1.2 m and thickness of 12 mm. Determine the maximum internal pressure it can sustain so that neither its circumferential nor its longitudinal stress component exceeds 140 MPa. Under the same conditions, what is the maximum internal pressure that a similar-size spherical vessel 13 EXAMPLE 8.1 (SOLN) Cylindrical pressure vessel Maximum stress occurs in the circumferential direction. From Eqn 8-1, we have pr σ 1 = ; t p(600 mm) 140 N/mm 2 = 12 mm p = 2.8 N/mm 2 Note that when pressure is reached, from Eqn 8-2, stress in the longitudinal direction will be σ 2 = 0.5(140 MPa) = 70 MPa. 14 7

8 EXAMPLE 8.1 (SOLN) Cylindrical pressure vessel Furthermore, maximum stress in the radial direction occurs on the material at the inner wall of the vessel and is (σ 3 ) max = p = 2.8 MPa. This value is 50 times smaller than the circumferential stress (140 MPa), and as stated earlier, its effects will be neglected. 15 EXAMPLE 8.1 (SOLN) Spherical pressure vessel Here, the maximum stress occurs in any two perpendicular directions on an element of the vessel. From Eqn 8-3, we have pr σ 2 = ; 2t p(600 mm) 140 N/mm 2 = 2(12 mm) p = 5.6 N/mm 2 Although it is more difficult to fabricate, the spherical pressure vessel will carry twice as much internal pressure as a cylindrical vessel. 16 8

10 8.2 STATE OF STRESS CAUSED BY COMBINED LOADINGS q This is necessary to ensure that the stress produced by one load is not related to the stress produced by any other load STATE OF STRESS CAUSED BY COMBINED LOADINGS Procedure for analysis Internal loading q Section the member perpendicular to its axis at the pt where the stress is to be determined q Obtain the resultant internal normal and shear force components and the bending and torsional moment components q Force components should act through the centroid of the x-section, and moment components should be computed about centroidal axes, which represent the principal axes of inertia for x-section 20 10

11 8.2 STATE OF STRESS CAUSED BY COMBINED LOADINGS Procedure for analysis Average normal stress q Compute the stress component associated with each internal loading. q For each case, represent the effect either as a distribution of stress acting over the entire x- sectional area, or show the stress on an element of the material located at a specified pt on the x- section STATE OF STRESS CAUSED BY COMBINED LOADINGS Procedure for analysis Normal force q Internal normal force is developed by a uniform normal-stress distribution by σ = P/A Shear force q Internal shear force in member subjected to bending is developed from shear-stress distribution determined from the shear formula, τ = VQ/It. Special care must be exercised as highlighted in section

12 8.2 STATE OF STRESS CAUSED BY COMBINED LOADINGS Procedure for analysis Bending moment q For straight members, the internal bending moment is developed by a normal-stress distribution that varies linearly from zero at the neutral axis to a maximum at outer boundary of the member. q Stress distribution obtained from flexure formula, τ = My/I. q For curved member, stress distribution is nonlinear and determined from σ = My/[Ae(R y)] STATE OF STRESS CAUSED BY COMBINED LOADINGS Procedure for analysis Torsional moment q For circular shafts and tubes, internal torsional moment is developed by a shear-stress distribution that varies linearly from the central axis of shaft to a maximum at shaft s outer boundary q Shear-stress distribution is determined from the torsional formula,τ = Tρ/J. q If member is a closed thin-walled tube, use τ = T/2A m t 24 12

13 8.2 STATE OF STRESS CAUSED BY COMBINED LOADINGS Procedure for analysis Thin-walled pressure vessels q If vessel is a thin-walled cylinder, internal pressure p will cause a biaxial state of stress in the material such that the hoop or circumferential stress component is σ 1 = pr/t and longitudinal σ 2 = pr/2t. q If vessel is a thin-walled sphere, then biaxial state of stress is represented by two equivalent components, each having a magnitude of σ 2 = pr/2t STATE OF STRESS CAUSED BY COMBINED LOADINGS Procedure for analysis Superposition q Once normal and shear stress components for each loading have been calculated, use the principle of superposition and determine the resultant normal and shear stress components q Represent the results on an element of material located at the pt, or show the results as a distribution of stress acting over the member s x-sectional area 26 13

14 EXAMPLE 8.2 A force of 15,000 N is applied to the edge of the member shown. Neglect the weight of the member and determine the state of stress at pts B and C. 27 EXAMPLE 8.2 (SOLN) Internal loadings Member is sectioned through B and C. For equilibrium at section, there must be an axial force of 15,000 N acting through the centroid and a bending moment of 750,000 N mm about the centroidal or principal axis

15 EXAMPLE 8.2 (SOLN) Stress components 1. Normal force Uniform normal-stress distribution due to normal force is shown. σ = P/A = = 3.75 MPa 2. Bending moment Normal stress distribution due to bending moment is shown. σ max = Mc/I = = MPa 29 EXAMPLE 8.2 (SOLN) Superposition If above normal-stress distributions are added algabraically, resultant stress distribution is shown. Although not needed here, the location of the line of zero stress can be determined by proportional triangles, i.e., 7.5 MPa x = 15 MPa (100 mm x) x= 33.3 mm 30 15

16 EXAMPLE 8.2 (SOLN) Superposition Elements of material at B and C are subjected only to normal or uniaxial stress as shown. Hence σ B = 7.5 MPa σ C = 15 MPa 31 EXAMPLE 8.3 Tank shown has inner radius of 600 mm and a thickness of 12 mm. It is filled to the top with water having a specific weight of γ st = 78 kn/m 3. Determine the state of stress at pt A. Tank is open at the top

17 EXAMPLE 8.3 (SOLN) Internal loadings Free-body diagram of section of both the tank and water above pt A is shown. Notice that the weight of the water is supported by the water surface just below the section, not by the walls of the tank. In this vertical direction, the walls simply hold up the weight of the tank. W st = γ st V st = = 3.56 kn 33 EXAMPLE 8.3 (SOLN) Internal loadings The stress in the circumferential direction is developed by the water pressure at level A. To obtain this pressure, we use Pascal s law, which states that the pressure at a pt located at depth z in the water is p = γ w z. Consequently, pressure on tank at level A is p = γ w z = (10 kn/m 3 )(1 m) = 10 kn/m

18 EXAMPLE 8.3 (SOLN) Stress components 1. Circumferential stress Applying Eqn 8-1, using inner radius of r = 600 mm, we have σ 1 = pr/t = = 500 kpa 2. Longitudinal stress Since weight of tank is supported uniformly by the walls, we have σ 2 = W st /A st = = 77.9 kpa 35 EXAMPLE 8.3 (SOLN) Stress components Note that Eqn 8-2, σ 2 = pr/2t does not apply here, since tank is open at the top and therefore, as stated previously, the water cannot develop a loading on the walls in the longitudinal direction. Pt A is therefore subjected to the biaxial stress as shown

19 EXAMPLE 8.5 The solid rod shown has a radius of 0.75 cm. If it is subjected to the loading shown, determine the stress at pt A. 37 EXAMPLE 8.5 (SOLN) Internal loadings Rod is sectioned through pt A. Using free-body diagram of segment AB, the resultant internal loadings can be determined from the six equations of equilibrium

20 EXAMPLE 8.5 (SOLN) Internal loadings The normal force (500 N) and shear force (800 N) must act through the centroid of the x- section and the bending-moment components (8000 N cm) and 7000 N cm) are applied about centroidal (principal) axes. In order to better visualize the stress distributions due to each of these loadings, we will consider the equal but opposite resultants acting on AC. 39 EXAMPLE 8.5 (SOLN) Stress components 1. Normal force Normal stress distribution is shown. For pt A, we have σ A = P/A = = 2.83 MPa 40 20

21 EXAMPLE 8.5 (SOLN) Stress components 2. Shear force Shear-stress distribution is shown. For pt A, Q is determined from the shaded semicircular area. Using tables provided in textbook, we have Q = y A = = cm 3 τ A = VQ/It = = 6.04 MPa 41 EXAMPLE 8.5 (SOLN) Stress components 3. Bending moments For the 8000 N cm component, pt A lies on the neutral axis, so the normal stress is σ A = 0 For the 7000 N cm component, c = 0.75 cm, so normal stress at pt A, is σ A = Mc/I = = MPa 42 21

22 EXAMPLE 8.5 (SOLN) Stress components 4. Torsional moment At pt A, ρ A = c = 0.75 cm, thus τ A = Tc/J = = MPa 43 EXAMPLE 8.5 (SOLN) Superposition When the above results are superimposed, it is seen that an element of material at pt A is subjected to both normal and shear stress components 44 22

23 CHAPTER REVIEW A pressure vessel is considered to have a thin wall provided r/t 10. For a thin-walled cylindrical vessel, the circumferential or hoop stress is σ 1 = pr/t. This stress is twice as great as the longitudinal stress, σ 2 = pr/2t. Thinwalled spherical vessels have the same stress within their walls in all directions so that σ 1 = σ 2 = pr/2t Superposition of stress components can be used to determine the normal and shear stress at a pt in a member subjected to a combined loading. 45 CHAPTER REVIEW To solve, it is first necessary to determine the resultant axial and shear force and the resultant torsional and bending moment at the section where the pt is located. Then the stress components are determined due to each of these loadings. The normal and shearstress resultants are then determined by algebraically adding the normal and shear-stress components

### 6. Bending CHAPTER OBJECTIVES

CHAPTER OBJECTIVES Determine stress in members caused by bending Discuss how to establish shear and moment diagrams for a beam or shaft Determine largest shear and moment in a member, and specify where

### Chapter 8. Chapter Spherical Pressure Vessels

Chapter 8 Chapter 8-1-8-3 Spherical Pressure Vessels Here are examples of spherical pressure vessels. Let s determine how to analyze them. Consider Figures 8-2 and 8-3 below. 1 Figure 8-2 illustrates the

### Samantha 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

### PURE BENDING. If a simply supported beam carries two point loads of 10 kn as shown in the following figure, pure bending occurs at segment BC.

BENDING STRESS The effect of a bending moment applied to a cross-section of a beam is to induce a state of stress across that section. These stresses are known as bending stresses and they act normally

### 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

### 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

### 2012 MECHANICS OF SOLIDS

R10 SET - 1 II B.Tech II Semester, Regular Examinations, April 2012 MECHANICS OF SOLIDS (Com. to ME, AME, MM) Time: 3 hours Max. Marks: 75 Answer any FIVE Questions All Questions carry Equal Marks ~~~~~~~~~~~~~~~~~~~~~~

PURWANCHAL UNIVERSITY III SEMESTER FINAL EXAMINATION-2002 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

### Stress Analysis Lecture 4 ME 276 Spring Dr./ Ahmed Mohamed Nagib Elmekawy

Stress Analysis Lecture 4 ME 76 Spring 017-018 Dr./ Ahmed Mohamed Nagib Elmekawy Shear and Moment Diagrams Beam Sign Convention The positive directions are as follows: The internal shear force causes a

### Advanced Structural Analysis EGF Cylinders Under Pressure

Advanced Structural Analysis EGF316 4. Cylinders Under Pressure 4.1 Introduction When a cylinder is subjected to pressure, three mutually perpendicular principal stresses will be set up within the walls

### 3 Hours/100 Marks Seat No.

*17304* 17304 14115 3 Hours/100 Marks Seat No. Instructions : (1) All questions are compulsory. (2) Illustrate your answers with neat sketches wherever necessary. (3) Figures to the right indicate full

### Chapter 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.

### 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

### CHAPTER -6- BENDING Part -1-

Ishik University / Sulaimani Civil Engineering Department Mechanics of Materials CE 211 CHAPTER -6- BENDING Part -1-1 CHAPTER -6- Bending Outlines of this chapter: 6.1. Chapter Objectives 6.2. Shear and

### QUESTION 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,

### 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

### QUESTION 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

### Determine the resultant internal loadings acting on the cross section at C of the beam shown in Fig. 1 4a.

E X M P L E 1.1 Determine the resultant internal loadings acting on the cross section at of the beam shown in Fig. 1 a. 70 N/m m 6 m Fig. 1 Support Reactions. This problem can be solved in the most direct

### Chapter 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

### R13. II B. Tech I Semester Regular Examinations, Jan MECHANICS OF SOLIDS (Com. to ME, AME, AE, MTE) PART-A

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 (Part-A and Part-B)

### 9. Stress Transformation

9.7 ABSOLUTE MAXIMUM SHEAR STRESS A pt in a body subjected to a general 3-D state of stress will have a normal stress and shear-stress components acting on each of its faces. We can develop stress-transformation

### SRI CHANDRASEKHARENDRA SARASWATHI VISWA MAHAVIDHYALAYA

SRI CHANDRASEKHARENDRA SARASWATHI VISWA MAHAVIDHYALAYA (Declared as Deemed-to-be University under Section 3 of the UGC Act, 1956, Vide notification No.F.9.9/92-U-3 dated 26 th May 1993 of the Govt. of

### 7. Design of pressure vessels and Transformation of plane stress Contents

7. Design of pressure vessels and Transformation of plane stress Contents 7. Design of pressure vessels and Transformation of plane stress... 1 7.1 Introduction... 7. Design of pressure vessels... 7..1

### KINGS 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)

### CHAPTER THREE SYMMETRIC BENDING OF CIRCLE PLATES

CHAPTER THREE SYMMETRIC BENDING OF CIRCLE PLATES * Governing equations in beam and plate bending ** Solution by superposition 1.1 From Beam Bending to Plate Bending 1.2 Governing Equations For Symmetric

### Sub. Code:

Important Instructions to examiners: ) The answers should be examined by key words and not as word-to-word as given in the model answer scheme. ) The model answer and the answer written by candidate may

### 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.

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

### (Refer Slide Time: 2:43-03:02)

Strength of Materials Prof. S. K. Bhattacharyya Department of Civil Engineering Indian Institute of Technology, Kharagpur Lecture - 34 Combined Stresses I Welcome to the first lesson of the eighth module

### MECE 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

### PERIYAR 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

### 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

### Spherical 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

### 18.Define the term modulus of resilience. May/June Define Principal Stress. 20. Define Hydrostatic Pressure.

CE6306 STREGNTH OF MATERIALS Question Bank Unit-I STRESS, STRAIN, DEFORMATION OF SOLIDS PART-A 1. Define Poison s Ratio May/June 2009 2. What is thermal stress? May/June 2009 3. Estimate the load carried

### (Refer Slide Time: 01:00 01:01)

Strength of Materials Prof: S.K.Bhattacharya Department of Civil Engineering Indian institute of Technology Kharagpur Lecture no 27 Lecture Title: Stresses in Beams- II Welcome to the second lesson of

### OUTCOME 1 - TUTORIAL 3 BENDING MOMENTS. You should judge your progress by completing the self assessment exercises. CONTENTS

Unit 2: Unit code: QCF Level: 4 Credit value: 15 Engineering Science L/601/1404 OUTCOME 1 - TUTORIAL 3 BENDING MOMENTS 1. Be able to determine the behavioural characteristics of elements of static engineering

### Chapter Objectives. Copyright 2011 Pearson Education South Asia Pte Ltd

Chapter Objectives To generalize the procedure by formulating equations that can be plotted so that they describe the internal shear and moment throughout a member. To use the relations between distributed

### STRESS STRAIN AND DEFORMATION OF SOLIDS, STATES OF STRESS

1 UNIT I 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

### ME325 EXAM I (Sample)

ME35 EXAM I (Sample) NAME: NOTE: COSED BOOK, COSED NOTES. ONY A SINGE 8.5x" ORMUA SHEET IS AOWED. ADDITIONA INORMATION IS AVAIABE ON THE AST PAGE O THIS EXAM. DO YOUR WORK ON THE EXAM ONY (NO SCRATCH PAPER

### CHAPTER 6: Shearing Stresses in Beams

(130) CHAPTER 6: Shearing Stresses in Beams When a beam is in pure bending, the only stress resultants are the bending moments and the only stresses are the normal stresses acting on the cross sections.

### Strength 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

[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

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

### 7.6 Stress in symmetrical elastic beam transmitting both shear force and bending moment

7.6 Stress in symmetrical elastic beam transmitting both shear force and bending moment à It is more difficult to obtain an exact solution to this problem since the presence of the shear force means that

### IDE 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

### (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

### CHAPTER OBJECTIVES Use various methods to determine the deflection and slope at specific pts on beams and shafts: 2. Discontinuity functions

1. Deflections of Beams and Shafts CHAPTER OBJECTIVES Use various methods to determine the deflection and slope at specific pts on beams and shafts: 1. Integration method. Discontinuity functions 3. Method

### The 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

### 7.3 Design of members subjected to combined forces

7.3 Design of members subjected to combined forces 7.3.1 General In the previous chapters of Draft IS: 800 LSM version, we have stipulated the codal provisions for determining the stress distribution in

### CHAPTER 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

### 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,

### NAME: 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.

### INSTITUTE OF AERONAUTICAL ENGINEERING (Autonomous) Dundigal, Hyderabad

INSTITUTE OF AERONAUTICAL ENGINEERING (Autonomous) Dundigal, Hyderabad -00 04 CIVIL ENGINEERING QUESTION BANK Course Name : STRENGTH OF MATERIALS II Course Code : A404 Class : II B. Tech II Semester Section

### UNIT-I 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

UNIT-I 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

### Stress and Strain ( , 3.14) MAE 316 Strength of Mechanical Components NC State University Department of Mechanical & Aerospace Engineering

(3.8-3.1, 3.14) MAE 316 Strength of Mechanical Components NC State Universit Department of Mechanical & Aerospace Engineering 1 Introduction MAE 316 is a continuation of MAE 314 (solid mechanics) Review

### 2014 MECHANICS OF MATERIALS

R10 SET - 1 II. Tech I Semester Regular Examinations, March 2014 MEHNIS OF MTERILS (ivil Engineering) Time: 3 hours Max. Marks: 75 nswer any FIVE Questions ll Questions carry Equal Marks ~~~~~~~~~~~~~~~~~~~~~~~~~

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

EMA 3702 Mechanics & Materials Science (Mechanics of Materials) Chapter 3 Torsion Introduction Stress and strain in components subjected to torque T Circular Cross-section shape Material Shaft design Non-circular

### 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

### Chapter 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. El-kashif Civil Engineering Department, University

### Advanced Structural Analysis EGF Section Properties and Bending

Advanced Structural Analysis EGF316 3. Section Properties and Bending 3.1 Loads in beams When we analyse beams, we need to consider various types of loads acting on them, for example, axial forces, shear

### needed to buckle an ideal column. Analyze the buckling with bending of a column. Discuss methods used to design concentric and eccentric columns.

CHAPTER OBJECTIVES Discuss the behavior of columns. Discuss the buckling of columns. Determine the axial load needed to buckle an ideal column. Analyze the buckling with bending of a column. Discuss methods

### Stress 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

### and F NAME: ME rd Sample Final Exam PROBLEM 1 (25 points) Prob. 1 questions are all or nothing. PROBLEM 1A. (5 points)

ME 270 3 rd Sample inal Exam PROBLEM 1 (25 points) Prob. 1 questions are all or nothing. PROBLEM 1A. (5 points) IND: In your own words, please state Newton s Laws: 1 st Law = 2 nd Law = 3 rd Law = PROBLEM

### 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

### 공업역학/ 일반물리1 수강대상기계공학과 2학년

교과목명 ( 국문) 고체역학 ( 영문) Solid Mechanics 담당교수( 소속) 박주혁( 기계공학과) 학수번호/ 구분/ 학점 004510/ 전필/3(3) 1반 전화( 연구실/HP) 3408-3771/010-6214-3771 강의시간/ 강의실화목9:00-10:15/ 충무관106 선수과목 공업역학/ 일반물리1 수강대상기계공학과 2학년 jhpark@sejong.ac.kr

### Stress Analysis Lecture 3 ME 276 Spring Dr./ Ahmed Mohamed Nagib Elmekawy

Stress Analysis Lecture 3 ME 276 Spring 2017-2018 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

### CE6306 STRENGTH OF MATERIALS TWO MARK QUESTIONS WITH ANSWERS ACADEMIC YEAR

CE6306 STRENGTH OF MATERIALS TWO MARK QUESTIONS WITH ANSWERS ACADEMIC YEAR 2014-2015 UNIT - 1 STRESS, STRAIN AND DEFORMATION OF SOLIDS PART- A 1. Define tensile stress and tensile strain. The stress induced

### Mechanical 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

### Only for Reference Page 1 of 18

Only for Reference www.civilpddc2013.weebly.com Page 1 of 18 Seat No.: Enrolment No. GUJARAT TECHNOLOGICAL UNIVERSITY PDDC - SEMESTER II EXAMINATION WINTER 2013 Subject Code: X20603 Date: 26-12-2013 Subject

### Samantha Ramirez, MSE

Samantha Ramirez, MSE Centroids The centroid of an area refers to the point that defines the geometric center for the area. In cases where the area has an axis of symmetry, the centroid will lie along

### 4. BEAMS: CURVED, COMPOSITE, UNSYMMETRICAL

4. BEMS: CURVED, COMPOSITE, UNSYMMETRICL Discussions of beams in bending are usually limited to beams with at least one longitudinal plane of symmetry with the load applied in the plane of symmetry or

### Two Dimensional State of Stress and Strain: examples

Lecture 1-5: Two Dimensional State of Stress and Strain: examples Principal stress. Stresses on oblique plane: Till now we have dealt with either pure normal direct stress or pure shear stress. In many

### Torsion 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

### 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

### Strength of Materials II (Mechanics of Materials) (SI Units) Dr. Ashraf Alfeehan

Strength of Materials II (Mechanics of Materials) (SI Units) Dr. Ashraf Alfeehan 2017-2018 Mechanics of Material II Text Books Mechanics of Materials, 10th edition (SI version), by: R. C. Hibbeler, 2017

### Russell C. Hibbeler. Chapter 1: Stress

Russell C. Hibbeler Chapter 1: Stress Introduction Mechanics of materials is a study of the relationship between the external loads on a body and the intensity of the internal loads within the body. This

### Introduction to Aerospace Engineering

Introduction to Aerospace Engineering Lecture slides Challenge the future 1 Aircraft & spacecraft loads Translating loads to stresses Faculty of Aerospace Engineering 29-11-2011 Delft University of Technology

### STRESS, STRAIN AND DEFORMATION OF SOLIDS

VELAMMAL COLLEGE OF ENGINEERING AND TECHNOLOGY, MADURAI 625009 DEPARTMENT OF CIVIL ENGINEERING CE8301 STRENGTH OF MATERIALS I -------------------------------------------------------------------------------------------------------------------------------

### Structural 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

### For more Stuffs Visit Owner: N.Rajeev. R07

Code.No: 43034 R07 SET-1 JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD II.B.TECH - I SEMESTER REGULAR EXAMINATIONS NOVEMBER, 2009 FOUNDATION OF SOLID MECHANICS (AERONAUTICAL ENGINEERING) Time: 3hours

### Lecture 15 Strain and stress in beams

Spring, 2019 ME 323 Mechanics of Materials Lecture 15 Strain and stress in beams Reading assignment: 6.1 6.2 News: Instructor: Prof. Marcial Gonzalez Last modified: 1/6/19 9:42:38 PM Beam theory (@ ME

### MTE 119 STATICS FINAL HELP SESSION REVIEW PROBLEMS PAGE 1 9 NAME & ID DATE. Example Problem P.1

MTE STATICS Example Problem P. Beer & Johnston, 004 by Mc Graw-Hill Companies, Inc. The structure shown consists of a beam of rectangular cross section (4in width, 8in height. (a Draw the shear and bending

### Engineering Science OUTCOME 1 - TUTORIAL 4 COLUMNS

Unit 2: Unit code: QCF Level: Credit value: 15 Engineering Science L/601/10 OUTCOME 1 - TUTORIAL COLUMNS 1. Be able to determine the behavioural characteristics of elements of static engineering systems

### MECHANICS 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

### 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

### Mechanics of Materials MENG 270 Fall 2003 Exam 3 Time allowed: 90min. Q.1(a) Q.1 (b) Q.2 Q.3 Q.4 Total

Mechanics of Materials MENG 70 Fall 00 Eam Time allowed: 90min Name. Computer No. Q.(a) Q. (b) Q. Q. Q.4 Total Problem No. (a) [5Points] An air vessel is 500 mm average diameter and 0 mm thickness, the

### AE3610 Experiments in Fluid and Solid Mechanics TRANSIENT MEASUREMENTS OF HOOP STRESSES FOR A THIN-WALL PRESSURE VESSEL

Objective AE3610 Experiments in Fluid and Solid Mechanics TRANSIENT MEASUREMENTS OF OOP STRESSES FOR A TIN-WA PRESSURE VESSE This experiment will allow you to investigate hoop and axial stress/strain relations

### PROBLEM #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

### STRENGTH OF MATERIALS-I. Unit-1. Simple stresses and strains

STRENGTH OF MATERIALS-I Unit-1 Simple stresses and strains 1. What is the Principle of surveying 2. Define Magnetic, True & Arbitrary Meridians. 3. Mention different types of chains 4. Differentiate between

### 3. BEAMS: STRAIN, STRESS, DEFLECTIONS

3. BEAMS: STRAIN, STRESS, DEFLECTIONS The beam, or flexural member, is frequently encountered in structures and machines, and its elementary stress analysis constitutes one of the more interesting facets

### Pressure Vessels Stresses Under Combined Loads Yield Criteria for Ductile Materials and Fracture Criteria for Brittle Materials

Pressure Vessels Stresses Under Combined Loads Yield Criteria for Ductile Materials and Fracture Criteria for Brittle Materials Pressure Vessels: In the previous lectures we have discussed elements subjected

### 2. Determine the deflection at C of the beam given in fig below. Use principal of virtual work. W L/2 B A L C

CE-1259, Strength of Materials UNIT I STRESS, STRAIN DEFORMATION OF SOLIDS Part -A 1. Define strain energy density. 2. State Maxwell s reciprocal theorem. 3. Define proof resilience. 4. State Castigliano

### Members 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

### CHAPTER 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

### 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

### 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.