column and strut

column and strut

1. COLUMN & STRUTS

2. STRUTS A structural member subjected to axial compressive force is called strut. Strut may be vertical, horizontal or inclined. The cross – sectional dimensions of strut are small. Normally, struts carry smaller compressive loads. Struts are used in roof truss and bridge trusses.

3. COLUMN When strut is vertical it is known as column. The cross – sectional dimensions of column are large. Normally, columns carry heavy compressive loads. Columns are used in concrete and steel buildings.

4. STRUT COLUMN

5. Radius of Gyration ( K ) : The distance from the given axis at which if all small elements of lamina placed, the M.I. Of the lamina about the axis does not changed. This distance is called radius of gyration. k = √(I/A) or I = AK² K=radius of gyration I = Moment of Inertia (mm 4 ) A = Area of Section (mm 2 )

6. Slenderness ratio ( λ ) : Slenderness Ratio = effective length of column/Minimum radius of gyration λ = le/ k min If λ is more , its load carrying capacity will be less.

7. Long Column :- When length of column is more as compared to its c/s dimension, it is called long column. Le/ k min > 50 For mild steel λ > 80 is called long column. Short Column:- When length of column is less as compared to its c/s dimension, it is called Short column. Le/ k min <50

8. Crushing Load : The load at which, short column fails by crushing is called crushing load. Crippling Load and Buckling Load : The load at which, long column starts buckling(bending) is called buckling load or crippling load. It depends upon the following factors. 1. Amount of load. 2. Length of column 3. End condition of column 4. C/s dimensions of column 5. Material of column.

9. Column End Condition And Effective Length 1.Both end hinged. 2.Both end fixed. 3.One end fixed and other hinged. 4.One end fixed and other free.

10. COLUMNS HAVING VARIOUS TYPES OF SUPPORTS Effective length 10

11. Euler’s Formula Euler’s Crippling Load, P E = ∏ ²EI /le² Where, E = is Modulus of Elasticity ( Mpa ) I = is MOI or 2 nd Moment of area (mm 4 ) Le =is Effective length (mm) Also known as Critical Buckling Load

12. Euler’s Formula for both end of the columns are hinged The Bending Moment at the Section is given by General solution is Since x = 0 at y = 0, then C 2 = 0. Since x = l at y = , then 12

13. IDEAL COLUMN WITH HINGED SUPPORTS for C 1 = 0, we get Which is satisfied if or 13

14. Assumption of Euler’s formula : The material is elastic, homogeneous, isotropic. The column is long. The load is truely axial. Failure is due to buckling. The cross section is uniform throughout. The Hook’s law is valid. The column is straight before application of load.

15. P cr = ( π 2 EI) / L e 2 But I =Ak 2 ∴ P cr /A= π 2 E/(L e /K) 2 σ cr = π 2 E/(L e /K) 2 Where σ cr is crippling stress or critical stress or stress at failure Limitation Of Euler’s Formula

16. ∴ σ c = π 2 E/(L e /K) 2 L e /K= √ ( π 2 E / σ c ) For steel σ c = 320N/mm 2 and E =2 x 10 5 N/mm 2 Limiting value (L e /K) is given by ( L e /K) lim =√ ( π 2 E / σ c ) = √ π 2 × 2 × 10 5 /320) = 78.54 Hence, the limiting value of crippling stress is the crushing stress. The corresponding slenderness ratio may be found by the relation σ cr = σ c Hence if L e /k < (Le /k) lim Euler's formula will not be valid.

17. P R = crippling load by Rankine’s formula P c = crushing load = σ c .A P E = buckling load= P E = ( π 2 EI) / L e 2 We know that, Euler’s formula for calculating crippling load is valid only for long columns. But the real problem arises for intermediate columns which fails due to the combination of buckling and direct stress. The Rankine suggested an empirical formula which is valid for all types of columns. The Rankine’s formula is given by, 1/ P R = 1/ Pc + 1/ P E Rankine Formula

18. where a = Rankine’s constant = σ c / π 2 EI and λ = slenderness ratio = L e / k P R = σ c A / (1+a. λ 2 )

19. Eccentric Loading Short Column σ max = P/A + P.e /Z = P/A (1 + ey c /k 2 ) Z = Ak 2 / y c Long Column Rankine’s Formula σ c = P/A (1 + ey c /k 2 ) (1 + α l e /k) Euler’s Formula σ max = P/A + Pe v /Z σ min = P/A – Pe v /Z v = sec {(l e /2) /√[P/(EI)]}

20. THANK YOU