Buckling

Critical Load Considerations

Structural members and machine elements must meet specific strength, deflection, and stability requirements.

Members such as columns that are long and slender subjected to an axial compressive force can deflect laterally or sideways if the force exceeds a critical value. This is called buckling.

Ideal Column with Pin Supports

Assumptions for “Ideal Column”:

• perfectly straight before loading;

• made from homogeneous material;

• load applied at centroid of cross-section;

• linear-elastic behavior;

• buckling occurs in a single plane.

To determine the critical load and buckled shape of the column, the simplified deflection equation for bending can be used:

Buckling Equation for Pin-Supported Slender Column

2

2

L

EIPcr

2

2

)r/L(

Ecr

Critical buckling stress for pin-supported columns as a function of the slenderness ratio for structural steel & aluminum with 36 & 27 ksi yield strengths, respectively.

Effective Length Factors, K, can be used in modified Euler’s Equations for other end conditions:

2

2

)KL(

EIPcr

2

2

)r/KL(

Ecr

13-6. The rod is made from an A-36 steel rod. Determine the smallest diameter of the rod, to the nearest 1/16 in., that will support the load of P = 5 kip without buckling. The ends are roller-supported.

13-11. The W10 X 45 is made of A-36 steel and is used as a column that has a length of 15 ft. If the ends of the column are fixed supported, can the column support the critical load without yielding?

Design codes are typically used for column design. These codes have been developed using experimental data, but rely on the Euler equation for relevant parameters. The Euler equation also tends to be used for long columns.

Empirical data and curve fit for a wide-flange steel column subjected to buckling

Steel ColumnsThe following equations have been developed for building construction by the Structural Stability Research Council (SSRC) and adopted as specifications by the American Institute of Steel Construction (AISC):

2

2

allowr/KL23

E12

200r

KL

r

KL

c

3c

3c

y2c

2

allow )r/KL(8/)r/KL()r/KL/()r/KL)(8/3()3/5(

r/KL2

r/KL1

cr

KL

r

KL0

Aluminum ColumnsColumn design for structural aluminum is specified by the Aluminum Association for specific alloys. Equations for a common alloy used in building construction (2014-T6) are shown below:

12r

KL0 ksi 28allow

55r

KL12 ksi

r

KL23.07.30allow

r

KL55 2allow

r/KL

ksi 54000

Timber ColumnsColumns used in timber construction are designed using formulas published by the National Forest Products Association (NFPA) or the American Institute of Timber Construction (AITC). NFPA formulas are shown below for columns with dimensions b and d where d is the smallest dimension:

11d

KL0 ksi 20.1allow

26d

KL11 ksi

0.26

d/KL

3

1120.1

2

allow

50d

KL26 2allow

d/KL

ksi 540

d

b

13-88. The bar is made of a 2014-T6 aluminum alloy. Determine its thickness b if its width is 5b. Assume that it is pin connected at its ends.

13-99. The timber column has a square cross section and is assumed to be pin connected at its top and bottom. If it supports an axial load of 50 kip, determine its side dimensions a to the nearest ½ in. Use the NFPA formulas.