There are three types of Materials. They are
If two equal and parallel forces F, not in the same line act on parallel faces of a member, then the member is said to be loaded in Shear. Consider the rectangular block shown in the figure, a force F is applied tangentially along the top and bottom face. ( The force is called as shear force ). The shear stress formula indicates only the average shear stress. In reality the distribution of shear stress is far from uniform. In reality it varies parabolically from zero at the edges to a maximum at the center.
Shear stress = Shear force / Area = P / ( L x H )
Shear strain = Transverse displacement / Distance form lower face.
Thermal Stress :
size of a body will change as the ambient temperature fluctuates, expanding as
it rises and contracting as it falls. If the
natural change ( +ve or - ve ) in the length of the rod is not prevented, then
the stress is not induced. The increase in length of a rod = α
α = Coefficient of linear expansion.
T - Temperature rise.
L - Actual change in length.
When bending a piece of metal, one surface of the material stretches in tension while the opposite surface compresses. It follows that there is a line or region of zero stress between the two surfaces, called the neutral axis. Make the following assumptions in simple bending theory:
It is the ratio of change in length to the original length. It has no units. It is expressed with the Greek word (epsilon) e = dl / l
Within in elastic limits, the ratio of stress to strain is constant. This constant is called Young's modulus of elasticity. In case of shear force, if the ratio of shear stress to shear strain is also constant. That constant is called as shear modulus of rigidity. This young's modulus of elasticity is a measure of stiffness. The greater the Young's modulus for a material, the better it can withstand greater forces. More about the relation between, stress and strain is discussed in the following paragraphs.
Young's modulus ( E ) = Stress / Strain.
The relationship between the stress and strain that a material displays is known as a Stress-Strain curve. Stress-strain diagrams can be generated for axial tension and compression, and shear loading conditions.
Tension specimens have a narrow region in the middle along the so-called gage length.
Compression specimens are much thicker and shorter than tension specimens with no cross-sectional variations.
In either case, data are collected in terms of applied force and the change in the gage length. The normal stress is obtained by dividing the applied force by the cross-sectional area of the specimen, and the normal strain is obtained by dividing the change in gage length by its original value. The plot of stress versus strain gives the stress-strain diagram.. These curves reveal many of the properties of a material (including data to establish the Modulus of Elasticity, E). A typical stress-strain diagram for a ductile metal undergoing tension is given below.
During the first portion of the curve (up to a strain of less than 1%), the stress and strain are proportional. The greatest stress at which a material is capable of sustaining the applied load without deviating from proportionality or stress to strain. This holds until the point 'a', the proportional limit, is reached. Stress and strain are proportional because this segment of the line is straight.
From a to b on the diagram, stress and strain are not proportional, but nevertheless, if the stress is removed at any point between O and b, the curve will be retraced in the opposite direction and the material will return to its original shape and length. In other words, the material will spring back into shape in a reverse order to the way it sprung out of shape to begin with. In the region Ob, then, the material is said to be elastic or to exhibit elastic behavior and the point b is called the elastic limit. The point on the stress strain curve beyond which the material permanently deforms, upon removal of the external load.
If the material is stressed further, the strain increases rapidly, but when the stress is removed at some point beyond b, say c, the material does not come back to its original shape or length but returns along a different path to a different point, shown along the dashed line in figure. The length of the material at zero stress is now greater than the original length and the material is said to have a permanent set.
Further increase of stress beyond c produces a large increase in strain until point d is reached at which fracture takes place. From b to d, the metal is said to undergo plastic deformation. If large plastic deformation takes place between the elastic limit and the fracture point, the metal is said to be ductile. Such materials are capable of being drawn out like a wire or hammered thin like gold leaf. If, however, fracture occurs soon after the elastic limit is passed, the metal is said to be brittle.
The maximum stress that a material withstands when subjected to an applied load. Dividing the load at failure by the cross sectional area determines the value.
This is the point at which the material exceeds the elastic limits and will not return to the original shape, if stress is removed. This value is determined by evaluating a stress-strain diagram produced during a tensile test.
The stress-strain curve for different material is different. The figure below shows the comparison of the curves for mild steel, cast iron and concrete. It can be seen that the concrete curve is almost a straight line. There is an abrupt end to the curve. This, and the fact that it is a very steep line, indicate that it is a brittle material. The curve for cast iron has a slight curve to it. It is also a brittle material. Both of these materials will fail with little warning once their limits are surpassed. Notice that the curve for mild steel seems to have a long gently curving "tail". This indicates a behavior that is distinctly different than either concrete or cast iron. The graph shows that after a certain point mild steel will continue to strain (in the case of tension, to stretch) as the stress (the loading) remains more or less constant. The steel will actually stretch like taffy. This is a material property which indicates a high ductility.
If the original cross-sectional area is used to calculate the stress for every value of applied force, then the resulting diagram is known as the Engineering Stress-Strain Diagram. However, if the applied force is divided by the actual value of the cross-sectional area, then the resulting diagram is known as the True Stress-Strain Diagram. Therefore, in engineering stress-strain diagram the ultimate and failure strength points do not coincide whereas in the true diagram they do. The difference in the two diagrams becomes apparent in the inelastic region of the curve where the change in the cross-sectional area of the specimen becomes very significant.
Yielding occurs when the design stress exceeds the material yield strength. Design stress is typically maximum surface stress (simple loading) or Von Mises stress (complex loading conditions). The Von Mises yield criterion states that yielding occurs when the Von Mises stress, sv exceeds the yield strength in tension. Often, Finite Element Analysis stress results use Von Mises stresses. Von Mises stress is
s2, s3 are principal
stresses. Safety factor is a function
of design stress and yield strength. The following
equation denotes safety factor,
fs. Where Y S
is the Yield Strength and D S is the
Poisson Ratio: Volumetric Strain ( ev ):
( s1- s2
)2 + ( s2-
)2 + ( s1-
Y S fs
--- D S
Within elastic limits the ratio of lateral strain to longitudinal strain is constant and is equal to Poisson ratio. When a load is applied on a body, there is a dimensional increase along the longitudinal direction and dimensional decrease in lateral direction. Poisson ratio is constant for a given material.
Because of increase in length, and decrease in breadth and depth, there is a change in volume. Volumetric strain is defined as the ratio of change in Volume to original volume.
It is the capability of a material to be drawn into wires. There are two methods used for its measurement. One based on total elongation produced and other based on total reduction in sectional area.
% increase in elongation = ( L- l ) / l
% reduction in cross sectional area = ( A - a ) / A x 100
This test is used to find out the resistance of a body against shock load. This is called as Izod impact test. The test specimen is a 10 mm square rod and notched at a face. The notch is at a depth of 2 mm and a radius of 0.25 mm at the bottom. It is fixed in a vice. The pendulum is raised and the value stored is around 165 joules.
Sometimes members are subjected to loads that vary in magnitudes. They may be even reversible loading. ( The member is subjected to repeated tensile and compressive stress ). These members fail at point lower than ultimate stress. This property is called fatigue of materials. At a certain range of applied stress, the number of cycles becomes infinite. That limit is called as Endurance limit.
s2, s3 are principal
stresses. Safety factor is a function
of design stress and yield strength. The following
equation denotes safety factor,
fs. Where Y S
is the Yield Strength and D S is the
Poisson Ratio: Volumetric Strain ( ev ):
Volumetric Strain ( ev ):
It is the energy stored on a member when work is done on it to deform it.
Types of Loading:
There are three types of loading. They are
These are structural members in which the load is applied at right angles to the axis. The following are the different types of beams.
Column and Struts:
These are members that are subjected to compressive load along the axis. Short columns fail by crushing. Thus we have to take care of crushing load. But long columns fail by buckling or bending, hence we have to take care of crippling load. This buckling load is less than the crushing load.
This value of bucking load is low for long members and vice versa. Thus buckling load depends on
It is the ratio of applied Stress to volumetric strain.
Sometimes the cross section of a member changes abruptly because of presence of a hole, notch, groove or shoulder. In regions close to the abrupt change the stress is of high magnitude. This change in section is called discontinuity or stress raisers. Following are the causes.
Disc spring / Belleville spring, Occupy small space and gives high spring rates. Parallel arrangement takes a higher load for a given deflection and series arrangement gives a larger deflection. Leaf springs are used in automobiles. They are energy absorbing devices. There are two types. Constant width and constant strength springs.
Hardness of a substance is the resistance that a body offers or indentation by other bodies. For testing hardness, there are two tests. They are Scratch test and Indentation test. The greater the hardness of the metal, the greater resistance it has to deformation. In mineralogy the property of matter commonly described as the resistance of a substance to being scratched by another substance. In metallurgy hardness is defined as the ability of a material to resist plastic deformation.
Hardness measurement methods:
These hardness tests measure a metal's hardness is to determine the metal's resistance to the penetration of a non-deformable ball or cone. The tests determine the depth which such a ball or cone will sink into the metal, under a given load, within a specific period of time. The followings are the most common hardness test methods used in today's technology. More details about each test is given in subsequently.
Rockwell Hardness Test:
The Rockwell Hardness test is based on the net increase in depth of impression as a load is applied. Hardness numbers have no units and are indicated R, L, M, E and K scales. The higher the number in each of the scales means the harder the material. The type of indenter and the test load determine the hardness scale (A, B, C, etc). In the Rockwell method of hardness testing, the depth of penetration of an indenter under certain arbitrary test conditions is determined. The indenter may either be a steel ball of some specified diameter or a spherical diamond-tipped cone of 120° angle and 0.2 mm tip radius, called Brale.
The Rockwell test uses two loads, one applied directly after the other. The first load, known as the "minor", load of 10 kilograms is applied to the specimen to help seat the indenter and remove the effects, in the test, of any surface irregularities. In essence, the minor load creates a uniformly shaped surface for the major load to be applied to. A minor load causes an initial penetration and holds the indenter in place. Then, the dial is set to zero and the major load is applied. Upon removal of the major load, the depth reading is taken while the minor load is still on. The difference in the depth of the indentation between the minor and major loads provides the Rockwell hardness number. The hardness number may then be read directly from the scale.
The Rockwell hardness tester to measure the hardness of metal measures resistance to penetration like the Brinell test, but in the Rockwell case, the depth of the impression is measured rather than the diametric area. With the Rockwell tester, the hardness is indicated directly on the scale attached to the machine. This dial like scale is really a depth gauge, graduated in special units.
For soft materials a 1/16" diameter steel ball is used with a 100-kilogram load and the hardness is read on the "B" scale. In testing harder materials, a 120 degrees diamond cone is used with up to a 150 kilogram load and the hardness is read on the "C" scale. There are several Rockwell scales other than "B" & "C" scales, (which are called the common scales). The other scales also use a letter for the scale symbol prefix, and many use a different sized steel ball indenter. A properly used Rockwell designation will have the hardness number followed by "HR" (Hardness Rockwell), which will be followed by another letter which indicates the specific Rockwell scale. An example is 60 HRB, which indicates that the specimen has a hardness reading of 60 on the B scale.
Brinell Hardness Test:
Brinell hardness is determined by forcing a hard steel or carbide sphere of a specified diameter under a specified load into the surface of a material and measuring the diameter of the indentation left after the test. The Brinell hardness number, is obtained by dividing the load used, in kilograms, by the actual surface area of the indentation, in square millimeters. The result is a pressure measurement, but the units are rarely stated. The BHN is calculated according to the following formula
BHN = the Brinell hardness number
F = the imposed load in kg
D = the diameter of the spherical indenter in mm
Di = diameter of the resulting indenter impression in mm
The Brinell hardness test uses a desk top machine to press a 10 mm diameter, hardened steel ball into the surface of the test specimen. The machine applies a load of 500 kilograms for soft metals such as copper, brass and thin stock. A 1500 kilogram load is used for aluminum castings, and a 3000 kilogram load is used for materials such as iron and steel.
The load is usually applied for 10 to 15 seconds. After the impression is made, a measurement of the diameter of the resulting round impression is taken. It is measured to plus or minus 0.05mm using a low-magnification portable microscope. The hardness is calculated by dividing the load by the area of the curved surface of the indention, (the area of a hemispherical surface is arrived at by multiplying the square of the diameter by 3.14159 and then dividing by 2 -As shown in the formula above). A well structured Brinell hardness number reveals the test conditions, and looks like this, "75 HB 10/500/30" which means that a Brinell Hardness of 75 was obtained using a 10mm diameter hardened steel with a 500 kilogram load applied for a period of 30 seconds. On tests of extremely hard metals a tungsten carbide ball is substituted for the steel ball.
Vickers Hardness Test:
It is the standard method for measuring the hardness of metals, particularly those with extremely hard surfaces, the surface is subjected to a standard pressure for a standard length of time by means of a pyramid-shaped diamond. The diagonal of the resulting indention is measured under a microscope and the Vickers Hardness value read from a conversion table.
The indenter employed in the Vickers test is a square-based diamond pyramid whose opposite sides meet at the apex at an angle of 136º. The diamond material of the indenter has an advantage over other indenters because it does not deform over time and use The diamond is pressed into the surface of the material at loads ranging up to approximately 120 kilograms-force, and the size of the impression (usually no more than 0.5 mm) is measured with the aid of a calibrated microscope. The Vickers number (HV) is calculated using the following formula
Vickers hardness test.
Knoops Hardness Test
This test method was devised in 1939 by F. Knoop at the National Bureau of Standards in the United States. By using lower indentation pressures than the Vickers hardness test, which had been designed for measuring metals, the Knoop test allowed the hardness testing of brittle materials such as glass and ceramics. In this test, a pyramid-shaped diamond indenter with apical angles of 130° and 172°30 (called a Knoop indenter) is pressed against a material. Making a thombohedral impression with one diagonal seven times longer than the other. The hardness of the material is determined by the depth to which the Knoop indenter penetrates.
The diamond indenter employed in the Knoop test is in the shape of an elongated four-sided pyramid, with the angle between two of the opposite faces being approximately 170º and the angle between the other two being 130º. Pressed into the material under loads that are often less than one kilogram-force, the indenter leaves a four-sided impression about 0.01 to 0.1 mm in size. The length of the impression is approximately seven times the width, and the depth is 1/30 the length. Given such dimensions, the area of the impression under load can be calculated after measuring only the length of the longest side with the aid of a calibrated microscope. The final Knoop hardness (HK) is derived from the following formula
The shore scleroscope measures hardness in terms of the elasticity of the material. A diamond-tipped hammer in a graduated glass tube is allowed to fall from a known height on the specimen to be tested, and the hardness number depends on the height to which the hammer rebounds; the harder the material, the higher the rebound. Shore hardness is a measure of the resistance of material to indentation by 3 spring-loaded indenter. The higher the number, the greater the resistance. The Shore hardness is measured with an apparatus known as a Durometer and consequently is also known as 'Durometer hardness'. The hardness value is determined by the penetration of the Durometer indenter foot into the sample. Because of the resilience of rubbers and plastics, the hardness reading may change over time - so the indentation time is sometimes reported along with the hardness number.
Shore Hardness, using either the Shore A or Shore D scale, is the preferred method for rubbers/elastomers and is also commonly used for 'softer' plastics. The Shore A scale is used for 'softer' rubbers while the Shore D scale is used for 'harder' ones. The shore A Hardness is the relative hardness of elastic materials such as rubber or soft plastics can be determined with an instrument called a Shore A durometer. If the indenter completely penetrates the sample, a reading of 0 is obtained, and if no penetration occurs, a reading of 100 results. The reading is dimensionless.
EXPERIMENTAL STRESS ANALYSIS
When mathematical methods become too cubersome or impossible for application as in the case of determining stress concentration around openings ( discontinuity ) or member with unusual cross section, experimental methods are used to determine the stresses. This methods are known as "Experimental stress analysis". Number of methods are available to obtain stress or strain distribution in loaded members. Often it is necessary to know either the stress or strain distribution in the whole field on the stresses or strains at selected points. The stress distribution in the entire field is obtained by the following methods.
Whole field method:
This method gives the overall (entire) stress distribution in the loaded member. Two techniques, namely, the photo elastic method and the brittle lacquer technique are available to evaluate the stress distribution in the entire field.
Point by point method:
This gives the stress or strain at selected points, usually strain gauges are used to obtain stresses at selected points.
Photo elastic method:
This method is one which is extensively used to solve the problem in practical way. Basic principle involved in this method is "double refraction" or "Bi-refringence". So, this method depends upon the property of certain transparent solids by which they become doubly refractive under the action of stress, the magnitude of optical effect bearing a definite relation to that of the stress. This optical phenomenon is called as photo elastic effect. The photo elastic method of evaluating the stress in a stress field is based on the following two photo elastic laws.
"The light on passing through a stressed model becomes polarized in the direction of principle stress axes and is transmitted only on the plane of principal stress"
"The velocity of transmission in each principal plane is dependent on the intensity of the principal stress in these planes.
When a ray of light is incident on certain crystals, it is split at entry into components which generally, are transmitted through the crystal in different directions with different velocities. This phenomenon is known as "Natural double refraction" or "Bi-refringence". One of the component which is not deviated is known as the "Ordinary ray" and the other ray, which is always deviated is called the "Extra ordinary ray". For some crystalline transparent materials such as mica, calcite, the property of double refraction is a permanent property of material. Certain transparent materials such as perpix, bakelite, araldite are optically sensitive and exhibit the property of double refraction when external loads are applied. This optical effect disappears when the external loads are removed. In other words these materials are ordinarily isotropic, optically but become optically anisotropic when loaded and display double refraction characteristics temporarily. Such materials are called as photo elastic materials.
When a material is subjected to external loads, it develops principal stresses P1 and P2 at any point 'O' along two mutually perpendicular directions. Because of this property of the material to exhibit double refraction when stressed, the refractive index of the material which is n1 in the direction of the principal stress P1 changes to n2 in the direction of the principal stress P2. The changes in the refractive indices is fount to be linearly proportional to the stresses.
The color of light is due to the frequency of waves and each frequency produces a different color (VIBGYOR). A monochromatic light may be considered as a light corresponding a particular wave length and color. Thus, light from mercury vapor lamp produces green and violet color with different wave frequencies. However, when a suitable filter is used, violet waves may be absorbed and only green light may be obtained. "Monochromatic light' is a light corresponding to a particular wave length or color and are obtained by using suitable filters. "Polarization" denotes the ability to extinguish light in all direction except one.
"Polariscopes" are devised to produce polarization of light. For the photo elastic investigations two types of polariscope are used.
Plane polariscope, plane polarized light is used,
Circular polariscope, circular polarized light is used.
A plane polariscope consists of a light source, to emit monochromatic light and white light, condensing lens to collect white rays, the field lens to give a parallel light beam, the polarize to produce plane polarized light, the loading frame by which external load can be applied as the model itself made out of photo elastic material such as Epoxy resins, columbic resin (CR - 39), homolite 100, Bakelite (carsalin 61 - 893), glass etc. An analyzer which would combine the two beams emerging out of the model to produce "Interference fringes". The projection lens projects the image or the stress pattern on the screen. A camera may also be used in pace of the screen to get the permanent record of the fringe patterns. A typical arrangement of a plane polariscope is shown in the figure below.
To analyzing the stress pattern, a scale model of the loaded member is made using a photo elastic material. The model is subjected to loads similar to the one that might be applied on the original member. Light, on passing through the polarizer, will be plane polarized, on entering the stress model, the light vector decomposes into two vectors along the two principal plane directions. As the result of this optical effect stress patterns known as "fringes" are developed. Fringes represent the loci or points of equal "Phase difference" produced by "temporary double refraction".
There are points of equal brightness or darkness. "Phase difference" is proportional to the difference of principal stresses or maximum shear stress. The fringe pattern related to the principal stress difference is called "Isochromatic fringe pattern" the fringe pattern consists of 'isoclinic and isochromatic' fringes.
Using law of elasticity and stress-optic law the stress pattern produced are transformed into stress differences and ultimately the state of stress at all points of the model are obtained. One such advanced automatic polariscope is shown in the figure.
Brittle coating method:
In this method, a brittle coating is sprayed over the surface of the specimen for about 0.1 to 0.25 mm thick. The coating is allowed to dry completely. The loads are applied on the sample. Since the coating is very ting the strains on the surface of the specimen are totally transmitted to the coating without any increase or decrease. White wash on walls represent the very common example of brittle coating, but these coatings require large strains to cause them crack. When the specimen is stressed, the coating cracks in a direction perpendicular to the maximum principal stress. Stresses in the specimen and the stresses in the coating are related using the theory of elasticity.
This method represent on the non-destructive methods of stress determination and the coating fails at very low stresses and the specimen is not over stressed. Commonly used coating is known as "Stress coat" and consist of a zing resinate as base, carbon di sulphide as solvent and dibutyl pthlate as plasticizer to control the plasticity of the coating and to vary the degree of brittleness of the coating.
This method is inexpensive and stress evaluation is easy and quick and also provides a simple and direct approach for failure analysis or is service components, determining the location and direction of stress sensors such as strain gauges. It is also useful in determining areas of stress concentration, measurements of thermal and residual strains in members and estimating the magnitude and directions of principal stresses in a stress fields. Following are the advantage of this method.
Enables stress in the whole field to be determined,
Directly applied to a prototype, no need for a model,
This technique may be applied to an actual machine component while working, and hence no need for simulation.
Analyzing the specimen stressed from coating stresses is simple and easy.
Behavior of coating depends upon temperature and humidity.
Behavior of coating should be properly understood as a number of variables affect the behavior.
The technique is more qualitative than quantitative.
A strain gauge may be defined a any instrument or device that is employed to measure the linear deformation over a given gauge length, occurring in the material of a structure during the loading of structure. Depending upon the magnification system, the strain gauges may be classified as follows
Mechanical gauges ( wedge and screw, lever spindle and compound, rack and pinion, combination of lever and rack and pinion and dial indicators)
Optical strain gauges.
Electrical (Inductance, Capacitance, Resistance, Pieze electric and Piezo resistive)
Photo stress gauge.
Mechanical strain gauge:
To meet the demand for greater sensitivity while retaining the advantage of relative ease of applying the mechanical gauges, mechanical magnification is used. Two commercially available gauges are Berry gauge and Tinius olsen strain gauge.
TINIUS AND OLESON GAUGE
Rack and Pinion:
The rack and pinion principle along with various types of gear trains is employed in gauges in which the magnification system is incorporated in an indicating dial. In general a dial indicator consists of an encased gear train actuated by a rack cut in the spindle, which follows the motion to be measured. A spring imposes sufficient spindle force to maintain a reasonable uniform and positive contact with moving part. The gear train terminates with a light weight pointer which indicates spindle travel on a graduated dial. Lost motion in the gear train is minimized by the positive force of a small coil spring. Dial gauges are permanently attached to structure to indicate the deflection on deformation obtained under working conditions These gauges then indicate excessive deformation due to either an overload or damage to the structure.
Electrical strain gauges:
These are usually measured on a small area. A very thin wire, usually 20 - 25 microns in diameter and having a considerable initial resistance is used to measure the strain at a point. This think wire is attached to the specimen surface using a suitable adhesive at the point where strain is to be measured in such a way that the strains on the surface of the specimen is totally transmitted to the wire. When the specimen suffers a tensile strain, the length of the wire increases, thus its area decreases and consequently the resistance will increase. This change in resistance is proportional to the tensile strain suffered by the specimen therefore, by measuring the change in resistance the strain at the surface of the specimen may be evaluated.
Most commonly used alloys as strain gauges are constantan (Nickel copper alloy), Nichrome (Nickel chromium alloy) and Isoelastic (Nickel, chromium, Molybdenum and Iron alloy). Resin adhesives are commonly used.
BOOKS ON STRENGTH OF MATERIALS
Schaum's Outline of Statics and Strength of Materials by by John H. Jackson
Problem Solver in Strength of Materials and Mechanics of Solids by James R. Ogden
Advanced Mechanics of Materials by Robert Cook
Practical Stress Analysis in Engineering Design by Alexander Blake
Advanced Strength of Materials by J. P. Den Hartog
Practical Stress Analysis in Engineering Design by Alexander Blake
Mechanics of Materials by James M. Gere
Advanced Mechanics of Materials by Richard J. Schmidt, Arthur P. Boresi
Mechanics of Materials by Anthony Bedford, Kenneth M. Liechti
Applied Statics and Strength of Materials (3rd Edition) by Leonard Spiegel, George F. Limbrunner
Strength of Materials and Mechanics of Solids Problem Solver by James Ogden