Factor Influencing The Shear Strength of Cohesionless Soil

The principal factors effecting the shearing strength of cohesion less soil are

1) Shape of particles

2) Gradation

3) Denseness

4) Confining pressure

5) Deviator stress

6) Intermediate principal stress

7) Loading

8) Vibration and repeated loading

9) Type of minerals

10) Capillary moisture

These parameters, stated above, are discussed below:

1) Shape of Particles

When other parameters is identical,angular shape particles of sand produce shear strength more than that of rounded shape particles.

2) Gradation

It is observed that a uniform graded sand have less shearing strength than that of well graded.

3) Denseness

When density increases interlocking increases to some extent. Consequently, the greater the denseness, the greater the strength.

Though the ultimate value of Ф′ is not affected by denseness, relative density(D_r) provides the value according to relation,

Ф′= 26^◦ +.2 D_r

4) Confining Pressure

Shear strength increases with an increase in confining pressure. But for the range of pressure in the common field problems, the effect of confining pressure on the angle of shearing resistance is not significance.

Shear Characteristics of Cohesionless Soil

Source of Shear strength of Cohesionless Soil

The friction between the particles cohesionless soil like sand and non-plastic silts provides shear strength. In case dense sands the friction as well as interlocking between particles also contributes to these strength.


Brittle Failure

Study on the stress-strain relationship shows that dense sands exhibits a relatively high initial tangent modulus. The stress reaches a maximum value at its peak at comparatively low strain and then decreases rapidly with an increasing strain and eventually becomes more or less constant.,


The dense sand shows initially a volume decrease in a drained test, but as the strain increases, the volume starts increasing.

In the case of dense sand, the specimen shows a clear failure plane and failure is known as brittle failure.


Plastic Failure

The stress-strain curve for a loose sand exhibits a relatively low initial tangent modulus. At large strains, the stress becomes more or less constant. The loose sand shows shows a volume decrease throughout increment of strains.

In the case loose sand, the specimen bulges (swell out) and ultimately fails by sliding simultaneously on numerous planes. The failure is known as plastic failure.


Determination of Failure Envelope

The failure envelope for dense sand can be derived either for the peak stresses or for the ultimate stresses. The value of the angle of shearing resistance (Φ^′) for the failure envelope for peak stress is considerably greater than for the ultimate stresses.

In the case of loose sands, as the peak stress and the ultimate stress are identical, there is only one failure envelope. The angle of shearing resistance in very loose state is approximately equal to the angle of repose. It should remember that the angle of repose is the angle at which a heap of dry sand stands without any support. It has been established that air-dry sand gives approximately the same value of Φ as the saturated sand. As it is easier to perform tests on dry sand, tests can be performed on dry sand instead of saturated sand.

If the failure envelope is slightly non-linear, a straight line may be drawn for the given pressure range and the angle of shearing resistance is taken as the slope of this line. The cohesion intercept, if any, is usually neglected.

Effect of Mixing Time on Properties of Concrete

The various components of a mix are proportioned so that the resulting concrete has adequate strength, proper workability for placing and cost effective. To achieve such properties the mixing should such that it produce an intimate mixture of cement, water, fine and coarse aggregate and suitable admixture of uniform consistency throughout each batch. The average strength of concrete increases with an increase in mixing time as it improve uniformity of mix.


Conventional Practice

In a construction site , it is common practice to mix concrete as rapidly as possible. But this time varies with type of mixture and basically it is not a function of time but of number of revolutions of the mixing. Generally, about 20 revolution are sufficient. As the manufacturer recommended certain speed of rotation, the number of revolutions and mixing time are independent.


Standard Value of Mixing


For a particular mixer, a relation between mixing time and uniformity is provided. Mixing less than 1 to 1.25 minutes result a significantly variable concrete but prolonging the time of mixing beyond these values render no significant improvement in uniformity. The length of mixing time also depends on quality of blending materials during charging of mixture.


The exact value of mixing time is also a function of size of mixture. A minimum mixing time of 1 minute for mixer size of 1 cum. yd (3/4 cum. m) and 15 additional seconds for each addition cubic yard giving satisfactory uniformity of mixing.

ASTM C 94-94


Mixing time is counted from the time when all the solid materials have been put in the mixer, and it is also required that all the water has to be added not later than after mixing time.

ACI 304R-89


Mixing time should count from the time when all the ingredient have been discharged into the mixer.



Exceptions


1) Many modern large mixer performs satisfactorily with a mixing time of 1 to 1.5 minutes.

2) In high speed pan mixers, the mixing time can be as short as 35 seconds.

3) When lightweight aggregate is used, the mixing time, less than 5 minutes, may hamper developments of strength for better result. Sometimes mixing of aggregate with water for 2 minutes is done first followed by 3 minutes mixing after cement is added.


Effect of Pro-longed Mixing Time

Generally, evaporation of water from the mix takes place resulting decrease in workability and increase in strength. The another effect is grinding of the aggregates, particularly if soft. The grinding makes aggregate more finer resulting lower workability. The friction effect also produces an increase in the temperature of the mix.

In case of air-entrained concrete, prolonged mixing reduces their content by about 1/6 per hour (depending on the type of air-entraining agent), while a delay in placing without continuous mixing causes a drop in air content by only about 1/10 per hour. On the other hand, a decrease in mixing time below 2 or 3 minutes may lead to inadequate entrainment of air.

Electrical Stabilization of Soil

Electrical stabilization of cohesive soil is performed using a process known as Electro-osmosis. A direct current (D.C) is supplied through clayey soil to migrate pore water to negative electrode (cathode). The strength of such soil is increased substantially as water is removed from it.

Electro-Osmosis

Electeo-osmosis is a method which drain water from cohesive soil with the help of direct current(D.C) . The cathode is a well point which collects the water, have to drained from the soil and discharges the water as in a conventional well-point system. 

Mechanism of Electro-Osmosis

Cation (positive ions) are formed in pore water when the dissolved minerals go into solution. These cations move towards the negatively charged surface of clay minerals to satisfy the electrical charge. As the water molecules act as dipoles, the cations also attract the negative end of dipoles. When the cations move to the cathode, they take with them the attached water molecules.

Installation of System

Anodes are in the form of steel rods located near toe of the slope of the excavation. Cathodes are in the form of perforated pipes, resembling well points, installed in the soil mass about 4~5 m away from the slope of cut. The electrodes are so arranged that the natural direction of flow of water is reversed and is directed away from the excavation. This arrangement is required to prevent sloughing of the slopes.

System Requirements

The system requires about 20~30 amperes of electricity per well at a voltage of 40~180v. The consumption of energy is between .5 to 10 KWh/Cum. of soil drained.

Significance of Electro-Osmosis

As considerable amount of water is removed from the soil mass, the strength properties are increased. It is also found that a small reversing of the direction of flow helps in increasing the stability of the slope even if there is no significant decrease in the water content of soil. So this process also increase the slope stability substantially.

Limitation of Electro-Osmosis

This process requires specialized and sophisticated equipment as well as electrical consumption of high amount is associated with this process. Thus it is a highly expensive drainage process compared with other method.

Suitability

This method should be used only in exception cases when other method can not be used. It is normally used to drain water in a cohesive soil of low permeability of

K=1 x 10^-5 to 1 x 10^-8 m/sec

Difference Between Magnitude and Intensity of Earthquakes

The stain energy released during an earthquake travels as seismic waves in all directions through the earth's layer, reflecting and refracting at each interface. These vibrations are more intense nearer the center of disturbance and these become feeble and ultimately die out, as the distance increased. Generally a common problem occur merging magnitude with intensity.

Magnitude:

Magnitude is a quantitative measure of the actual size of the earthquake. Based on maximum displacement, magnitude is determined and through magnitude energy released can be computed.


Intensity :

Intensity is a qualitative measure of the actual shaking at a location during an earthquake. Intensity is determined from three features of shaking -

1) perception by people and animal

2) performance of building

3) changes to natural surrounding.


Basic Difference

Magnitude defines size of a earthquake. As example, the size of an earthquake can be measured by the amount of strain energy released by the fault rupture. On the other hand, intensity is an indicator of the severity of shaking generated at given location. There is no doubt that the severity of shaking is much higher near epicenter than farther away.

This means that the magnitude of the earthquake is a single value for a given earthquake but different locations experience different degree of intensity for same quake.

Requirements For Frames in Regions of High Seismic Risk

In region of high seismic risk, structural frames proportioned to resist forces induced by earthquake motion should satisfy the requirements stated below :


1) Tensile steel ratio in flexural members should be well below balance steel ratio,Pb to ensure adequate rotation capacity at plastic hinges.


2) Use of high strength steel (having limited ductility) should be avoided.


3) Lateral reinforcement in column must serve not only their usual functions of column (ties or spiral) but also serve as shear reinforcement to provide adequate resistance to horizontal forces.

4) As much as possible longitudinal beam reinforcement should be carried through beam- column joints without interruption.

But when required special attention must be paid to bar anchorage .

5) At least minimum amount of flexural reinforcement should be used in both top and bottom throughout the length of all beams to allow for

a) possible shifts in points of inflection

b) load combination not accounted in design.


6) At least minimum amount of web reinforcement should be used throughout length of all beams and closed-hoop reinforcement should be provided in regions which can subject to plastic hinging to improve rotational capacity.

Determination of Effective Span Length

Effective Span length: A structure is generally represented by a simple line diagram using a center line distances between column and between floor beams. But, In practical cases, the column may have considerable width as well as the beam may have considerable depth leaving some modification of the respective length of these members. As width and depth of column and beam respectively, is accounted in these center line diagram, the derived results (moment, shear, torsion etc.) due to self weight and superimposed loads, from analysis is not practical and reasonable one. These difference between their center line distances and clear distances raised concept of effective of effective span length which results in a practical substantially accurate value of the results.

Significance of effective span length:In analysis of Beam as well as column the effective span length provides effective and sometimes economical results.

Beam Analysis

It is usually assumed that the members are prismatic having constant moment of inertia between center

lines,but a beam intersecting a column have a moment of inertia about infinite between column face to column center line. This is due to the consideration that the depth of the beam greatly increased in that region. Thus consideration of actual variation in member depth in analysis produce increased support moment followed by decreased span moment. In addition, it is apparent that critical section for design for negative bending would be at the center line, through an effective depth of unlimited value is available, in the region of support, in the beam.  

Column Analysis

The variation of width and moment of inertia stated above is also applicable for columns.

Nature of Variation

Beam

The slope of moment diagram for the beam is quite steep in the region of the support producing substantial difference between the support center line moment and face moment. If a section is designed considering center line moment, a unnecessary large section will result. It is economical also desirable to reduce support moments by elastic analysis to account for the finite width of the supports.

The difference in moment between support center line and support face is
equal to (Val)/3

Where , l=center line distance between support,

a l= column width

a= a ratio to column width to c/c distance between support.

Column

It is observed that,in the case of columns, the gradient of moment curve is not as steep as that for beam. So the difference between center line moment and the moment at the top and bottom face of the beam is small and in most cases it is disregarded.

Method of Approaches to analysis

Structural designers frequently uses two methods to analysis the frames. These are

1) The structure is simplified using simple line diagram and a deduction of (Val)/2 is done from center line moment without adjusting for the higher stiffness with in the thickness width of the column. This method is less realistic.

2) This involves the consideration of a rigid link within the width of the column, connecting the column center line with clear span of the flexural member. In case of column analysis, the portion of the column within the depth of the beam can also be represented using a rigid link. These is both realistic and easy to implement in matrix analysis programs.

Consequence of This Concept

Consequence of the concepts of effective span is reduction in congestion in the beam-column joint location where it is often difficult to place concrete because of the high quantity of reinforcing steel from the flexural members framing into the column (usually from two different directions) and from the column itself. But, a somewhat higher percentage of reinforcement required at midspan usually causes little difficulty in concrete placement.

Effective span for simply supported and continuous beams are as follows:

1) Simply supported beams

The effective span of a simply supported beam shall be taken as the smaller of the distance between the centers of bearing, or the clear distance between supports plus the effective depth.
2) Continuous beams

If the width of support is less than 1/12 of the clear span, the effective span shall be taken as stated (1) above. If the supports are wider than 1/12 of the clear span or 600 mm, whichever is less, the effective span shall be as follows:

a) For end span with one fixed and the other continuous or for intermediate spans, the effective span shall be the clear span between supports.

b) For end span with one end free and other continuous, the effective span shall be equal to the clear span plus half the effective depth of the beam or the clear span plus half the width of the discontinuous support, whichever is less.

3) Monolithic frames

In case of monolithic frames, the effective span shall be equal to the distance between intersections of the center lines of the connecting members.
4) Cantilever beams


The effective length of the cantilever shall be taken as its length to the face of the supports plus half its effective depth except where it forms the end of a continuous beam where the length to the center of the support shall be used.

Determination of Degree of Hydration of Cement

The degree of hydration of cement can be determined by different means. But, unfortunately, the application of these method to commercial cements is by no means simple. These methods are :

a) The amount of chemically combined water.

b) The amount of unhydrated cement present ( using X-ray quantitative analysis ).

c) The amount of Ca(OH)2 present in the paste.

d) The specific gravity of the paste.

e) The heat evolved by hydration.

f) Indirectly from the strength of the hydrated pasted.

g) Thermogravimetric techniques and continuous X -ray diffraction scanning of wet pastes undergoing hydration gives an idea about early reactions.

h) Back-scattered electron imaging in a scanning electron microscope also can help to study on the microstructure of hydrated cement paste.

SEISMIC WAVES

Earthquake

A type of earth movement that locally gives rise to engineering problems is called the earthquake.Whenever the earth is suddenly struck or disturbed vibration are produced. These vibrations are setup or start from a limited area and are propagated outward in all directions. Thus an earthquake may be defined as the passage of these vibrations in the earth.

Earthquake waves

Two types of waves are produced during earthquake. These are

1. Body waves
2. Surface waves
   

1. Body waves

Body waves consist of two waves. These are

a) Primary waves ( P-waves)

b) Secondary Waves (S-waves)

a) Primary waves

1) Nature : These are longitudinal or compressional in nature. Therefore it is known as longitudinal waves or compressional waves.

2) Direction of Particle Vibration : The rock particles vibrates in the direction of propagation of the waves, with a push and pull effect.

3) Speed : It is the fastest waves and therefore first to be recorded at the recording station. It travels with about the same speed as sound through same rock.

4) Example : In granites, P -waves have speed of about 4.8 Km/Sec.

5) Penetration Capacity : These waves are capable of passing through solids as well as liquides.

b) Secondary Waves :

1) Nature: These are transverse or distortional in nature. Therefore it is known as transverse waves, shear waves or shake waves.

2) Direction of Particle Vibration: The rock particles vibrate at right angles to the direction of propagation like light waves.

3) Speed: These travel slower than the P-waves and are second to be recorded.

4) Example: In granites, S-waves have speed of about 3 km/sec.

5) Penetration capacity: These can pass through solids but it is in capable of passing through liquids.

2. Surface Waves

These waves travel along the earth's surface having similarity in behavior with sea waves.These waves also known as Long waves (L-waves).

Surface waves consist of two waves. These are

a) Love Waves

b) Rayleigh Waves

a) Love Waves

Love waves cause surface motions similar to that by S-waves, but with no vertical component. S-waves in associated with effects of Love waves cause maximum damage to structure by their racking motion on the surface in both vertical and horizontal directions.

b) Rayleigh Waves

Rayleigh Wave makes a material particle oscillate in an elliptic path in the vertical plane (with horizontal motion along direction of energy transmission).

The speed of waves in decreasing sequence :


when P-waves and S-waves reach the earth's surface, most of their energy is reflected back. Some of the energy is returned back to the surface by reflections at different layers of soil and rock. Shaking is more severe (about twice as much) at the earth's surface than at substantial depths. This is often the basis for designing structures buried underground for similar levels of acceleration than those above the ground.

REQUIREMENTS FOR STRUCTURAL INTEGRITY

Structure may subject to adverse effect from local damage from severe local abnormal loads in addition to conventional design load, which are not considered to occur in design life. Such loads include explosion due to gas or industrial liquids, vehicle impact, impact of falling objects and local effect of very winds such as tornadoes. Improving redundancy and ductility of structures by providing minor changes in the detailing of the reinforcement , the overall integrity of reinforced concrete structure to withstand such abnormal loads can be enhanced substantially. This is achieved by providing, as a minimum, some continuity reinforcement or tie between horizontal framing members. In the event of damage to a major supporting element or an abnormal loading event, the integrity reinforcement is intended to continue any resulting damage to a relatively small area, thus improving stability.

General integrity to a structure

Improvement of integrity of a whole structure providing proper ties, creates certain differing observation among engineers for a particular framing system. So providing general structural integrity to a structure is a requirement that can not be stated in simple terms. The code, however , does set forth specific examples of certain reinforcing details for cast-in-situ joist, beams, and two-way slab construction.

Cast-in-place Joists and Beams

When a support is damaged, top reinforcement which is continuous over the support, but not confined by stirrups, will tend to tear out of the concrete and will not provide the catenary action needed to bridge the damaged support. Some catenary action can also be provided, providing a portion of the bottom reinforcement in beams continuous over the supports. By providing some continuous top and bottom reinforcement in edge or perimeter beams, an entire structure can be tied together; also, the continuous tie provided to perimeter beams of a structure will toughen the exterior portion of a structure, should an exterior column be severely damaged.

Fig-1 Continuity Reinforcement for Joist Construction.
Notes:

1. Larger of 1/4(+As1) or 1/4(+As2) continuous or spliced with Class A splices

2. Larger of 1/6(-As1) or 1/6(-As2) continuous or spliced with Class A splices


Fig-2 Continuity Reinforcement for Perimeter Beams.


Notes:

1. Larger of 1/4(+As1) or 1/4(+As2) continuous or spliced with Class A splices

2. Larger of 1/6(-As1) or 1/6(-As2) continuous or spliced with Class A splices

Fig-3 Continuity Reinforcement For Beams Without Closed Stirrups

The following specification should follow to improve integrity of the overall structure :

a) In one-way slab construction, at least one bottom bar shall be continuous or shall be spliced over the support with a class-A tension splice. At non-continuous supports, the bars may be terminated with a standard hook.

b) Beams at the perimeter of the structure shall have at least one-sixth of the tension reinforcement required for negative moment at the support and one-quarter of the positive moment reinforcement required at mid-span made continuous around the perimeter and tied with closed stirrups. Closed stirrups need not be extended through any joints. The required continuity may be provided with top reinforced spliced at mid-span and bottom reinforcement spliced at or near the support with class-A tension splices.

Application Of Sand Drains

Sand drains

Sand drains is a process of radial consolidation which increase rate of drainage in the rate of drainage in the embankment by driving a casing into the embankment and making vertical bore holes. These holes is back filled with suitable grade of sand.

Process of construction of drains

The driven casing is withdrawn after the sand has been filled. A sand blanket is placed over the top of the sand drains to connect all the sand drains. To accelerate the drainage, a surcharge load is placed on the sand blanket. The surcharge is usually in the form of dumped soil.

Mechanism of consolidation

The pore water pressure is increased by the applied surcharge load in the embankment. The drainage occur in the vertical and horizontal directions. The horizontal drainage occure because of sand drains. The sand drains accelerate the the process of dissipation of excess pore water created by the surcharge.

Spacing of drains

The drains are generally laid either in a square pattern or a triangular pattern. The spacing (s) of the drains is kept smaller than the thickness of the embankment (2H) in order to reduce the length of the radial drainage path.

Zone of influence

The zone of influence of each drain in a triangle pattern is hexagonal in plan, which can be approximated by an equivalent circle of radius R, where R = 0.525 S. In case of a square pattern, the radius of circle of influence R is equal to 0.554 S. The radius of the sand drain is represented by r_w.

Theory of sand drains

The theory of sand drains was given by Rendulic (1935) and Barron (1948). Later, Richart (1959) summarized the theories. Depending on the type of strain, there are two cases.

1)  Free strain case.

2)  Equal strain case.

1)  Free strain case

If the surcharge load placed over the sand blanket is flexible, free strain case occurs. In this case, there is uniform distribution of surface loads, but the settlement at the surface is uneven.

2) 2) Equal strain case

This case occurs when the surcharge applied is rigid, such as heavy steel plates. In this case, the settlements are uniform, but the distribution of pressure is non-uniform.

Limitation of sand drain application

Following consideration is not included in design of sand drains:

1) 1) Secondary consolidation is not taken into account in the design of sand drains. In fact, the sand drains are ineffective in controlling the secondary consolidation for highly plastic and organic soils.

2) 2) In case of deriving equation for effectiveness of sand drains, it is not considered that the excess pore water pressure developed, actually in soil where sand drains are exist, is generally less than that of the case having no sand drains. Sand drains tend to act as weak piles and reduce the stresses in the clay.

3) The typical design parameter for sand drain may vary as below :

a) Spacing of sand drains, S = ( 2 ~ 5) m

b) Depth of sand drains, 2 H = (3 ~ 35) m

c) Radius of sand drains well, r_{w = (0.2} ~ 0.3) m

d) Thickness of sand blanket = (0.6 ~ 1) m

STRUCTURE IMPORTANCE CATEGORY

Based on the level of necessity of remaining safe and functional during any post disaster period e.g. after a cyclone, or an earthquake, buildings, structures and related equipments are classified into five structure importance categories such as

Each building or structure shall be placed in one of the structure importance categories and provided with a structure importance coefficient for design against wind earthquake induced forces.  

Essential Facilities :

1. Hospital and other medical facilities having surgery and emergency treatment area.

2. Fire and police stations.

3. Tanks or other structures containing, housing or supporting water or other fire-suppression materials or equipment required for the protection of essential or hazardous facilities, or special occupancy structures.

4. Emergency vehicle shelters and garages.

5. Structures and equipment in emergency-preparedness centres, including cyclone and flood

shelters.

6. Standby power-generating equipment for essential facilities.

7. Structures and equipment in government communication centers and other facilities required for emergency response.

П) Hazardous facilities

Structures supporting or containing sufficient quantities of toxic or explosive substances to be dangerous to the safety of the general public if released.

Ш) Special occupancy structures

1) Covered structures whose primary occupancy is public assembly with capacity >300 persons.

2) Buildings for schools through secondary or day-care centers with capacity >250 students.

3) Buildings for colleges or adult education schools with capacity >500 students.

4) Medical facilities with 50 or more resident incapacitated patients, not included above.

5) Jails and detention facilities.

6) All structures with occupancy > 5,000 persons.

7) Structures and equipment in power-generating stations and other public utility facilities not included above, and required for continued operation.

ІV) Standard occupancy structures

All structures having occupancies or functions not listed above.

V) Low risk structures

Buildings and structures that exhibit a low risk to human life and property in the event of failure, such as agricultural buildings, minor storage facilities, temporary facilities, construction facilities, and boundary walls.

Stress- strain relationship of steel

Besides the strength of material, the stiffness of a material is frequently of equal importance. Other mechanical properties such as hardness, toughness, and ductility also determine the selection of a material but have a lesser degree of importance.

Experimental setup

A specimen, in tension test, is gripped between the jaws of a testing machine. The values of the load and elongation in a specified length, called the gage length, are observed simultaneously. These data are then plotted on a graph with the ordinate representing the load and the abscissa representing the elongation.

Here, plotting is not done for load against extension but for unit load or stress against unit elongation or strain. This facilitates to compare the properties of one specimen with those of other specimens and the resulting diagram is called a stress-strain diagram.

Stress

Stress is defined as load per unit area. Stress is expressed symbolically as
σ = P / A . . . . . . . . . . . . . (1)
where σ = sress
P= applied load
A= cross-sectional area
It is noticed that maximum stress in tension or compression occurs over a section normal to the load.

Units

In SI stress in expressed as MN/m2 or Mpa.
In U.S customary unit, stress is expressed as Ksi ( Kips/in2) and in certain applications, such as soil mechanics it is also common to measure stress in units of psf ( lb/ft2).
Although the equation (1) is fairly simple, it requires care discussion. Dividing load by area does not gives the stress at all points in the cross-sectional area; it merely determines the average stress. A more precise definition of stress is obtained by dividing the differential load dp by the differential area over which it acts:
σ = dp/dA . . . . . . . . . . . . . . (2)
Strain
To obtain the unit deformation or strain, Є, we devise the elongation ∂ by the length L in which it was measured, thereby obtaining

Є =∂/L . . . . . . . . . . . . . . (3)

The strain so computed, however, measures only the average value of strain. The correct expression for strain at any position is

Є = d∂/dL . . . . . . . . . . . . . . (4)

Where, d∂ is the differential elongation of the differential length dL. Thus equation (4) determines the average strain in a length so small that the strain must be constant over the length. However, under certain conditions the strain may be assumed constant and its value is computed from equation (3). These conditions are As follows:

1) The specimen must be of constant cross section.
2) The materials must be homogeneous.
3) The load must be axial, that is, produce uniform stress.

Units

Though strain represents a change in length divided by the original length, strain is a dimensionless quantity. However, it is common to use a units of meters per meter (m/m) or inches per inch (in/in) to represent strain. In engineering work, strains of the order of 1.0 x 10-3 are frequently encountered.
Different concepts developed from the stress –strain curve are as follows :

1) Proportional limit.
2) Elastic limit.
3) Yield point.
4) Ultimate stress or ultimate strength.
5) Rupture strength.


1) Proportional limit

The initial straight line portion from origin o shows proportionality between stress and strain. The point beyond which proportionality ends is called proportional limit. It has been noticed that this proportionality does not extend throughout the diagram. Beyond this point, the stress is no longer proportional to strain. The proportional limit is very important as all subsequent theory involving the behavior of elastic bodies is based on this proportionality, providing a upper limit of usable stress of a material.

2) Elastic limit


This is the stress beyond which the material will not return to its original shape when unloaded but will retain a permanent deformation called permanent set.

3) Yield point


This is the point at which there is an appreciable elongation or yielding of material without any corresponding increase of load; indeed, the load may actually decrease while yielding occurs.

Determination of yield point

The phenomenon of yielding is peculiar to structural steel; other grades of steel and steel-alloy or other materials do not posses it, as in indicated by the typical stress-strain curves of these materials shown in figure

These curves, incidentally, are typical for a first loading of materials that contain appreciable residual stresses produced by manufacturing or aging processes. After repeated loading, these residual stresses are removed and the stress-strain curves become practically straight, as can be demonstrated in the testing laboratory. The yield strength is closely associated with the yield point. For materials that do not have a well defined yield point, yield strength is determined by the offset method.

Offset Method

This consists of drawing a line parallel to the initial tangent of the stress-strain curve, this line being started at an arbitrary offset strain, usually of 0.2% (0.002 m/m or 0.002 in/in). as shown in figure, the intersection of this line, with the stress-strain curves is called yield strength.


Ultimate strength

The ultimate stress or ultimate strength as it is more commonly called is the highest ordinates on the stress-strain curve.

Rupture strength

This is the stress at failure. For structural steel it is somewhat lower than ultimate strength because the rupture strength is computed by dividing the rupture load by the original cross-sectional area, which although convenient, is incorrect. The error is caused the phenomenon known as necking. As failure occur, the material stretches very rapidly and simultaneously narrows down, as shown in figure; so that the rupture load is actually distributed over a smaller area.
If the rupture area is measured after failure occurs and divided into the rupture load, the result is truer value of the actual failure stress. Although this is considerably higher than the ultimate strength, the ultimate is commonly taken as the maximum stress of the material.