# Shear Forces and Bending Moments

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Chapter 4 Shear Forces and Bending Moments 4.1 Introduction Consider a beam subjected to transverse loads as shown in figure, the deflections occur in the plane same as the loading plane, is called the plane of bending. In this chapter we discuss shear forces and bending moments in beams related to the loads. 4.2 Types of Beams, Loads, and Reactions Type of beams a. simply supported beam (simple beam) b. cantilever beam (fixed end beam) c. beam with an overhang 1

2 Type of loads a. concentrated load (single force) b. distributed load (measured by their intensity) : uniformly distributed load (uniform load) linearly varying load c. couple Reactions consider the loaded beam in figure equation of equilibrium in horizontal direction F x = 0 H A - P 1 cos = 0 H A = P 1 cos M B = 0 - R A L + (P 1 sin ) (L - a) + P 2 (L - b) + q c 2 / 2 = 0 (P 1 sin ) (L - a) P 2 (L - b) q c 2 R A = CCCCCCC + CCCC + CC L L 2 L (P 1 sin ) a P 2 b q c 2 R B = CCCCC + CC + CC L L 2 L for the cantilever beam F x = 0 H A = 5 P 3 / 13 F y = 0 12 P 3 (q 1 + q 2 ) b R A = CC + CCCCC

3 M A = 0 12 P 3 q 1 b q 1 b M A = CC + CC (L 2b/3) + CC (L b/3) for the overhanging beam M B = 0 - R A L + P 4 (L a) + M 1 = 0 M A = 0 - P 4 a + R B L + M 1 = 0 P 4 (L a) + M 1 P 4 a - M 1 R A = CCCCCC R B = CCCC L L 4.3 Shear Forces and Bending Moments Consider a cantilever beam with a concentrated load P applied at the end A, at the cross section mn, the shear force and bending moment are found F y = 0 V = P M = 0 M = P x sign conventions (deformation sign conventions) the shear force tends to rotate the material clockwise is defined as positive the bending moment tends to compress the upper part of the beam and elongate the lower part is defined as positive 3

4 Example 4-1 a simple beam AB supports a force P and a couple M 0, find the shear V and bending moment M at (a) at x = (L/2) _ (b) at x = (L/2) + 3P R A = C - M 0 C P R B = C + M 0 C 4 L 4 L (a) at x = (L/2) _ F y = 0 R A - P - V = 0 V = R A - P = - P / 4 - M 0 / L M = 0 - R A (L/2) + P (L/4) + M = 0 M = R A (L/2) + P (L/4) = P L / 8 - M 0 / 2 (b) at x = (L/2) + [similarly as (a)] V = - P / 4 - M 0 / L M = P L / 8 + M 0 / 2 Example 4-2 a cantilever beam AB subjected to a linearly varying distributed load as shown, find the shear force V and the bending moment M q = q 0 x / L F y = 0 - V - 2 (q 0 x / L) (x) = 0 V = - q 0 x 2 / (2 L) V max = - q 0 L / 2 4

5 M = 0 M + 2 (q 0 x / L) (x) (x / 3) = 0 M = - q 0 x 3 / (6 L) M max = - q 0 L 2 / 6 Example 4-3 an overhanging beam ABC is supported to an uniform load of intensity q and a concentrated load P, calculate the shear force V and the bending moment M at D from equations of equilibrium, it is found R A = 40 kn R B = 48 kn at section D F y = x 5 - V = 0 V = - 18 kn M = 0-40 x x x 5 x M = 0 M = 69 kn-m from the free body diagram of the right-hand part, same results can be obtained 4.4 Relationships Between Loads, Shear Forces, and Bending Moments consider an element of a beam of length dx subjected to distributed loads q equilibrium of forces in vertical direction 5

6 F y = 0 V - q dx - (V + dv) = 0 or dv / dx = - q integrate between two points A and B B dv = - q dx A A B i.e. B V B - V A = - q dx A = - (area of the loading diagram between A and B) the area of loading diagram may be positive or negative moment equilibrium of the element M = 0 - M - q dx (dx/2) - (V + dv) dx + M + dm = 0 or dm / dx = V maximum (or minimum) bending-moment occurs at dm / dx = 0, i.e. at the point of shear force V = 0 integrate between two points A and B B dm = A B V dx A i.e. M B - M A B = V dx A = (area of the shear-force diagram between A and B) this equation is valid even when concentrated loads act on the beam between A and B, but it is not valid if a couple acts between A and B 6

7 concentrated loads equilibrium of force V - P - (V + V 1 ) = 0 or V 1 = - P i.e. an abrupt change in the shear force occurs at any point where a concentrated load acts equilibrium of moment - M - P (dx/2) - (V + V 1 ) dx + M + M 1 = 0 or M 1 = P (dx/2) + V dx + V 1 dx j 0 since the length dx of the element is infinitesimally small, i.e. M 1 is also infinitesimally small, thus, the bending moment does not change as we pass through the point of application of a concentrated load loads in the form of couples equilibrium of force V 1 = 0 i.e. no change in shear force at the point of application of a couple equilibrium of moment - M + M 0 - (V + V 1 ) dx + M + M 1 = 0 or M 1 = - M 0 the bending moment changes abruptly at a point of application of a couple 7

8 4.5 Shear-Force and Bending-Moment Diagrams concentrated loads consider a simply support beam AB with a concentrated load P R A = P b / L R B = P a / L for 0 < x < a V = R A = P b / L M = R A x = P b x / L note that dm / dx = P b / L = V for a < x < L V = R A - P = - P a / L M = R A x - P (x - a) = P a (L - x) / L note that dm / dx = - P a / L = V with M max = P a b / L uniform load consider a simple beam AB with a uniformly distributed load of constant intensity q 8

9 R A = R B = q L / 2 V = R A - q x = q L / 2 - q x M = R A x - q x (x/2) = q L x / 2 - q x 2 / 2 note that dm / dx = q L / 2 - q x / 2 = V M max = q L 2 / 8 at x = L / 2 several concentrated loads for 0 < x < a 1 V = R A M = R A x M 1 = R A a 1 for a 1 < x < a 2 V = R A - P 1 M = R A x - P 1 (x - a 1 ) M 2 - M 1 = (R A - P 1 )(a 2 - a 1 ) similarly for others M 2 = M max because V = 0 at that point Example 4-4 construct the shear-force and bending -moment diagrams for the simple beam AB R A = q b (b + 2c) / 2L R B = q b (b + 2a) / 2L for 0 < x < a V = R A M = R A x 9

10 for a < x < a + b V = R A - q (x - a) M = R A x - q (x - a) 2 / 2 for a + b < x < L V = - R B M = R B (L - x) maximum moment occurs where V = 0 i.e. x 1 = a + b (b + 2c) / 2L M max = q b (b + 2c) (4 a L + 2 b c + b 2 ) / 8L 2 for a = c, x 1 = L / 2 M max = q b (2L - b) / 8 for b = L, a = c = 0 (uniform loading over the entire span) M max = q L 2 / 8 Example 4-5 construct the V- and M-dia for the cantilever beam supported to P 1 and P 2 R B = P 1 + P 2 M B = P 1 L + P 2 b for 0 < x < a V = - P 1 M = - P 1 x for a < x < L V = - P 1 - P 2 M = - P 1 x - P 2 (x - a) 10

11 Example 4-6 construct the V- and M-dia for the cantilever beam supporting to a constant uniform load of intensity q R B = q L M B = q L 2 / 2 then V = - q x M = - q x 2 / 2 alternative method V max = - q L M max = - q L 2 / 2 x V - V A = V - 0 = V = - q dx = - q x 0 M - M A = M - 0 = M = x - V dx = - q x 2 / 2 0 Example 4-7 an overhanging beam is subjected to a uniform load of q = 1 kn/m on AB and a couple M 0 = 12 kn-m on midpoint of BC, construct the V- and M-dia for the beam R B = 5.25 kn R C = 1.25 kn shear force diagram V = - q x V = constant on AB on BC 11

12 bending moment diagram M B = - q b 2 / 2 = - 1 x 4 2 / 2 = - 8 kn-m the slope of M on BC is constant (1.25 kn), the bending moment just to the left of M 0 is M = x 8 = 2 kn-m the bending moment just to the right of M 0 is M = 2-12 = - 10 kn-m and the bending moment at point C is M C = x 8 = 0 as expected 12

### Stresses in Beam (Basic Topics)

Chapter 5 Stresses in Beam (Basic Topics) 5.1 Introduction Beam : loads acting transversely to the longitudinal axis the loads create shear forces and bending moments, stresses and strains due to V and

### 2. Axial Force, Shear Force, Torque and Bending Moment Diagrams

2. Axial Force, Shear Force, Torque and Bending Moment Diagrams In this section, we learn how to summarize the internal actions (shear force and bending moment) that occur throughout an axial member, shaft,

### Chapter 9 Deflections of Beams

Chapter 9 Deflections of Beams 9.1 Introduction in this chapter, we describe methods for determining the equation of the deflection curve of beams and finding deflection and slope at specific points along

### 4 Shear Forces and Bending Moments

4 Shear Forces and ending oments Shear Forces and ending oments 8 lb 16 lb roblem 4.3-1 alculate the shear force and bending moment at a cross section just to the left of the 16-lb load acting on the simple

### Strength of Materials Prof: S.K.Bhattacharya Dept of Civil Engineering, IIT, Kharagpur Lecture no 23 Bending of Beams- II

Strength of Materials Prof: S.K.Bhattacharya Dept of Civil Engineering, IIT, Kharagpur Lecture no 23 Bending of Beams- II Welcome to the second lesson of the fifth module which is on Bending of Beams part

### Mechanics of Materials. Chapter 4 Shear and Moment In Beams

Mechanics of Materials Chapter 4 Shear and Moment In Beams 4.1 Introduction The term beam refers to a slender bar that carries transverse loading; that is, the applied force are perpendicular to the bar.

### Bending Stress and Strain

Bending Stress and Strain DEFLECTIONS OF BEAMS When a beam with a straight longitudinal ais is loaded by lateral forces, the ais is deformed into a curve, called the deflection curve of the beam. We will

### Chapter 7: Forces in Beams and Cables

Chapter 7: Forces in Beams and Cables 최해진 hjchoi@cau.ac.kr Contents Introduction Internal Forces in embers Sample Problem 7.1 Various Types of Beam Loading and Support Shear and Bending oment in a Beam

### 5 Solutions /23/09 5:11 PM Page 377

5 Solutions 44918 1/23/09 5:11 PM Page 377 5 71. The rod assembly is used to support the 250-lb cylinder. Determine the components of reaction at the ball-andsocket joint, the smooth journal bearing E,

### Recitation #5. Understanding Shear Force and Bending Moment Diagrams

Recitation #5 Understanding Shear Force and Bending Moment Diagrams Shear force and bending moment are examples of interanl forces that are induced in a structure when loads are applied to that structure.

### Analysis of Stresses and Strains

Chapter 7 Analysis of Stresses and Strains 7.1 Introduction axial load = P / A torsional load in circular shaft = T / I p bending moment and shear force in beam = M y / I = V Q / I b in this chapter, we

### MECHANICS OF SOLIDS - BEAMS TUTORIAL 2 SHEAR FORCE AND BENDING MOMENTS IN BEAMS

MECHANICS OF SOLIDS - BEAMS TUTORIAL 2 SHEAR FORCE AND BENDING MOMENTS IN BEAMS This is the second tutorial on bending of beams. You should judge your progress by completing the self assessment exercises.

### Problem 1: Computation of Reactions. Problem 2: Computation of Reactions. Problem 3: Computation of Reactions

Problem 1: Computation of Reactions Problem 2: Computation of Reactions Problem 3: Computation of Reactions Problem 4: Computation of forces and moments Problem 5: Bending Moment and Shear force Problem

### ENGR-1100 Introduction to Engineering Analysis. Lecture 13

ENGR-1100 Introduction to Engineering Analysis Lecture 13 EQUILIBRIUM OF A RIGID BODY & FREE-BODY DIAGRAMS Today s Objectives: Students will be able to: a) Identify support reactions, and, b) Draw a free-body

### Module 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method. Version 2 CE IIT, Kharagpur

Module Analysis of Statically Indeterminate Structures by the Matrix Force Method esson 11 The Force Method of Analysis: Frames Instructional Objectives After reading this chapter the student will be able

### ENGINEERING SCIENCE H1 OUTCOME 1 - TUTORIAL 3 BENDING MOMENTS EDEXCEL HNC/D ENGINEERING SCIENCE LEVEL 4 H1 FORMERLY UNIT 21718P

ENGINEERING SCIENCE H1 OUTCOME 1 - TUTORIAL 3 BENDING MOMENTS EDEXCEL HNC/D ENGINEERING SCIENCE LEVEL 4 H1 FORMERLY UNIT 21718P This material is duplicated in the Mechanical Principles module H2 and those

### 8.2 Continuous Beams (Part I)

8.2 Continuous Beams (Part I) This section covers the following topics. Analysis Incorporation of Moment due to Reactions Pressure Line due to Prestressing Force Introduction Beams are made continuous

### CHAPTER 3 SHEARING FORCE AND BENDING MOMENT DIAGRAMS. Summary

CHAPTER 3 SHEARING FORCE AND BENDING MOMENT DIAGRAMS Summary At any section in a beam carrying transverse loads the shearing force is defined as the algebraic sum of the forces taken on either side of

### Frame structure Basic properties of plane frame structure Simple open frame structure Simple closed frame structure

Statics of Building Structures I., RASUS Frame structure Basic properties of plane frame structure Simple open frame structure Simple closed frame structure Department of Structural echanics Faculty of

### BEAMS: SHEAR AND MOMENT DIAGRAMS (GRAPHICAL)

LECTURE Third Edition BES: SHER ND OENT DIGRS (GRPHICL). J. Clark School of Engineering Department of Civil and Environmental Engineering 3 Chapter 5.3 by Dr. Ibrahim. ssakkaf SPRING 003 ENES 0 echanics

### Shear and Moment Diagrams. Shear and Moment Diagrams. Shear and Moment Diagrams. Shear and Moment Diagrams. Shear and Moment Diagrams

CI 3 Shear Force and Bending oment Diagrams /8 If the variation of and are written as functions of position,, and plotted, the resulting graphs are called the shear diagram and the moment diagram. Developing

### MODULE 3 ANALYSIS OF STATICALLY INDETERMINATE STRUCTURES BY THE DISPLACEMENT METHOD. Version 2 CE IIT, Kharagpur

ODULE 3 ANALYI O TATICALLY INDETERINATE TRUCTURE BY THE DIPLACEENT ETHOD LEON 19 THE OENT- DITRIBUTION ETHOD: TATICALLY INDETERINATE BEA WITH UPPORT ETTLEENT Instructional Objectives After reading this

### Statics of Structural Supports

Statics of Structural Supports TYPES OF FORCES External Forces actions of other bodies on the structure under consideration. Internal Forces forces and couples exerted on a member or portion of the structure

### Chapter 2: Load, Stress and Strain

Chapter 2: Load, Stress and Strain The careful text- books measure (Let all who build beware!) The load, the shock, the pressure Material can bear. So when the buckled girder Lets down the grinding span,

### Statics of Structural Supports

Statics of Structural Supports TYPES OF FORCES External Forces actions of other bodies on the structure under consideration. Internal Forces forces and couples exerted on a member or portion of the structure

### Structural Axial, Shear and Bending Moments

Structural Axial, Shear and Bending Moments Positive Internal Forces Acting Recall from mechanics of materials that the internal forces P (generic axial), V (shear) and M (moment) represent resultants

### 8.2 Shear and Bending-Moment Diagrams: Equation Form

8.2 Shear and ending-oment Diagrams: Equation Form 8.2 Shear and ending-oment Diagrams: Equation Form Eample 1, page 1 of 6 1. Epress the shear and bending moment as functions of, the distance from the

### Shear Force and Moment Diagrams

C h a p t e r 9 Shear Force and Moment Diagrams In this chapter, you will learn the following to World Class standards: Making a Shear Force Diagram Simple Shear Force Diagram Practice Problems More Complex

### HØGSKOLEN I GJØVIK Avdeling for teknologi, økonomi og ledelse. Løsningsforslag for kontinuasjonseksamen i Mekanikk 4/1-10

Løsningsforslag for kontinuasjonseksamen i 4/1-10 Oppgave 1 (T betyr tension, dvs. strekk, og C betyr compression, dvs. trykk.) Side 1 av 9 Leif Erik Storm Oppgave 2 Løsning (fra http://www.public.iastate.edu/~statics/examples/vmdiags/vmdiaga.html

### ENGINEERING MECHANICS STATIC

EX 16 Using the method of joints, determine the force in each member of the truss shown. State whether each member in tension or in compression. Sol Free-body diagram of the pin at B X = 0 500- BC sin

### Approximate Analysis of Statically Indeterminate Structures

Approximate Analysis of Statically Indeterminate Structures Every successful structure must be capable of reaching stable equilibrium under its applied loads, regardless of structural behavior. Exact analysis

### TUTORIAL FOR RISA EDUCATIONAL

1. INTRODUCTION TUTORIAL FOR RISA EDUCATIONAL C.M. Uang and K.M. Leet The educational version of the software RISA-2D, developed by RISA Technologies for the textbook Fundamentals of Structural Analysis,

### Module 3. Analysis of Statically Indeterminate Structures by the Displacement Method. Version 2 CE IIT, Kharagpur

odule 3 Analysis of Statically Indeterminate Structures by the Displacement ethod Lesson 21 The oment- Distribution ethod: rames with Sidesway Instructional Objectives After reading this chapter the student

### Stress and Deformation Analysis. Representing Stresses on a Stress Element. Representing Stresses on a Stress Element con t

Stress and Deformation Analysis Material in this lecture was taken from chapter 3 of Representing Stresses on a Stress Element One main goals of stress analysis is to determine the point within a load-carrying

### 8.2 Elastic Strain Energy

Section 8. 8. Elastic Strain Energy The strain energy stored in an elastic material upon deformation is calculated below for a number of different geometries and loading conditions. These expressions for

### Deflections. Question: What are Structural Deflections?

Question: What are Structural Deflections? Answer: The deformations or movements of a structure and its components, such as beams and trusses, from their original positions. It is as important for the

### Mechanics of Materials. Chapter 6 Deflection of Beams

Mechanics of Materials Chapter 6 Deflection of Beams 6.1 Introduction Because the design of beams is frequently governed by rigidity rather than strength. For example, building codes specify limits on

### Sign Conventions. Applied and Reaction Force Sign Conventions Are Static Sign Conventions

Sign Conventions Sign conventions are a standardized set of widely accepted rules that provide a consistent method of setting up, solving, and communicating solutions to engineering mechanics problems--statics,

### Common Beam Formulas (http://structsource.com/analysis/types/beam.htm)

Common Beam Formulas (http://structsource.com/analysis/types/beam.htm) Beam formulas may be used to determine the deflection, shear and bending moment in a beam based on the applied loading and boundary

### Statics and Mechanics of Materials

Statics and Mechanics of Materials Chapter 4-1 Internal force, normal and shearing Stress Outlines Internal Forces - cutting plane Result of mutual attraction (or repulsion) between molecules on both

### DESIGN OF SLABS. 3) Based on support or boundary condition: Simply supported, Cantilever slab,

DESIGN OF SLABS Dr. G. P. Chandradhara Professor of Civil Engineering S. J. College of Engineering Mysore 1. GENERAL A slab is a flat two dimensional planar structural element having thickness small compared

### EML 5526 FEA Project 1 Alexander, Dylan. Project 1 Finite Element Analysis and Design of a Plane Truss

Problem Statement: Project 1 Finite Element Analysis and Design of a Plane Truss The plane truss in Figure 1 is analyzed using finite element analysis (FEA) for three load cases: A) Axial load: 10,000

### Prof. Dr. Hamed Hadhoud. Design of Water Tanks: Part (1)

Design of Water Tanks: Part (1) 1 Types of Tanks Elevated Tanks Resting on Soil & Underground Tanks Tank Walls Walls Shallow Medium Deep L/ L/ < & L/ >0.5 /L L L L 3 Shallow Walls L 1 m L/ (for same continuity

### Chapter 8. Shear Force and Bending Moment Diagrams for Uniformly Distributed Loads.

hapter 8 Shear Force and ending Moment Diagrams for Uniformly Distributed Loads. 8.1 Introduction In Unit 4 we saw how to calculate moments for uniformly distributed loads. You might find it worthwhile

### Announcements. Moment of a Force

Announcements Test observations Units Significant figures Position vectors Moment of a Force Today s Objectives Understand and define Moment Determine moments of a force in 2-D and 3-D cases Moment of

### Unit M4.3 Statics of Beams

Unit M4.3 Statics of Beams Readings: CD 3.2-3.6 (CD 3.8 -- etension to 3-D) 16.003/004 -- Unified Engineering Department of Aeronautics and Astronautics Massachusetts Institute of Technology EARNING OBJECTIVES

### 1. a) Discuss how finite element is evolved in engineering field. (8) b) Explain the finite element idealization of structures with examples.

M.TECH. DEGREE EXAMINATION Branch: Civil Engineering Specialization Geomechanics and structures Model Question Paper - I MCEGS 106-2 FINITE ELEMENT ANALYSIS Time: 3 hours Maximum: 100 Marks Answer ALL

### Introduction, Method of Sections

Lecture #1 Introduction, Method of Sections Reading: 1:1-2 Mechanics of Materials is the study of the relationship between external, applied forces and internal effects (stress & deformation). An understanding

### P4 Stress and Strain Dr. A.B. Zavatsky MT07 Lecture 4 Stresses on Inclined Sections

4 Stress and Strain Dr. A.B. Zavatsky MT07 Lecture 4 Stresses on Inclined Sections Shear stress and shear strain. Equality of shear stresses on perpendicular planes. Hooke s law in shear. Normal and shear

### The Mathematics of Beam Deflection

The athematics of eam Deflection Scenario s a structural engineer you are part of a team working on the design of a prestigious new hotel comple in a developing city in the iddle East. It has been decided

### Bending Stress in Beams

936-73-600 Bending Stress in Beams Derive a relationship for bending stress in a beam: Basic Assumptions:. Deflections are very small with respect to the depth of the beam. Plane sections before bending

### Analysis of Statically Determinate Trusses

Analysis of Statically Determinate Trusses THEORY OF STRUCTURES Asst. Prof. Dr. Cenk Üstündağ Common Types of Trusses A truss is one of the major types of engineering structures which provides a practical

### 6.3 Trusses: Method of Sections

6.3 Trusses: Method of Sections 6.3 Trusses: Method of Sections xample 1, page 1 of 2 1. etermine the force in members,, and, and state whether the force is tension or compression. 5 m 4 kn 4 kn 5 m 1

### Module 4. Analysis of Statically Indeterminate Structures by the Direct Stiffness Method. Version 2 CE IIT, Kharagpur

Module Analysis of Statically Indeterminate Structures by the Direct Stiffness Method Version CE IIT, haragpur esson 7 The Direct Stiffness Method: Beams Version CE IIT, haragpur Instructional Objectives

### Modeling Beams on Elastic Foundations Using Plate Elements in Finite Element Method

Modeling Beams on Elastic Foundations Using Plate Elements in Finite Element Method Yun-gang Zhan School of Naval Architecture and Ocean Engineering, Jiangsu University of Science and Technology, Zhenjiang,

### Type of Force 1 Axial (tension / compression) Shear. 3 Bending 4 Torsion 5 Images 6 Symbol (+ -)

Cause: external force P Force vs. Stress Effect: internal stress f 05 Force vs. Stress Copyright G G Schierle, 2001-05 press Esc to end, for next, for previous slide 1 Type of Force 1 Axial (tension /

### Chapter 5: Indeterminate Structures Slope-Deflection Method

Chapter 5: Indeterminate Structures Slope-Deflection Method 1. Introduction Slope-deflection method is the second of the two classical methods presented in this course. This method considers the deflection

### BEAMS: DEFORMATION BY SUPERPOSITION

ETURE EMS: EFORMTION Y SUPERPOSITION Third Edition. J. lark School of Engineering epartment of ivil and Environmental Engineering 19 hapter 9.7 9.8 b r. Ibrahim. ssakkaf SPRING 00 ENES 0 Mechanics of Materials

### P4 Stress and Strain Dr. A.B. Zavatsky MT07 Lecture 3 Statically Indeterminate Structures

4 Stress and Strain Dr... Zavatsky MT07 ecture 3 Statically Indeterminate Structures Statically determinate structures. Statically indeterminate structures (equations of equilibrium, compatibility, and

### CHAP 4 FINITE ELEMENT ANALYSIS OF BEAMS AND FRAMES INTRODUCTION

CHAP FINITE EEMENT ANAYSIS OF BEAMS AND FRAMES INTRODUCTION We learned Direct Stiffness Method in Chapter imited to simple elements such as D bars we will learn Energ Method to build beam finite element

### 6 1. Draw the shear and moment diagrams for the shaft. The bearings at A and B exert only vertical reactions on the shaft.

06 Solutions 46060_Part1 5/27/10 3:51 PM Page 329 6 1. Draw the shear and moment diagrams for the shaft. The bearings at and exert only vertical reactions on the shaft. 250 mm 800 mm 24 kn 6 2. Draw the

### Design MEMO 54a Reinforcement design for RVK 41

Page of 5 CONTENTS PART BASIC ASSUMTIONS... 2 GENERAL... 2 STANDARDS... 2 QUALITIES... 3 DIMENSIONS... 3 LOADS... 3 PART 2 REINFORCEMENT... 4 EQUILIBRIUM... 4 Page 2 of 5 PART BASIC ASSUMTIONS GENERAL

### Structural Steel Design Project

Job No: Sheet 1 of 1 Rev Job Title: Eccentrically Loaded Bolt Group Worked Example 1 Checked by Date Design Example 1: Design a bolted connection between a bracket 8 mm thick and the flange of an ISHB

### Aluminium systems profile selection

Aluminium systems profile selection The purpose of this document is to summarise the way that aluminium profile selection should be made, based on the strength requirements for each application. Curtain

### FTOOL Tutorial for creation and analysis of a simple frame and result visualization

FTOOL Tutorial for creation and analysis of a simple frame and result visualization Educational Version 3.00 August 2012 http://www.tecgraf.puc-rio.br/ftool This file: http://www.tecgraf.puc-rio.br/ftp_pub/lfm/ftool300tutorialframe.pdf

### INTRODUCTION TO BEAMS

CHAPTER Structural Steel Design LRFD Method INTRODUCTION TO BEAMS Third Edition A. J. Clark School of Engineering Department of Civil and Environmental Engineering Part II Structural Steel Design and Analysis

### Stress Analysis, Strain Analysis, and Shearing of Soils

C H A P T E R 4 Stress Analysis, Strain Analysis, and Shearing of Soils Ut tensio sic vis (strains and stresses are related linearly). Robert Hooke So I think we really have to, first, make some new kind

### www.mathsbox.org.uk Displacement (x) Velocity (v) Acceleration (a) x = f(t) differentiate v = dx Acceleration Velocity (v) Displacement x

Mechanics 2 : Revision Notes 1. Kinematics and variable acceleration Displacement (x) Velocity (v) Acceleration (a) x = f(t) differentiate v = dx differentiate a = dv = d2 x dt dt dt 2 Acceleration Velocity

### Two-Force Members, Three-Force Members, Distributed Loads

Two-Force Members, Three-Force Members, Distributed Loads Two-Force Members - Examples ME 202 2 Two-Force Members Only two forces act on the body. The line of action (LOA) of forces at both A and B must

### MECHANICS OF SOLIDS - BEAMS TUTORIAL 3 THE DEFLECTION OF BEAMS

MECHANICS OF SOLIDS - BEAMS TUTORIAL THE DEECTION OF BEAMS This is the third tutorial on the bending of beams. You should judge your progress by completing the self assessment exercises. On completion

### Page 1 of 18 28.4.2008 Sven Alexander Last revised 1.3.2010. SB-Produksjon STATICAL CALCULATIONS FOR BCC 250

Page 1 of 18 CONTENT PART 1 BASIC ASSUMPTIONS PAGE 1.1 General 1. Standards 1.3 Loads 1. Qualities PART ANCHORAGE OF THE UNITS.1 Beam unit equilibrium 3. Beam unit anchorage in front..1 Check of capacity..

### Lesson 4 Rigid Body Statics. Taking into account finite size of rigid bodies

Lesson 4 Rigid Body Statics When performing static equilibrium calculations for objects, we always start by assuming the objects are rigid bodies. This assumption means that the object does not change

### superimposing the stresses and strains cause by each load acting separately

COMBINED LOADS In many structures the members are required to resist more than one kind of loading (combined loading). These can often be analyzed by superimposing the stresses and strains cause by each

### M x (a) (b) (c) Figure 2: Lateral Buckling The positions of the beam shown in Figures 2a and 2b should be considered as two possible equilibrium posit

Lateral Stability of a Slender Cantilever Beam With End Load Erik Thompson Consider the slender cantilever beam with an end load shown in Figure 1. The bending moment at any cross-section is in the x-direction.

### Finite Element Simulation of Simple Bending Problem and Code Development in C++

EUROPEAN ACADEMIC RESEARCH, VOL. I, ISSUE 6/ SEPEMBER 013 ISSN 86-48, www.euacademic.org IMPACT FACTOR: 0.485 (GIF) Finite Element Simulation of Simple Bending Problem and Code Development in C++ ABDUL

### Basic principles 41. Fig. 2.2: Force in a bar along the bar axis.

Truss Structures 2 Truss structures constitute a special class of structures in which individual straight members are connected at joints. The members are assumed to be connected to the joints in a manner

### STRESS AND DEFORMATION ANALYSIS OF LINEAR ELASTIC BARS IN TENSION

Chapter 11 STRESS AND DEFORMATION ANALYSIS OF LINEAR ELASTIC BARS IN TENSION Figure 11.1: In Chapter10, the equilibrium, kinematic and constitutive equations for a general three-dimensional solid deformable

### Chapter 5: Indeterminate Structures Force Method

Chapter 5: Indeterminate Structures Force Method 1. Introduction Statically indeterminate structures are the ones where the independent reaction components, and/or internal forces cannot be obtained by

### SOLUTION 6 6. Determine the force in each member of the truss, and state if the members are in tension or compression.

6 6. etermine the force in each member of the truss, and state if the members are in tension or compression. 600 N 4 m Method of Joints: We will begin by analyzing the equilibrium of joint, and then proceed

### When the fluid velocity is zero, called the hydrostatic condition, the pressure variation is due only to the weight of the fluid.

Fluid Statics When the fluid velocity is zero, called the hydrostatic condition, the pressure variation is due only to the weight of the fluid. Consider a small wedge of fluid at rest of size Δx, Δz, Δs

### 2) When labeling bends that will be rotated, refer to the amount of rotation as the horizontal and/or vertical deflection angle.

13. Rotation of Fittings. a. Rotation of Bends. 1) Bends can be rotated about the pipe axis to produce a simultaneous deflection or combined bend. This section covers the design of horizontal bends rotated

### Design MEMO 60 Reinforcement design for TSS 102

Date: 04.0.0 sss Page of 5 CONTENTS PART BASIC ASSUMTIONS... GENERAL... STANDARDS... QUALITIES... 3 DIMENSIONS... 3 LOADS... 3 PART REINFORCEMENT... 4 EQUILIBRIUM... 4 Date: 04.0.0 sss Page of 5 PART BASIC

### Reinforced Concrete Design to BS8110 Structural Design 1 Lesson 5

Lesson 5: Deflection in reinforced concrete beams Content 4.1 Introduction 4. Definitions 4..1 Tension 4.. Compression 4.3 Initial sizing 4.3.1 Worked example 4.4 Reinforcement details 4.5 Anchorage at

### A beam is a structural member that is subjected primarily to transverse loads and negligible

Chapter. Design of Beams Flexure and Shear.1 Section force-deformation response & Plastic Moment (M p ) A beam is a structural member that is subjected primarily to transverse loads and negligible axial

### Indeterminate Analysis Force Method 1

Indeterminate Analysis Force Method 1 The force (flexibility) method expresses the relationships between displacements and forces that exist in a structure. Primary objective of the force method is to

### Optimum proportions for the design of suspension bridge

Journal of Civil Engineering (IEB), 34 (1) (26) 1-14 Optimum proportions for the design of suspension bridge Tanvir Manzur and Alamgir Habib Department of Civil Engineering Bangladesh University of Engineering

### CHAPTER 6 SIMPLY SUPPORTED BEAMS

CHPTE 6 SIMPLY SUPPOTED BEMS EXECISE 40, Page 87 1. Determine the moment of a force of 5 N applied to a spanner at an effective length of 180 mm from the centre of a nut. Moment, M = force distance = 5

### Module 5 (Lectures 17 to 19) MAT FOUNDATIONS

Module 5 (Lectures 17 to 19) MAT FOUNDATIONS Topics 17.1 INTRODUCTION Rectangular Combined Footing: Trapezoidal Combined Footings: Cantilever Footing: Mat foundation: 17.2 COMMON TYPES OF MAT FOUNDATIONS

### MECHANICS OF MATERIALS

T dition CHTR MCHNICS OF MTRIS Ferdinand. Beer. Russell Johnston, Jr. John T. DeWolf ecture Notes: J. Walt Oler Texas Tech University Stress and Strain xial oading - Contents Stress & Strain: xial oading

### Module 6 : Design of Retaining Structures. Lecture 28 : Anchored sheet pile walls [ Section 28.1 : Introduction ]

Lecture 28 : Anchored sheet pile walls [ Section 28.1 : Introduction ] Objectives In this section you will learn the following Introduction Lecture 28 : Anchored sheet pile walls [ Section 28.1 : Introduction

### Unit 4: Science and Materials in Construction and the Built Environment. Chapter 14. Understand how Forces act on Structures

Chapter 14 Understand how Forces act on Structures 14.1 Introduction The analysis of structures considered here will be based on a number of fundamental concepts which follow from simple Newtonian mechanics;

### PLANE TRUSSES. Definitions

Definitions PLANE TRUSSES A truss is one of the major types of engineering structures which provides a practical and economical solution for many engineering constructions, especially in the design of

### Basics of Reinforced Concrete Design

Basics of Reinforced Concrete Design Presented by: Ronald Thornton, P.E. Define several terms related to reinforced concrete design Learn the basic theory behind structural analysis and reinforced concrete

### F. P. Beer et al., Meccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn 9788838668579, 2014 McGraw-Hill Education (Italy) srl

F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn 9788888579, 04 cgraw-hill Education (Italy) srl Reactions: Σ = 0: bp = 0 = Pb Σ = 0: ap = 0 = Pa From to B: 0

CHAPTER 2.0 ANSWER 1. A tank is filled with seawater to a depth of 12 ft. If the specific gravity of seawater is 1.03 and the atmospheric pressure at this location is 14.8 psi, the absolute pressure (psi)

Beam Analysis in Matlab (For simply supported & Cantilever; For Point Load & UDL) 1 Gulab Pamnani; 2 Dageshwar Singh Rajput; 3 Nikhil Tiwari; 4 Amit Gajendra 1 Assistant Professor, Department of Mechanical

### MECHANICS OF SOLIDS - BEAMS TUTORIAL TUTORIAL 4 - COMPLEMENTARY SHEAR STRESS

MECHANICS OF SOLIDS - BEAMS TUTORIAL TUTORIAL 4 - COMPLEMENTARY SHEAR STRESS This the fourth and final tutorial on bending of beams. You should judge our progress b completing the self assessment exercises.

### Mechanics of Materials. Chapter 5 Stresses In Beams

Mechanics of Materials Chapter 5 Stresses In Beams 5.1 Introduction In previous chapters, the stresses in bars caused by axial loading and torsion. Here consider the third fundamental loading : bending.