COLUMNS & STRUTS

COLUMNS & STRUTS

Column is a vertical member under axial compressive load. Type of column is decided by a slenderness ratio defined later. For various types of columns, value of slenderness ratio differ with different materials of the column.

For example, Values of slenderness ratio

Type of column

 STEEL

 WOOD

 RCC

Short

≤30

  ≤ 10

   ≤  30

 Medium

≥ 30 & ≤ 120

  ≥ 10 & ≤ 30

  ≥ 30 & ≤ 50

Long

  ≥ 120

  ≥ 30

  ≥  50

Firstly short column

Secondly Medium column

Thirdly long column

SHORT COLUMN

Fail by contraction in case of ductile column. These fail by crushing in case of brittle materials.

Formula used is P = σyp A               ductile materials

P = σ ultimate A                                  brittle materials

Short column have the maximum load carrying capacity.

LONG COLUMNS

These columns fail by lateral displacement at a much lower stress than the yield stress. This failure due to lateral displacement is called buckling. Buckling is also referred as instability. Long columns have the least load carrying capacity. If possible, long column may be avoided. Euler equation governs long columns.

EULER’S FORMULA FOR LONG COLUMNS

PEULER=P critical=P crippling=P buckling= π2EA/(Le/k min)2

PEULER=P cr = P crippling=P buckling ,       N (NEWTONS)

E = Young’s modulus, N/mm2

Take E steel= 200 x 1000 N/mm2

A Area of cross section in mm2

Le is effective length. It is the length in which BOW is completed as shown.

 

 

 

 

 

 

 

Le/k min = Slenderness ratio, no units

Le = L         Both ends are hinged/pivoted

Le = L/2,    both ends are Fixed/ En-castered/Built in

Le = L/2,    one end is fixed and the other is hinged

Le = 2L,      one end is fixed and the other is free (CANTILEVER)

k min = (db3/12),      rectangular column, d  is larger side and b is smaller side of the rectangle

k min = (a/(12)0.5),   square section column, where a is the side of the square cross section

k min = (I/A)0.5 ,        circular SOLID column, = d/4

k min = (IH/A),            Hollow circular column= (d02 + di2)0.5/4

STRUTS

Struts are long compression members. These are horizontal or are in inclined position.

These are not vertical. Their analysis is carried by Euler formula.

MEDIUM COLUMN

These columns fail by a combination of little contraction followed by buckling. Rankine-Gordon formula is used for medium columns.

RANKING GORDON FORMULA IS AN EMPIRICAL FORMULA FOR MEDIUM COLUMN

P Rankine=P cr = P crippling=P buckling= σypA/(1+ α(Le/k min)2)

α= σyp/  π2E

Where P Rankine=P cr = P crippling=P buckling = (N) NEWTONS

σ yp,=  YIELD STRESS IN COMPRESSION, N/mm2 =Also called Rankin’s constant

α is RANKINE’s CONSTANT, No units

Le/k min = Slenderness ratio, no units

JOHNSON’s EMPIRICAL FORMULA FOR MEDIUM COLUMNS

It is for medium columns. It is applicable for columns with slenderness ratio equal or less than critical slenderness ratio.

Critical slenderness ratio =(Le/k min)critical = (2π2E/σyp)0.5

Critical slenderness ratio will be different for different materials.

cr)JOHN  = σyp[1— (σyp (Le/ k min)2)/ 4π2E]

(P cr)JOHN  = σyp A[1— (σyp (Le / k min)2)/ 4π2E]

P safe = P allow=P cr /FOS

σsafe = σallow = σcr/FOS