Lecture 42: Failure Analysis – Buckling of Columns Joshua Pribe Fall 2019 Lecture Book: Ch

ME 323 – Mechanics of Materials

Lecture 42: Failure analysis – Buckling of columns Joshua Pribe Fall 2019 Lecture Book: Ch. 18 Stability and equilibrium

What happens if we are in a state of unstable equilibrium?

Stable Neutral Unstable

2 Buckling experiment

There is a critical stress at which buckling occurs depending on the material and the geometry How do the material properties and geometric parameters influence the buckling stress?

3 Euler buckling equation

Consider static equilibrium of the buckled pinned-pinned column

4 Euler buckling equation

We have a differential equation for the deflection with BCs at the pins: d 2v EI+= Pv( x ) 0 v(0)== 0and v ( L ) 0 dx2

The solution is: P  P  A = 0 v(s x) = Aco x+ Bsin x     with  EI EI PP Bsin L= 0  L = n , n = 1, 2, 3, ... EI EI

5 Effect of boundary conditions

Critical load and critical stress for buckling: EI EA P = 22= cr L2 2 e (Legr )

2 E cr = 2 (Lreg)

I r = Pinned- Pinned- Fixed- where g Fixed- A pinned fixed fixed is the “radius of gyration” free LLe = LLe = 0.7 LLe = 0.5 LLe = 2

Modifications to Euler buckling theory

Euler buckling equation: works well for slender rods

Needs to be modified for smaller “slenderness ratios” (where the critical stress for Euler buckling is at least half the yield strength)

7 Summary

L  2 E Critical slenderness ratio: e = r 0.5 gYc Euler buckling (high slenderness ratio): LL     2 E EI If ee    :  = or P =  2 rr    cr 2 cr L2 gg   c (Lreg) e

Johnson buckling (low slenderness ratio): 2 LL    (Lreg) If  ee     : =−1 rr    cr2 Y gg   c 2 Lr ( eg)c

I with radius of gyration rg = A 8 Summary

Effective length from the boundary conditions:

Pinned- Pinned- Fixed- Fixed- pinned fixed fixed free LLe = LLe = 0.7 LLe = 0.5 LLe = 2 9 Example 18.1

Determine the critical buckling load Pcr of a steel pipe column that has a length of L with a tubular cross section of inner radius ri and thickness t. The material has Young’s modulus E and yield strength σY. Use pinned-fixed boundary conditions.

Example 18.2

The steel compression strut BC of the frame ABC is a tube with an outer diameter of d = 48 mm and a wall thickness of t = 5 mm. Determine the factor of safety against elastic buckling if a distributed load of 10 kN/m is applied to AB. Let

E = 210 GPa and σY = 340 MPa.

Example 18.8 (additional examples)

Members (1), (2), and (3) have Young’s modulus E = 107 psi 3 and σY = 60 × 10 psi. Each member has a solid circular cross section with diameter d = 1 in. A force P = 10 kips is applied to joint C. Determine the maximum length L that can be used without buckling based on Euler’s theory of buckling.

Real-world example: Railroad track buckling

What would cause the railroad track to buckle as shown in the picture?

G. Yang and M.A. Bradford, Engineering Failure Analysis 92 (2018), pp. 107-120.