Met201 ktu syllabus

1. Module 5
• Column Theory-Euler’s Formula, Rankine’s formula

2. Columns and struts
• member of a structure which is subjected to axial compressive load
•Column-vertical member
•Strut-Not vertical(inclined)

3. Struts in suspension system of automobiles

4. Struts in aircrafts

5. •A column can be:
•Short column-When length of column to its least lateral dimension is <12
-When Slenderness ratio<80
•Long column-When length of column to its least lateral dimension is >12
-When slenderness ratio>120

6. Failure of a column
•By direct compressive stress
•Buckling
•Combined buckling and compressive stress
•Buckling load/crippling load-Load at which a long column just buckles

7. Euler’s theory for long columns
Euler postulated a theory for long columns based on the following assumptions:
• Column is very long in proportion to its cross sectional dimensions
• Column is initially straight and the compressive load is applied axially
• Material of the column is elastic, homogeneous and isotropic
• Effect of direct stress is very small in comparison with bending stress.
• Column shall fail by buckling alone.
• Effect of self-weight of column is negligible

8. Four end conditions in Euler’s theory

9. Derivation of Euler’s formula(for both ends hinged/pinned)

10. Derivation of Euler’s formula(for both ends hinged/pinned)

11. Effective length of column
•It is the length of an equivalent pin-ended column having the same
load-carrying capacity as the member under consideration.
•The smaller the effective length of a particular column, the smaller its danger
of lateral buckling and the greater its load carrying capacity.

12. Effective length of column

13. Crippling stress

14. Slenderness ratio
•It is the ratio of effective length of a column to its least radius of gyration
•So Slenderness ratio=le
/k

15. Problem

18. Problem

23. Rankine’s formula for columns
•It has been shown that Euler’s formula is valid for long column
having l/k ratio greater than a certain value for a particular material. Euler’s
formula does not give a reliable result for short column and length of column
intermediate between very long to short.
•An empirical formula has been proposed by Rankine for columns of all
lengths.

25. Derivation of Rankine’s formula

27. Problem

29. Problem