undefined | unit 5 axially and eccentrically loaded columns

MOS

Unit - 5

Axially and Eccentric loaded column

Q1) Explain axially loaded column and concept of critical and buckling load.

A2)

Axially Loaded Column:

Column is a vertical structural member carrying an axial compressive load.

It is the main member of the structure It is usually long as compared to its cross section. There is possibility of buckling of the column.

Strut is a structural member carrying an axial compressive load.

It is a component member of truss or mechanism. It is usually short in length. There is no possibility of buckling of struts. Load carried by strut is small as compared to that by a column.

It may be horizontal, inclined or even vertical.

2. Concept of critical load and buckling:

The load at which the member just buckle is called as buckling load or crippling load or critical load as shown in fig.

Q2) Explain Euler’s formula for buckling load with hinged ends.

A2) Euler’s formula for buckling load with hinged ends

Consider a column AB of length L with uniform cross-section area A as shown in fig.

Let P = Crippling load at which column is just buckled

Consider a section at a distance x from lower end A

Let y be the lateral displacement at this section.

Moment due to critical load P about x, i.e. M = - Py

The bending moment equation at any point is given by EI

EI = -Py

EI + Py = 0

+ = 0

The solution of the above differential equation is,

Y = +

Where and are the integral constants.

Applying boundary conditions,

At A, x = 0, y = 0

2. At B, x = L, y = 0

Since is zero, the will not be zero.

If and both are zero, the column will not ends.

= 0 = 0,

Considering the least practical value

= = P =

Q3) Explain Rankin’s formula in detail.

A3) Rankin’s Formula:

It is commonly known as Rankin-Gordon Formula.

Rankin devised an empirical formula applicable for both short and long column.

Short column fails by crushing load

A =

where

= Crushing stress

Long column fails by buckling load

In practice, the column or struts are fails due to combined effect of crushing stress and bending stress.

= + =

Rankin’s constant is property of the material of the column. Value of rankines constant a and crushing value for some materials are given below:

Sr.No.	Material	Crushing Stress	Rankines Constant
1	Wrought Iron	250 MPa
2	Cast Iron	550 MPa
3	Mild Steel	320 MPa
4	Timber	50 MPa

Q4) Explain limitations of Euler’s formula.

A4) Limitations of Euler’s Formula:

Crippling stress =

The critical stress is directly proportional to the modulus of elasticity of material and it is the inversely proportional to the square of slenderness ratio of the column.

The graph plotted between critical stress and slenderness ratio as shown in fig

When the slenderness ratio decreases, the critical stress increases.

The value of

at the point c is called as critical value of slenderness ratio.

For any slenderness ratio above this value, column fail due to buckling and for any value of slenderness ratio less than this value, the column fails by crushing and not by buckling.

Euler’s formula not applicable for short column, this is the limitation of Eulers formula.

Mathematically, Euler’s formula is applicable,

If crushing stress Buckling stress

When slenderness ratio is greater then , the Eulers formula is applicable. This is Eulers limitation.

Q5) What is the safe load on column and explain what do you mean by eccentricity?

A5)

Safe Load on Column:

An axially loaded column, the load is applied at the centroid of the section and in a direction parallel to the longitudinal axis of the column.

The term centrally loaded and eccentrically loaded are also used for axially loaded column.

An ideal column assumes to be perfectly straight and is centrally loaded.

The axial load which is sufficient to keep the column in such a slight deflect shape is called a critical load.

A column is defined as a structural member subjected to compressive force in a direction parallel to its longitudinal axis. Columns re structural members that are loaded parallel to their length. Most columns are vertical and are used to carry load from the higher level to the lower level

The column carries a structures load without buckle or fail or deflect is a safe load of a column.

2. Eccentricity:

It is a horizontal distance between the line of action of force and the centroidal axis.

It is denoted by e.

It produces bending stresses in the member.

Q6) Explain Direct Load and Eccentric Load in detail

A6)

Direct Load:

The load which is coincide with the centroidal axis of the column is called as direct load. It produces direct stress only.

2. Eccentric Loading:

The load whose line of action does not coincide with the axis of column is called as an eccentric loading. It produces buckling of column.

Due to eccentric load, both direct stress and bending stress induced.

Combined maximum stress = direct stress bending stress

Maximum stress = + ( Compressive)

Minimum Stress = - (Tensile)

Q7) Explain the various loads on chimney.

A7) Chimney:

Let a chimney having cross-sectional area A and Height h subjected to uniform wind pressure

Weight of chimney = W = Weight density Volume

Direct stress at base,

= =

Let wind pressure act on the surface area comes in contact produced bending of structure.

Total wind pressure = Coefficient of wind pressure Wind pressure Projected area

P = C

Where C = Coefficient of wind pressure

For plane surface C = 1 and for circular surface C = 2/3

Bending moment at base

M = total wind pressure distance of its application from base.

= P

Bending Stress = =

Stress at base

= Direct stress Bending stress =

Maximum stress at base = +

Minimum stress at base = -

Q8) Explain the effect of various forces on dam.

A8) Dam:

Consider a trapezoidal dam having top width ‘a’ and bottom width ‘b’ and height ‘H’.

Weight of dam per meter length = Weight density Area Length

= H

The weight of the dam act at a distance x from vertical face.

X =

Consider dam is full of water on its vertical face.

Water pressure at the base p =

Horizontal force exerted by water = Area of pressure diagram

P = base height =

P =

It is acting a distance from base i.e. at the C.G. of pressure triangle.

Let R be the resultant reaction of Horizontal force P and weight of dam W and passes through point D at a distance Z from A.

Moment about point C

P = = Wx

X =

Resultant cuts the base of dam at C from A

Z = + x

Eccentricity of weight component

e = Z -

Moment on the base section

M = W.e

Section modulus of the base section

Z = = =

Bending stress = =

Maximum stress =

Minimum stress =

Q9) Explain the effect of self-weight and lateral force in detail.

A9)

Effect of Lateral Force:

Lateral forces are live loads that’s are applied parallel to the ground: that is, they are horizontal forces acting on the structure.

They are different to gravity loads for example which are vertical downward forces.

The most common types are wind load, seismic load, water and earth pressure.

Lateral loads such are wind load, water and earth pressure have the potential to become an uplift force.

2. Effect of self-weight:

To study the effect of self-weight on the optimal topologies weight of the structure is also include in algorithm and the value of point load is varied from 200 to 10% of self-weight of structure.

The value compliance for static load 200% of the self-weight is 433.57Nm whereas for only self –Weight is 15.43Nm.

Q10) Explain the stress distribution for rectangular section.

A10) A short column of rectangular section of crosses sectional area A and axial load P is as shown in fig. The intensity of stress is uniform. The stress produced is called as direct stress.

Direct stress =

In figure, a short column of rectangular cross-section is subjected to an eccentric load P.

Let P = Load acting on the column

e = Eccentricity of the load

b = width of column

d = depth of column

Moment M = P.e

Area A = b d

M.I. of section about an axis through its c.g. and parallel to axis about which the load is eccentric, is

I =

Modulus of section, z = =

Now, Direct stress, = =

Total stress at extreme fibre

= ………… In terms of Eccentricity

= ………….. In terms of Modulus of section

Maximum stress at A and B is

= ………… In terms of Eccentricity

= ………….. In terms of Modulus of section

Minimum stress at C and D is

= ………… In terms of Eccentricity

= ………….. In terms of Modulus of section

Q11) Explain conditions for stress distribution at base.

A11) Condition for stress distribution at base:

, the stress throughout the section will be of same nature i.e. compressive. Refer fig. (a)

, the stress throughout the section will be same nature i.e. compressive.

, the stress will be partly tensile and partly compression.

Q12) Explain the condition for no tension.

A12)

Condition for no tension:

When the eccentric load acting on a column, I produced both direct stress and bending stress.

If the

, the resultant stress is compressive.

, the resultant stress in the section is partly compressive and partly tensile.

, the resultant stress still In the compressive such that

and

= 0

Tensile stress at the base of the column is harmful as it produce tensile crack

To avoid this tensile crack or for no tension condition, direct stress should be greater than or equal to bending stress.

For no tension condition,

Hence, for no tension condition, eccentricity should be less than or equal to

2. Rectangular Section

For no tension e = Area A = bd

Sectional modulus

= =

e and e

2e and 2e

For no tension condition, the load must lies within the middle third shaded area of eccentricity 2e shown in fig. The central shaded portion is known core or kernel of section.

3. Circular Section:

Sectional modulus Z=