IDEAL COLUMN
 2 EI
Smallest critical load to cause buckling is Pcr  2
L
IDEAL COLUMN
Pcr  2 EI
Smallest critical stress to cause buckling is  cr  
A AL2
IDEAL COLUMN
 2 EI
 2 Er2

Smallest critical stress to cause buckling is  cr 
2
A( KL)
( KL)2
IDEAL COLUMN
 2E
Smallest critical stress to cause buckling is  cr 
( KL / r )2
Where r = (I/A)1/2 is the “radius of gyration”
and (L/r) is the “slenderness ratio”.
The critical stress
curve is
hyperbolic
and valid
only when σcr
is below the
yield stress σY.
σY
 cr 
 2E
KL / r 2
COLUMNS HAVING VARIOUS END-CONDITIONS
• K for various end conditions:
 cr 
 2E
KL / r 2
EXAMPLE 1
The A-36 steel W200x46 member shown is to be used as a
pin-connected column. Determine the largest axial load it can
support before it either begins to buckle or the steel yields.
E  200 GPa
 Y  250 MPa
K 1
 cr 
 2E
KL / r 2
EXAMPLE 1 (continued)
• Column properties from reference table,
• Consider buckling about the “weak axis” (y–y axis).
• When fully loaded, the average compressive stress in the column is
• Since this stress exceeds the yield stress of
250 MPa, let’s investigate this a little further.
 cr 
 2E
KL / r 2
EXAMPLE 1 (continued)
• The critical stress can also be calculated using the formula  cr 
 2E
KL / r 2
• Notice the slenderness ratio is less than 89.
• This indicates the column is “short” and
it will yield before it has a chance to buckle.
• Thus, the 320.5 MPa can never be achieved
before the whole cross-section yields at
250 MPa.
 cr 
 2E
KL / r 2
EXAMPLE 1 (continued)
• The load at which the column yields is PY   Y A
• Notice this load is less than
250MPa
EXAMPLE 2)
The 3 in. x 1 in. A-992 steel member BC is used as a pin-connected brace.
Determine the maximum load P that can be supported before the
brace either begins to buckle or it yields.
E  29,000 ksi
 Y  50 ksi
Kx  1
K y  0.5
EXAMPLE 2 (continued))
• The compressive force in member BC as a function of P is
M
A
0
• The critical stress in the brace is
 cr 
 2E
KL / r 2
EXAMPLE 2 (continued))
• The critical stress is controlled by buckling about the___axis
• The critical stress is below the
yield stress of ______.
• The brace will buckle before it has a chance to yield,
thus cr controls the design of the brace.
• The maximum load P can now be determined from