Macaulay’s Method Lecture No-2 J P Supale Mechanical Engineering Department SKN SITS LONAVALA Strength of Materials.

Slides:



Advertisements
Similar presentations
Overview of Loads ON and IN Structures / Machines
Advertisements

CHAPTER OBJECTIVES Use various methods to determine the deflection and slope at specific pts on beams and shafts: Integration method Discontinuity functions.
AERSP 301 Bending of open and closed section beams
Circular Plates Moments acting on an element of a deformed circular plate.
CHAPTER 6 BENDING.
UNIT –III Bending Moment and Shear Force in Beams
Column Design ( ) MAE 316 – Strength of Mechanical Components
Matrix Methods (Notes Only)
AERSP 301 Structural Idealization
Sean Dalton Chapter 3 Slope and deflection of beams.
Copyright © 2011 Pearson Education South Asia Pte Ltd
BFC (Mechanics of Materials) Chapter 4: Deformation of Statically Determinate Structure (Beam Deflection) Shahrul Niza Mokhatar
BEAMS SHEAR AND MOMENT.
CHAPTER #3 SHEAR FORCE & BENDING MOMENT
9 Deflection of Beams.
Mechanics of Materials – MAE 243 (Section 002) Spring 2008 Dr. Konstantinos A. Sierros.
NAZARIN B. NORDIN What you will learn:
Bending Shear and Moment Diagram, Graphical method to construct shear
Eng Ship Structures 1 Hull Girder Response Analysis
Moment Area Theorems: Theorem 1:
9 Deflection of Beams.
7.4 Cables Flexible cables and chains are used to support and transmit loads from one member to another In suspension bridges and trolley wheels, they.
Copyright © 2010 Pearson Education South Asia Pte Ltd
CIVL3310 STRUCTURAL ANALYSIS
Bending of Beams MECHENG242 Mechanics of Materials 2.3 Combined Bending and Axial Loading 2.0 Bending of Beams 2.4 Deflections in Beams 2.5 Buckling (Refer:
CHAPTER OBJECTIVES Use various methods to determine the deflection and slope at specific pts on beams and shafts: Integration method Discontinuity functions.
Analysis of Beams and Frames Theory of Structure - I.
UNIT - II Shear Force Diagrams and Bending Moment Diagrams Lecture Number -1 Prof. M. J. Naidu Mechanical Engineering Department Smt. Kashibai Navale College.
ME16A: CHAPTER FIVE DEFLECTION OF BEAMS. CHAPTER FIVE - DEFLECTION OF BEAMS P A yAyA B x v.
UNIT-01. SIMPLE STRESSES & STRAINS
Mechanics of Materials – MAE 243 (Section 002) Spring 2008 Dr. Konstantinos A. Sierros.
MAE 314 – Solid Mechanics Yun Jing
Standard Cases for Slope and Deflection (Cantilever) Lecture No-4 J P Supale Mechanical Engineering Department SKN SITS LONAVALA Strength of Materials.
A simply supported beam of span 8 m carries two concentrated loads of 32 kN and 48 kN at 3m and 6 m from left support. Calculate the deflection at the.
Strain Energy Due to Shear, Bending and Torsion Lecture No-6 J P Supale Mechanical Engineering Department SKN SITS LONAVALA Strength of Materials.
Chapter 9 Deflection of Beams.
Unit-5. Torsion in Shafts and Buckling of Axially Loaded Columns Lecture Number-3 Mr. M.A.Mohite Mechanical Engineering S.I.T., Lonavala.
☻ 1.0 Axial Forces 2.0 Bending of Beams M M
Unit-5. Torsion in Shafts and Buckling of Axially Loaded Columns Lecture Number-5 Mr. M. A.Mohite Mechanical Engineering S.I.T., Lonavala.
Tarek Hegazy, Univ. of Waterloo 2 t / m 2 t 1m 2m Xa Ya Yb Xb Yc Step 1: Stability Check No. of Equilibrium Equations: 3 No. of Extra.
Shear Force Diagram (SFD): The diagram which shows the variation of shear force along the length of the beam is called Shear Force Diagram (SFD). The diagram.
Stresses in Machine Elements Lecture Number - 03 Venkat S Mechanical Engineering SINHGAD COLLEGE OF ENGG,vadgaon Strength of Materials.
CED, UET, Taxila 1 Arches Theory of Structure - I.
Chapter 6: Bending.
60 kN 100 kN 130 kN Q.1 Determine the magnitude, sense, and direction of the resultant of the concurrent force system below
Slope Deflection Method
BME 315 – Biomechanics Chapter 4. Mechanical Properties of the Body Professor: Darryl Thelen University of Wisconsin-Madison Fall 2009.
Strain Energy Lecture No-5 J P Supale Mechanical Engineering Department SKN SITS LONAVALA Strength of Materials.
CIVL471 DESIGN OF RC STRUCTURES
Eng Ship Structures 1 Hull Girder Response Analysis
Structural analysis 2 Enrollment no
Mechanics of Materials Dr. Konstantinos A. Sierros
Relation Between BM and Slope
Solid Mechanics Course No. ME213.
shear force and bending moment diagram
3. Stresses in Machine Elements
Overview of Loads ON and IN Structures / Machines
Horizontal Shear Stress in Beam
Standard Cases for Slope and Deflection (SSB)
9 Deflection of Beams.
Structure II Course Code: ARCH 209 Dr. Aeid A. Abdulrazeg
Numerical Analysis of a Beam
Centroid 1st Moment of area 2nd Moment of area Section Modulus
CHAPTER 8 SHEAR FORCE & BENDING MOMENT
Examples.
Structural Analysis II
Beam relationships Moment sign conventions: Deflection: ν
Various Types of Beam Loading and Support
CHAPTER SIX DEFLECTION OF BEAMS.
BUCKLING OF COLUMNS. AIM To study the failure analysis of buckling of columns.
Presentation transcript:

Macaulay’s Method Lecture No-2 J P Supale Mechanical Engineering Department SKN SITS LONAVALA Strength of Materials

Step 1 – Calculate the reaction at supports. Ler R A be the reaction at support A and R B be the reaction at support B. Strength of Materials A B W1 W2W3 L1 L2L3L4 L

Step 2 – Take a section x-x at a distance X from left support, in last part of beam. Strength of Materials A B W1 W2W3 L1 L2L3L4 X X X

Step 3 – Take moment of all forces about section x-x Step 4 – Separate the moment of each force by using compartment line Strength of Materials

Step 5 – Use the differential deflection equation Step 6 – Equate the left hand side of it with step 4 equation Strength of Materials

Step 7 – Integrate the above equation Note:- 1. while integrating keep constant of integration generally in first compartment only. 2. take as a single term only and integrate it. Strength of Materials

Step 8 – Again integrate the above equation. Step 9 – Calculate the values of constants of integration C 1 and C 2, by keeping the boundary condition values. Put values of C 1 and C 2, in step 7 to get Slope Equation and in step 8 to get Deflection equation. Strength of Materials

Example-1 A Simply supported beam subjected to central point load W. Determine maximum slope and deflection. Strength of Materials W L A B L/2

Example-1 Step 1 – Calculate the reaction at supports. R A = R B =W/2 Step 2- Take a section x-x at a distance X from left support, in last part of beam. Strength of Materials W L A B L/2 X X X

Example-1 Step 3- Take moment about section X-X. And put compartment line for each force. Step 4 – Equate it with differential deflection equation and put value of R A. Step 5 – Integrate above equation. Strength of Materials

Example-1 Step 6 – Again integrate above equation Step 7 – Apply boundary conditions, we know that at A X=0 and y=0 put these values in step 6, we get Step 8 – Put C2 =0, and at B, X=L and y=0 in step 6 again, we get Strength of Materials

Example-1 Step 9 – Put values of C 1 and C 2 in step 5 and 6 to get following equations: Slope Equation Deflection Equation Strength of Materials

Example-1 Step 10 – Calculate slope at A, put X=0,in slope eq. Calculate slope at B, put X=L in slope eq. Strength of Materials

Example-1 Step 11 – Calculate deflection under load, put X = L/2 Strength of Materials

x y P B L A M xz Q xy x P P.L Example:

P To find C 1 and C 2 : Boundary conditions: x=0 x=0 Equation of the deflected shape is: v Max occurs at x=L

ab L Macaulay’s Notation y x Example: Q xy M xz P x P

Boundary conditions: x=0 x=L From (i): From (ii): Since (L-a)=b Equation of the deflected shape is:

This value of x is then substituted into the above equation of the deflected shape in order to obtain v Max. To find v Max : v Max occurs where (i.e. slope=0) Assuming y Max will be at x<a, when P v Max Note: if

Workout Numerical A cast iron beam 40 mm wide and 80 mm deep is simply supported on a span of 1.2 m. The beam carries a point load of 15 kN at the centre. Find the deflection at the centre. Take E = N/mm 2.  b=40mm, d=80mm, L=1.2m  W=15kN, E= N/mm 2 Strength of Materials