COLUMNS & STRUTS CLASS NOTES FOR MECHANICAL ENGINEERING

COLUMNS & STRUTS CLASS NOTES

FOR MECHANICAL ENGINEERING

Column is a vertical member under

axial compressive load. Type

of column is decided by a

slenderness ratio. For various

types of columns, value of

slenderness ratio depends on

the end conditions and material

of the column. Horizontal and

inclined long columns are struts.

There are three types of columns.

Fig. Types of columns, short, medium & long (depends on

slenderness ratio and material)

Firstly short column

Secondly Medium column

Thirdly long column

STRUTS

Struts are long compression members. These are horizontal

or  inclined position. These are not vertical. Their analysis

is carried by Euler formula.

DIFFERENT END FIXING OF COLUMNS AND STRUTS

(i) Hinged at both ends

(ii) Hinged at one end & free at the other end

(iii) Fixed at one end while free at the other end

(iv) Fixed at both ends

MINIMUM RADIUS OF GYRATION, k_min

k_min = (I_min/A)^0.5

Rectangular section column

  k _min = (db³/12)

(d  is larger side and b is smaller side of the rectangle)

Square Section Column

  k _min = (a/(12)^0.5)  where a is the side of the square cross section

Solid Circular Section column

   k _min = (I/A)^0.5  = d/4

Hollow Circular Hollow Section Column

   k _min = (I_H/A) = (d₀² + d_i²)^0.5/4

 Equivalent length of a column

Fig. Equivalent length of long columns with different end fixings

It depends on the end fixing condition of the column.

The symbol of equivalent symbol is L_e.

(a) Column with both ends hinged

Its entire length buckles to form a bow. Therefore

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.

Equivalent lengths forming a bow are:

(i) Both ends hinged L_e= L,

(ii) One end fixed and other free L_e= 2L

(iii) One end fixed other hinged L_e=  L / √2

(iv) Both ends fixed  L_e= L/2

Minimum Radius of Gyration

k_min is least of ( I_xx /A)^0.5 and ( I_yy/A)^0.5

 Slenderness ratio

 Columns and struts use it. It is the ratio of equivalent

length L_e and the least radius of gyration k _min.

Slenderness ratio = L_e/k _min. It has no units.

Table: 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

For STEEL COLUMNS

Firstly L_e / k_min ≤ 30, short column

Secondly L_e/k_min >30 but<120, medium column

Thirdly L_e / k_min > 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 = σ_yp 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

P_EULER=P _critical=P _crippling=P _buckling= π²EA/(L_e/k _min)²

P_EULER=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= σ_ypA/(1+ α(L_e/k _min)²)

α= σ_yp/  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

L_e/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 =(L_e/k _min)_critical = (2π²E/σ_yp)^0.5

Critical slenderness ratio will be different for different materials.

(σ_cr)_JOHN  = σ_yp[1— (σ_yp (L_e/ k _min)²)/ 4π²E]

(P _cr)_JOHN  = σ_yp A[1— (σ_yp (L_e / k _min)²)/ 4π²E]

P _safe = P _allow=P _cr /FOS

σ_safe = σ_allow = σ_cr/FOS

Column and struts are structural

members. Slenderness ratio and

end fixing play crucial role. The

 strength decreases with the

increase of slenderness ratio.

Fixed end increases strength.

Eccentric loading decreases

load carrying capacity of the

columns and struts. Equilibrium

of a column is stable, unstable and neutral.

The load at which buckling starts is

buckling /crippling/critical/Euler load

Eccentric Loaded Columns

Fig. Axial & eccentrically loaded columns

It was assumed that the column is under true axial load. But in actual practice, load is not truly axial. It is always under an eccentric load. The loading is not acting along the axis. The eccentricity produces bending stresses in addition to the axial compressive stress. The resultant stress increases. Thus loading carrying capacity of eccentrically loaded columns decreases. Maximum stress under eccentrically loaded columns is

σ_max = (P/A) [1± (e(d/2)/k²_min) Sec[(L_e/2)(P/EI)^0.5]

e = eccentricity

P = Axial load

_Le = Equivalent length

E = Young’s modulus

I = Moment of inertia of the column

 Difference between a column and a strut

Column is a vertical compression member. It is a short or medium or long column on the basis of slenderness ratio. Whereas STRUT is only a long compression member and which is horizontal or inclined to horizontal.

 Assumptions used in deriving Euler formula for long columns

σ (buckling) = π² E/ (le/k min)²

(i)  Stresses and strains are within elastic limit.

(ii) Load applied is truly axial.

(iii)   A transverse plane remains a transverse plane before and after the load application.

(iv) Material is homogeneous, isotropic and continuous.

(v)  Initially column is straight

(vi) Negligible shortening due to axial compression

Effect of lateral load on the bucking of columns

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.

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