PRESENTATION ON THEORY OF STRUCTURE

Subject code- 1615501UNIT-01

Direct And Bending Stresses

INTRODUCTION

Whenever a body is subjected to an axial tension orcompression, a direct stress comes into play at every section ofbody. We also know that whenever a body is subjected to abending moment a bending moment a bending stress comes intoplay .

A little consideration will show that since both these stressesas normal to a cross-section, therefore the two stresses may bealgebraically added into a single resultant stress.

STRESSES

Every material is elastic in nature. That is why, whenever some external system of forces acts on a body, it undergoes some deformation. As the body undergoes deformation, its molecules set up some resistance to deformation. This resistance per unit area to deformation is known as stress.

σ = P/A

Where, P - load or force acting on the body, and A - Cross-sectional area of the

body.

In S.I system, the unit of stress is Pascal (Pa) which is equal to 1 N/m2.

Combined Stress

• We have studied a number of separate situations (tension, compression, direct, bending, torsion, pressure in cylinders and spheres.)

• In order to find the combined effect we have to look at an element of material at particular locations, where both effects determine the stresses. We calculate the stresses as though they occurred separately, and then combine them to find the overall effect expressed as Principle stresses.

Torsion and Bending

Tension and Compression

.

Combined bending and direct of

a stocky strut:

• Consider a short column of rectangular cross

section. The column carries an axial

compressive load P, together with bending

moment M, at some section, applied about

the centroidal axis Cx

• The area of the column is A, and Ix is the second moment of the area about Cx . If P acts alone, the average longitudinal stress over thesection is

(–P/A)

• The stress being compressive. If the couple M acts alone, and if the material remains elastic, the longitudinal stress in any fiber a distance from Cx is (-My/Iy)

Clearly the greatest compressive stress occurs in the upper extreme

fibers, and has the value,

Eccentric Loading:

A load, whose line of action does

not coincide with the axis of a column or a strut, is known as an

eccentric load.

Ex:

A bucket full of water, carried by a person in his hand,

then in addition to his carrying bucket, he has also to lean or

bend on the other side of the bucket, so as to counteract any

possibility of his falling towards the bucket. Thus we say that he

is subjected to

Direct load, due to the weight of bucket

Moment due to eccentricity of the load.

Beam Mode

Limit of Eccentricity• When an eccentric load is acting on a column, it

produces direct stress as well as bending stress. On

one side of the neutral axis there is maximum stress

and on the other side of the neutral axis there is a

minimum stress.

• A little consideration will show that so long as the

bending stress remains less than direct stress, the

resultant stress is compressive. If the bending stress

is equal to the direct stress, then there will be a

tensile stress on one side.

• Though cement concrete can take up a small tensile

stress, yet it is desirable that no tensile stress should

come into play

• e ≤ Z/A

• It means that for tensile condition, the eccentricity

should be less than (Z/A) or equal to (Z/A). Now we

shall discuss the limit for eccentricity in the

following cases,

LIMITS

• Limit of eccentricity for a rectangular section

• No tension condition,

• e ≤ d/6

• Limit of eccentricity of a hollow rectangular section

• No tension condition,

• Limit of eccentricity of a circular section, e ≤ d/8 Limit of

eccentricity for hollow circular section

• e ≤

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PRESENTATION ON THEORY OF STRUCTURE

Subject code- 1615501

UNIT-02

Slope And Deflection

Slope And Deflection

• SLOPE: It is angular shift at any point of the

beam between no load condition and loaded beam.

Its value is different at different points on the

length of the beam. It is represented by dy/dx or θ.

Its units are radians. There is a maximum limit for

slope for any loaded beam.

• Deflection: It is the vertical shift of a point on the

beam between no load condition and loaded beam.

Its value is different at different points on the

length of the beam. It is represented by y or 𝜹. Its

units are mm. There is a limit for maximum

• METHODS TO FIND SLOPE AND

DEFLECTION

1. Double Integration Method: It is valid for finding

slope and deflection for one load at a time. Thus it

is time consuming.

2. Macaulay’s Method: Uses SQUARE BRACKETS.

It is applicable for any number and any types of

loads.

STIFFNESS OF BEAM

• In structural engineering, beam stiffness is a beam’s ability to resist deflection, or bending, when a bending moment is applied. A bending moment results when a force is applied somewhere in the middle of a beam fixed at one or both ends. It will also occur if a torque is applied to the beam, although this is less common in real-world applications. Beam stiffness is affected by both the material of the beam and the shape of the beam’s cross section.

Slope and Deflection of Beam

Deflection of Beam

DOUBLE- INTEGRATION

METHOD• Double- integration method is that it produces the equation for

the deflection everywhere along the beams. semigraphical

procedure that utilizes the properties of the area under the

bending moment diagram.

THE DOUBLE INTEGRATION

METHOD

• The Double Integration Method, also

known as Macaulay’s Method is a powerful

tool in solving deflection and slope of a

beam at any point because we will be able

to get the equation of the elastic curve. In

calculus, the radius of curvature of a curve

y = f(x) is given by

MACAULAY'S METHOD

• Macaulay's method (the double

integration method) is a technique used in

structural analysis to determine

the deflection of Euler-Bernoulli beams.

Use of Macaulay's technique is very

convenient for cases of discontinuous

and/or discrete loading.

• When the loads on a beam do not conform to standard cases, the

solution for slope and deflection must be found from first

principles. Macaulay developed a method for making the

integrations simpler.

The basic equation governing the slope and deflection

of beams is

= M Where M is a function of x.

1. Write down the bending moment equation placing x on the extreme right hand end of the

beam so that it contains all the loads. write all terms containing x in a square bracket.

= M = R1[x] - F1[x - a] - F2 [x - b] - F3 [x - c]

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PRESENTATION ON THEORY OF STRUCTURE

Subject code- 1615501

UNIT-03

Fixed Beam

Fixed beamA fixed beam is one with ends restrained from rotation.

In reality a beams ends are never completely fixed, as

they are often modeled for simplicity. However, they

can easily be restrained enough relative to the stiffness

of the beam and column to be considered fixed.

ADVANTAGE &

DISADVANTAGE• The advantages are that you reduce the saging

moment in the beam thus also reducing the deflection.

• The disadvantages are that you are causing moment at

the top over supports thus you will need some

reinforcing in the top of the beam.

Principle of superpositionThe principle of superposition simply states that on a

linear elastic structure, the combined effect of several

loads acting simultaneously is equal to the algebraic

sum of the effects of each load acting individually.

Sf and bm diagram

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PRESENTATION ON THEORY OF STRUCTURE

Subject code- 1615501

UNIT-04

Continuous Beam

CONTINUOUS BEAM

• A continuous beam is a structural component that

provides resistance to bending when a load or force

is applied. These beams are commonly used in

bridges. A beam of this type has more than two

points of support along its length.

CONTINUOUS BEAM

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PRESENTATION ON THEORY OF

STRUCTURE

Subject code- 1615501

UNIT-05

Moment Distribution Method

INTRODUCTION

The end moments of a redundant framed

structure are determined by using the

classical methods, viz. Clapeyron’s

theorem of three moments, strain energy

method and slope deflection method.

These methods of analysis require a

solution of set of simultaneous equations.

Solving equations is a laborious task if the

unknown quantities are more than three in

number. In such situations, the moment

distribution method developed by

Professor Hardy Cross is useful. This

method is essentially balancing the

moments at a joint or junction. It can be

described as a method which gives

solution by successive approximations of

slope deflection equations.

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In conclusion, when a positive moment M is applied to

the hinged end of a beam a positive moment of Ê1ˆ M

will be transferred to the fixed end. Á̃

Consider a two span continuous beam ACB as shown in Fig. 2.2(a). A and B are fixed supports with a prop at C. A moment is applied at C and it is required to know how much moment is distributed between spans AC and CB. Let this moment M be decomposed and distributed as M1 to CA and M2 to CB as shown in Fig. 2.2(b).

i.e.

M1 +M2 =M

As the ends A and B are fixed; the slope between A and B is zero. That is, the area of the bending moment diagram between A and B is zero.

BASIC DEFINITIONS OF

TERMS IN THE MOMENT

DISTRIBUTION METHODStiffness

Rotational stiffness can be defined as the moment required to rotate through a unit angle (radian) without translation of either end.

(b) Stiffness Factor

• (i) It is the moment that must be applied at one end of a constant section member (which is unyielding supports at both ends) to produce a unit rotation of that end when the other end is fixed, i.e. k

= 4EI/l.

• (ii) It is the moment required to rotate the near end of a prismatic member through a unit angle without translation, the far end being hinged is k = 3EI/l.

(b)Carry Over Factor

It is the ratio of induced moment to the applied moment (Theorem 1). The carry over factor is always (1/2) for members of constant moment of inertia (prismatic section). If the end is hinged/pin connected, the carry over factor is zero. It should be mentioned here that carry over factors values differ for non-prismatic members. For non-prismatic beams (beams with variable moment of inertia); the carry over factor is not half and is different for both ends.

(d) Distribution Factors

Consider a frame with members OA, OB, OC and OD rigidly connected at O as shown in Fig. 2.6. Let M be the applied moment at joint O in the clockwise direction. Let the joint rotate through an angle . The members OA,OB,OC and OD also rotate by the same angle θ.

BASIC STAGES INTHEMOMENT

DISTRIBUTION METHODThe moment distribution method can be illustrated with the following example.

It is desired to draw the bending moment diagram by computing the bending moments at salient points of the given beam as shown below.

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NEW GOVERNMENT

POLYTECHNIC,PATNA-13

• PRESENTATION ON THEORY OF STRUCTURE

• Subject code- 1615501

• UNIT-06

• COLUMNS

COLUMN

A column is a structural element that transmits, the weight

of the structure above to other structural elements or

foundations through compression .

MECHANISM

1.The vertically gravity load acts on aslab.

2.Which transfer the load to thebeams.

3. Which in turn transfer the load tothecolumn.

4. Then down to the foundations.

TYPES OF COLUMN

.

Rectangle column

Square column

Circular column

Polygon column

Based on pattern of lateral reinforcement

Tied columns Spiral columns

Based on materials

RCC Column Steel Column Stone Column Timber Column

PRECAUTIONS OF COLUMN

CONSTRUCTION

1. Size of column.

2. Clear covering.

3. Proper curing of RCC column.

Assumptions made in Euler's Theory

• The column is initially perfectly straight and is axially loaded.

• The section of the column is uniform.

• The column material is perfectly elastic, homogeneous and isotropic and obeys Hooke's Law.

• The length of the column is very large compared to the lateral dimensions.

• The direct stress is very small compared with the bending stress corresponding to the buckling condition.

• The self-weight of the column is neglected.

• The column will fail by buckling only.

RANKINE’S THEORY

• Rankine's Theory assumes that failure will occur when the maximum

principal stress at any point reaches a value equal to the tensile stress in a

simple tension specimen at failure.

CRIPPLING LOAD

• The crippling load, or more frequently called Buckling load, is the load over which a column prefers to deform laterally rather than compressing itself. Buckling is not about going over the maximum compressive stress, it is rather about the structure finding a geometrically stable alternative to being compressed.