engr 220 section 13.1~13.2. buckling of columns
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ENGR 220Section 13.1~13.2
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Buckling of Columns
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Column Buckling
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Ideal Column Buckling
Ideal Column : Pin supports at both endsHomogenous materialLoad applied through centroidLinearly elasticColumn bends in a single plane
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Stable or Unstable
Ability of the column to restore itself
Resistance to Bending
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Stable or Unstable
Ability of the column to restore itself
Resistance to Bending
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Relate v to Moment via Deflection Equation
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Maximum Axial Load : Euler Load
2
2
L
EIPcr
Deflected shape is a sine curve
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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
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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
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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
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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.
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Critical Stress
2
2
cr AL
EI
A
Pcr
2
2
cr)/(
rL
E
A
Pcr
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Radius of Gyration
AI r
r
L Ratio sSlendernes
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Short Columns (posts) : Yielding and direct fracture (no buckling)
Intermediate Columns: Inelastic instability
Long Columns: Buckling (Euler equation).
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Types of Supports
• Pinned – Pinned (Euler Column) • Fixed – Free• Fixed – Fixed• Fixed – Pinned
Moment equation and Boundary Conditions
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Effective Length Le = K L
Effective Length Slenderness Ratio = Le/r
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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
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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.
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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.
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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.
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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.