ENGR 220 Section 13.1~13.2

Buckling of Columns

Column Buckling

Ideal Column Buckling Ideal Column : Pin supports at both ends Homogenous material Load applied through centroid Linearly elastic Column 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 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.

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.

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.

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

Radius of Gyration

Short Columns (posts) : Yielding and direct fracture (no buckling) Intermediate Columns: Inelastic instability Long Columns: Buckling (Euler equation).

Moment equation and Boundary Conditions 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.