ENGR 220Section 13.1~13.2

Buckling of Columns

Column Buckling

Ideal Column Buckling

Ideal Column : Pin supports at both endsHomogenous materialLoad applied through centroidLinearly elasticColumn bends in a single plane

Stable or Unstable

Ability of the column to restore itself

Resistance to Bending

Stable or Unstable

Ability of the column to restore itself

Resistance to Bending

Relate v to Moment via Deflection Equation

Maximum Axial Load : Euler Load

2

2

L

EIPcr

Deflected shape is a sine curve

Column Length

• Shorter is better

• If you double the length then the maximum axial load decreases by a factor of four.

2

2

L

EIPcr

Modulus of Elasticity

• Maximum axial load is independent of the yield strength of the material.

• High strength steel is no advantage over low strength steel.

2

2

L

EIPcr

Moment of Inertia I = y2 dA

= Second moment of Area

• The load carrying capability of a column increases as the moment of inertia of a column increases.

• Columns will buckle about the principal axis having the least moment of inertia.

2

2

L

EIPcr

Efficient Columns have cross sectional area located as far as possible from centroidal axes.

This column buckles about a-a

not b-b

Circular tubes, Square tubes

Better than

Solid sections.

Critical Stress

2

2

cr AL

EI

A

Pcr

2

2

cr)/(

rL

E

A

Pcr

Radius of Gyration

AI r

r

L Ratio sSlendernes

Short Columns (posts) : Yielding and direct fracture (no buckling)

Intermediate Columns: Inelastic instability

Long Columns: Buckling (Euler equation).

Types of Supports

• Pinned – Pinned (Euler Column) • Fixed – Free• Fixed – Fixed• Fixed – Pinned

Moment equation and Boundary Conditions

Effective Length Le = K L

Effective Length Slenderness Ratio = Le/r

Example 1: The Al column is fixed at the bottom and is braced at the top by cables to prevent movement along X axis. Determine the largest allowable P that can be applied.

F.S. = 3, EAl = 70 GPa, yield stress = 215 MPa,

A = 7.5 x 10-3 m2, Ix = 61.3 x 10-6 m4, Iy = 23.2 x 10-6 m4

Example 2: The column consists of a Rigid member pinned at the bottom and attached to a spring at the top. When the column is in vertical position, the spring is unstretched. Determine the critical load P that can be placed on the column.

Example 3: Member BD in the truss below, is an A-36 steel rod of radius 2 inches. Determine maximum load that can be supported by the truss without causing member BD to buckle.All members are pin connected.

Example 4: The linkage is made using two A-36 steel rods, each having a circular cross section. Determine the diameter of each rod to the nearest 1/8th in. that will support the 900 lb load without buckling. Factor of Safety 1.8.

Example 5: The 50 mm diameter C86100 Bronze rod is fixed at A and has a gap of 2 mm from the wall at B. Determine the increase in temperature that will cause rod AB to buckle. Contact at B acts as a pin.