Buckling

Buckling:

Buckling (Discession of Stability of Structures) is defined as, "A deformation occurred due to compressive loads in vertical members". 

• Example: mainly columns, vertical members, etc.
• Buckling is due to instability (change in configuration) NOT a failure through yielding.
• The analysis and design of vertical prismatic members supporting axial compressive loads is carried out.
• Failure of structures are presented by designing structures so that the maximum stresses and maximum displacements remain tolerable limits.

There are two primary concerns:

1. The strength of the structure i.e. its ability to support a load without experiencing excessive stresses.
2. The ability of the structure to support a specified load without undergoing unacceptable deformations.

Important Points:

• Stability  ↣  material's ability to support a given load without experiencing a sudden change in its configuration (or load carrying capability without no sudden change).
• Failure  ↣  is due to instability of structures (design), geometric failure.
• If column buckles at load  ↣  require supports.
• Column collapse  ↣  if stresses > critical load.
• Buckling must be avoided in Structural design. For example: in earthquakes, building moves left and right which changes its configuration, so collapse.

Buckling Derivation:

Conditions of equilibrium are defined on the basis of above load equation

Ideal Columns With Pin Support:

Ideal columns are defined as, "one that is perfectly straight before loading and is made of homogeneous material".

•  Load is applied through the centroid of cross-section.
• When critical load is reached, column will be on the verge of becoming unstable.

Euler Equation for Critical Load:

Critical Load is defined as, "the maximum axial load a column can support when it is on the verge of buckling".

Different Modes of Buckling is given by:

• Critical load is independent of strength of material, depends on material stiffness.
• Load carrying capability of column increases with moment of inertia.
• Efficient column  ↣  are those in which column's cross-section is located as far away as possible from centroidal axes (like hollow columns).
• Columns will buckle about the principal axis of cross-section having least MOI.
• Slenderness Ratio (L/r)  ↣  is the measure of column's flexibility and help in classifying columns as short, long and intermediate.
• Columns having slenderness ratio < minimum ratio  ↣  are failed by yielding (not buckling) and are called Short Columns.

Critical Stress to Slenderness Ratio Diagram:

• Columns not to fail by either yielding or buckling, its stress must remain underneath the curve.
• It tells minimum slenderness ratio (which is the yielding point).
• Euler Formula is not valid below min. slenderness ratio (yield value exceeded before buckling can occur). 

Effective Length:

It represent the distance between zero-moment points is called Effective Length (Le). Many design codes provide column formula that use a dimensionless coefficient K is called Effective Length Factor.

1. Determine by Euler formula for critical load.
2. By using other supports.

Secant Formula:

• Euler formula is derived assuming load applied through centroid and column are perfectly straight.
• In reality columns are not always straight and generally load is applied eccentrically.

Now, 

P/A to KL/r Curves:

• Plot of ec/r^2 against stress and slenderness ratio (for A-36 steel i.e. structural steel).
• If eccentricity increases, failure occur before it should be.
• Graph shows big changes in load carrying capacity of a column with respect to e.
• Column having high values of slenderness ratio fails at or near to Euler critical load.
• Selection of 'e' is very important in column design.

References:

• Material from Class Lectures + Book named Engineering Mechanics of Materials by R.C. Hibbeler (10th Edition) + my knowledge. 
• Pics and GIF from Google Images.  
• Videos from YouTube.