Buckling of Column With Two Intermediate Elastic Restraints

1. Buckling of Column With Two Intermediate Elastic Restraints Thesis Presentation 15.11.2007 Author: Md. Rayhan Chowdhury Mohammad Misbah Uddin Md. Abu Zaed Khan Md. Monirul Islam Masud Supervisor: Dr. Mohammad Nazmul Islam Presidency University

2. Introduction • Background • In a laterally loaded cross-bracing system, one bracing member will be under compression while the other member subjected to tension. The tension brace may be modeled as a discrete, lateral elastic spring attached to the compression member. Thus, the prediction of the elastic buckling loads of columns with two intermediate elastic restraints is therefore of practical interest. • Objectives • The main objective of this theoretical research is to find a set of stability criteria for Euler columns with two intermediate elastic restraints. • Scope • The scope of this thesis is the derivation of Euler column buckling theory.

3. P Spring 2 L c2 a2L Spring 1 c1 a1L x z, w Model • Here, • Column Length, L • Flexural Rigidity, EI • Spring 1 • Stiffness, c1 • Located at a1L (a1 < L) • Spring 2 • Stiffness, c2 • Located at a2L (a1 ≤ a2 ≤ L) • The entire column can be divided into three segments as • Segment-1: 0 ≤ x ≤ a1L • Segment-1: a1L ≤ x ≤ a2L • Segment-1: a2L ≤ x ≤ L Fig. Column with two intermediate elastic restraints

4. Governing equation for column buckling • Where i = 1, 2, 3 denote the quantity belonging to segment 1, segment 2 and segment 3. • and General Solution: for (1) for (2) for (3)

5. Continuity condition at a1 (For deflection, slope, bending moment and shear force ) (4) (5) (6) (7) Where,

6. Continuity condition at a2 (For deflection, slope, bending moment and shear force ) (8) (9) (10) (11) Where,

7. Continuation… Substituting Eqs. (1) and (2) into Eqs. (4)-(7), a set of homogeneous equations is obtained which may be expressed in forms of Bi in terms of Ai, i.e., (12) (13) (14) (15)

8. Continuation… Substituting Eqs. (2) and (3) into Eqs. (8)-(11), another set of homogeneous equations is obtained which may be expressed in forms of Ci in terms of Bi, i.e., (16) (17) (18) (19)

9. Continuation… Hence, Substituting Eqs. (12) - (15) into Eqs. (16)-(19), we get Ci in terms of Ai (20) (21) (22) (23)

10. P Spring 2 L c2 a2L Spring 1 c1 a1L x z, w Boundary Condition (fixed – free) At the fixed end: (24) (25) At the free end: (26) (27) Fig. Boundary condition, fixed - free

11. B. C. (fixed – free) continuation a. b. Differentiating the boundary equation: Hence, we can develop the Eigen value equation from the boundary equation in form (24) (25) Where {A} = (A1, A2, A3, A4) and [M] is the coefficient matrix of {A}. (26) (27) Finally the determinant of matrix[M] yields the stability criteria. If we substitute the value of Ci into Eqs. (26) and (27), the buckling problem involves only four constants Ai (i = 1,2,3,4).

12. B. C. (fixed – free) continuation The following tables present the buckling load parameter for different locations and stiffness of the intermediate restraints:

13. Conclusions • Exact stability criteria for columns with two intermediate elastic restraints at arbitrary location along the column length are derived. • This stability criteria can be used to determine the buckling capacity of compressive member in a cross-bracing system.

14. ?

15. Thank you all. Presidency University