Rotational Stiffness Calculator - Moment Over Angle Formula

A rotational stiffness calculator determines a body's rotational stiffness k from the applied moment M and the rotation angle theta that moment produces, using the linear relationship k = M / theta. Rotational stiffness is the rotational cousin of axial stiffness in Hooke's law: it tells you how much moment is required per radian of rotation, and it shows up in foundation rocking, shaft design, vibration analysis, and structural support modelling.

• • Foundation rocking analysis: Estimate the rotational stiffness of a circular foundation or pile cap subjected to a rocking moment in geotechnical work.
• • Shaft and bearing design: Compare candidate drive shafts or bearing supports by their moment per radian response.
• • Structural support stiffness: Quantify how much moment a column base, beam connection, or pinned support resists before it rotates.
• • Vibration and modal analysis: Build the rotational stiffness term that feeds into natural frequency and mode-shape calculations for torsional vibration.

In SI units k is reported in newton-metres per radian (N*m/rad). US customary references sometimes use pound-force-inches per degree. The relationship is linear only in the small-angle, elastic regime that engineering textbooks assume.

For a circular foundation on elastic soil, a geotechnical formula gives k = 8 * G * R^3 / (3 * (1 - nu)), where G is the shear modulus, R is the foundation radius, and nu is Poisson's ratio. That is a different expression for the same rotational stiffness concept this page is built around.

For the linear Hooke's law case F = k * x, the Spring Constant and Deflection Calculator handles axial springs with the same solve-direction pattern this page uses for M = k * theta.

Pick the variable you want to solve for and the calculator rearranges M = k * theta to fit that branch. Solving for k reads the moment and the angle; solving for M reads the stiffness and the angle; solving for theta reads the stiffness and the moment.

• k: Rotational stiffness in newton-metres per radian (N*m/rad). The primary output regardless of the chosen branch.
• M: Applied moment (torque) about the rotation axis, in newton-metres (N*m).
• theta: Rotation angle in radians internally; the form accepts degrees and converts them by multiplying by pi / 180.

The page recalculates as you type, so flipping the Solve for selector updates the result panel without reloading. The angle echo in radians and degrees lets you cross-check the unit conversion against a textbook entry without leaving the page.

The linear relationship only holds in the elastic, small-angle regime. Past the yield moment or under large rotations the effective k drops and this linear expression underestimates the true response.

According to HyperPhysics: Description of Motion (R. Nave, Georgia State University), angular quantities in mechanics follow the same linear convention as their translational counterparts, and a body's rotational stiffness is the rotational analogue of the linear spring constant in newton-metres per radian.

According to HyperPhysics: Elasticity and Hooke's Law, the rotational form M = k * theta is the rotational analogue of Hooke's law F = k * x, with k reported in moment per radian as long as the deformation stays inside the elastic regime.

Once you know the rotational stiffness of a shaft, the Angle of Twist Calculator computes the deformation theta directly from GJ / L so you can compare the linear elastic prediction against your moment input.

Four mechanical concepts drive every rotational stiffness calculation. Understanding them keeps the formula honest about when it applies.

These concepts reappear across mechanics of materials, structural dynamics, and foundation engineering, and they form the basis for the rotational stiffness values tabulated for shafts, bearings, and pile caps.

When the rotational spring is one of several degrees of freedom at a node, the Stiffness Matrix Calculator assembles the full M x M stiffness matrix so the rotational entry is consistent with the translational ones.

Pick the variable you want to solve for, fill in the other two, and read the answer from the results panel. The page recalculates as you type.

1. 1 Choose what to solve for: Use the Solve for menu to pick rotational stiffness k, applied moment M, or rotation angle theta. The chosen variable is the one the calculator will return.
2. 2 Enter the two known values: Type k and theta for Solve for M; type k and M for Solve for theta; or type M and theta for Solve for k. The third field can stay blank or be ignored for that branch.
3. 3 Match units to the labels: Moments go in as newton-metres (enter kN*m by typing the value in N*m). Angles go in as degrees; the calculator multiplies by pi / 180 internally because the SI definition uses radians.
4. 4 Read the result panel: The black box always shows k in N*m/rad. The lines below show the applied moment and the rotation angle in both radians and degrees so you can confirm the conversion.
5. 5 Switch solve directions freely: Flip the Solve for menu between k, M, and theta without clearing the other two inputs. The equation rearranges to keep the value you already entered unchanged.
6. 6 Stay inside the linear regime: Treat the result as a small-angle, elastic-stiffness estimate. If theta climbs past a few degrees or M approaches the yield torque, switch to a finite-element or tangent-stiffness model instead.

To size a footing under a rocking load, pick Solve for k, enter the design moment M = 250 kN*m and the design rocking angle theta = 0.5 deg. The calculator returns theta_rad = 0.008727 rad and k = 28,648,000 N*m/rad, which you can compare against the geotechnical 8 G R^3 / (3 (1 - nu)) estimate to confirm the foundation is stiff enough.

When the rotation is dominated by bending in a beam or shaft, the Beam Bending Stress Calculator gives the bending stress so you can decide whether the rotation you entered is still inside the elastic limit.

Putting the M / theta relationship behind a single rotational stiffness calculator saves time on homework, lab reports, and design checks where you need one of the three variables quickly.

• • Three solve directions from one page: Switch between k, M, and theta without retyping the formula or restarting the calculation.
• • Automatic degree-to-radian conversion: Enter angles in degrees the way most textbooks and lab reports quote them and the calculator handles the pi / 180 conversion internally.
• • Worked example for sanity checking: The 16 N*m over 40.1 degree example reproduces a published reference value, so you can confirm the calculator before trusting new inputs.
• • Ready for design checks: Use the result directly when comparing a calculated rotational stiffness against geotechnical, structural, or mechanical code recommendations.

For a quick homework problem you only need two numbers and a solve direction. For a real design check the result returns the rotational stiffness in N*m/rad, which matches the unit used by foundation and shaft references.

Once you know the rotational stiffness of the driveline, the Torque Power Speed Calculator converts the steady-state torque into shaft power so you can size the motor or coupling alongside the stiffness check.

Three numbers feed every solve direction. Two are physical quantities of the loaded body, and one is a choice of angle unit that affects the numerical value of k.

• • Linear assumption: M = k * theta assumes elastic, small-angle behaviour and ignores plasticity, large rotations, and geometric stiffening or softening.
• • Single degree of freedom: the calculator handles one rotational spring at a time. For coupled translational and rotational DOFs use the stiffness matrix instead.

For a homogeneous circular shaft of length L and torsional constant J in a material of shear modulus G, the rotational stiffness is k = G * J / L, the design formula the M / theta relationship simplifies to in the elastic regime. The same M / theta ratio appears in foundation rocking, where k depends on shear modulus and footing radius.

According to The Engineering Toolbox: Stiffness, stiffness is the resistance of an elastic body to deflection by an applied force and is defined as k = F / delta in newtons per metre; the rotational form M = k * theta extends the same resistance-of-deflection idea to moments in newton-metres per radian.

When the angle gets large enough to violate the small-angle assumption, the Angular Displacement Calculator shows the actual arc length and chord length so you can decide whether to stay with M = k * theta or switch to a more detailed model.