Langford Analytic · Knowledge Base
Engineering Calculators
Transparent engineering calculations for everyday analysis and design — stress, beams, section properties, buckling, bolts, welds, fatigue, fracture, dynamics, pressure, thermal, composites, materials and unit conversions. Each calculator shows the governing equation, assumptions and method so the result can be independently checked.
Stress & Strength
Axial Stress CalculatorCalculate direct axial stress from force and cross-sectional area. Shows the governing equation, substituted values and unit conversion.Bending Stress CalculatorCalculate bending stress from bending moment, distance from neutral axis and second moment of area using the flexure formula.von Mises Stress CalculatorCalculate von Mises equivalent stress from principal stresses or stress components. Compares with yield to give a margin of safety.Principal Stress CalculatorCalculate principal stresses and maximum shear stress from a 2D stress state (σx, σy, τxy). Shows Mohr’s circle orientation.Torsional Shear Stress CalculatorCalculate torsional shear stress in a circular shaft from applied torque, radius and polar second moment of area.Margin of Safety CalculatorCalculate margin of safety and reserve factor from allowable and applied stress, with interaction ratio for combined loading.Bearing Stress CalculatorCalculate bearing stress at a fastener hole from bolt load, hole diameter and plate thickness.
Beams & Deflection
Section Properties
Rectangular Section Properties CalculatorCalculate area, centroid, second moment of area, section modulus and radius of gyration for a solid or hollow rectangular section.Circular Section Properties CalculatorCalculate area, polar and rectangular second moment of area, section modulus and radius of gyration for solid or hollow circular sections.
Buckling & Stability
Euler Buckling CalculatorCalculate Euler critical buckling load and stress for columns with standard end conditions. Includes slenderness ratio and regime check.Euler Critical Load CalculatorTransparent calculator for the Euler critical buckling load of a slender column, showing the equation, variables, substitution and result.Column Slenderness Ratio CalculatorTransparent calculator for the slenderness ratio L_e/r and the transition slenderness, determining whether Euler or inelastic buckling governs.Radius of Gyration CalculatorTransparent calculator for the radius of gyration of a cross-section, showing the equation, variables and result for both principal axes.Simple Plate Buckling Stress CalculatorTransparent calculator for the critical buckling stress of a simply supported plate under uniaxial compression, showing equation, variables and result.Effective-Length Column Buckling CalculatorTransparent calculator for the Euler critical load with arbitrary end conditions, showing the effect of the effective length factor K.Beam-Column Second-Order Amplification CalculatorTransparent calculator for the moment amplification factor in a beam-column, showing the equation, variables and the effect of axial load on bending.
Bolts & Fasteners
Bolt Strength CalculatorCalculate bolt shear capacity, tensile capacity, and combined tension-shear interaction for NAS, MS, and metric ISO bolts.Bolt Preload and Joint Separation CalculatorCalculate bolt preload from torque, joint stiffness factor, additional bolt load under external force, and the separation load.Fastener Group Load CalculatorCalculate load distribution in a rectangular fastener group under shear and moment, identifying the critical fastener.
Welded Joints
Fatigue
Goodman Fatigue Diagram CalculatorCalculate equivalent stress amplitude using Goodman, Gerber and Soderberg mean stress corrections. Compares with the fatigue endurance limit.Miner’s Rule Cumulative Damage CalculatorCalculate cumulative fatigue damage using Miner’s linear damage rule for multiple load blocks. Enter stress level, applied cycles and allowable cycles per block.S-N Fatigue Life CalculatorEstimate fatigue life using the Basquin S-N method with Goodman mean stress correction and stress concentration factor.
Fracture Mechanics
Stress Intensity Factor CalculatorCalculate stress intensity factor K for common crack configurations (centre crack, edge crack, surface crack) from stress, crack length and geometry factor.Paris Law Crack Growth CalculatorCalculate crack growth rate and cycles to grow a crack from initial to final size using the Paris law with user-entered material constants.
Dynamics & Vibration
Natural Frequency CalculatorCalculate natural frequency of a single-degree-of-freedom mass-spring system. Shows angular frequency, frequency in Hz and period.Damping Ratio and Logarithmic Decrement CalculatorCalculate damping ratio from logarithmic decrement of free vibration, or from critical damping and actual damping values.RMS from PSD CalculatorCalculate RMS response by integrating a PSD over a frequency band. Preserves g²/Hz units and explains the square-root relationship.Dynamic Magnification Factor CalculatorCalculate the dynamic magnification factor for a single-degree-of-freedom system under harmonic excitation, including frequency ratio and damping effects.
Pressure Vessels
Thermal
Composites
Materials
Units & Conversions
Seismic
Seismic Frequency to Period CalculatorConvert between natural frequency (Hz), natural period (s) and angular frequency (rad/s) for seismic analysis applications.g to SI Acceleration CalculatorConvert acceleration expressed in multiples of g to SI units (m/s^2) and imperial units (in/s^2) for seismic analysis input.SRSS Combination CalculatorCombine modal responses using the square root of sum of squares method. Enter modal response values and obtain the SRSS combined peak response.Directional SRSS Combination CalculatorCombine seismic responses from two horizontal and one vertical direction using SRSS or 100-40-40 directional combination methods.Effective Modal Mass CalculatorCalculate effective modal mass from the participation factor and generalised mass, and check cumulative mass participation against the 90% threshold.Seismic SDOF Natural Frequency CalculatorCalculate the natural frequency, period and angular frequency of a single-degree-of-freedom system from mass and stiffness for seismic analysis.Seismic Base Shear CalculatorCalculate seismic base shear from total mass and spectral acceleration, or from equivalent lateral force method parameters per code provisions.Seismic Overturning Moment CalculatorCalculate the seismic overturning moment from base shear and effective height, and compare with the stabilising moment from self-weight.Seismic Anchor Load Distribution CalculatorCalculate tension and shear loads per anchor in a rectangular anchor pattern under seismic overturning moment and base shear.
Turbomachinery
Centrifugal Acceleration CalculatorCalculate centrifugal acceleration from angular velocity and radius — the fundamental body force driver in rotating turbomachinery.Centrifugal Force CalculatorCalculate the centrifugal force on a rotating mass from mass, angular velocity and radius — the blade root load.Simple Blade Centrifugal Root Load CalculatorCalculate the nominal root centrifugal stress for a turbine blade from mass, CG radius, speed and root area.Simple Rotating Disc Stress Illustration CalculatorIllustrative calculation of the bore hoop stress for an annular rotating disc from geometry, material and speed.Radial Growth Estimate CalculatorEstimate the centrifugal radial growth of a rotating ring from density, angular velocity, radius and elastic modulus.Engine Order Frequency CalculatorCalculate the excitation frequency for a given engine order and rotational speed, and the resonance speed for a given natural frequency.Blade Passing Frequency CalculatorCalculate the blade passing frequency from the number of blades and the rotational speed — the excitation frequency for stationary components.
Pressure & Containment
Pressure × Area Resultant Force CalculatorCalculate the resultant force from a uniform pressure acting on a flat surface area. Shows the governing equation, variables, substitution and result.Thin-Wall Hoop Stress CalculatorCalculate the hoop (circumferential) stress in a thin-walled cylindrical pressure vessel. Shows the governing equation, variables and assumptions.Thin-Wall Cylinder Longitudinal Stress CalculatorCalculate the longitudinal (axial) stress in a thin-walled closed-ended cylindrical pressure vessel.Spherical Thin-Wall Membrane Stress CalculatorCalculate the membrane stress in a thin-walled spherical pressure vessel under internal pressure.Pressure End Thrust CalculatorCalculate the pressure end thrust force on a vessel closure, blind flange or pipe termination.Basic Thick-Walled Cylinder Stress CalculatorCalculate the inner and outer surface hoop stress for a thick-walled cylinder under internal pressure using the Lame equations.Differential Pressure CalculatorCalculate the differential pressure across a wall from internal and external pressure values, converting between gauge and absolute conventions.Pressure Stress / Allowable Margin CalculatorCalculate the margin of safety for a pressure stress against a user-supplied allowable stress. The user provides the allowable — no code-specific factors are embedded.Pipe Pressure Thrust CalculatorCalculate the pressure thrust force at a pipe bend, reducer or closed end from the internal pressure and pipe dimensions.
Probabilistic & Reliability
Normal Distribution Probability CalculatorCalculate the probability of a normally distributed variable falling below or above a given value. Shows the z-score, the equation and the result.Standard Normal Z-Score CalculatorConvert a value to a z-score and compute the corresponding standard normal probability. Shows the conversion equation and the result.Coefficient of Variation CalculatorCalculate the coefficient of variation (CoV) from mean and standard deviation. Shows the equation, substitution and result as a percentage.Reliability Index / Failure Probability Conversion CalculatorConvert between a reliability index (beta) and a failure probability (Pf) using the standard normal CDF relationship.Weibull CDF / Reliability CalculatorCalculate the Weibull CDF (failure probability) and reliability (survival probability) for given shape, scale and stress values.Simple Monte Carlo Sampling DemonstrationA demonstration calculator that generates random samples from a normal distribution and shows the sample mean, standard deviation and histogram convergence. Illustrative only.Mean, Variance & Standard Deviation CalculatorCalculate the sample mean, variance and standard deviation from a set of data values. Shows the equations and the computed results.Simple Correlation Coefficient CalculatorCalculate the Pearson correlation coefficient between two sets of data values. Shows the equation, variables and the computed result.
Ballistics & Penetration
Kinetic Energy from Mass & Velocity CalculatorCalculate the kinetic energy of a specified mass and velocity. Shows the equation, variables, substitution and result.Momentum from Mass & Velocity CalculatorCalculate the momentum of a specified mass and velocity. Shows the equation, variables and result.Constant-Force Impact Impulse CalculatorCalculate the impulse from a user-specified force and duration. Shows the equation, variables and result.Wave Transit Time CalculatorCalculate the stress-wave transit time through a specified material thickness. Shows the equation, variables and result.Longitudinal Wave Speed CalculatorCalculate the longitudinal wave speed in a material from its elastic modulus and density. Shows the equation, variables and result.
Creep & High-Temperature
Norton Creep Rate CalculatorCalculate the steady-state (secondary) creep rate from the Norton power-law relationship. Shows the equation, variables, substitution and result.Creep Strain from Norton-Bailey CalculatorCalculate creep strain from a Norton-Bailey relationship that includes primary and secondary creep. Shows the equation, variables and result.Larson-Miller Parameter CalculatorCalculate the Larson-Miller parameter from temperature and rupture time, or estimate rupture time from a known LMP. Shows the equation and result.Simple Stress Relaxation CalculatorA simplified illustration of stress relaxation under imposed strain, using a secondary creep law to estimate the stress decline. Illustrative only.Thermal Expansion at Elevated Temperature CalculatorCalculate the thermal expansion strain and displacement for a material at a specified temperature change. Shows the equation, variables and result.
Random Vibration
PSD Integration to RMS CalculatorCalculate the RMS from a general PSD by numerical integration. Shows the equation, variables, substitution and result.Constant-PSD RMS CalculatorCalculate the RMS from a flat (constant) PSD over a frequency range. Shows the equation, variables, substitution and result.Piecewise PSD RMS CalculatorCalculate the RMS from a piecewise log-log PSD defined by breakpoints. Shows the segment integration formula and result.Miles Equation CalculatorCalculate the RMS response of an SDOF system using Miles equation. Shows the equation, variables, substitution and result.Natural Frequency / Damping Random Response IllustrationAn illustrative calculator showing how natural frequency and damping affect the Miles equation RMS response, demonstrating the sensitivity.Sigma / RMS Peak IllustrationAn illustrative calculator showing the relationship between RMS, sigma levels and expected peak over a duration.
Shock & Transient Dynamics
Force–Time Impulse & Momentum Change CalculatorCalculate impulse from a rectangular or average force–time input and show the resulting momentum change.Momentum Change CalculatorCalculate change in momentum and velocity from mass and impulse.Free-Fall Impact Velocity CalculatorCalculate ideal free-fall impact velocity from drop height, for qualification screening only.Half-Sine Pulse Impulse CalculatorCalculate ideal velocity change and equivalent impulse per unit mass for a half-sine acceleration pulse.Shock SDOF Natural Frequency CalculatorCalculate SDOF natural frequency or stiffness for a mass–spring screening model.Shock Isolation Frequency Ratio CalculatorIllustrate linear SDOF frequency ratio and transmissibility for harmonic screening of an isolation concept.
Advanced Damage & Failure
Composite Failure Index Comparison HelperCompare failure indices from different criteria (maximum stress, Tsai-Hill, Hashin) for the same stress state. User-supplied stresses and strengths.First-Ply Failure Load Scaling HelperConvert a failure index to a load scaling factor (strength ratio) for both linear and quadratic criteria.Simple Fracture Energy from Traction–Separation LawCalculate the fracture energy from a triangular traction-separation law given peak traction and failure displacement.Mixed-Mode Energy Release Ratio HelperCalculate the mode mixity and check the propagation criterion for mixed-mode delamination.Cohesive Initial Stiffness Sensitivity IllustrationAn illustrative helper showing how cohesive initial stiffness affects the artificial compliance and the trade-off with numerical conditioning.Impact Kinetic Energy CalculatorCalculate the incident kinetic energy of an impactor from mass and velocity, and estimate absorbed energy from coefficient of restitution.Residual Strength Reduction Percentage HelperCalculate the residual strength retention percentage from undamaged and damaged strengths.