Power Transmission
Machine Design Equations and Calculators

Power Screw Buckling and Deflection Equations and Calculator

Preview: Power Screw Buckling and Deflection Design Calculator

Power screws subjected to compressive loads may buckle. The Euler formula can be used to estimate the critical load F_c at which buckling will occur for relatively long screws of column length L_c and second moment of area:

I= πd⁴_r / 64

re-written as:

F_c = [( C π² E ) / L²_c ] · I

Second Moment of Inertia:

I = π d⁴_r / 64

Therfore:

F_c = [( C π² E ) / L²_c ] · ( π d⁴_r / 64 )

Buckling End-Condition Constants
End condition	C
Fixed-free	1/4
Rounded-rounded	1
Fixed-rounded	2
Fixed-fixed	4

A column of length L_c and radius of gyration k is considered long when its slenderness ratio L_c / k is larger than the critical slenderness ratio:

L_c / k > ( L_c / k )_critical

L_c / k > [ ( 2 · π² C · E ) / S_y ]^1/2

The radius of gyration k, cross-sectional area A, and second moment of area I are related by I = A k², simplifying the above expression to:

L_c / d_r > .25 [ ( 2 · π² · C · E ) / S_y ]^1/2

For a steel screw whose yield strength is 60,000 psi and whose end-condition constant is 1.0, the critical slenderness ratio is about 100, and L_c / d_r is about 25. For steels whose slenderness ratio is less than critical, the Johnson parabolic relation can be used:

F_c / A = S_y - ( 1 / C · E ) [ ( S_y · L_c ) / ( 2 · π · k ) ]²

which can be solved for a circular cross section of minor diameter d_r as:

d_r = [ ( f_c / ( π S_y ) + ( S_y L² L²_c ) / ( π² C E ) ]^1/2

The load should be externally guided for long travels to prevent eccentric loading. Axial compression or extension δ can be approximated by:

δ = ( F L_c ) / ( A E ) = ( 4 F L_c ) / ( π d⁴_r G )

And similarly, angle of twist Φ, in radians, can be approximated by:

Φ = ( T L_c ) / ( J G ) = ( 32 T L_c ) / ( π d⁴_r G )

Where:

C = Theoretical end-condition constant - see table Buckling End-Condition Constants
E = Modulus of Elasticity (psi)
L_c = Length of Power Screw Column (in)
d_r = Root or minor diameter (in)
K = Length Factor
k = Radius of gyration
A = Cross Sectional Area (in²)
S_y = Yield strength (psi)
F_c = Critical Load at Buckling, lbf
Φ = Angle of twist, Radians
G = Shear Modulus, psi
T = Torque Peak, lb-in
J = Polar Second Moment of Inertia, in³

References:

ANSI B1.7M-1984 (R1992), "Screw Threads, Nomenclature, Definitions, and Letter Symbols," American Society of Mechanical Engineers, New York, 1992.
ANSI Bl.5-1977, "Acme Screw Threads," American Society of Mechanical Engineers, New York, 1977.
ANSI Bl.8-1977, "Stub Acme Screw Threads," American Society of Mechanical Engineers, New York, 1977.
ANSI Bl.9-1973 (R1979), "Buttress Screw Threads," American Society of Mechanical Engineers, New York, 1973.