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 P RESIDENCY U NIVERSITY

2. Top right corner for field-mark, customer or partner logotypes. See Best practice for example. Slide title 40 pt Slide subtitle 24 pt Text 24 pt Bullets level 2-5 20 pt Department of Civil EngineeringUndergraduate Thesis Presentation2007-11-152 P RESIDENCY U NIVERSITY 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. Top right corner for field-mark, customer or partner logotypes. See Best practice for example. Slide title 40 pt Slide subtitle 24 pt Text 24 pt Bullets level 2-5 20 pt Department of Civil EngineeringUndergraduate Thesis Presentation2007-11-153 P RESIDENCY U NIVERSITY Model Fig. Column with two intermediate elastic restraints P z, w x c1c1 c2c2 L a2La2L a1La1L Spring 1 Spring 2 Here,  Column Length, L  Flexural Rigidity, EI  Spring 1 –Stiffness, c 1 –Located at a 1 L (a 1 < L)  Spring 2 –Stiffness, c 2 –Located at a 2 L (a 1 ≤ a 2 ≤ L)  The entire column can be divided into three segments as –Segment-1: 0 ≤ x ≤ a 1 L –Segment-1: a 1 L ≤ x ≤ a 2 L –Segment-1: a 2 L ≤ x ≤ L

4. Top right corner for field-mark, customer or partner logotypes. See Best practice for example. Slide title 40 pt Slide subtitle 24 pt Text 24 pt Bullets level 2-5 20 pt Department of Civil EngineeringUndergraduate Thesis Presentation2007-11-154 P RESIDENCY U NIVERSITY Governing equation for column buckling  Where i = 1, 2, 3 denote the quantity belonging to segment 1, segment 2 and segment 3.  and (1)for (2)for (3)for General Solution:

5. Top right corner for field-mark, customer or partner logotypes. See Best practice for example. Slide title 40 pt Slide subtitle 24 pt Text 24 pt Bullets level 2-5 20 pt Department of Civil EngineeringUndergraduate Thesis Presentation2007-11-155 P RESIDENCY U NIVERSITY Continuity condition at a 1 (4) (5) Where, (6) (For deflection, slope, bending moment and shear force ) (7)

6. Top right corner for field-mark, customer or partner logotypes. See Best practice for example. Slide title 40 pt Slide subtitle 24 pt Text 24 pt Bullets level 2-5 20 pt Department of Civil EngineeringUndergraduate Thesis Presentation2007-11-156 P RESIDENCY U NIVERSITY Continuity condition at a 2 (8) (9) Where, (10) (For deflection, slope, bending moment and shear force ) (11)

7. Top right corner for field-mark, customer or partner logotypes. See Best practice for example. Slide title 40 pt Slide subtitle 24 pt Text 24 pt Bullets level 2-5 20 pt Department of Civil EngineeringUndergraduate Thesis Presentation2007-11-157 P RESIDENCY U NIVERSITY Continuation… (12) (13) (14) (15) Substituting Eqs. (1) and (2) into Eqs. (4)-(7), a set of homogeneous equations is obtained which may be expressed in forms of B i in terms of A i, i.e.,

8. Top right corner for field-mark, customer or partner logotypes. See Best practice for example. Slide title 40 pt Slide subtitle 24 pt Text 24 pt Bullets level 2-5 20 pt Department of Civil EngineeringUndergraduate Thesis Presentation2007-11-158 P RESIDENCY U NIVERSITY Continuation… (16) (17) (18) (19) Substituting Eqs. (2) and (3) into Eqs. (8)-(11), another set of homogeneous equations is obtained which may be expressed in forms of C i in terms of B i, i.e.,

9. Top right corner for field-mark, customer or partner logotypes. See Best practice for example. Slide title 40 pt Slide subtitle 24 pt Text 24 pt Bullets level 2-5 20 pt Department of Civil EngineeringUndergraduate Thesis Presentation2007-11-159 P RESIDENCY U NIVERSITY Continuation… (20) (21) (22) (23) Hence, Substituting Eqs. (12) - (15) into Eqs. (16)-(19), we get C i in terms of A i

10. Top right corner for field-mark, customer or partner logotypes. See Best practice for example. Slide title 40 pt Slide subtitle 24 pt Text 24 pt Bullets level 2-5 20 pt Department of Civil EngineeringUndergraduate Thesis Presentation2007-11-1510 P RESIDENCY U NIVERSITY Boundary Condition (fixed – free) (24) (25) (26) (27) At the fixed end: Fig. Boundary condition, fixed - free P z, w x c1c1 c2c2 L a2La2L a1La1L Spring 1 Spring 2 At the free end:

11. Top right corner for field-mark, customer or partner logotypes. See Best practice for example. Slide title 40 pt Slide subtitle 24 pt Text 24 pt Bullets level 2-5 20 pt Department of Civil EngineeringUndergraduate Thesis Presentation2007-11-1511 P RESIDENCY U NIVERSITY B. C. (fixed – free) continuation (24) (25) (26) (27) Differentiating the boundary equation: If we substitute the value of C i into Eqs. (26) and (27), the buckling problem involves only four constants A i (i = 1,2,3,4). a. Hence, we can develop the Eigen value equation from the boundary equation in form b. Where {A} = (A 1, A 2, A 3, A 4 ) and [M] is the coefficient matrix of {A}. Finally the determinant of matrix[M] yields the stability criteria.

12. Top right corner for field-mark, customer or partner logotypes. See Best practice for example. Slide title 40 pt Slide subtitle 24 pt Text 24 pt Bullets level 2-5 20 pt Department of Civil EngineeringUndergraduate Thesis Presentation2007-11-1512 P RESIDENCY U NIVERSITY B. C. (fixed – free) continuation ξ 1 =20, ξ2=20: (ξ 1 =c 1 L 3 and ξ 2 =c 2 L 3 ) ξ 1 =20, ξ2=20: (ξ 1 =c 1 L 3 and ξ 2 =c 2 L 3 ) λ 2 = PL 2 /(EI) a2 α = 0 α = 0.2 α = 0.4 α = 0.6 α = 0.8 α = 1 0.14.71014.71014.714.70984.70914.7078 0.34.56674.56634.56094.53954.48884.4004 0.54.14764.14464.1073.98553.7943.642 0.74.10444.09624.00593.8073.69133.7816 0.94.46024.43964.24323.99094.07484.3574 The following tables present the buckling load parameter for different locations and stiffness of the intermediate restraints:

13. Top right corner for field-mark, customer or partner logotypes. See Best practice for example. Slide title 40 pt Slide subtitle 24 pt Text 24 pt Bullets level 2-5 20 pt Department of Civil EngineeringUndergraduate Thesis Presentation2007-11-1513 P RESIDENCY U NIVERSITY 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. Top right corner for field-mark, customer or partner logotypes. See Best practice for example. Slide title 40 pt Slide subtitle 24 pt Text 24 pt Bullets level 2-5 20 pt Department of Civil EngineeringUndergraduate Thesis Presentation2007-11-1514 P RESIDENCY U NIVERSITY ?

15. Top right corner for field-mark, customer or partner logotypes. See Best practice for example. Slide title 40 pt Slide subtitle 24 pt Text 24 pt Bullets level 2-5 20 pt Department of Civil EngineeringUndergraduate Thesis Presentation2007-11-1515 P RESIDENCY U NIVERSITY Thank you all. P RESIDENCY U NIVERSITY