intro & how to salve numericals

1. COLUMNS
• In their simplest form, columns are long, straight, prismatic bars
subjected to compressive, axial loads.
• If a column begins to deform laterally, the deflection may become
large and lead to catastrophic failure. This situation, called buckling
• Buckling, can be defined as the sudden large deformation of a
structure due to a slight increase of an existing load under which
the structure had exhibited little, if any, deformation before the
load was increased
• Buckling failure is a stability failure, the column
has transitioned from a stable equilibrium to an unstable one.

2. If the column model returns to its initial configuration, the
system is said to be stable. If pin B moves farther to the
right, then the system is said to be unstable.

3. • The magnitude of axial load P at which the restoring
moment equals the upsetting moment is called the
critical load Pcr
• To determine the critical load for the column model,
consider the moment equilibrium of bar BC in Figure
for the load P = Pcr :

4. • If the load P applied to the column model is less than
Pcr , then the restoring moment is greater than the
upsetting moment and the system is stable. However,
if P = Pcr , then the system is unstable.
• At the point of transition where P = Pcr , the system is
neither stable nor unstable, but rather, it is said to be in
neutral equilibrium.
• For long, slender columns, the critical buckling load
occurs at stress levels that are much lower than the
proportional limit for the material, which indicates that
this type of buckling is an elastic phenomenon

5. Buckling of Pin-Ended Columns

6. • The column is loaded by a compressive load P
that passes through the centroid of the cross
section at the ends.
• The pins at each end are frictionless,and the load
is applied to the column by the pins. The column
itself is perfectlystraight and made of a linearly
elastic material that follows Hooke’s Law.
• Since the column is assumed to have no
imperfections, it is termed an ideal column.

8. Solution of the Differential Equation

11. Euler Buckling Stress

13. Buckling of a fixed-pinned column