Mechanics of Materials_Demo Class

MECHANICS OF MATERIALS

Demo ClassBy

Bharath V G

• Many structural elements like bars, tubes, beams, columns, trusses, cylinders, spheres, shafts are used for the benefit of the mankind.

• They may be made up of timber, steel, copper, aluminum, concrete or any other materials.

• The application of the laws of mechanics to find the support reactions due to the applied forces is normally covered under the subject of ENGINEERING MECHANICS.

INTRODUCTION

• In transferring, these forces from their point of application to supports the material of the structure develops the resistive forces and it undergoes deformation. The effect of these resisting forces, on the structural elements, is treated under the subject STRENGTH OF MATERIALS “OR” MECHANICS OF SOLIDS.

• The Strength Of Materials is an interdisciplinary subject.

INTRODUCTION

• Architects and civil engineers like to see that the trusses, slabs, beams, columns, etc. of the buildings and bridges are safe.

• Aeronautical engineers need this subject for the design of the component of the aircraft.

• Mechanical engineers and the Chemical engineers must know this subject for the design of the machine components and the pressure vessels.

INTRODUCTION

• Mining engineers need it to design safe mines.

• Metallurgist must understand this subject well so that he can think for further improvement of the mechanical properties of the materials.

• Electrical, Electronics and Computer Engineers need the basic knowledge of this subject because of several mechanical components they need in their products.

INTRODUCTION

• When a member is subjected to load, it develops resisting forces; i.e. it is the force of resistance offered by the material from which the member is manufactured.

• To find the resisting force developed by a member, we will use the method of section. In this method a section plane may be passed through the member and equilibrium of any part of the member can be studied.

CONCEPT OF INTERNAL FORCES

Force/Moment can be applied in the following ways:-

• Axial ( Push / Pull )• Flexural ( Bending)• Torsion (Twisting )• Shear ( Slicing )

Axis of the memberAxial Force

Axial Force:- As it’s name suggests, it is the force which is acting along the axis of the member. In other words, it’s line of action is passing through to the axis of the member.

Push / comp.

Pull / Tens.

Axis of the member

Flexural Force:- It is the force whose line of action is perpendicular to the axis of the member.

Flexural Forces

Shear Force:- Any force which tries to shear-off the member, is termed as shear force.

Torsion:- Any moment which tries to twist the member, is termed as Torsion.

Fixed end of the member

Axis of the member

Torsion

In this subject we will derive the relationship between

FORCE, STRESS, STRAIN & DEFORMATION

To design any structure, our first aim is to find out the type, nature and magnitude of forces acting on it. Accordingly we will design the structure.

Our next aim is to ensure that the structure designed by us remain safe and serviceable.

To ensure safety, the stresses developed in the member must remain within the permissible limits specified by the standards.

To ensure Serviceability, the deformations developed in the member must remain within the permissible limits specified by the standards.

TYPES OF SUPPORTS• There are mainly three types of supports:1) Simple Support: It restrains movement of the

beam in only one direction, i.e. movement perpendicular to the base of the support. It is also known as Roller support.

Reaction

2) Hinged support: It restrains movement of the beam in two directions i.e. movement perpendicular to the base of the support and movement parallel to the base of the support.

Reactions

3) Fixed support: It restrains all the three possible movements of the beam. i.e. movement perpendicular to the base of the support and movement parallel to the base of the support and the rotation at the support.

Reactions:

There are mainly five types of beams:1) Cantilever beam: It is a beam which has one end, as fixed, and the other end as free.

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fixed endfree end

2) Simply- supported beam: It is a beam, which has it’s ends, supported freely on walls or the columns. {Out of it’s two simple supports, one support will be hinged support and the other support will be roller support, then only the beam will be determinate}

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3) Over-hang beam: When the beam is continued beyond the support and behave as a cantilever then the combined beam is known as an over-hang beam.

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L L1L2

4) Fixed Beam: A beam whose both the ends are fixed or built-in

in the walls or in the columns, then that beam is known as the fixed beam.

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5) Continuous Beam:A beam which is supported on more than two

supports that, it is called a continuous beam.

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POINT LOAD:- If a comparatively large load acts on a very small area, then that load is called a point load. It is expressed in N or kN.

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point loadW kN

UNIFORMLY DISTRIBUTED LOAD:- When the load is uniformly distributed over some length, then that load is called a uniformly distributed load. It is expressed in N/m or kN/m.

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w kN/m

Total Load = w kN/m *L m = w*L kN

UNIFORMLY VARYING LOAD:- When the load Intensity is varying uniformly over some length, then that load is called a uniformly varying load. In this case total load will be the area covered by the triangle.

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Total Load = ½ *w * L = (w*L)/2 kN

w kN/m

CONCENTRATED MOMENT ( moment acting at any point):- If, at a point, a couple forms a moment, then that is called Concentrated Moment. It is expressed in Nm or kNm.

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M

EQUILIBRIUM OF A RIGID BODY :-•A rigid body can be in equilibrium if the resultant force and moment of all forces at any point is zero.•A beam is noting but a rigid body.

•For a rigid body in equilibrium, the condition of static equilibrium are three, Viz; Fx = 0 ; Fy = 0 ; M = 0 ;

A beam is said to be statically determinate if the total no. of unknown reactions are equal to the no. of conditions of static equilibrium.

Total no. of unknown reactions will depend upon the type of beam and the type of support.

The no. of conditions of static equilibrium, for a rigid body, are 3 (three):

Fx = 0 ; Fy = 0 ; M = 0

A B

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VA VB

conditions of static equilibrium = 3,No. of unknown reactions = 3, SoThe beam is statically determinate.

A beam is said to be statically indeterminate if the total no. of unknown reactions are more than the no. of conditions of static equilibrium.

A B

HA

VA VB

HB

Conditions of static equilibrium = 3,No. of unknown reactions = 4, Sothe beam is statically indeterminate.

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HA

MA VB

VA

HA

MAVB

HB

VA

HA

MA VB

HB

MB

We will limit our study to shear force and bending moment diagrams of STATICALLY DETERMINATE BEAMS.

Commonly encountered statically determinate beams are,

a) Cantilever Beam,b) Simply Supported Beam,c) Over-hanging Beam.

These beams are usually subjected to the following types of loading;

a) Point Load,b) Uniformly Distributed Load,c) Uniformly Varying Load,d) Concentrated Moment.

The beam transfers the applied load to the supports. The effect of applied load is to create bending moment and shear force at each cross-section. In transferring the applied load to the supports, the beam develops resistance against moments and shear force at all of it’s cross-sections.

The effect of applied load is to create bending moment and shear force at each cross-section. We will determine the shear force and bending moment caused by loads at each section, for various given loading condition. Then we will plot the Variation in the shear force and bending moment across the length of the beam.

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