Engineering mechanics: statics, strength and motion
Calculate forces, stresses, deformations, motion, vibrations and hydrostatics with transparent models.
Choose the relevant topic and load model, enter quantities with units, then check the result and model limits.
Calculators in this category
Statics and beam loads
Combine forces, replace distributed loads and find internal bending moments.
Resultant of two forces with an included angle
Calculate the magnitude of the resultant of two forces with a common point of application from both force magnitudes and their included angle using the cosine rule: R = √(F₁²+F₂²+2F₁F₂cos α). Solvable for each individual force and for the included angle.
Open calculator →Bar forces in a two-bar system (bracket, tripod)
Calculate the bar force in a two-bar system (bracket, jib, tripod) from the suspended load and the two bar angles to the load direction: S₁ = G·sin α₂/sin(α₁+α₂). Solvable for the load, so a limiting bar force yields the maximum load that may be suspended.
Open calculator →Resultant of a trapezoidal or triangular line load
Calculate the resultant (total force) of a linearly varying line load from its two edge ordinates and the loaded length: R = (q₁+q₂)/2 · l. One formula covers rectangular, triangular and trapezoidal loads, and it is solvable for every input.
Open calculator →Position of the resultant of a trapezoidal or triangular line load
Calculate the distance of the resultant from the start of a linearly varying line load: xs = l/3 · (q₁+2q₂)/(q₁+q₂). This line of action is needed for the moment equilibrium – only for a uniform load does it lie at mid-length, whereas for a triangular load it sits at two thirds of the length.
Open calculator →Centroid from two part areas (composite method)
Calculate the centroid of an area composed of two parts with the composite method: xs = (A₁x₁ + A₂x₂)/(A₁+A₂). Cut-outs and holes are entered as a negative part area. The same works for masses if masses are entered instead of areas.
Open calculator →Maximum beam moment under an off-centre point load
Calculate maximum bending moment of a simply supported beam under one off-centre point force from load, span and force location.
Open calculator →Strength and deformation
Assess buckling limits, material properties, stresses and elastic deformation.
Euler buckling load of a compression member
Calculate the critical buckling load Fkrit of a slender compression member from elastic modulus, smallest second moment of area and buckling length: Fkrit = π²EI/lk². The buckling length lk covers all four Euler cases through its factor, and any of the four quantities can be the target.
Open calculator →Slenderness ratio and radius of gyration of a compression member
Calculate the slenderness ratio λ = lk/i of a compression member from buckling length, second moment of area and cross-sectional area; the radius of gyration i = √(I/A) is determined along the way. λ decides whether a member buckles elastically according to Euler or must be treated as a stocky member.
Open calculator →Euler buckling stress of a compression member
Calculate the critical buckling stress σkrit = π²E/λ² of a compression member from elastic modulus and slenderness ratio. Comparing σkrit with the material's proportional limit shows immediately whether the Euler formula is valid for the member at all.
Open calculator →Limit slenderness: validity boundary of the Euler formula
Calculate the limit slenderness λ₀ = π·√(E/|σP|) from elastic modulus and proportional limit. λ₀ separates slender members that buckle elastically according to Euler from stocky members for which the Euler formula is inadmissible – for structural steel S235 it gives λ₀ ≈ 104.
Open calculator →Shear modulus from Young's modulus and Poisson's ratio
Calculate the shear modulus G = E/(2·(1+ν)) from Young's modulus and Poisson's ratio – also rearranged for E or ν. An isotropic elastic material has only two independent elastic constants, so the third always follows from the other two.
Open calculator →Transverse contraction: lateral strain and diameter change
Calculate the lateral strain εq = −ν·εl and the resulting diameter or width change Δd of a loaded bar. Solvable for Poisson's ratio ν, so that ν can be determined directly from measured longitudinal and transverse deformation in a tensile test.
Open calculator →Boiler formula: hoop stress in a thin-walled cylinder
Calculate the hoop stress (circumferential stress) σφ = p·r/t of a thin-walled cylindrical vessel or pipe under internal pressure – solvable for pressure, radius and wall thickness. Hoop stress is twice the longitudinal stress σx = p·r/(2t), which is why pipes and vessels under internal pressure split lengthwise.
Open calculator →Hoop stress in a thick-walled cylinder
Calculate hoop stress at the inner wall of a thick cylindrical tube under internal pressure from pressure and inner and outer radii.
Open calculator →Stress at the centre of a rotating solid disk
Calculate the maximum radial and hoop stress of a free rotating solid disk of constant thickness from density, rotational speed, radius and Poisson ratio.
Open calculator →Cantilever tip rotation under an end load
Calculate the slope angle at the free end of a cantilever beam from end force, length, Young modulus and second moment of area.
Open calculator →Extension of a hanging rod under its own weight
Calculate elongation of a uniform vertical rod fixed at its top due solely to self-weight from density, length, Young modulus and gravity.
Open calculator →Motion, kinetics and impact
Calculate trajectories, rolling and sliding motion, and momentum and angular-momentum balances.
Oblique projectile motion: range at any launch angle
Calculate the range of an oblique projectile from launch speed, launch angle and launch height – and conversely the launch speed needed for a required range. Unlike horizontal projectile motion the launch angle is free, and when launching above the landing point the optimum angle is below 45°.
Open calculator →Straight central impact: velocity after impact
Calculate the velocity of a body after a straight central impact from both masses, both velocities before impact and the coefficient of restitution k. Through k one calculator covers every case: k = 0 perfectly plastic, k = 1 perfectly elastic, in between the real impact. Solvable for k to determine the coefficient of restitution from a measured velocity.
Open calculator →Kinetic energy loss in a straight central impact
Calculate the loss of kinetic energy in a straight central impact from both masses, both velocities before impact and the coefficient of restitution: ΔT = ½(1−k²)·m₁m₂/(m₁+m₂)·(v₁−v₂)². This energy goes into deformation, heat and sound – in hammering and pile driving it is precisely the useful part.
Open calculator →Parallel-axis theorem: mass moment of inertia about a parallel axis
Calculate the mass moment of inertia of a body about any axis from its value about the parallel axis through the centre of mass: JA = JS + m·rS². Solvable for the centroidal inertia, the mass and the axis distance, so a measured value can also be reduced back to the centroidal one.
Open calculator →Gyroscopic moment: bearing load from a guided rotor
Calculate the gyroscopic moment MK = J·ωK·ωF that the bearings of a rotating body must additionally carry when its axis of rotation is itself being swivelled. This simplified form holds for mutually perpendicular axes of rotation – the usual case in machine dynamics.
Open calculator →Rolling acceleration of a body on an inclined plane
Calculate the acceleration of a body rolling without slip down an inclined plane from the slope angle and the inertia ratio: a = g·sin α/(1 + JS/(m·R²)). A solid cylinder reaches two thirds, a sphere five sevenths and a thin-walled tube only half the acceleration of a sliding body.
Open calculator →Common speed after two rotating masses are coupled
Calculate the common speed of two rotating masses after they are coupled, from both inertias and both initial speeds: ω = (J₁ω₁ + J₂ω₂)/(J₁+J₂). Angular momentum is conserved, rotational energy is not – the difference is converted into heat as friction work in the clutch.
Open calculator →Exit speed on an inclined slide with friction
Calculate the exit speed of a sliding body at the end of an inclined slide from the slide length, the slope angle and the kinetic friction coefficient using the energy principle: v = √(2·L·g·(sin α − μ·cos α)). Mass and material cancel out – only geometry and the friction coefficient matter.
Open calculator →Vibrations
Assess natural periods and transmission of oscillating forces.
Period of a simple pendulum
Calculate a simple pendulum's period from its length and gravity for small swings, or find the length needed for a desired period.
Open calculator →Force transmissibility of a vibration isolator
Calculate the ratio of harmonic force reaching a foundation to applied excitation force from frequency ratio and damping ratio.
Open calculator →Hydrostatics
Determine pressure forces in still liquids and initial stability of floating bodies.
Hydrostatic force on a vertical rectangular gate
Calculate the resultant hydrostatic force on a fully submerged vertical rectangular gate from liquid density, top depth, width and height.
Open calculator →Centre of pressure of a vertical rectangular gate
Calculate the depth where the resultant hydrostatic force acts on a fully submerged vertical rectangular gate from top-edge depth and gate height.
Open calculator →Metacentric height and initial stability
Calculate metacentric height from waterplane second moment, displaced volume and centre separation for small heel angles.
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