Power Screw Strength Equations – Stress & Buckling

Power screw strength is determined by considering the stresses developed in the screw and nut while transmitting axial load. Important checks include thread bearing pressure, thread bending and shear, direct tensile or compressive stress, torsional stress and, for sufficiently slender screws in compression, column buckling.

Need help choosing a machine element? See the Machine Element Selection Guide for guidance on cams, flywheels, power screws, shock absorbers, springs and mechanisms, and selecting the appropriate RoyMech engineering reference.

Engineering note: The equations on this page provide general mechanical design guidance. Actual power-screw capacity also depends on thread geometry, material properties, manufacturing tolerances, surface finish, lubrication, nut design, load distribution, duty cycle and the manufacturer's design data. Proprietary power screws should normally be selected using the supplier's rated capacities and technical information.

Power Screw Strength Considerations

Simple calculations normally assume that the axial load is distributed across all of the engaged threads. In practice the load is not shared uniformly: the first engaged threads normally carry a greater proportion of the load because of the elastic deformation and relative stiffness of the screw and nut.

Local yielding can redistribute some of this load. Thread finish, clearance, geometry, material combination and lubrication can also influence the actual stress distribution. Consequently, the equations below should be regarded as engineering design models rather than an exact representation of the local thread stresses.

The principal strength considerations are:

Bearing stress between the screw and nut threads
Bending stress at the thread root
Shear stress in the screw and nut threads
Direct tensile or compressive stress in the screw
Torsional and combined stress
Column buckling when a sufficiently slender screw is loaded in compression

Power Screw Symbols and Definitions

Symbol	Definition
σ_B	Bearing stress
σ_b	Bending stress
σ_c	Compressive stress
σ_t	Tensile stress
σ_co	Maximum compressive stress on the concave side of a column
τ	Shear stress
τ_max	Maximum shear stress under combined loading
A	Stress area
b	Width of screw thread at the root
d_m	Mean screw diameter
d_r	Screw root diameter
d_p	Pitch diameter
d_o	Outside diameter of screw
d_t	Diameter used in approximate stress-area calculation
E	Modulus of elasticity
e	Load eccentricity for column calculation
h	Thread depth
I	Second moment of area
J	Polar second moment of area
k	Least radius of gyration, √(I/A)
K	Column end-restraint load multiplier used in the legacy Ritter formulation below
L	Unsupported length of screw between effective supports
l	Thread lead
n	Number of threads in engagement
p	Thread pitch
P	Column load
S	Legacy Ritter-equation parameter; see note in the buckling section
T	Torque transmitted by screw
W	Axial force generated or supported by screw

Bearing Stress

Bearing stress results from the compressive contact force between the mating screw and nut thread surfaces while supporting or transmitting the axial load W.

σ_{B} = \frac{W}{π d_{m} h n}

where n is the number of engaged threads.

Legacy Safe Bearing Pressure Data

Legacy reference data: The following values are retained from the original RoyMech reference. Their original published source is not identified on the legacy page and they should therefore not be treated as current manufacturer design limits. For new designs, confirm allowable bearing pressure against the screw/nut materials, lubrication, operating speed, duty and supplier data.

Screw Material	Nut Material	Bearing Pressure	Operating Condition
Steel	Bronze	17–24 N/mm²	Low speed
Steel	Bronze	11–17 N/mm²	3 m/min
Steel	Cast iron	13–17 N/mm²	2.5 m/min
Steel	Bronze	5–10 N/mm²	6–12 m/min
Steel	Cast iron	7–7 N/mm²	6–12 m/min
Steel	Bronze	1–2 N/mm²	15 m/min

Note: The value 7–7 N/mm² is reproduced exactly from the original RoyMech table. The original source has not been established, so the value has not been silently altered.

Bending Stress at the Thread Root

An approximate thread-root bending stress can be obtained by treating the engaged thread as a cantilever fixed at its root, with the load assumed to act at the midpoint of the thread projection.

The ordinary bending relationship is:

σ = \frac{M y}{I}

For the simplified power-screw model, the total length of the resisting thread section is πd_mn and its width at the root is b.

I = \frac{π d_{m} n b^{3}}{12}

y = \frac{b}{2}

M = \frac{W h}{2}

The resulting maximum thread-root bending stress is:

σ_{b} = \frac{3 W h}{π d_{m} n b^{2}}

The original RoyMech approximation uses b = 3p/4 for an ISO metric thread and b = p/2 for a square single-start thread. These values relate to the simplified thread-section model and should not replace dimensions taken from the applicable thread standard when a detailed stress assessment is required.

Thread Shear Stress

Both the screw and nut threads are subjected to transverse shear resulting from the transmitted axial load. For the rectangular-section approximation used here, the maximum shear stress occurs at the neutral axis.

Screw Thread

τ = \frac{3 W}{2 π d_{r} n b}

Nut Thread

τ = \frac{3 W}{2 π d_{o} n b}

Tensile and Compressive Stress

A power screw carrying an axial load is subjected to direct tensile or compressive stress:

σ_{t / c} = \frac{W}{A}

For preliminary conservative calculations, the area based on the screw root diameter may be used:

A = \frac{π {d_{r}}^{2}}{4}

The original RoyMech approximation for the stress-area diameter uses the mean of the pitch and root diameters:

d_{t} = \frac{d_{p} + d_{r}}{2}

A = \frac{π {d_{t}}^{2}}{4}

For standard ISO metric threads, use the appropriate specified tensile stress area where available. See the RoyMech ISO metric thread stress-area information.

Combined Axial and Torsional Stress

A power screw transmitting torque while carrying an axial load is subjected simultaneously to direct stress and torsional shear stress.

Using the maximum shear stress (Tresca) relationship for a state consisting of direct stress σ and shear stress τ:

τ_{\max} = \sqrt{{(\frac{σ}{2})}^{2} + τ^{2}}

For a solid circular screw section of root diameter d_r, the torsional shear stress produced by torque T is:

τ = \frac{T (d_{r} / 2)}{J} = \frac{16 T}{π {d_{r}}^{3}}

With the direct axial stress σ = W/A, the combined maximum shear stress becomes:

τ_{\max} = \sqrt{{(\frac{W}{2 A})}^{2} + {(\frac{16 T}{π {d_{r}}^{3}})}^{2}}

For a screw in tension this combined-stress check can be applied directly. For a screw in compression, column stability must also be considered when the screw is sufficiently long and slender.

Power Screw Buckling

A long power screw carrying compressive load can behave as a column and may fail by buckling before the material reaches its simple compressive strength.

Design guidance: The original RoyMech notes use a screw length greater than approximately eight times the root diameter as a trigger for considering column behaviour, and a slenderness ratio of approximately 100 when distinguishing shorter from longer columns. These should be regarded as preliminary rules of thumb rather than universal boundaries. Column behaviour depends on geometry, material, end restraint and the applicable design method.

The slenderness ratio is based on the effective unsupported length and the least radius of gyration:

λ = \frac{L}{k}

For sufficiently slender columns, Euler buckling theory may be appropriate. See the RoyMech Simple Struts and Column Buckling reference for the general treatment of columns and end conditions.

Legacy Ritter / Gordon-Rankine Form

The original RoyMech power-screw notes use the following form of the Ritter equation for machine-tool screw design:

σ_{co} = \frac{P}{A} [1 + {(\frac{L}{k})}^{2} \frac{S}{π^{2} K E}]

Legacy notation: The original RoyMech equation uses the symbol S, but the surviving page does not define this parameter. The equation is therefore retained as legacy reference information rather than silently assigning a modern interpretation to S. Before using this particular form for design, establish the definition and units of S from the original design method or another authoritative source.

The original equation uses the following values of K to represent the influence of column end restraint:

End Condition	K
One end fixed, one end free	0.25
Both ends pin-connected	1
One end pin-connected, one end fixed	2
Both ends fixed	4

These values are best understood here as column buckling-load multipliers relative to the pin-ended case. They should not be confused with the effective-length factor convention used in many modern column calculations, where the numerical factors are expressed differently.

Eccentrically Loaded Screw

The preceding legacy expression assumes a purely axial column load. Where the load is eccentric to the screw centreline by distance e, the original RoyMech notes give the following variation:

σ_{co} = \frac{P}{A} [1 + {(\frac{L}{k})}^{2} \frac{S}{π^{2} K E} + \frac{y e}{k^{2}}]

The maximum compressive stress on the concave side of the screw column should remain within the allowable compressive stress for the screw material, using an appropriate design factor and a buckling method suitable for the application.

Practical Power Screw Strength Checks

A complete power-screw assessment should not rely on a single calculated stress. Depending on the application, the designer should consider the thread bearing pressure, thread-root bending and shear, screw tensile or compressive stress, torsional stress, combined loading and column stability.

The calculation should also reflect the actual operating conditions. Repeated motion, reversals, shock loading, contamination, lubrication, temperature, wear and the required service life can be as important as the static stress calculation.

Safety-critical applications: For lifting equipment, presses, personnel-supporting systems or other applications where failure could cause injury or major damage, use the relevant machinery and lifting standards and the power-screw manufacturer's rated data. Do not rely solely on the simplified equations on this page.

Power Screw Strength Equations – Stress and Buckling