Plasticity, Theory of Plasticity ,Creep in concrete ,Creep, Stiffness ,Elasticity

,Plasticity ,Euler-Bernoulli beam equation, Buckling, Ductility VS Malleability,

Ductile Materials, Brittle Materials ,Modulus of Elasticity, Plastic Strain, Tensile

Strength, Yield Strength, Ultimate Strength, Solid mechanics or Mechanics of solids,

Strength of Materials, Types of forces, Normal forces, Fatigue , Resilience, Unit of

Resilience, Modulus of rigidity , Modulus of Resilience, Modulus of Toughness

Poisson’s Ratio.

ENGINEERING MATERIALS RELATED

TERMS

1

ENGINEERING

MATERIALS

RELATED TERMS

CIVIL ENGINEERING TERMS.

It is very important to know the terms described in this book .These

terms are mainly on the “CIVIL “& based on the characteristics of

different types of materials used in construction process.

TAPON CHAKRABARTI

DEDICATING

THE

STUDENTS

OF CIVIL

ENGINEERIN

G, ALL OVER

THE WORLD.

2

Engineering Materials related Terms

PLASTICITY is concerned with the mechanics of materials deformed beyond their elastic limit.

A strong knowledge of plasticity is essential for engineers dealing with a wide range of engineering

problems, such as those encountered in the forming of metals, the design of pressure vessels, the

mechanics of impact, civil and structural engineering, as well as the understanding of fatigue and the

economical design of structures.

THEORY OF PLASTICITY is the most comprehensive reference on the subject as well as the

most up to date -- no other significant Plasticity reference has been published recently, making this of

great interest to academics and professionals. This new edition presents extensive new material on the

use of computational methods, plus coverage of important developments in cyclic plasticity and soil

plasticity, and is accompanied by a fully worked solutions manual.

* A complete plasticity reference for graduate students, researchers and practicing engineers; no other

book offers such an up to date or comprehensive reference on this key continuum mechanics subject

* Updates with new material on computational analysis and applications, new end of chapter exercises

and a worked solutions manual

* Plasticity is a key subject in all mechanical engineering disciplines, as well as in manufacturing

engineering and civil engineering.

CREEP IN CONCRETE is defined as: deformation of structure under sustained load. Basically,

long term pressure or stress on concrete can make it change shape. This deformation usually occurs in

the direction the force is being applied. Like a concrete column getting more compressed, or a beam

bending. Creep does not necessarily cause concrete to fail or break apart. Creep is factored in when

concrete structures are designed

Factors Affecting Creep

1. Aggregate

2. Mix Proportions

3. Age of concrete

1. Influence of Aggregate

Aggregate undergoes very little creep. It is really the paste which is responsible for the creep.

However, the aggregate influences the creep of concrete through a restraining effect on the magnitude

of creep. The paste which is creeping under load is restrained by aggregate which do not creep. The

stronger the aggregate the more is the restraining effect and hence the less is the magnitude of creep.

3

The modulus of elasticity of aggregate is one of the important factors influencing creep. It can be

easily imagined that the higher the modulus of elasticity the less is the creep. Light weight aggregate

shows substantially higher creep than normal weight aggregate.

2. Influence of Mix Proportions:

The amount of paste content and its quality is one of the most important factors influencing creep. A

poorer paste structure undergoes higher creep. Therefore, it can be said that creep increases with

increase in water/cement ratio. In other words, it can also be said that creep is inversely proportional

to the strength of concrete. Broadly speaking, all other factors which are affecting the water/cement

ratio are also affecting the creep.

3. Influence of Age:

Age at which a concrete member is loaded will have a predominant effect on the magnitude of creep.

This can be easily understood from the fact that the quality of gel improves with time. Such gel creeps

less, whereas a young gel under load being not so stronger creeps more. What is said above is not a

very accurate statement because of the fact that the moisture content of the concrete being different at

different age also influences the magnitude of creep.

EFFECTS OF CREEP ON CONCRETE AND REINFORCED CONCRETE:

In reinforced concrete beams, creep increases the deflection with time and may be a

critical consideration in design.

In eccentrically loaded columns, creep increases the deflection and can load to buckling.

In case of statically indeterminate structures and column and beam junctions creep may relieve the

stress concentration induced by shrinkage, temperatures changes or movement of support. Creep

property of concrete will be useful in all concrete structures to reduce the internal stresses due to

non-uniform load or restrained shrinkage.

In mass concrete structures such as dams, on account of differential temperature conditions at the

interior and surface, creep is harmful and by itself may be a cause of cracking in the interior of

dams. Therefore, all precautions and steps must be taken to see that increase in temperature does

not take place in the interior of mass concrete structure.

Loss of prestress due to creep of concrete in prestressed concrete structure.

CREEP:

Deformation that occurs from stresses over long period of time, typically materials exposed to

constant heat will be susceptible to creep.

DEFORMATION:

Change in shape (i.e. length, width, diameter, etc.) of a material due to an applied force.

4

Also fatigue is another form of deformation. Fatigue describes deformation which is caused by

repetitive stress.

STIFFNESS

Stiffness depends upon material properties and geometry. The stiffness of a structural element of a

given material is the product of the material's Young's modulus and the element's second moment of

area. Stiffness is measured in force per unit length (newtons per millimetre or N/mm), and is

equivalent to the 'force constant' in Hooke's Law.

The deflection of a structure under loading is dependent on its stiffness. The dynamic response of a

structure to dynamic loads (the natural frequency of a structure) is also dependent on its stiffness.

In a structure made up of multiple structural elements where the surface distributing the forces to the

elements is rigid, the elements will carry loads in proportion to their relative stiffness - the stiffer an

element, the more load it will attract. This means that load/stiffness ratio, which is deflection, remains

same in two connected (jointed) elements. In a structure where the surface distributing the forces to

the elements is flexible (like a wood framed structure), the elements will carry loads in proportion to

their relative tributary areas.

A structure is considered to fail the chosen serviceability criteria if it is insufficiently stiff to have

acceptably small deflection or dynamic response under loading.

The inverse of stiffness is flexibility.

ELASTICITY

Much engineering design is based on the assumption that materials behave elastically. For most

materials this assumption is incorrect, but empirical evidence has shown that design using this

assumption can be safe. Materials that are elastic obey Hooke's Law, and plasticity does not occur.

For systems that obey Hooke's Law, the extension produced is directly proportional to the load:

Where

x is the distance that the spring has been stretched or compressed away from the equilibrium position,

which is the position where the spring would naturally come to rest [usually in meters],

F is the restoring force exerted by the material [usually in newtons], and

k is the force constant (or spring constant). This is the stiffness of the spring. The constant has units

of force per unit length (usually in newtons per meter)

PLASTICITY

Some design is based on the assumption that materials will behave plastically. A plastic material is

one which does not obey Hooke's Law, and therefore deformation is not proportional to the applied

5

load. Plastic materials are ductile materials. Plasticity theory can be used for some reinforced concrete

structures assuming they are under reinforced, meaning that the steel reinforcement fails before the

concrete does.

Plasticity theory states that the point at which a structure collapses (reaches yield) lies between an

upper and a lower bound on the load, defined as follows:

If, for a given external load, it is possible to find a distribution of moments that satisfies

equilibrium requirements, with the moment not exceeding the yield moment at any location, and

if the boundary conditions are satisfied, then the given load is a lower bound on the collapse load.

If, for a small increment of displacement the internal work done by the structure, assuming that

the moment at every plastic hinge is equal to the yield moment and that the boundary conditions

are satisfied, is equal to the external work done by the given load for that same small increment of

displacement, then that load is an upper bound on the collapse load.

If the correct collapse load is found, the two methods will give the same result for the collapse load.

Plasticity theory depends upon a correct understanding of when yield will occur. A number of

different models for stress distribution and approximations to the yield surface of plastic materials

exist:

EULER-BERNOULLI BEAM EQUATION

The Euler-Bernoulli beam equation defines the behavior of a beam element (see below). It is based on

five assumptions:

(1) Continuum mechanics is valid for a bending beam

(2) the stress at a cross section varies linearly in the direction of bending, and is zero at the centroid of

every cross section.

(3) The bending moment at a particular cross section varies linearly with the second derivative of the

deflected shape at that location.

(4) The beam is composed of an isotropic material

(5) the applied load is orthogonal to the beam's neutral axis and acts in a unique plane.

A simplified version of Euler-Bernoulli beam equation is:

Here is the deflection and is a load per unit length. is the elastic modulus and is

the second moment of area, the product of these giving the stiffness of the beam.

This equation is very common in engineering practice: it describes the deflection of a uniform, static

beam.

Successive derivatives of u have important meaning:

6

Is the deflection.

Is the slope of the beam.

Is the bending moment in the beam.

Is the shear force in the beam.

A bending moment manifests itself as a tension and a compression force, acting as a couple in a beam.

The stresses caused by these forces can be represented by:

Where is the stress, is the bending moment, is the distance from the neutral axis of the beam

to the point under consideration and is the second moment of area. Often the equation is simplified

to the moment divided by the section modulus (S), which is I/y. This equation allows a structural

engineer to assess the stress in a structural element when subjected to a bending moment.

BUCKLING

A column under a centric axial load exhibiting the characteristic deformation of buckling.

When subjected to compressive forces it is possible for structural elements to deform significantly due

to the destabilizing effect of that load. The effect can be initiated or exacerbated by possible

inaccuracies in manufacture or construction.

The Euler buckling formula defines the axial compression force which will cause a strut (or column)

to fail in buckling.

Where

= maximum or critical force (vertical load on column),

= modulus of elasticity,

= area moment of inertia, or second moment of area

= unsupported length of column,

7

= column effective length factor, whose value depends on the conditions of end support of the

column, as follows.

For both ends pinned (hinged, free to rotate), = 1.0.

For both ends fixed, = 0.50.

For one end fixed and the other end pinned, 0.70.

For one end fixed and the other end free to move laterally, = 2.0.

This value is sometimes expressed for design purposes as a critical buckling stress.

Where

= maximum or critical stress

= the least radius of gyration of the cross section

Other forms of buckling include lateral torsional buckling, where the compression flange of a beam in

bending will buckle, and buckling of plate elements in plate girders due to compression in the plane of

the plate.

DUCTILITY VS MALLEABILITY

Ductility is a solid material's ability to deform under tensile stress; this is often characterized by the

material's ability to be stretched into a wire.

Malleability, a similar property, is a material's ability to deform under compressive stress; this is often

characterized by the material's ability to form a thin sheet by hammering or rolling.

Both of these mechanical properties are aspects of plasticity, the extent to which a solid material can

be plastic-ally deformed without fracture.

8

9

10

11

DUCTILE MATERIALS:

Ductile materials will withstand large strains before the specimen ruptures.

Ductile materials often have relatively small Young’s moduli and ultimate stresses.

Ductile materials exhibit large strains and yielding before they fail.

Steel and aluminum usually fall in the class of Ductile Materials

12

BRITTLE MATERIALS:

Brittle materials fracture at much lower strains.

Brittle materials often have relatively large Young’s moduli and ultimate stresses.

Brittle materials fail suddenly and without much warning.

Glass and cast iron fall in the class of Brittle Materials.

MODULUS OF ELASTICITY:

In solid mechanics, Young’s modulus (E) is a measure of the stiffness of a material. It is defined

as the ratio of stress over strain in the region in which Hooke’s Law is obeyed for the material.

Units: Young’s modulus is the ratio of stress, which has units of pressure to strain, which is

dimensionless; therefore Young’s modulus itself has units of pressure.

PROPORTIONAL LIMIT:

The point up to which the stress and strain are linearly related is called the proportional limit.

ULTIMATE STRESS:

The largest stress in the stress strain curve is called the ultimate stress.

RUPTURE STRESS:

The stress at the point of rupture is called the fracture or rupture stress

ELASTIC REGION:

The region of the stress-strain curve in which the material returns to the undeformed state when

applied forces are removed is called the elastic region.

PLASTIC REGION:

The region in which the material deforms permanently is called the plastic region.

YIELD POINT:

The point demarcating the elastic from the plastic region is called the yield point. The stress at yield

point is called the yield stress.

PLASTIC STRAIN:

The permanent strain when stresses are zero is called the plastic strain.

The off-set yield stress is a stress that would produce a plastic strain corresponding to the specified off-

set strain. A material that can undergo large plastic deformation before fracture is called a ductile

material. A material that exhibits little or no plastic deformation at failure is called a brittle material.

13

Hardness is the resistance to indentation. The raising of the yield point with increasing strain is

called strain hardening.

Necking Phenomenon:

The sudden decrease in the area of cross-section after ultimate stress is called necking.

TENSILE STRENGTH

Tensile strength is the stress at the maximum on the stress-strain curve. Or The greatest longitudinal or

axial stress a material can bear without breaking.

Tensile strength is the maximum amount of tensile stress that a material can take before failure. There

are three definitions of tensile strength

Yield strength

Ultimate strength

Breaking strength

YIELD STRENGTH

The stress at which material strain changes from elastic deformation to plastic deformation, causing it

to deform permanently. This point is not a well-defined point. We can say that yield strength is

a strength which a material can bear without permanent deformation.

Yield strength of Structural steel A36 is 250 MPa.

ULTIMATE STRENGTH

The maximum stress a material can withstand. Ultimate tensile strength of Structural steel A36 is 400

MPa.

BREAKING STRENGTH

The stress coordinate on the stress-strain curve at the point of rupture.

DIRECT STRESSES:

The Stresses which are acting normal to the plane of the body are called as normal or direct stresses.

These are called normal because these are acting perpendicular to the plane of the body.

SHEARING STRESSES:

The Stresses which act parallel to the stressed surfaces are called as shearing stresses.

TENSILE STRESSES:

If a straight bar is subjected to a pair of collinear forces acting in opposite direction and coinciding with

the ends of the bar and directed away from the bar, then the bar tends to increase in length and

the stresses developed in the bar will be tensile.

14

COMPRESSIVE STRESSES:

If a straight bar is subjected to a pair of collinear forces acting in opposite direction and coinciding with

the ends of the bar and directed towards the bar, then the bar tends to shorter in length and

the stresses developed will be compressive stresses.

SOLID MECHANICS OR MECHANICS OF SOLIDS

Solid mechanics or Mechanics of solids is the branch of Mechanics, Physics, and Mathematics that

concerns the behavior of solid matter under external actions (e.g., external forces, temperature changes,

applied displacements, etc.). Or it is a branch of science which deals with the internal effects of the

forces on the bodies when they are loaded, from initial point to rupture or break.

STRENGTH OF MATERIALS

Strength of materials deals with the relations between externally applied loads and their internal effects

on the bodies. Purpose of studying strength of materials to ensure that the structure used will be safe

against the maximum internal effects that may be produced by any combination of external loading.

TYPES OF FORCES

Forces that are acting on the body, may be of the following types. Effects of these forces are studied in

solid mechanics or mechanics of solids.

NORMAL FORCES

The forces that are acting perpendicular to the body are known as normal forces.

Normal forces are of two types:

1. Tensile

2. Compressive.

TANGENTIAL FORCES

The forces that are acting parallel to the body are known as tangential forces. Shear forces and frictional

forces are example of tangential forces.

INDEPENDENT FORCES

Active forces are independent forces.

DEPENDENT FORCES

Reactive forces are dependent forces.

CONCENTRATED FORCES

15

If forces are acting on an area of a body which is negligible as compared to the body, then these are

known as concentrated forces.

DISTRIBUTED FORCES

If forces are acting on an area of a body which is not negligible as compared to the body, then these are

known as distributed forces. These may be of uniformly distributed load (UDL) or uniformly varying

load (UVL) or any kind of general loading.

EFFECTS OF FORCES

The effects of forces which are studied in solid mechanics or mechanics of solids are

1. Translation

2. Rotation (moment and torque)

FATIGUE

In materials science, fatigue is the weakening of a material caused by repeatedly applied loads. It is the

progressive and localized structural damage that occurs when a material is subjected to cyclic loading.

The nominal maximum stress values that cause such damage may be much less than the strength of the

material typically quoted as the ultimate tensile stress limit, or the yield stress limit.

Fatigue occurs when a material is subjected to repeated loading and unloading. If the loads are above a

certain threshold, microscopic cracks will begin to form at the stress concentrators such as the surface,

persistent slip bands (PSBs), and grain interfaces.[1] Eventually a crack will reach a critical size, the

crack will propagate suddenly, and the structure will fracture. The shape of the structure will

significantly affect the fatigue life; square holes or sharp corners will lead to elevated local stresses

where fatigue cracks can initiate. Round holes and smooth transitions or fillets will therefore increase

the fatigue strength of the structure.

RESILIENCE

Resilience is the ability of a material to absorb energy when it is deformed elastically, and release that

energy upon unloading. Proof resilience is defined as the maximum energy that can be absorbed within

the elastic limit, without creating a permanent distortion. The modulus of resilience is defined as the

maximum energy that can be absorbed per unit volume without creating a permanent distortion. It can

be calculated by integrating the stress-strain curve from zero to the elastic limit. In uniaxial tension,

Where Ur is the modulus of resilience, σy is the yield strength, and E is the Young's modulus.

16

UNIT OF RESILIENCE

Resilience (Ur) is measured in a unit of joule per cubic meter (J·m–3) in the SI system, i.e. elastical

deformation energy per volume of test specimen (merely for gage-length part). Like the unit of

tensile toughness (UT), the unit of resilience can be easily calculated by using area underneath the

stress–strain (σ–ε) curve, which gives resilience value, as given below.

Fig: The area under the linear portion of a stress-strain curve is the resilience of the material Typical

Stress vs. Strain diagram for a ductile material (e.g. Steel).

MODULUS OF RIGIDITY

Is defined as the ratio of (a) longitudinal stress to longitudinal strain (b) shear stress to shear strain (c)

stress to strain (d) stress to volumetric strain just as the modulus of elasticity, E, relates tensile stress to

tensile strain, the modulus of rigidity, G, and relates shear stress to shear strain.

The modulus of rigidity, G, is, for isotropic materials, related to E as G = E/ (2(1+u)) where u = Poisson

ratio which varies from 0 to 0.5 and is usually 0.25-0.33 for many metals. Tensile stress = Ee e = tensile

strain

shear stress = Gk k = shear strain

WHAT IS A BULK MODULUS FOR A PERFECTLY RIGID BODY?

It is infinity. Now Bulk Modulus= Stress/Strain Now Strain in a perfectly rigid body=0Therefore Bulk

Modulus= Infinite for a perfectly rigid body

WHAT IS THE DIFFERENCE BETWEEN MODULUS OF ELASTICITY AND RIGIDITY?

We knew from Hook's law- "stress is proportional to strain."

So, stress = k * strain [here, k is a constant]

or, stress/strain= k

Now, if the stress and strain occurs due to axial force

then k is known as modulus of elasticity and it is denoted by E.

if the stress and strain occurs due to shear force

then k is known as modulus of rigidity and it is denoted by G.

RELATION BETWEEN YOUNG'S MODULUS SHEAR MODULUS AND BULK MODULUS?

Let young's modulus = E, Shear modulus = G, Bulk Modulus = K and poisson's ratio = v E = 3K(1-

2v) E = 2G(1+v)

17

WHAT IS THE MODULUS OF ELASTICITY OF WATER?

Pure de-aired water has a bulk modulus equal to approximately 2.2 GPa.

There is a common misconception that fluids are totally incompressible, however as can be seen from

the above this is not true (if it were, the bulk modulus would be infinitely high). It is reasonable to state

that water is highly resistant to compression however.

It should also be noted that the presence of dissolved gasses in water can significantly reduce this value

so consider carefully the application or system being modelled before choosing an elastic modulus for

water or any other fluid.

WHAT IS THE ELASTIC MODULUS OF LIMESTONE?

Intact Limestone can have a Young's modulus (E) ranging from:

9 GPa - 80 GPa.A,B

Poisson's ratio (v) for intact specimens varies from between:

0.2 - 0.3.B

Bulk modulus (K) derived from the above values using the following relation:

K = E / (3(1-(2v))

Bulk Modulus ranges from:

5 GPa - 66.67 GPa Shear modulus (G) derived from the above values using the following relation:

G = E / 2(1+v)

Shear modulus ranges from:

3.5 GPa to 33.33 GPa

Limestone is a natural earth material and so significant variability in stiffness properties may occur, as

such ranges of values are commonly quoted for the strength of rock and other geotechnical materials.

Ideally therefore the user would have access to specific lab test data for the rock type in question.

MODULUS OF RESILIENCE

The work done on a unit volume of material, as a simple tensile force is gradually increased from zero

to such a value that the proportional limit of the material is reached, is defined as the modulus of

resilience. This may be calculated as the area under the stress-strain curve from the origin up to the

proportional limit and is represented as the shaded area in the figure below.

18

MODULUS OF TOUGHNESS

The work done on a unit volume of material as a simple tensile force is gradually increased from zero

to the value causing rupture is defined as the modulus of toughness.

POISSON’S RATIO

When a bar is subject to a simple tensile loading there is an increase in length of the bar in the direction

of the load, but a decrease in the lateral dimensions perpendicular to the load. The ratio of the strain

in the lateral direction to that in the axial direction is defined as Poisson’s ratio.