# COLUMNS & STRUTS CLASS NOTES FOR MECHANICAL ENGINEERING

## FOR MECHANICAL ENGINEERING

### There are three types of columns. #### equivalent length of a column with both ends hinged is ‘L’.

Equivalent length takes care of the end fixing conditions.

Different equivalent length for different end fixing conditions.

#### Slenderness ratio = Le/k min. It has no units.

##### RCC
###### >  50

For STEEL COLUMNS

Firstly Le / kmin ≤ 30, short column

Secondly Le/kmin >30 but<120, medium column

Thirdly Le / kmin > 120, long column

### Buckling in columns

It is similar to bending in appearance. But it occurs under an AXIAL load ( load parallel to length). Buckling is bending due to the axial load. Bending occurs due to a load perpendicular to length in a loaded beam. Medium and long columns have buckling. A long column fail by buckling alone. A medium column first contracts then buckles. A short column only contracts.

### SHORT COLUMNS

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. Buckling is instability.

### 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.

### 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 , 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

### The load at which buckling starts is Fig. Axial & eccentrically loaded columns

#### (v)  Initially column is straight

(vi) Negligible shortening due to axial compression

#### It increases the chances of buckling. Thus decreases the bucking strength of the material.

Limitations of Euler formula

(i) It assumes initial straight column.

There is always small curvature in the column.

(ii) it assumes truly axial load. There is always some eccentricity in load application.

Therefore results may not agree with the experimental data.