Unit III Theory of columns. Dr.P.Venkateswara Rao, Associate Professor, Dept. of Civil Engg., SVCE, Sriperumbudir

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Unit III Theory of columns. Dr.P.Venkateswara Rao, Associate Professor, Dept. of Civil Engg., SVCE, Sriperumbudir"

Transcription

1 Unit III Theory of columns 1

2 Unit III Theory of Columns References: Punmia B.C.,"Theory of Structures" (SMTS) Vol II, Laxmi Publishing Pvt Ltd, New Delhi Rattan.S.S., "Strength of Materials", Tata McGraw Hill Education Pvt. Ltd., New Delhi, Rajput R.K., "Strength of Materials (Mechanics of Solids)", S.Chand & company Ltd., New Delhi, Ramamrutham S., Theory of structures Dhanpat Rai & Sons, New Delhi

3 Columns Columns are compression members. There are various examples of members subjected to compressive loads. Various names of compression members as per application: Post is a general term applied to a compression member. Strut is a compression member whose lateral dimensions are small compared to it s length. A strut may be horizontal, inclined or vertical and this term is used in trusses. (Tie is a tension member in a truss) But a vertical strut, used in buildings or frames is called column. Columns, pillars and stanchions are vertical members used in building frames. 3

4 Classification of Columns 1.Short column: Short column fails by crushing (compressive yielding) of the material. 2. Long column: Long column fails by buckling or bending. ( geometric or configuration failure) 3. Intermediate column: Intermediate column fails by combined buckling and crushing. (failure due to both material crushing and geometrical instability) 4

5 Buckling When a slender member is subjected to an axial compressive load, it may fail by a condition called buckling. Buckling is a geometric instability in which - - -:original shape : Buckled shape the lateral displacement of the axial member can suddenly become very large. 5

6 Buckled R.C.C. Columns 6

7 Buckled steel columns 7

8 Buckling examples of structural members 1. Building columns that transfer loads to the ground 2. Truss members in compression 3. Machine elements 4. Submarine hulls subjected to water pressure 8

9 Equilibrium States Three types of equilibrium: 1. Stable equilibrium 2. Unstable equilibrium 3. Neutral equilibrium 9

10 Buckling Mechanism 1. Stable equilibrium: If the load P is sufficiently small, when the force F is removed, the column will go back to its original straight condition. Gravity is the restoring force Elasticity of the column is the restoring force. 10

11 Buckling Mechanism (contd ) 2. Neutral equilibrium: When the column carries critical load P cr (Increased value of the load P) and a lateral force F is applied and removed, the column will remain in the slightly deflected position. Deflection amount depends on magnitude of force. Elastic restoring force is sufficient to prevent excessive deflection. 11

12 Buckling Mechanism (contd ) 3. Unstable equilibrium: When the column carries a load which is more than critical load P cr (Increased value of the load P) and a lateral force F is applied and removed, the column will bend considerably and it grows into excessively large deflection. Even small disturbance causes unstable. Elastic restoring force is insufficient to prevent excessive deflection. 12

13 Buckling Mechanism (contd ) Conclusion: Depending on the magnitude of force P, either column remains in straight position or in slight bent position or collapse due to crack extension. 13

14 Euler s long column theory The direct stress f 0 due to direct load is very small compared to bending stress f b due to buckling in long column. Euler derived an equation, for the buckling load of long column based on bending stress (neglecting the effect of direct stress). Buckling load cannot be used in short column. 14

15 Assumptions in the Euler s theory 1. The column is initially straight. 2. The cross section is uniform throughout. 3. The ends of the column are frictionless. 4. The material is homogeneous and isotropic. 5. The self weight of the column is neglected. 6. The line of thrust coincides exactly with the axis of the column. 7. The shortening of column due to axial compression is negligible. 8. The column failure occurs due to buckling only. 15

16 Cases of long columns based on end conditions 1. Both end pinned 2. Both ends fixed 3. One end fixed and the other end pinned 4. One end fixed and the other end free 16

17 Sign conventions for bending moments Convexity towards centre line, M x is +ve Concavity towards centre line, M x is -ve 17

18 End conditions of column Three important end conditions based on support types. (i) Pinned end: End is fixed in position only. Deflection, y = 0. (ii) Fixed end: End is fixed in position and direction. Deflection y = 0 and slope, dy dx = 0. (iii) Free end: Neither fixed in position nor in direction. 18

19 Case 1. Both ends hinged Consider a column AB of length l with its both ends free to rotate around frictionless pins and carrying a critical load P. As a result of critical loading, let the column deflect into a buckled shape AX 1 B as shown in the figure. Prior to this critical load, the column is straight. Smallest force at which buckled shape is possible is known as critical force. P 19

20 Case 1. Both ends hinged (contd ) 20

21 Case 1. Both ends hinged (contd ) 21

22 Case 1. Both ends hinged (contd ) 22

23 Case 1. Both ends hinged (contd ) 23

24 2. Both ends fixed 24

25 2. Both ends fixed (contd ) 25

26 2. Both ends fixed (contd ) 26

27 2. Both ends fixed (contd ) 27

28 2. Both ends fixed (contd ) 28

29 3. One end fixed and other hinged M A = Fixed end moment at A and H= Horizontal reaction at A and B as shown in Figure. Moment at X due to critical load P, M = Py + H(l x) 29

30 3. One end fixed and other hinged (contd ) 30

31 3. One end fixed and other hinged (contd ) 31

32 3. One end fixed and other hinged (contd ) 32

33 3. One end fixed and other hinged (contd ) 33

34 3. One end fixed and other hinged (contd ) 34

35 4. One end fixed, other free 35

36 4. One end fixed, other free (contd ) 36

37 4. One end fixed, other free (contd ) 37

38 4. One end fixed, other free (contd ) 38

39 4. One end fixed, other free (contd ) 39

40 Equivalent Length of a column 40

41 Equivalent length (Effective Length) 41

42 Effective Length Physically, the effective length is the distance between points on the buckled column where the moment goes to zero, i.e., where the column is effectively pinned. Considering the deflected shape, the moment is zero where the curvature is zero (from beam theory). Zero curvature corresponds to an inflection point in the deflected shape (where the curvature changes sign). 42

43 Effective length for columns with common end conditions l e = l l l 0.5 l l 0.7l l 2l 43

44 Effective length for columns with common end conditions 44

45 Effective length for columns with common end conditions 45

46 Critical stress of a column 46

47 Critical stress of a column (contd ) 47

48 Critical stress of a column (contd ) 48

49 Limitations of Euler s formula 1. It is applicable to an ideal strut only and in practice, there is always crookedness in the column and the load applied may not be exactly co-axial. 2. It takes no account of direct stress. It means that it may give a buckling load for struts, far in excess of load which they can be withstand under direct compression. 49

50 Problems Problem. A hollow circular column of internal diameter 20 mm and external diameter 40 mm has a total length of 5m. One end of the column is fixed and the other end is hinged. Find out the crippling stress of the column if E = N/mm 2. Also findout the shortest length of this column for which Euler s formula is valid taking the yield stress equal to 250 N/mm 2. 50

51 Solution. Problems d=20 mm; D=40 mm; l = 5 mm; E= N/mm 2. Euler s crippling load for one end fixed and the Hinged, P = 2π2 EI l 2 Euler s crippling stress, p c = P c = 2π2 EI A Al 2 Area of the column, A = π(d2 d 2 ) Moment of inertia, I = π(d4 d 4 ) 4 = π( ) 4 = mm 2 64 = π( ) 64 = mm 4 51

52 Solution. Problems Euler s crippling stress, p c = P c = 2π2 EI A = N/mm 2 Yield stress= 250 N/mm 2. = 2π Al l k = 2π2 E 250 = 2π I = k = I A = = l = k = = mm Shortest length of this column= 1.4 m. 52

53 Problems Problem1. A T-section 150 mm x 120 mm x 20 mm is used as a strut of 4 m long with hinged at its both ends. Calculate the crippling load if modulus of elasticity for the material be 2.0 x 10 5 N/mm 2. 53

54 Problem figure: Problems 150 mm 20 mm 120 mm 20 mm 54

55 Problems Solution : 55

56 Solution (contd ) Problems 150 mm y =34 mm 20 mm 120 mm 20 mm 56

57 Solution (contd ) Problems 150 mm Y y =34 mm X X 20 mm 20 mm 120 mm Y 57

58 Solution (contd ) Problems 58

59 Problems Problem 2: Compare the ratio of the strength of a solid steel column to that of a hollow of the same cross-sectional area. The internal diameter of the hollow column is ¾ of the external diameter. Both the column have the same length and are pinned at both ends. 59

60 Problems Solution: 60

61 Solution (contd ): Problems 61

62 Solution (contd ): Problems 62

63 Intermediate columns: Empirical Formulae Euler s formula is valid only for long columns, i.e. for columns having L ratio greater than a certain value for a particular K material. Euler s formula though valid for L > 89 for mild steel column, k doesn t take into account the direct compressive stress. For intermediate columns, empirical formulae i.e., Rankine s formula, Gordan s formula and Johnson s formulae are appropriate. 63

64 Rankine s Formula Rankine proposed an empirical formula for very short to very long columns. 64

65 Rankine s Formula (contd ) 65

66 Rankine s Formula (contd ) 66

67 Rankine s constant for various materials 67

68 Problems Problem 3: A hollow cast iron column 4.5 m long with both ends fixed, is to carry an axial load of 250 kn under working conditions. The internal diameter is 0.8 times the outer diameter of the column. Using Rankine Gordon s formula, determine the diameters of the column adopting a factor of safety of 4. Assume f c, the compressive strength to be 550 N/mm 2 and Rankine s constant a=1/

69 d=0.8 D L e = L 2 = Problems = 2.25 m = 2250 mm Area of the column, A = π(d2 0.8D 2 ) = 0.28 D 2 mm 2 4 = mm 2 Moment of inertia, I = π(d4 0.8 D 4 ) 64 k = I A Given: f c =550 N/mm 2, a=1/1600 = D 4 mm D4 = = 0.32 D mm 0.28 D2 69

70 Problems P safe load = P = F.S 4 Rankine Load, P R = f c A 1+a l e k = N = D D = 154D D D 2 = 154D D = 154D 4 70

71 Problems D D = 0 D 2 = ± D=61.24 mm d = = mm Diameters of the hollow cylindrical column ; d=48.99 mm D= mm. 71

72 Problems Problem 4: A round steel rod of diameter 15 mm and length 2m is subjected to a gradual increasing axial compressive load. Using Euler s formula find the buckling load. Find also the maximum lateral deflection corresponding to the buckling condition. Both ends of the rod may be taken as hinged. Take E = 2.1 x 10 5 N/mm 2 and the yield stress of steel = 240 N/mm 2. 72

73 Solution: Problems (contd ) 73

74 Solution (contd ) Problems (contd ) 74

75 Solution (contd ) Problems (contd ) 75

76 Problems (contd ) Problem 5: A cast iron hollow cylindrical column 3 m in length when hinged at both ends, has a critical buckling load of P kn. When this column is fixed at both the ends, its critical load rises to P+300 kn. If the ratio of external diameter to internal diameter is 1.25 and E = 1.0x10 5 N/mm 2. Determine the external diameter of the column. 76

77 Solution: Problems (contd ) 77

78 Solution (contd ) Problems (contd ) 78

79 Problems (contd ) Problem 5: A hollow cylindrical cast iron column is 4 m long, both ends being fixed. Design the column to carry an axial load of 250 kn. Use Rankine s formula and adopt a factor of safety of 5. The internal diameter may be taken as 0.8 times the external diameter. Take f c = 550 N/mm 2 and a =

80 Problems (contd ) Solution: 80

81 Solution (contd ) Problems (contd ) 81

82 Solution (contd ) Problems (contd ) 82

83 Problems (contd ) Problem 6: A column of 9 m long has a cross section shown in figure. The column is pinned at both ends. If the column is subjected to an axial load equal in value ¼ of the Euler s critical load for the column. Determine the factor of safety on the Rankine s ultimate stress value. Take f c = 326 N/mm 2, Rankine s constant a = 1, E =200 Gpa. Properties of one RSJ area = mm 2. I xx = cm 4, I yy = cm 4, Thickness of the web=6.7 mm. 83

84 Problems (contd ) 84

85 Solution: Problems (contd ) 85

86 Solution (contd ) Problems (contd ) l e for ends pinned = l = 9000 mm 86

87 Solution (contd ) Problems (contd ) 87

88 Solution (contd ) Problems (contd ) 88

89 Problems (contd ) Problem 7: A compound stanchion is made up of two ISMC 250 placed back to back with a gap between adjacent flat surfaces. Two 360 mm x 12 mm plates are riveted to the flanges, so as to form a symmetrical box section. Detremine the amount of gap if the column is to carry the maximum load. Properties of one ISMC 250 are: Area = 3867 mm 2, Max MI (I xx1 ) = mm 4 Max MI (I yy1 ) = mm 4 Distance of the centroid from the back = 23 mm. 89

90 Problems (contd ) If the effective length of the stanchion is 8.5 m, calculate the safe maximum load, the working stress being interpolated from the following table. 90

91 Problems (contd ) 91

92 Solution: Problems (contd ) 92

93 Solution (contd ) Problems (contd ) 93

94 Solution (contd ) Problems (contd ) 94

95 Short Columns P Axial loading A compression member such as column may be subjected to an axial load which pass through the geometrical axis of the member, thus causing direct stress. The axial force will cause a direct compressive stress given by, f 0 = P A f 0 Direct stress distribution 95

96 Eccentric Loading: Short Columns Some times a compression member such as column may be subjected to an axial load which may not pass through the geometrical axis of the member, thus causing bending as well as direct stress. In all such cases position of neutral layer change or altogether vanishes. The axial force will cause a direct compressive stress given by, f 0 = P A The bending couple will cause longitudinal tensile and compressive stresses. 96

97 Eccentric Loading: Short Columns The fiber stress f b at any distance y from the N.A. is given by, f b = M Y = P e y ( tensile or compressive). I x I x The extreme fibre stress due to bending is given by, f b = M Z x = P e Z x Therefore the total stress at any section of the column is given by, f = f 0 + f b = P A ± P e y I x Hence the maximum total stresses are given by, f max = f 0 + f b = P A + M Z x and f min = f 0 f b = P A M Z x 97

98 Short Columns P Eccentric loading e f max = f 0 + f b = P A + M Z x f min = f 0 f b = P A M Z x e f 0 f b f 0 f 0 < f b f 0 = f b f 0 > f b Stress distribution f b 98

99 Eccentric loading: Short Columns If f 0 is greater than f b the stress throughout the section will be of the same sign. When f 0 = f b, f max = 2f 0 and f min = 0 If, however, f 0 is less than f b, the stress will change sign, being partly tensile and partly compressive across the section. f 0 f b f 0 f 0 < f b f 0 = f b f 0 > f b Stress distribution f b 99

100 Middle third Rule: Short Columns In masonry and concrete structures, the development of tensile stress in the section is not desirable, as they are weak in tension. This limits the eccentricity e to a certain value which will be investigated for different sections as follows. In order that the stress may not change sign from compression to tensile, we have f 0 f b P A My I P Ped ; A 2Ak 2 2k2 e ; d where K is radius of gyration of the section with regard to N.A. and d is the depth of the section. f 0 f b f 0 f 0 < f b f 0 = f b f 0 > f b Stress distribution f b 100

101 Short Columns Ex: Rectangular section: e 2k2 I = bd3 12 and A = bd k 2 = I bd3 = 12 = d2 A bd 12 Since e 2k2 d e 2 d2 12 d e d 6 d b d and hence, e max = d 6 101

102 Ex: Rectangular section: Short Columns e max = d 6 The stress will be of the same sign throughout the b section if the load line is within the middle third of the section. In the case of rectangular section, the maximum intensities of extreme stresses are given by f = P A ± Pe Z = p bd ± 6Pe bd 2 f = P bd 1 ± 6e d d 102

103 Short Columns The core of a section: If the line of action of the stress is on neither of the centre lines of the sections, the bending is unsymmetrical. x A D d However, there is certain area within which the line of action of the force P must cut the cross section if the stress is not to become tensile. The area is known as core or kernal of the section. y y P B C x b 103

104 Short Columns The core of a Rectangular section : Let the point of application of the load P have the coordinates (x,y), with reference to the axis shown in above figure in which x is positive when measured to the right of origin o and y is positive when measured upwards. The stress at any point having coordinates (x,y ) is given by, f = P bd + Pxx bd 3 + Pyy db x A o y P B x b D d y C 104

105 Short Columns The core of a Rectangular section (contd..): f = P bd + Pxx bd 3 + Pyy db = 12P 1 bd 12 + xx d 2 + yy b 2 At D, x = d and 2 y = b 2 at D, we have f = 6P bd 1 6 x d y b and therefore f will be minimum. Thus x A y o B P x b D d y C 105

106 Short Columns The core of a Rectangular section (contd..): f = 6P 1 bd 6 x d y b The value of f reaches zero when x + y = 1 or 6x + 6y = 1 or d b 6 d b x d + y b = 1, which is a straight line equation, whose intercept 6 6 on the axes are respectively d 6 and b 6. Similar limits will apply in other quadrants and the stress will be y wholly compressive throughout the section. A B P b x x D d y C 106

107 Short Columns The core of a Rectangular section (contd..): If the line of action of P fall within rhombus ghjk, the diagonals of which are of length d and b respectively. 3 3 x A D d y d/3 y b/3 B C x b This rhombus is called the core of the rectangular section. 107

108 Short Columns The core of a solid circular section: For no tension, e 2k2 d I = πd4 64, A = πd2 4 k 2 = I πd 4 A = 64 πd 2 4 e 2 d2 16 d = d2 16 d/4 e d 8 The core is the circle with the same centre and diameter d/4 108

109 Short Columns The core of a hollow circular section: For no tension, e 2k2 d I = π(d4 d 4 ) 64, A = π(d2 d 2 ) 4 k 2 = I π(d 4 d 4 ) A = 64 π(d 2 d 2 ) 4 e 2 (D2 + d 2 ) 16 D = (D2 + d 2 ) 16 e (D2 + d 2 ) 8D The core is the circle with the same centre and diameter (D2 +d 2 ) 4D 109

110 A.U. Question Paper Problems State the Euler s assumption in column theory. And derive a relation for the Euler s crippling load for a columns with both ends hinged (Nov/Dec 2014). 110

111 A.U. Question Paper Problems A short length of tube having Internal diameter and external diameter are 4 cm and 5 cm respectively, which failed in compression at a load of 250 kn. When a 1.8 m length of the same tube was tested as a strut with fixed ends, the load failure was 160 kn. Assuming that σ c in Rankin s formula is given by the first test, find the value of the constant α in the same formula. What will be the crippling load of this tube if it is used as a strut 2.8 m long with one end fixed and the other hinged? (Nov/Dec 2014) 111

112 2 marks questions and Answers 1. State the middle third rule? 2. What is known as crippling load? 112

MAHALAKSHMI ENGINEERING COLLEGE

MAHALAKSHMI ENGINEERING COLLEGE MAHALAKSHMI ENGINEERING COLLEGE TIRUCHIRAALLI - 6113. QUESTION WITH ANSWERS DEARTMENT : CIVIL SEMESTER: V SUB.CODE/ NAME: CE 5 / Strength of Materials UNIT 3 COULMNS ART - A ( marks) 1. Define columns

More information

Columns and Struts. 600 A Textbook of Machine Design

Columns and Struts. 600 A Textbook of Machine Design 600 A Textbook of Machine Design C H A P T E R 16 Columns and Struts 1. Introduction.. Failure of a Column or Strut. 3. Types of End Conditions of Columns. 4. Euler s Column Theory. 5. Assumptions in Euler

More information

ENGINEERING SCIENCE H1 OUTCOME 1 - TUTORIAL 4 COLUMNS EDEXCEL HNC/D ENGINEERING SCIENCE LEVEL 4 H1 FORMERLY UNIT 21718P

ENGINEERING SCIENCE H1 OUTCOME 1 - TUTORIAL 4 COLUMNS EDEXCEL HNC/D ENGINEERING SCIENCE LEVEL 4 H1 FORMERLY UNIT 21718P ENGINEERING SCIENCE H1 OUTCOME 1 - TUTORIAL COLUMNS EDEXCEL HNC/D ENGINEERING SCIENCE LEVEL H1 FORMERLY UNIT 21718P This material is duplicated in the Mechanical Principles module H2 and those studying

More information

Mechanical Design in Optical Engineering

Mechanical Design in Optical Engineering OPTI Buckling Buckling and Stability: As we learned in the previous lectures, structures may fail in a variety of ways, depending on the materials, load and support conditions. We had two primary concerns:

More information

INSTITUTE OF AERONAUTICAL ENGINEERING (Autonomous) Dundigal, Hyderabad

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

More information

Elastic Stability Of Columns

Elastic Stability Of Columns Elastic Stability Of Columns Introduction: Structural members which carry compressive loads may be divided into two broad categories depending on their relative lengths and cross-sectional dimensions.

More information

Comparison of Euler Theory with Experiment results

Comparison of Euler Theory with Experiment results Comparison of Euler Theory with Experiment results Limitations of Euler's Theory : In practice the ideal conditions are never [ i.e. the strut is initially straight and the end load being applied axially

More information

Only for Reference Page 1 of 18

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

More information

MECHANICS OF MATERIALS

MECHANICS OF MATERIALS CHTER MECHNICS OF MTERILS 10 Ferdinand. Beer E. Russell Johnston, Jr. Columns John T. DeWolf cture Notes: J. Walt Oler Texas Tech University 006 The McGraw-Hill Companies, Inc. ll rights reserved. Columns

More information

March 24, Chapter 4. Deflection and Stiffness. Dr. Mohammad Suliman Abuhaiba, PE

March 24, Chapter 4. Deflection and Stiffness. Dr. Mohammad Suliman Abuhaiba, PE Chapter 4 Deflection and Stiffness 1 2 Chapter Outline Spring Rates Tension, Compression, and Torsion Deflection Due to Bending Beam Deflection Methods Beam Deflections by Superposition Strain Energy Castigliano

More information

COLUMNS: BUCKLING (DIFFERENT ENDS)

COLUMNS: BUCKLING (DIFFERENT ENDS) COLUMNS: BUCKLING (DIFFERENT ENDS) Buckling of Long Straight Columns Example 4 Slide No. 1 A simple pin-connected truss is loaded and supported as shown in Fig. 1. All members of the truss are WT10 43

More information

Chapter 5 Compression Member

Chapter 5 Compression Member Chapter 5 Compression Member This chapter starts with the behaviour of columns, general discussion of buckling, and determination of the axial load needed to buckle. Followed b the assumption of Euler

More information

Chapter 6: Cross-Sectional Properties of Structural Members

Chapter 6: Cross-Sectional Properties of Structural Members Chapter 6: Cross-Sectional Properties of Structural Members Introduction Beam design requires the knowledge of the following. Material strengths (allowable stresses) Critical shear and moment values Cross

More information

DETAILED SYLLABUS FOR DISTANCE EDUCATION. Diploma. (Three Years Semester Scheme) Diploma in Architecture (DARC)

DETAILED SYLLABUS FOR DISTANCE EDUCATION. Diploma. (Three Years Semester Scheme) Diploma in Architecture (DARC) DETAILED SYLLABUS FOR DISTANCE EDUCATION Diploma (Three Years Semester Scheme) Diploma in Architecture (DARC) COURSE TITLE DURATION : Diploma in ARCHITECTURE (DARC) : 03 Years (Semester System) FOURTH

More information

Structural Mechanics Column Behaviour

Structural Mechanics Column Behaviour Structural Mechanics Column Behaviour 008/9 Dr. Colin Caprani, 1 Contents 1. Introduction... 3 1.1 Background... 3 1. Stability of Equilibrium... 4. Buckling Solutions... 6.1 Introduction... 6. Pinned-Pinned

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

Civil Engineering Design (1) Design of Reinforced Concrete Columns 2006/7

Civil Engineering Design (1) Design of Reinforced Concrete Columns 2006/7 Civil Engineering Design (1) Design of Reinforced Concrete Columns 2006/7 Dr. Colin Caprani, Chartered Engineer 1 Contents 1. Introduction... 3 1.1 Background... 3 1.2 Failure Modes... 5 1.3 Design Aspects...

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

Structural Steelwork Eurocodes Development of A Trans-national Approach

Structural Steelwork Eurocodes Development of A Trans-national Approach Structural Steelwork Eurocodes Development of A Trans-national Approach Course: Eurocode Module 7 : Worked Examples Lecture 0 : Simple braced frame Contents: 1. Simple Braced Frame 1.1 Characteristic Loads

More information

CHAPTER 4. ANALYSIS AND DESIGN OF COLUMNS

CHAPTER 4. ANALYSIS AND DESIGN OF COLUMNS 4.1. INTRODUCTION CHAPTER 4. ANALYSIS AND DESIGN OF COLUMNS A column is a vertical structural member transmitting axial compression loads with or without moments. The cross sectional dimensions of a column

More information

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. UNIT 1 STRESS STRAIN AND DEFORMATION OF SOLIDS, STATES OF STRESS 1. Define stress. When an external force acts on a body, it undergoes deformation. At the same time the body resists deformation. The magnitude

More information

ENCE 455 Design of Steel Structures. III. Compression Members

ENCE 455 Design of Steel Structures. III. Compression Members ENCE 455 Design of Steel Structures III. Compression Members C. C. Fu, Ph.D., P.E. Civil and Environmental Engineering Department University of Maryland Compression Members Following subjects are covered:

More information

3. BEAMS: STRAIN, STRESS, DEFLECTIONS

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

More information

Question 1. Ignore bottom surface. Solution: Design variables: X = (R, H) Objective function: maximize volume, πr 2 H OR Minimize, f(x) = πr 2 H

Question 1. Ignore bottom surface. Solution: Design variables: X = (R, H) Objective function: maximize volume, πr 2 H OR Minimize, f(x) = πr 2 H Question 1 (Problem 2.3 of rora s Introduction to Optimum Design): Design a beer mug, shown in fig, to hold as much beer as possible. The height and radius of the mug should be not more than 20 cm. The

More information

12 Bending stresses and direct stresses corn bined

12 Bending stresses and direct stresses corn bined 12 Bending stresses and direct stresses corn bined 12.1 Introduction Many instances arise in practice where a member undergoes bending combined with a thrust or pull. If a member carries a thrust, direct

More information

Compression Members. ENCE 455 Design of Steel Structures. III. Compression Members. Introduction. Compression Members (cont.)

Compression Members. ENCE 455 Design of Steel Structures. III. Compression Members. Introduction. Compression Members (cont.) ENCE 455 Design of Steel Structures III. Compression Members C. C. Fu, Ph.D., P.E. Civil and Environmental Engineering Department University of Maryland Compression Members Following subjects are covered:

More information

Symmetric Bending of Beams

Symmetric Bending of Beams Symmetric Bending of Beams beam is any long structural member on which loads act perpendicular to the longitudinal axis. Learning objectives Understand the theory, its limitations and its applications

More information

Properties of Sections

Properties of Sections ARCH 314 Structures I Test Primer Questions Dr.-Ing. Peter von Buelow Properties of Sections 1. Select all that apply to the characteristics of the Center of Gravity: A) 1. The point about which the body

More information

STRESS. Bar. ! Stress. ! Average Normal Stress in an Axially Loaded. ! Average Shear Stress. ! Allowable Stress. ! Design of Simple Connections

STRESS. Bar. ! Stress. ! Average Normal Stress in an Axially Loaded. ! Average Shear Stress. ! Allowable Stress. ! Design of Simple Connections STRESS! Stress Evisdom! verage Normal Stress in an xially Loaded ar! verage Shear Stress! llowable Stress! Design of Simple onnections 1 Equilibrium of a Deformable ody ody Force w F R x w(s). D s y Support

More information

BOOK OF COURSE WORKS ON STRENGTH OF MATERIALS FOR THE 2 ND YEAR STUDENTS OF THE UACEG

BOOK OF COURSE WORKS ON STRENGTH OF MATERIALS FOR THE 2 ND YEAR STUDENTS OF THE UACEG BOOK OF COURSE WORKS ON STRENGTH OF MATERIALS FOR THE ND YEAR STUDENTS OF THE UACEG Assoc.Prof. Dr. Svetlana Lilkova-Markova, Chief. Assist. Prof. Dimitar Lolov Sofia, 011 STRENGTH OF MATERIALS GENERAL

More information

VTU EDUSAT PROGRAMME Lecture Notes on Design of Columns

VTU EDUSAT PROGRAMME Lecture Notes on Design of Columns VTU EDUSAT PROGRAMME 17 2012 Lecture Notes on Design of Columns DESIGN OF RCC STRUCTURAL ELEMENTS - 10CV52 (PART B, UNIT 6) Dr. M. C. Nataraja Professor, Civil Engineering Department, Sri Jayachamarajendra

More information

PLATE GIRDERS II. Load. Web plate Welds A Longitudinal elevation. Fig. 1 A typical Plate Girder

PLATE GIRDERS II. Load. Web plate Welds A Longitudinal elevation. Fig. 1 A typical Plate Girder 16 PLATE GIRDERS II 1.0 INTRODUCTION This chapter describes the current practice for the design of plate girders adopting meaningful simplifications of the equations derived in the chapter on Plate Girders

More information

Chapter 5 Elastic Strain, Deflection, and Stability 1. Elastic Stress-Strain Relationship

Chapter 5 Elastic Strain, Deflection, and Stability 1. Elastic Stress-Strain Relationship Chapter 5 Elastic Strain, Deflection, and Stability Elastic Stress-Strain Relationship A stress in the x-direction causes a strain in the x-direction by σ x also causes a strain in the y-direction & z-direction

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

7.4 The Elementary Beam Theory

7.4 The Elementary Beam Theory 7.4 The Elementary Beam Theory In this section, problems involving long and slender beams are addressed. s with pressure vessels, the geometry of the beam, and the specific type of loading which will be

More information

14. *14.8 CASTIGLIANO S THEOREM

14. *14.8 CASTIGLIANO S THEOREM *14.8 CASTIGLIANO S THEOREM Consider a body of arbitrary shape subjected to a series of n forces P 1, P 2, P n. Since external work done by forces is equal to internal strain energy stored in body, by

More information

Mechanics of Solids notes

Mechanics of Solids notes Mechanics of Solids notes 1 UNIT II Pure Bending Loading restrictions: As we are aware of the fact internal reactions developed on any cross-section of a beam may consists of a resultant normal force,

More information

7.5 Elastic Buckling Columns and Buckling

7.5 Elastic Buckling Columns and Buckling 7.5 Elastic Buckling The initial theory of the buckling of columns was worked out by Euler in 1757, a nice example of a theory preceding the application, the application mainly being for the later invented

More information

BE Semester- I ( ) Question Bank (MECHANICS OF SOLIDS)

BE Semester- I ( ) Question Bank (MECHANICS OF SOLIDS) BE Semester- I ( ) Question Bank (MECHANICS OF SOLIDS) All questions carry equal marks(10 marks) Q.1 (a) Write the SI units of following quantities and also mention whether it is scalar or vector: (i)

More information

FIXED BEAMS IN BENDING

FIXED BEAMS IN BENDING FIXED BEAMS IN BENDING INTRODUCTION Fixed or built-in beams are commonly used in building construction because they possess high rigidity in comparison to simply supported beams. When a simply supported

More information

Mechanics of Solids. Mechanics Of Solids. Suraj kr. Ray Department of Civil Engineering

Mechanics of Solids. Mechanics Of Solids. Suraj kr. Ray Department of Civil Engineering Mechanics Of Solids Suraj kr. Ray (surajjj2445@gmail.com) Department of Civil Engineering 1 Mechanics of Solids is a branch of applied mechanics that deals with the behaviour of solid bodies subjected

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

MECHANICS LAB AM 317 EXP 5 COLUMN BEHAVIOR BUCKLING

MECHANICS LAB AM 317 EXP 5 COLUMN BEHAVIOR BUCKLING MECHANICS LAB AM 317 EX 5 COLUMN BEHAVIOR BUCKLING I. OBJECTIVES I.1 To determine the effect the slenderness ratio has on the load carrying capacity of columns of varying lengths. I. To observe short,

More information

D e s i g n o f R i v e t e d J o i n t s, C o t t e r & K n u c k l e J o i n t s

D e s i g n o f R i v e t e d J o i n t s, C o t t e r & K n u c k l e J o i n t s D e s i g n o f R i v e t e d J o i n t s, C o t t e r & K n u c k l e J o i n t s 1. Design of various types of riveted joints under different static loading conditions, eccentrically loaded riveted joints.

More information

Module 4 : Deflection of Structures Lecture 4 : Strain Energy Method

Module 4 : Deflection of Structures Lecture 4 : Strain Energy Method Module 4 : Deflection of Structures Lecture 4 : Strain Energy Method Objectives In this course you will learn the following Deflection by strain energy method. Evaluation of strain energy in member under

More information

2014 MECHANICS OF MATERIALS

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 ~~~~~~~~~~~~~~~~~~~~~~~~~

More information

Influence of residual stresses in the structural behavior of. tubular columns and arches. Nuno Rocha Cima Gomes

Influence of residual stresses in the structural behavior of. tubular columns and arches. Nuno Rocha Cima Gomes October 2014 Influence of residual stresses in the structural behavior of Abstract tubular columns and arches Nuno Rocha Cima Gomes Instituto Superior Técnico, Universidade de Lisboa, Portugal Contact:

More information

CHAPTER 3 THE EFFECTS OF FORCES ON MATERIALS

CHAPTER 3 THE EFFECTS OF FORCES ON MATERIALS CHAPTER THE EFFECTS OF FORCES ON MATERIALS EXERCISE 1, Page 50 1. A rectangular bar having a cross-sectional area of 80 mm has a tensile force of 0 kn applied to it. Determine the stress in the bar. Stress

More information

Fundamentals of Structural Design Part of Steel Structures

Fundamentals of Structural Design Part of Steel Structures Fundamentals of Structural Design Part of Steel Structures Civil Engineering for Bachelors 133FSTD Teacher: Zdeněk Sokol Office number: B619 1 Syllabus of lectures 1. Introduction, history of steel structures,

More information

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

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

More information

Automatic Scheme for Inelastic Column Buckling

Automatic Scheme for Inelastic Column Buckling Proceedings of the World Congress on Civil, Structural, and Environmental Engineering (CSEE 16) Prague, Czech Republic March 30 31, 2016 Paper No. ICSENM 122 DOI: 10.11159/icsenm16.122 Automatic Scheme

More information

Flexural-Torsional Buckling of General Cold-Formed Steel Columns with Unequal Unbraced Lengths

Flexural-Torsional Buckling of General Cold-Formed Steel Columns with Unequal Unbraced Lengths Proceedings of the Annual Stability Conference Structural Stability Research Council San Antonio, Texas, March 21-24, 2017 Flexural-Torsional Buckling of General Cold-Formed Steel Columns with Unequal

More information

LECTURE 14 Strength of a Bar in Transverse Bending. 1 Introduction. As we have seen, only normal stresses occur at cross sections of a rod in pure

LECTURE 14 Strength of a Bar in Transverse Bending. 1 Introduction. As we have seen, only normal stresses occur at cross sections of a rod in pure V. DEMENKO MECHNCS OF MTERLS 015 1 LECTURE 14 Strength of a Bar in Transverse Bending 1 ntroduction s we have seen, onl normal stresses occur at cross sections of a rod in pure bending. The corresponding

More information

A. Objective of the Course: Objectives of introducing this subject at second year level in civil branches are: 1. Introduction 02

A. Objective of the Course: Objectives of introducing this subject at second year level in civil branches are: 1. Introduction 02 Subject Code: 0CL030 Subject Name: Mechanics of Solids B.Tech. II Year (Sem-3) Mechanical & Automobile Engineering Teaching Credits Examination Marks Scheme Theory Marks Practical Marks Total L 4 T 0 P

More information

MECH 401 Mechanical Design Applications

MECH 401 Mechanical Design Applications MECH 401 Mechanical Design Applications Dr. M. O Malley Master Notes Spring 008 Dr. D. M. McStravick Rice University Updates HW 1 due Thursday (1-17-08) Last time Introduction Units Reliability engineering

More information

Materials: engineering, science, processing and design, 2nd edition Copyright (c)2010 Michael Ashby, Hugh Shercliff, David Cebon.

Materials: engineering, science, processing and design, 2nd edition Copyright (c)2010 Michael Ashby, Hugh Shercliff, David Cebon. Modes of Loading (1) tension (a) (2) compression (b) (3) bending (c) (4) torsion (d) and combinations of them (e) Figure 4.2 1 Standard Solution to Elastic Problems Three common modes of loading: (a) tie

More information

6. Bending CHAPTER OBJECTIVES

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

More information

Beam Bending Stresses and Shear Stress

Beam Bending Stresses and Shear Stress Beam Bending Stresses and Shear Stress Notation: A = name or area Aweb = area o the web o a wide lange section b = width o a rectangle = total width o material at a horizontal section c = largest distance

More information

Structures. Shainal Sutaria

Structures. Shainal Sutaria Structures ST Shainal Sutaria Student Number: 1059965 Wednesday, 14 th Jan, 011 Abstract An experiment to find the characteristics of flow under a sluice gate with a hydraulic jump, also known as a standing

More information

CHAPTER 5 Statically Determinate Plane Trusses

CHAPTER 5 Statically Determinate Plane Trusses CHAPTER 5 Statically Determinate Plane Trusses TYPES OF ROOF TRUSS TYPES OF ROOF TRUSS ROOF TRUSS SETUP ROOF TRUSS SETUP OBJECTIVES To determine the STABILITY and DETERMINACY of plane trusses To analyse

More information

CHAPTER 5 Statically Determinate Plane Trusses TYPES OF ROOF TRUSS

CHAPTER 5 Statically Determinate Plane Trusses TYPES OF ROOF TRUSS CHAPTER 5 Statically Determinate Plane Trusses TYPES OF ROOF TRUSS 1 TYPES OF ROOF TRUSS ROOF TRUSS SETUP 2 ROOF TRUSS SETUP OBJECTIVES To determine the STABILITY and DETERMINACY of plane trusses To analyse

More information

APPENDIX 1 MODEL CALCULATION OF VARIOUS CODES

APPENDIX 1 MODEL CALCULATION OF VARIOUS CODES 163 APPENDIX 1 MODEL CALCULATION OF VARIOUS CODES A1.1 DESIGN AS PER NORTH AMERICAN SPECIFICATION OF COLD FORMED STEEL (AISI S100: 2007) 1. Based on Initiation of Yielding: Effective yield moment, M n

More information

MECHANICS OF SOLIDS Credit Hours: 6

MECHANICS OF SOLIDS Credit Hours: 6 MECHANICS OF SOLIDS Credit Hours: 6 Teaching Scheme Theory Tutorials Practical Total Credit Hours/week 4 0 6 6 Marks 00 0 50 50 6 A. Objective of the Course: Objectives of introducing this subject at second

More information

7 STATICALLY DETERMINATE PLANE TRUSSES

7 STATICALLY DETERMINATE PLANE TRUSSES 7 STATICALLY DETERMINATE PLANE TRUSSES OBJECTIVES: This chapter starts with the definition of a truss and briefly explains various types of plane truss. The determinancy and stability of a truss also will

More information

CHAPTER -6- BENDING Part -1-

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

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. Analysis of Beams for Bending

MECHANICS OF MATERIALS. Analysis of Beams for Bending MECHANICS OF MATERIALS Analysis of Beams for Bending By NUR FARHAYU ARIFFIN Faculty of Civil Engineering & Earth Resources Chapter Description Expected Outcomes Define the elastic deformation of an axially

More information

For more Stuffs Visit Owner: N.Rajeev. R07

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

More information

TUTORIAL SHEET 1. magnitude of P and the values of ø and θ. Ans: ø =74 0 and θ= 53 0

TUTORIAL SHEET 1. magnitude of P and the values of ø and θ. Ans: ø =74 0 and θ= 53 0 TUTORIAL SHEET 1 1. The rectangular platform is hinged at A and B and supported by a cable which passes over a frictionless hook at E. Knowing that the tension in the cable is 1349N, determine the moment

More information

External Work. When a force F undergoes a displacement dx in the same direction i as the force, the work done is

External Work. When a force F undergoes a displacement dx in the same direction i as the force, the work done is Structure Analysis I Chapter 9 Deflection Energy Method External Work Energy Method When a force F undergoes a displacement dx in the same direction i as the force, the work done is du e = F dx If the

More information

two structural analysis (statics & mechanics) APPLIED ACHITECTURAL STRUCTURES: DR. ANNE NICHOLS SPRING 2017 lecture STRUCTURAL ANALYSIS AND SYSTEMS

two structural analysis (statics & mechanics) APPLIED ACHITECTURAL STRUCTURES: DR. ANNE NICHOLS SPRING 2017 lecture STRUCTURAL ANALYSIS AND SYSTEMS APPLIED ACHITECTURAL STRUCTURES: STRUCTURAL ANALYSIS AND SYSTEMS DR. ANNE NICHOLS SPRING 2017 lecture two structural analysis (statics & mechanics) Analysis 1 Structural Requirements strength serviceability

More information

Materials and Shape. Part 1: Materials for efficient structure. A. K. M. B. Rashid Professor, Department of MME BUET, Dhaka. Learning Objectives

Materials and Shape. Part 1: Materials for efficient structure. A. K. M. B. Rashid Professor, Department of MME BUET, Dhaka. Learning Objectives MME445: Lecture 27 Materials and Shape Part 1: Materials for efficient structure A. K. M. B. Rashid Professor, Department of MME BUET, Dhaka Learning Objectives Knowledge & Understanding Understand the

More information

MECHANICS OF MATERIALS. Prepared by Engr. John Paul Timola

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

More information

Strength Of Materials/Mechanics of Solids

Strength Of Materials/Mechanics of Solids Table of Contents Stress, Strain, and Energy 1. Stress and Strain 2. Change in length 3. Determinate Structure - Both ends free 4. Indeterminate Structure - Both ends fixed 5. Composite Material of equal

More information

CH. 4 BEAMS & COLUMNS

CH. 4 BEAMS & COLUMNS CH. 4 BEAMS & COLUMNS BEAMS Beams Basic theory of bending: internal resisting moment at any point in a beam must equal the bending moments produced by the external loads on the beam Rx = Cc + Tt - If the

More information

SERVICEABILITY OF BEAMS AND ONE-WAY SLABS

SERVICEABILITY OF BEAMS AND ONE-WAY SLABS CHAPTER REINFORCED CONCRETE Reinforced Concrete Design A Fundamental Approach - Fifth Edition Fifth Edition SERVICEABILITY OF BEAMS AND ONE-WAY SLABS A. J. Clark School of Engineering Department of Civil

More information

Longitudinal strength standard

Longitudinal strength standard (1989) (Rev. 1 199) (Rev. Nov. 001) Longitudinal strength standard.1 Application This requirement applies only to steel ships of length 90 m and greater in unrestricted service. For ships having one or

More information

Failure in Flexure. Introduction to Steel Design, Tensile Steel Members Modes of Failure & Effective Areas

Failure in Flexure. Introduction to Steel Design, Tensile Steel Members Modes of Failure & Effective Areas Introduction to Steel Design, Tensile Steel Members Modes of Failure & Effective Areas MORGAN STATE UNIVERSITY SCHOOL OF ARCHITECTURE AND PLANNING LECTURE VIII Dr. Jason E. Charalambides Failure in Flexure!

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

MECHANICS OF SOLIDS COMPRESSION MEMBERS TUTORIAL 2 INTERMEDIATE AND SHORT COMPRESSION MEMBERS

MECHANICS OF SOLIDS COMPRESSION MEMBERS TUTORIAL 2 INTERMEDIATE AND SHORT COMPRESSION MEMBERS MECHANICS O SOIDS COMPRESSION MEMBERS TUTORIA INTERMEDIATE AND SHORT COMPRESSION MEMBERS Yo shold jdge yor progress by completing the self assessment exercises. On completion of this ttorial yo shold be

More information

1-1 Locate the centroid of the plane area shown. 1-2 Determine the location of centroid of the composite area shown.

1-1 Locate the centroid of the plane area shown. 1-2 Determine the location of centroid of the composite area shown. Chapter 1 Review of Mechanics of Materials 1-1 Locate the centroid of the plane area shown 650 mm 1000 mm 650 x 1- Determine the location of centroid of the composite area shown. 00 150 mm radius 00 mm

More information

DESIGN OF STORAGE AND MATERIAL HANDLING SYSTEM FOR PIPE INDUSTRY A CASE STUDY

DESIGN OF STORAGE AND MATERIAL HANDLING SYSTEM FOR PIPE INDUSTRY A CASE STUDY DESIGN OF STORAGE AND MATERIAL HANDLING SYSTEM FOR PIPE INDUSTRY A CASE STUDY Mr.S.S.Chougule 1, Mr.A.S.Chavan 2, Miss.S.M.Bhujbal 3, Mr.K.S.Falke 4 Prof.R.R.Joshi 5, Prof.M.B.Tandle 6 1,2,3,4 Student,

More information

Steel Structures Design and Drawing Lecture Notes

Steel Structures Design and Drawing Lecture Notes Steel Structures Design and Drawing Lecture Notes INTRODUCTION When the need for a new structure arises, an individual or agency has to arrange the funds required for its construction. The individual or

More information

4. SHAFTS. A shaft is an element used to transmit power and torque, and it can support

4. SHAFTS. A shaft is an element used to transmit power and torque, and it can support 4. SHAFTS A shaft is an element used to transmit power and torque, and it can support reverse bending (fatigue). Most shafts have circular cross sections, either solid or tubular. The difference between

More information

Chapter 9: Column Analysis and Design

Chapter 9: Column Analysis and Design Chapter 9: Column Analysis and Design Introduction Columns are usually considered as vertical structural elements, but they can be positioned in any orientation (e.g. diagonal and horizontal compression

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

structural analysis Excessive beam deflection can be seen as a mode of failure.

structural analysis Excessive beam deflection can be seen as a mode of failure. Structure Analysis I Chapter 8 Deflections Introduction Calculation of deflections is an important part of structural analysis Excessive beam deflection can be seen as a mode of failure. Extensive glass

More information

MECHANICAL PROPERTIES OF SOLIDS

MECHANICAL PROPERTIES OF SOLIDS Chapter Nine MECHANICAL PROPERTIES OF SOLIDS MCQ I 9.1 Modulus of rigidity of ideal liquids is (a) infinity. (b) zero. (c) unity. (d) some finite small non-zero constant value. 9. The maximum load a wire

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

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

Flexure: Behavior and Nominal Strength of Beam Sections

Flexure: Behavior and Nominal Strength of Beam Sections 4 5000 4000 (increased d ) (increased f (increased A s or f y ) c or b) Flexure: Behavior and Nominal Strength of Beam Sections Moment (kip-in.) 3000 2000 1000 0 0 (basic) (A s 0.5A s ) 0.0005 0.001 0.0015

More information

Degree Thesis Flexural Rigidity (D) in Beams. Author: Zious Karis. Instructor: Rene Herrmann

Degree Thesis Flexural Rigidity (D) in Beams. Author: Zious Karis. Instructor: Rene Herrmann Degree Thesis Flexural Rigidity (D) in Beams Author: Zious Karis Instructor: Rene Herrmann Degree Thesis Materials Processing Technology 2017 DEGREE THESIS Arcada University of Applied Sciences, Helsinki,

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

Basic Energy Principles in Stiffness Analysis

Basic Energy Principles in Stiffness Analysis Basic Energy Principles in Stiffness Analysis Stress-Strain Relations The application of any theory requires knowledge of the physical properties of the material(s) comprising the structure. We are limiting

More information

CHAPTER 2 STRUTS. Summary. The allowable stresses and end loads given by Euler s theory for struts with varying end conditions are given in Table 2.1.

CHAPTER 2 STRUTS. Summary. The allowable stresses and end loads given by Euler s theory for struts with varying end conditions are given in Table 2.1. ~ ~ ~ ~ CHAPTER 2 STRUTS Summary The allowable stresses and end loads given by Euler s theory for struts with varying end conditions are given in Table 2.1. End condition Euler load P, I Table 2.1. Fixed-free

More information

Mechanics of Materials

Mechanics of Materials Mechanics of Materials 2. Introduction Dr. Rami Zakaria References: 1. Engineering Mechanics: Statics, R.C. Hibbeler, 12 th ed, Pearson 2. Mechanics of Materials: R.C. Hibbeler, 9 th ed, Pearson 3. Mechanics

More 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

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

LIMITATIONS OF THE STANDARD INTERACTION FORMULA FOR BIAXIAL BENDING AS APPLIED TO RECTANGULAR STEEL TUBULAR COLUMNS

LIMITATIONS OF THE STANDARD INTERACTION FORMULA FOR BIAXIAL BENDING AS APPLIED TO RECTANGULAR STEEL TUBULAR COLUMNS LIMITATIONS OF THE STANDARD INTERACTION FORMULA FOR BIAXIAL BENDING AS APPLIED TO RECTANGULAR STEEL TUBULAR COLUMNS Ramon V. Jarquio 1 ABSTRACT The limitations of the standard interaction formula for biaxial

More information

MECH 401 Mechanical Design Applications

MECH 401 Mechanical Design Applications MECH 40 Mechanical Design Applications Dr. M. K. O Malley Master Notes Spring 008 Dr. D. M. McStravick Rice University Design Considerations Stress Deflection Strain Stiffness Stability Often the controlling

More information