1. Strength of Materials I EGCE201 กำลังวัสดุ 1 Instructor: ดร. วรรณสิริ พันธ์อุไร ( อ. ปู ) ห้องทำงาน : 6391 ภาควิชาวิศวกรรมโยธา E-mail: egwpr@mahidol.ac.thegwpr@mahidol.ac.th โทรศัพท์ : 66(0) 2889-2138 ต่อ 6391

2. Columns Members that support axial loads. Columns fail as a result of an instability. The column may satisfy the conditions for which stress and deformation do not result in failure, but failure can still result. Buckle – suddenly becomes sharply curved

3. Euler’s Formula Buckling is an instability related to deflection. One would like to determine the smallest value of P, the critical buckling load, which is known as Euler’s formula.

4. Derivation Taking moments about Q Use the relation between beam deflection and moment. This is a linear, homogeneous differential equation with constant coefficients.

5. Derivation (continued) By setting p 2 =P/EI, the relation above becomes The general solution for this equation is Using the B.C.’s for ends A and B, we find that for y=0 at x=0, B=0. Next, one consider the boundary condition y=0 at x=L, which yields The possible solutions are A=0 and sin pL=0. If A=0,y=0 the column is straight.

6. Derivation (continued) Examine the second solution which is satisfied if Using p 2 =P/EI and solving for P, the following relation is established. The smallest value of P occurs when n=1. Setting n=1, one obtains the critical buckling load, which is known as Euler’s formula.

7. Buckling axis The area moment of inertia (I) defines the axis about which buckling will occur.

8. Buckling axis (continued) Using the dimensions shown, we have

9. Critical stress The stress corresponding to P cr is called the critical stress and is denoted as  cr. The inertia can be represented in terms of the radius of gyration by I=Ar 2 where A is the cross-sectional area of the column and r is the radius of gyration. Using this definition for inertia, the critical stress is written as The quantity L / r is called the slenderness ratio of the column. The min r =I min and should be used when computing the critical stress.

10. Extension of Euler’s buckling

11. Example I

12. Example II