We have discussed a lot about materials. There are different types of materials such as

Metals Ceramics Polymers Composites ……………………………….!

Machine

Mechanic

Mechanics

Mechanical Engineering

Mechanism

In very simple words

Mechanics is the science of force and motion and hence science of machinery.

If further elaborate

Mechanics is the physical science that is concerned with the condition of rest or motion of bodies acted on by forces or by thermal disturbances.

It can further be explained as

Mechanics is the branch of Physics concerned with the behavior of bodies when subjected to forces or displacements, and the subsequent effects of the bodies on their environment.

The discipline has its roots in several ancient civilizations. During the early modern period, scientists such as Galileo, Kepler, and especially Newton, laid the foundation of mechanics and what is now known as classical mechanics.

The study of bodies at rest is known as Statics

The study of bodies while these are motion is called Dynamics

The practical application of science of mechanics is known as Applied Mechanics. Applied mechanics examines the response of bodies (solids and fluids) or systems of bodies to external forces. Some examples of mechanical systems include the flow of a liquid under pressure, and the fracture of a solid from an applied force.

Therefore, there are two major types of applied mechanics, Mechanics of Solids and Mechanics of Fluids respectively.

The basic principles of Statics and Dynamics are fundamental to Mechanics of Solids and Mechanics of Fluids.

Mechanics of Solids is the subject to be discussed in this semester. This is also known as Mechanics of Materials, Strength of Materials and also Mechanics of Deformable Bodies.

Classical Mechanics is a branch of classical physics which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces. It is one of the oldest and largest subjects in science, engineering and technology.

Classical mechanics describes the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects such as spacecraft, planet, stars and galaxies. Besides this, many specializations within the subject deal with gases, liquids, solids, and other specific sub-topics.

Classical mechanics provides extremely accurate results as long as the domain of study is restricted to large objects and the speeds involved do not approach the speed of light.

Quantum mechanics is a branch of Physics providing a mathematical description of the wave-particle duality of matter and energy. Quantum mechanics describes the time evolution of physical systems via a mathematical structure called the wave function.

Quantum mechanics differs significantly from classical mechanics in its predictions when the scale of observations becomes comparable to the atomic and sub-atomic scale.

However, many macroscopic properties of systems can only be fully understood and explained with the use of quantum mechanics. Phenomena such as superconductivity, the properties of materials such as semiconductors and nuclear reaction mechanisms observed as macroscopic behaviour, cannot be explained using classical mechanics.

Mechanics of Materials By Beer & DeWolf McGraw-Hill

Mechanics of Materials By R. Craig John Wiley

Mechanics of MaterialsBy James Gere Brooks

Mechanics of Materials By R. C. Hibbleler Pearson/Prentice Hall

Application of Mechanics of Materials is endless and can be found in every discipline of engineering. For example, buildings and bridges, machines and motors, submarines and ships, or aeorplanes and antennas.

The subjects of Statics and Dynamics are also important, but these deal with primarily with the forces and motions associated with particles and rigid bodies.

However, Mechanics of Materials go a step further to examine the stresses and strains that occur inside real bodies that deform under loads.

As we have discussed that Mechanics of Materials deals with response of solid bodies subjected to different types of loading conditions.

The solid bodies from engineering point of view are axially loaded members, shafts in torsion, thin shells, beams and columns, structures which are sub-assemblies and assemblies of these components.

Aim of Mechanics of Materials would be the determination of the stresses, strains, and deflections produced by the loads. A complete picture of the mechanical behaviour of the body can easily be determined if stress, strain and deflections are known for all values of load up to the failure load.

Both theoretical analyses and experimental results have equally important roles in the study of Mechanics of Materials.

However, it must be noted that theoretical derivations and formulas cannot be used in a realistic way unless certain properties of the materials are not determined.

The specific properties of materials are available only after conducting suitable experiments in the laboratories and extracting results.

Furthermore, many practical engineering problems of great importance cannot be handled efficiently by theoretical means, experimental results are quite essential.

Discussion about systems of units is also very important. If is not very clearly understood accurate results can be obtained. All systems are same, it all depends upon your choice. However, understanding must be quite clear.

Many systems of measurements have been devised over the centuries, but today there are two basic systems of measurements use in engineering and scientific work; SI system and USCS system.

International System of Units (SI) U.S. Customary System, Imperial System,

British System

The International System (SI) is based upon Mass (Kg), Length (Meter) and Time (Seconds). These three quantities are independent of the location and hence this system is known as Absolute System.

The U.S. Customary System (USCS) is based upon Force (Pound), Length (Foot) and Time (Seconds). This system is not independent as unit of force depends upon gravitational attraction and hence is also called Gravitational System.

The SI system is an improved and modernized version of well known metric system as SI system has important new features not previously part of metric system.

Stress is defined as force per nit area. In SI system the unit of force is Newton (N) and the unit of area is square meters. Hence unit of stress is Newtons per square Meter (N/m²). This is also known as Pascals (Pa).

In USCS system the unit of is Pound and hence unit of stress would be Pounds (P) per square Foot (Lb/ft²). Commonly stress is measured as Pounds per squre Inches (psi).

As Pascal is a very small value in SI units, it is taken as Mega Pascals (MPa) or Giga Pascals (GPa). In USCS system larger values of stresses are also measured as ksi.

Pound = 4.45 Newtons Kip = 4.45 Kilonewtons

psi = 6890 Pascals ksi = 1000 psi

ksi = 6.890 Pascals

psf = 47.9 Pascals ksf = 47.9 Kilopascals

Many names for Stress and Strain are used in literature such Normal Stress & Strain, Tensile Stress & Strain, Engineering Stress & Strain, True Stress & Strain, Nominal Stress & Strain, and Uniaxial Stress & Strain.

In fact all these terminologies are same; difference is that how you are manipulating and interpreting the condition of application of load.

Very simply stress is defined as the intensity of force that is, the force per unit area. Similarly strain is defined as the elongation per unit length of material. Stress has dimension while strain is dimensionless parameter.

In order to investigate the internal stresses produced in axial loaded bar, a section of prismatic bar is cut perpendicular to the longitudinal axis of this bar. The tensile load “P” acts at the right-hand end of free body.

At the other end are forces representing the action of the removed part of the bar upon the part that remains. These forces are uniformly distributed over the cross section.

From the equilibrium of the body resultant these forces must be equal in magnitude and opposite in direction to the applied load “P”. Hence stress are

σ = P/A

This equation (σ = P/A) holds for the uniform stress in an axially loaded, prismatic bar of arbitrary cross-sectional shape.

If the bar is stretched by the forces, the resulting stresses are known as tensile stresses and these bear positive sign, and if the bar is compressed the resulting stresses are known as compressive stresses and bear negative sign.

If the stress acts in a direction perpendicular to the cut surface, it is also know as normal stress. Normal stress may be both tensile or compressive stress. In contrast to normal stress there is another type of stress known as shear stress.

In order for the equation to be valid, the stress must be uniformly distributed over the cross-section of the bar. In case stress is not uniform, the equation for stress would yield the average normal stress.

Uniformity of stresses over the cross-section would only exist if the axial force acts through the centroid of the cross-sectional area. When the load does not act at the centroid, a more complicated situation will arise as bending would occur.

If the initial area of the bar is used to calculate stress, the resulting stress is called nominal stress, conventional stress and engineering stress. In case original area of bar is used a more accurate value of stress is obtained known as true stress.

If a prismatic bar is axially loaded it would extend in length in case of tension and would shorten in case of compression. In fact change in length is the cumulative result of the stretching and compression of all elements of the material.

In general the elongation or contraction of a segment would be equal to its length divided by the total length and multiplied by the total elongation. In this way we can conceive the concept of elongation per unit length, or strain. This is calculated as followed:

ε = δ/L Similar to stress if bar is in tension, the strain is

called tensile strain and if in compression it is called compressive strain.

The definitions of normal stress and strain are based upon purely statical and geometrical considerations and hence these two equations be used for any magnitude of load and for any material.

However, the principal requirement is that the deformation of the bar must be uniform (be prismatic), the load acts through the centroid of the cross-section and the material must be homogenous.

If all these conditions are fulfilled the resulting state of stress and strain is called uniaxial stress & strain.

The required mechanical properties of materials are determined by tests performed on small specimens with standard dimensions in laboratories.

The dimensions of test specimens and methods of applying loads have standarized by some organizations such as ASTM, ASA, ANSI, and BSI, PSI.

The most common test is the tension test in which tensile loads are applied to a cylindrical specimen. In a static test the load is applied very slowly, and in dynamic test the rate of loading may very high.

After performing tensile test and determining the stress and strain at various loads stress-strain diagram can be developed. A typical diagram is shown here.

FOLLOWING IS THE STRESS – STRAIN DIAGRAM FOR NON-FERROUS MATERIAL:

This stress–strain curve provide following properties of material:

1: True elastic limit2: Proportionality limit3: Elastic limit4: Offset yield strength

During tensile testing of a material sample, the stress–strain curve is a graphical representation of the relationship between stress and strain, i.e. elongation, compression, or distortion.

The slope of stress-strain curve at any point is called the tangent modulus; the slope of the elastic (linear) portion of the curve is a property used to characterize materials and is known as the Young’s Modulus or Modulus of Elasticity.

The linear relationship between stress and strain is defined by Hooke’s Law by the formula

σ = Eε

This equation applies only to ordinary tension compression; and for more complicated state of stress such as biaxial stress and plane stress, a more generalized Hooke’s Law is required.

ε = 1/E(σ - ν σ)

The modulus elasticity has relatively large values of materials that are very stiff such as steel. More flexible materials have a lower values. For most materials the values of E is same in tension and compression.

If a prismatic bar is loaded in tension, the axial elongation would be accompanied by lateral contraction, and it would be normal to the direction of the applied load.

Lateral changes can easily be observed in polymers and these not easily observable in metals. However, these can easily be detected with measuring devices.

Lateral strain is proportional to the axial strain in the linear elastic range if the material is both homogenous and isotropic. Most of the materials meet these requirements.

The ratio of the strain in the lateral direction to the strain in the axial direction is called the Poisson’s ratio, and is given by the following formula:

ν = - Lateral Strain/Axial Strain

For isotropic materials poisson’s ratio is 1/4, more recent calculation gives this value as 1/3, and both values are in the range of calculated values.

The lateral contraction of a bar in tension , or the expansion of a bar in compression is an example of strain without any corresponding stress.

The unit volume change in the materials is known as the “Dialatation”, and is calculated as the change in volume divided by the original volume. This is calculated as

℮ = ∆V/V = ε(1 - 2 ν) = σ/E(1- 2 ν)

This equation can be used to calculate the increase in volume in a bar in tension if the values of axial strain or stress and Poisson’s ratio are know.

In the elastic region the unit volume change would be in the range of 0.3ε to 0.5ε.

Shear stress is different from tensile strain in the sense that shear stress is always parallel or tangential to surface. These stresses are higher near the middle and become zero at certain locations on the edges.

The average shear stress on a x-sectional area is obtained by dividing the shear force V by the area A.

τ = V/A

Like normal stresses shear stresses also represent intensity of force, force per unit area hence it has the same units as of tensile stress.

Simple shear or direct shear arises in the design of bolts, pins, rivets, keys, welds, and glued joints. Shear stresses also arise in indirect manner when structures are subjected to tension, torsion, and bending.

Shear stresses on opposite faces of an element are equal in magnitude and opposite in direction. These stresses on perpendicular faces of an element are equal in magnitude and have directions such that both stresses point toward, or point away from, the line of intersection of the faces.

Under the action of shear stresses, shear strain, similar to tensile strain, is produced. It is to be noted that shear stresses have no tendency to elongate or shorten the member, rather these stresses produce a change in the element changing it from square shape to an oblique form.

The change in angle, ϒ, less than π/2 at one side and greater than π/2 on the other side, is a measure of the distortion, or change in shape of the element and is known as shear strain.

Shear stress acting on a positive face of an element is positive if it acts in the positive direction of one of the coordinate axes; and is negative if it acts in the negative direction of the axis.

Similarly shear stress acting on a negative face of an element is positive if it acts in the negative direction of an axis and negative if it acts in the positive direction.

In the same way shear strain in an element is positive when the angle between two positive or two negative faces is reduced. It is negative when the angle between two positive or two negative faces is increased.

As a rule of thumb, it must be remembered that positive shear stresses always produce positive shear strains.

Shear stress-strain diagram can be plotted in a similar way tests. In these tests hollow, circular tubes are twisting in order to produce a state of pure shear.

The material properties like proportional limit, the yield stress and the ultimate stress under shear are usually about half as large as in tension.

Similar to tensile stress, shear stress is directly proportional to shear strain in the elastic region and Hooke’s Law hold true as

τ = Gϒ

G is shear modulus of elasticity or modulus of rigidity exactly similar to linear modulus of elasticity, E.

Both E and G can be related by the following relationship:

G = E/2(1+ⱱ)

This relationship shows that E, G and ⱱ are not independent linear properties.

As the value of Poisson’s ratio is between zero and one-half for ordinary materials, the relationship also shows that value of G must be from one-third to one-half of E.

Structures such as building, bridges, machines, aircraft, vehicles , ships and many more are designed to support or transmit loads.

In order to avoid the failure of structures the loads that a structure actually can support must be greater than the loads it will be required to sustain during service. Factor of safety must always be greater than one.

In more simple words the actual strength of a structure must exceed the required strength; and the ratio of actual strength to required strength is known as Factor of Safety n.

For many structures, it is quite important that the materials remain within elastic region to avoid any type of permanent deformation. In such cases factor of safety to structures is applied with respect to yield stress to get allowable or working stress.

In some cases ultimate strength is also taken in to account and is considered in the same way as considered for yield stress.

The determination of a factor of safety must also be considered for following matters:

◦ Probability of accidental overloading◦ Types of loads◦ Accuracy of determination of loads◦ Possibilities for fatigue failures◦ Inaccuracies in construction◦ Variability in the quality of workmanship◦ Variations in the properties of materials

A steel rod 1m long and 13 mm in diameter carries a tensile load of 13.5kN. The bar increases in length by 0.05 cm when load is applied. Determine the stress and strain in the rod.

A tensile test is performed on a brass specimen 1 cm in diameter using a gauge length of 50mm. When applying a load of 25kN we observe that the distance between the gauge marks is increased by 0.0152cm. Calculate the modulus of elasticity.

A round rod of diameter 3.8cm is loaded in tension by a force P. The change in diameter is measured 0.08mm. Assuming E = 2.8 GPa and Poisson’s ratio 0.4, calculate the axial force in the rod.

A steel bar of length of 2.5m with a square X-section 100 mm on each side sis subjected to an axial tensile force of 1300kN. Assuming E=200 GPa and Poisson’s ratio 0.3, find the elongation of the bar, the change in the X-sectional dimensions and the change in volum.

Drive the formula for the increase of ∆V in the volume of a prismatic bar of length L hanging vertically under its own weight (W = total weight of the bar)

A solid cylinder of some with E=86GPa, ν = 0.30 of diameter 60mm and length 37cm is compressed by an axial force P = 200kN. Find (a) increase in the diameter of the bar, (b) find the decrease ∆V in the volume of the bar.

QUESTIONS AND QUERIESIF ANY!

IF NOT THENGOOD BYE

SEE ALL OF YOU IN NEXT LECTURE

ON shear STRESS AND STRAIN, allowable stresses & axially

loaded members