Lecture 42 – Buckling of columns

Instructor: Prof. Marcial Gonzalez

Spring, 2020ME 323 – Mechanics of Materials

Reading assignment: 10.1 – 10.3

News: __

Last modified: 1/17/20 12:59:41 PM

+ Video due Friday, December 6th at 11 a.m. (6 extra points)

+ HW12 due Thursday, December 5th

+ Final Exam (comprehensive)Monday, December 9th,

3:30 p.m. – 5:30 p.m., WTHR 200

+ Formula sheet will be uploaded to the blog.

+ Those of you who require extra accommodation

received an email with indications.

(please arrive 15 minutes before the exam)

+ Course evaluation survey (6 extra points, need proof)- Please take a few minutes to complete the evaluation.

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Announcements

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Cylindrical body with hemispherical end caps

axial stress inthe cylinder

Hoop stress inthe cylinder

normal stress inthe sphere

Thin wall pressure vessels (Lecture 30)

Collapse due to external pressure(no need for vacuum, many times a differential pressure is enough!)

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Design of deformable bodies (Lecture 4)

Material properties from stress-strain diagrams:- Buckling under compression

Q: Does the buckling load Pbuckling depend on the geometry of the bar?

A: Yes!

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Buckling of columns

Stability of equilibrium:

- Note:+ Work-Energy Principle … implies equilibrium+ Castigliano’s Second Theorem … implies stable equilibrium+ Principle of minimum potential energy … implies stable equilibrium

Stable equilibrium Neutral equilibrium Unstable equilibrium

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Buckling of columns

Pin-ended column:

Before buckling(stable configuration)

Buckled configuration(neutral configuration)

Critical load:

Euler buckling load

Q1: Is this the only critical load?A1: No!

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Buckling of columns

Pin-ended column:

Critical load:

Euler buckling load

Q1: Is this the only critical load?

A1: No!

If we increase the load even further,

there will be another neutral

configuration (and thus another

critical load) that will induce a

second-buckling mode!

+ first buckling mode (Euler buckling load),

+ second buckling mode,

+ third buckling mode, …

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Buckling of columns

Pin-ended column:

Before buckling(stable configuration)

Buckled configuration(neutral configuration)

Critical load:

Euler buckling load

Q1: Is this the only critical load?A1: No!If we increase the load even further, there will be another neutralconfiguration (and thus another critical load) that will induce a second-buckling mode!

Q2: Are boundary conditions important?A2: Yes!

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Buckling of columns

Pin-ended column:

Before buckling(stable configuration)

Equilibrium (in the deformed configuration)

Euler-Bernoulli beam theory:

Boundary conditions:

Solution(s):

with

from B.C.Buckled

configuration

FBD

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Buckling of columns

Effect of end conditions:

Critical load:

Fixed-fixed Fixed-freeFixed-pinnedPinned-pinned

: effective length factor

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Buckling of columns

Problem 85: Effect of end conditions …

Problem 86: Effect of cross-section size (e.g., of D) ...

Problem 87: … asymmetric cross section (e.g., rectangular)

Problem 88: Effect of length (e.g., of L) ...

Problem 89: Effect of plastic deformations! (see next ...)

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Buckling of columns

Plastic deformation? Euler’s column vs. Johnson’s column:

Euler’s formula

Johnson’s formula

Slenderness ratio

Critical stress

with aSafety Factor

If :

If :

Any questions?

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Buckling of columns