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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= πd^{4}_{r} / 64
re-written as:
F_{c} = [( C π^{2} E ) / L^{2}_{c} ] · I
Second Moment of Inertia:
I = π d^{4}_{r} / 64
Therfore:
F_{c} = [( C π^{2} E ) / L^{2}_{c} ] · ( π d^{4}_{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 · π^{2} 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 k^{2}, simplifying the above expression to:
L_{c} / d_{r} > .25 [ ( 2 · π^{2} · 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} L^{2} L^{2}_{c} ) / ( π^{2} 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 d^{4}_{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 d^{4}_{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 (in^{2})
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, in^{3}
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.
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