Columns and struts

Definitions:

• Columns and Stanchions: Vertical compression members in buildings.
• Struts: compression members in roof trusses
• Beam: Jib of a crane.
• Beam column: Co beam that is acted on by an axial compressive force in addition to transversely applied loads.

Classification:

• Short Column: A column that fails essentially by direct crushing at ultimate load.

Crushing load Pc = fc.A,

fc= ultimate crushing stress.

• Long columns:

1. Members considerably long in comparison of lateral dimensions.
2. The member essentially fails by buckling or crippling to bending.

Radius of gyration: 

Slenderness Ratio : Effective length/least radius of gyration.

Significance: As slenderness ratio increases, permissible stress or critical stress reduces, consequently, load carrying capacity also reduces.

• Radius of gyration will be least along major axis of cross section.

Eg: for a rectangular column along yy -axis

• For a given area, Tubular section will have maximum radius of gyration.
• H-Section is more efficient than I-Section.
  

Equilibrium of a column: A column is said to have buckled or failed when it reaches “Neutral Equilibrium”.

Euler’s Theory:

• Critical load: The smallest force at which a buckled shape is possible. Prior to this load the column remains straight. The columns buckle in the plane of the major axis of the cross section as shown below:
  
• Assumptions:

1. Column is initially perfectly straight and is axially loaded.
2. Section of column is uniform
3. The material is perfectly elastic, homogeneous, isotropic and obeys hook’s law.
4. Length of column is very large compared to lateral dimension.
5. Direct stress is small compared to bending stress corresponding to buckling condition.
6. Self weight of column is ignorable.
7. The column will fail by buckling alone.

• Euler’s formula for general case: For a general case critical load,
  

Where

l= Effective length

I = Moment of Inertia of section about the axis of least resistance.

E = Young’s Modulus.

• Effective length and critical loads for various boundary conditions compared to a column whose both ends are hinged.

l = Eff. Length

L = actual length

• Limitations of Euler’s formula:

1. Euler’s formula can also be written as
   
   As f and E are constant for a particular material, Euler’s formula is valid for a particular range of slenderness ratio, for e.g. for mild steel whose fc = 3300 Kg/cm2 and E = 2.1 x 106 Kg/cm2 Euler formula is not valid for slenderness ratio less than 80.
2. Euler’s formula are valid only up to proportional limit i.e., in linear inelastic zone

Note:

• The relation between slenderness ratio and corresponding critical stress is hyperbolic.
• According to Euler formulas the critical load does not depend upon strength property of material the only material property involved is the elastic modules ‘E’ which physically represents the stiffness characteristics of the material.

Rankine’s formula:

• It is an empirical formula.
• Takes into account both direct crushing (Pa) load and Euler critical load( PE)
  

Basic Formula :

Rankine’s Co-efficient: is independent of geometry and end conditions, can be modified to incorporate imperfections

Material	fy	Rankine’s Constant
Mild steel	3200	1/7500
Wrough Iron	2500	1/9000
Cast Iron	5500	1/1600

• Rankine’s formula is valid for any type of column.
• No limitations for slenderness ratio.

Straight line formula : (Johnson’s straight line formula)(Empirical)

• It is assumed that allowable stress depends or L/r (slenderness ratio) and varies in a straight line fashion. Applicable to small slenderness ranges

P=A[f-n( λ)]

P = safe load on the column.
A = cross sectional area of column
f= allowable stress in column material
n = constant depends on the material

λ= slenderness ratio

Parabolic formula: (Johnson’s straight line formula) (Empirical)

• Allowable stress is assumed to vary as (L/r)2

P= A[f - Bλ2]

Where:

P = safe load on the column

A= cross sectional area of column

F = allowable stresses in the column material

λ = slenderness ratio

Eccentrically loaded columns:

• Euler’s formula
  

Where

Where σmax = critical stress in the column

P = axial load on the column

e = eccentricity of the column load

I = effective length of column

EI= flexural rigidity

• Rankine’s method:
  

P=Rankine’s load

f= allowable crushing strength of material

e = eccentricity of loading

Yc = distance of compression fibre from centriod

λ = slenderness ratio

r = least radius of gyration with respect to minor axis

• Secant formula:

For standard pinned column

σmax = maximum stress is located at the extreme compression fiber of the middle point (x = L/2)

of the column.

Validity: It applies to column of any length provided the maximum stress does not exceed

the elastic limit.

NOTE:

• In the above equation (‘r’ may not be minimum since it is obtained from the value of ‘I’ associated with the axis around which bending occurs.
• The relation between σmax and ‘P’ is not linear σmax increases faster than ‘p’ . Therefore the solutions for maximum stresses in columns caused by different axial forces cannot be super posed instead, the forces must be superposed first, and then the stresses can be calculated.

Perry’s formula: (approximate formula)

Where

σ = permissible stress in the material(given)

From the above equation ao can be calculated with that safe load is equal to P = σ0A

σ = maximum permissible stress

σo = stress due to direct load

σE = stress due to Euler’s critical load

Yc = distance to extreme compression fibre.

Core of a cross section: The area with in which a direct load to act, so as not to cause tension in any part of cross section.

• Rectangle or square: Middle third rule Core is a rhombus whose diagonals are d/3 and b/3 Are of core = bd/18 i.e., 1/18th of total area.
  
• >> Solid circular section:

1. middle fourth rule
2. core dia is d/4 (i.e., eccentricity limit is d/8 to avoid tension) and core area is 1/16th  of area of circular section.

• >> Hollow circular section

External dia = D

Internal dia = d