Torsion of Bars Calculators

Torsion of Circular Cross Section Bar

Loading of a bar, where of all internal force factors acting in its cross-sections, only the moment whose vector is directed along the bar axis is not equal to zero is called torsion. When a cylindrical bar is twisted, only shear stresses arise across its cross sections. At that, the cross-sections remain plane and do not change their size in the radial direction. Also, the distances between the sections do not change, but at that moment they turn relative to each other by some angle φ to be calculated using the following equation:

dφ = ( T / G*I_p )*dz

T - Torque;
G - Shear modulus;
I_p - Polar moment of inertia of the bar cross-section.

In the general case, the maximum shear stresses occur at the edge of the cross-section, except for the outer corners, where the shear stresses are equal to zero. A bar of non-circular cross-section undergoes warping where the points of its cross-section go out of the plane and move along the axis. The shear stresses in the bar may be found by the following equation:

τ = T*ρ / I_p

ρ - distance from centroid to calculated point.

In this calculation, a circular bar of length L and diameter D, fixed at one end is considered. The bar is under torque T, applied to the opposite end. Following the calculations, the total twist angle φ and the maximum shear stresses τ in the section are determined.

For the calculation, the elastic modulus E and Poisson's ratio ν of the bar should be specified.

Bar of circular section under torsion

Calculation of Bar of circular section under torsion

INITIAL DATA

D - Bar outer diameter;

L - Bar length;

Т - Torque;

ν - Poisson's ratio;

Е - Young's modulus.

RESULTS DATA

τ - Maximum shear stress in the cross-section;

φ - Twist angle.

Metric Imperial

Outer diameter (D)

Bar length (L)

Torque (Т)

Poisson's ratio (ν)

Young's modulus (Е)

Shear stress (τ)

Twist angle (φ)

BASIC FORMULAS

Maximum shear stress:

τ = 2Т / π*r³;

Twist angle:

φ = 2T*L / (π*r⁴*G),

G - Shear modulus.

INITIAL DATA

D - Bar outer diameter;

L - Bar length;

Т - Torque;

ν - Poisson's ratio;

Е - Young's modulus.

RESULTS DATA

τ - Maximum shear stress in cross-section;

φ - Twist angle.

MATERIALS PROPERTIES

Material	Young’s modulus Pa (psi)	Poisson’s ratio
Steel	1.86÷2.1×10¹¹ (2.7÷3.05×10⁷)	0.25÷0.33
Cast iron	0.78÷1.47×10¹¹ (1.1÷2.1×10⁷)	0.23÷0.27
Copper	1.0÷1.3×10¹¹ (1.45÷1.9×10⁷)	0.34
Tin bronze	0.74÷1.22×10¹¹ (1.1÷1.8×10⁷)	0.32÷0.35
Brass	0.98÷1.08×10¹¹ (1.4÷1.6×10⁷)	0.32÷0.34
Aluminum alloy	0.7×10¹¹ (1.0×10⁷)	0.33
Magnesium alloy	0.4÷0.44×10¹¹ (5.8÷6.4×10⁶)	0.34
Nickel	2.5×10¹¹ (3.6×10⁷)	0.33
Titanium	1.16×10¹¹ (1.7×10⁷)	0.32
Lead	0.15÷0.2×10¹¹ (2.2÷2.9×10⁶)	0.42
Zinc	0.78×10¹¹ (1.1×10⁷)	0.27
Glass	4.9÷5.9×10¹⁰ (7.1÷8.5×10⁶)	0.24÷0.27
Concrete	1.48÷2.25×10¹⁰ (2.1÷3.3×10⁶)	0.16÷0.18
Wood (along the grain)	8.8÷15.7×10¹⁰ (12.8÷22.8×10⁶)	-
Wood (across the grain)	3.9÷9.8×10¹⁰ (5.7÷14.2×10⁶)	-
Nylon	1.03×10¹⁰ (1.5×10⁶)	-

OTHER CALCULATORS

AREA MOMENTS OF INERTIA

BEAM CALCULATORS

TORSION OF BARS

CIRCULAR FLAT PLATES

BUCKLING

ELASTIC CONTACT

IMPACT LOADS

NATURAL FREQUENCIES

PRESSURED SHELLS

FLUID DYNAMIC

COMPOSITES

SPRINGS

THREAD CONNECTIONS

SHAFT CONNECTIONS

BEARINGS

DRIVES

FATIGUE

HEAT TRANSFER