Langford Analytic · Knowledge Base

Worked Examples

Step-by-step engineering calculations showing assumptions, mathematics, results, and interpretation.

108 articles & resources

Statics & Strength

Combined Stress and von Mises Check — Worked ExampleA complete stress analysis of an L-bracket under combined axial, bending and shear loading, calculating principal stresses, von Mises equivalent stress and margin of safety.Margin of Safety Calculation — Worked ExampleA step-by-step margin of safety calculation for a tension-loaded aluminium plate, showing limit load, ultimate load and the interaction between yield and ultimate margins.Bearing Stress in a Loaded Plate — Worked ExampleA bearing stress and net-section check for a single-bolt aluminium lug, including edge margin assessment and comparison with MMPDS bearing allowables.

Beams & Deflection

Simply Supported Beam Deflection — Worked ExampleA complete deflection calculation for a simply supported steel beam under a central point load, with stress check and FEA comparison.

Buckling & Stability

Euler Buckling of a Pinned Column — Worked ExampleA step-by-step Euler column buckling calculation for an aluminium tube, including slenderness check, critical load, margin of safety and comparison with FE eigenvalue buckling.Euler Column Buckling Worked ExampleStep-by-step calculation of the Euler critical load for a pinned-pinned steel column, showing geometry, slenderness check, formula application and interpretation.Effective-Length Worked ExampleCalculation of the Euler critical load for a fixed-pinned column, showing the effect of end restraint on the effective length and buckling load.Plate Buckling Worked ExampleStep-by-step calculation of the critical buckling stress for a simply supported plate under uniaxial compression, with aspect ratio and boundary condition considerations.Shear Buckling Worked ExampleCalculation of the critical shear buckling stress for a simply supported plate, showing the buckling coefficient, formula and comparison with applied shear.Eigenvalue Buckling FEA ExampleWorked example of an eigenvalue buckling FEA workflow for a stiffened panel, showing setup, mode extraction, load factor interpretation and hand-calculation verification.Shell Buckling Imperfection-Sensitivity ExampleWorked example showing how geometric imperfections reduce the collapse load of a cylindrical shell, with sensitivity study across multiple amplitudes.Nonlinear Collapse ExampleWorked example of a nonlinear buckling analysis for a stiffened panel, showing load-displacement response, limit point, collapse load and comparison with eigenvalue.Stiffened Panel Buckling ExampleComplete worked example for a stiffened panel — skin buckling stress, stiffener column check, crippling assessment, post-buckling reserve and collapse estimation.

Loads & Load Paths

Free-Body Diagram and Support Reactions — Worked ExampleA complete free-body diagram analysis of a loaded bracket, determining support reactions and verifying equilibrium.

Bolted Joints & Fasteners

Bolt Joint Strength — Worked ExampleA complete margin of safety calculation for a tension-shear loaded NAS bolt in a single-shear aluminium fitting, following MMPDS interaction methodology.Preloaded Bolted Joint Separation — Worked ExampleA complete joint stiffness, load-sharing and separation calculation showing how external load splits between bolt and clamped members, and the separation load.

Welded Joints

Fillet Weld Throat Shear Check — Worked ExampleA complete fillet weld throat stress calculation for a bracket-to-plate welded joint under combined shear and torsion loading.

Fatigue

S-N Fatigue Life Calculation — Worked ExampleA complete S-N fatigue analysis of a 2024-T3 aluminium panel with a circular hole, applying stress concentration, surface finish correction, Goodman mean stress correction and Basquin life prediction.Goodman Mean Stress Correction — Worked ExampleA worked example applying the Goodman, Gerber and Soderberg corrections to a fatigue loading with non-zero mean stress, comparing the three methods.

Fracture & Damage Tolerance

Stress Intensity Factor and Critical Crack Size — Worked ExampleA complete LEFM calculation of stress intensity factor K for a centre crack in a plate, comparison with fracture toughness KIC, and determination of critical crack size.

Dynamics & Vibration

Modal Analysis — Natural Frequency and FEA ComparisonAnalytical natural frequency calculation for a simply supported beam and comparison with FEA, demonstrating setup, result interpretation and mesh sensitivity.Random Vibration RMS Response — Worked ExampleA complete random vibration analysis example: PSD input, response PSD, RMS integration, and 3-sigma stress assessment for a cantilever bracket.

Composites

Composite Laminate ABD Matrix — Worked ExampleA hand calculation of the extensional stiffness matrix A for a [0/±45/90]s quasi-isotropic CFRP laminate, with effective engineering constants and a Tsai-Wu failure index check.

Thermal & Thermal-Stress

Constrained Thermal Stress — Worked ExampleA complete thermal stress calculation for a constrained aluminium bar heated uniformly, showing the difference between free expansion and fully constrained thermal stress.

Pressure & Structural Loading

Thin-Walled Cylinder Hoop Stress — Worked ExampleA complete thin-walled pressure vessel calculation: hoop stress, longitudinal stress, combined stress check and margin of safety for an aluminium cylinder.

FEA Verification

FEA Mesh Convergence — Worked ExampleA complete mesh convergence study on a notched plate in tension, showing how stress converges with mesh refinement and distinguishing a real stress concentration from a singularity.

Seismic

Response Spectrum Analysis of a Single-Degree-of-Freedom SystemA worked example showing how spectral acceleration is obtained from a response spectrum for a single-degree-of-freedom system, and how peak displacement and equivalent static force are calculated.Modal Response Spectrum Analysis of a Simple FrameA worked example performing modal response spectrum analysis on a two-storey equipment frame, showing modal extraction, spectral acceleration lookup, modal combination by SRSS and directional combination.Floor Response Spectrum Applied to an Equipment FrameA worked example showing how a floor response spectrum is applied to equipment mounted on a floor, including spectrum interpolation, damping selection and peak response calculation.SRSS Modal Combination Worked ExampleA worked example showing how modal peak responses are combined using the Square Root of Sum of Squares method, with explicit calculation of the combined result.CQC Modal Combination Worked ExampleA worked example showing how modal peak responses are combined using the Complete Quadratic Combination method, including correlation coefficient calculation.SRSS vs CQC Comparison ExampleA worked example directly comparing SRSS and CQC results for the same structure with varying modal spacing, showing when the difference becomes significant.Effective Modal Mass CalculationA worked example calculating effective modal mass for each mode of a three-storey frame and verifying modal completeness.Cumulative Modal Mass ExampleA worked example showing how cumulative modal mass participation is used to judge modal truncation, with a case where the threshold is not met.Directional Seismic Combination ExampleA worked example showing how responses from two orthogonal horizontal seismic directions are combined using both 100-40-40 and SRSS directional methods.Cabinet Seismic Assessment ExampleA worked example assessing a steel electrical cabinet for seismic loading, including modal analysis, FRS application, anchorage check and frame stress assessment.Seismic Base Shear and Overturning ExampleA worked example calculating base shear and overturning moment for a simple equipment support structure under seismic loading.Anchor Bolt Seismic Load ExampleA worked example calculating anchor bolt loads for a tall equipment cabinet under seismic loading, including combined tension-shear interaction.Response Spectrum Stress Combination ExampleA worked example showing how stress components from individual modes are combined, and why SRSS of von Mises stresses differs from von Mises of SRSS-combined stress components.Structure With a Resonance Inside the Peak Spectrum RegionA worked example showing how a structure with its natural frequency in the peak region of the response spectrum experiences maximum amplification, and how this drives the seismic demand.Structure With Natural Frequency Outside the Spectrum PeakA worked example showing how a structure with its natural frequency beyond the peak region of the response spectrum experiences low amplification, and how this reduces seismic demand.

Turbomachinery

Blade Centrifugal Load Worked ExampleStep-by-step calculation of the centrifugal force and root stress for a turbine blade, showing assumptions, equations, substitution, result and verification.Rotating Disc Stress Worked ExampleAnalytical calculation of radial and hoop stress in an annular rotating disc, showing the governing equations, substitution and interpretation.Blade Radial-Growth Worked ExampleCalculation of the centrifugal and thermal radial growth of a turbine blade, showing the combined growth and the effect on tip clearance.Turbine Disc Thermal-Stress ExampleEstimation of the thermal stress in a turbine disc from a radial temperature gradient between the hot rim and the cooler bore.Pre-Stressed Modal Analysis ExampleIllustrative example showing the frequency shift from centrifugal stiffening for a turbine blade from zero speed to operating speed.Engine-Order Resonance ExampleIllustrative example identifying a resonance crossing from a Campbell diagram and calculating the resonance speed for a specific engine order.Blade Tip-Clearance ExampleIllustrative example combining blade growth, casing expansion and rotor motion to compute the minimum running tip clearance.Turbomachinery Load-Mapping Verification ExampleIllustrative example verifying a CFD-to-FEA pressure load transfer by comparing total force and moment before and after mapping.

Pressure & Containment

Thin-Walled Cylinder Worked ExampleStep-by-step calculation of hoop and longitudinal stress in a thin-walled cylindrical pressure vessel, showing assumptions, equations, substitution and verification.Spherical Vessel Stress Worked ExampleCalculation of membrane stress in a thin-walled spherical pressure vessel and comparison with an equivalent cylindrical vessel.Pressure End-Thrust Worked ExampleCalculation of pressure end thrust on a vessel closure, bolt load in a flanged joint and support reaction at a fixed end.Thick-Walled Cylinder Worked ExampleLame equation calculation for a thick-walled cylinder under internal pressure, showing inner and outer surface stresses and comparison with the thin-wall prediction.Nozzle Pressure-Load Worked ExampleCalculation of the pressure stress at a nozzle-to-shell junction, including the stress concentration factor and comparison with the nominal shell stress.Flanged Closure Load-Path Worked ExampleCalculation of the pressure thrust, bolt load, flange moment and gasket compression for a bolted flange closure under internal pressure.External-Pressure Buckling Worked ExampleCalculation of the elastic buckling pressure for a cylindrical shell under external pressure and comparison with an eigenvalue FEA prediction.Pressure-Shell Ovality Imperfection-Sensitivity ExampleCalculation showing the effect of ovality on the collapse pressure of a cylindrical shell, demonstrating the sensitivity of buckling resistance to geometric imperfections.Stress-Linearisation Worked ExampleStep-by-step extraction of membrane, bending and peak stress components from a through-thickness stress distribution in a pressure vessel wall.Pipe-Bend Pressure-Thrust Worked ExampleCalculation of the pressure thrust force at a 90-degree pipe bend and the resulting support reaction.Pressure FEA Equilibrium Worked ExampleVerification of a pressure vessel FEA model by checking the pressure resultant, reaction forces and hoop stress against analytical calculations.

Probabilistic & Reliability

Normal Distribution Engineering ExampleA worked example applying the normal distribution to dimensional tolerance analysis — computing the probability of a manufactured dimension falling outside the tolerance band.Weibull Strength Distribution ExampleA worked example fitting a Weibull distribution to strength test data and computing the probability of failure below a design stress.Monte Carlo Structural Response ExampleA worked example of Monte Carlo simulation on a simple structural model — propagating load and material uncertainty to obtain the stress distribution and failure probability.Correlated Variables ExampleA worked example showing the effect of correlation between yield strength and ultimate strength on the failure probability estimate.Limit-State Probability-of-Failure ExampleA worked example defining a limit-state function and computing the failure probability using both direct integration and Monte Carlo estimation.FORM Reliability ExampleA worked example of FORM analysis on a simple limit state — transforming to standard-normal space, finding the design point and computing the reliability index.Probabilistic Fatigue-Life ExampleA worked example computing the probability of fatigue failure before the design life, considering S-N scatter and load uncertainty.Bayesian Updating ExampleA worked example of Bayesian updating — combining a prior material property distribution with test data to produce an updated posterior distribution.Reliability Sensitivity ExampleA worked example identifying which input uncertainties contribute most to the failure probability using sensitivity analysis.

Ballistics & Penetration

Kinetic Energy & Momentum Comparison ExampleA worked example comparing two impacts with the same momentum but different kinetic energy, showing the different structural responses.Stress-Wave Reflection ExampleA worked example computing the wave transit time through a steel plate and assessing whether stress-wave effects are significant.Structural Panel Impact ExampleA worked example estimating the response of a steel panel to a specified impact using energy balance and simplified membrane theory.High-Strain-Rate Material Response ExampleA worked example estimating the temperature rise from adiabatic heating during high-rate deformation of a steel specimen.Explicit Energy-Balance ExampleA worked example interpreting the energy balance from an explicit FEA impact analysis and identifying a potential problem.Mesh-Sensitivity ExampleA worked example showing how a penetration result changes with mesh density and how to assess mesh convergence.Residual Deformation Assessment ExampleA worked example assessing the residual deformation and remaining structural margin of a panel after a non-perforating impact.Impact Test-Correlation ExampleA worked example comparing an FEA impact prediction with a physical test result and assessing the correlation.

Creep & High-Temperature

Norton Creep Law Worked ExampleA worked example applying the Norton creep law to estimate the steady-state creep rate and creep strain for a component at a specified stress and temperature.Creep Strain Accumulation ExampleA worked example computing creep strain accumulation over a service period with a step change in stress, comparing time-hardening and strain-hardening approaches.Stress Relaxation Worked ExampleA worked example computing stress relaxation in a bolt held at fixed displacement at elevated temperature, showing the declining stress over time.Larson-Miller Parameter Worked ExampleA worked example using the Larson-Miller parameter to estimate the rupture time at a specified temperature and stress from a master curve.Creep Damage Accumulation ExampleA worked example computing creep damage using the Robinson time-fraction rule for a component with varying operating conditions.High-Temperature Bolt Relaxation ExampleA worked example assessing bolt preload loss at elevated temperature, considering thermal expansion mismatch and creep relaxation.Creep FEA Verification ExampleA worked example verifying a creep FEA model by comparing single-element results with an analytical Norton creep solution.

Random Vibration

PSD Integration & RMS Worked ExampleA worked example computing the Grms from a piecewise PSD specification, demonstrating segment-by-segment integration and the total RMS.Piecewise PSD Worked ExampleA worked example interpreting a piecewise PSD specification — computing slopes, interpolating between breakpoints and verifying the Grms.Miles Equation Worked ExampleA worked example applying Miles equation to estimate the RMS response of an SDOF system at resonance under a flat input PSD.SDOF Random Response ExampleA worked example computing the response PSD and RMS of an SDOF system to a broadband random input, showing the transfer function and output PSD.Random Vibration Stress PSD ExampleA worked example showing how a stress PSD is computed from the modal stress response and how it feeds into fatigue damage estimation.Correlated Input ExampleA worked example showing how correlated multiple-input excitation produces a different response than the uncorrelated (SRSS) case.Random Vibration Fatigue ExampleA worked example estimating fatigue life from a stress PSD using the narrow-band approximation, showing the damage calculation and life prediction.Analysis–Test Correlation ExampleA worked example comparing predicted and measured random vibration response — modal frequencies, response PSD and RMS — and diagnosing discrepancies.

Shock & Transient Dynamics

Half-Sine Shock Worked ExampleIllustrative worked example showing assumptions, equations, units, calculation and engineering interpretation.Impulse & Momentum Worked ExampleIllustrative worked example showing assumptions, equations, units, calculation and engineering interpretation.SDOF Shock Response ExampleIllustrative worked example showing assumptions, equations, units, calculation and engineering interpretation.Shock Response Spectrum ExampleIllustrative worked example showing assumptions, equations, units, calculation and engineering interpretation.Direct vs Modal Transient ExampleIllustrative worked example showing assumptions, equations, units, calculation and engineering interpretation.Drop Impact-Velocity ExampleIllustrative worked example showing assumptions, equations, units, calculation and engineering interpretation.Shock Isolation ExampleIllustrative worked example showing assumptions, equations, units, calculation and engineering interpretation.Analysis–Test SRS Correlation ExampleIllustrative worked example showing assumptions, equations, units, calculation and engineering interpretation.

Advanced Damage & Failure

First-Ply vs Progressive Failure ExampleA worked example comparing first-ply failure load with progressive failure ultimate load for a multi-angle laminate under tension.Hashin Progressive Damage ExampleA worked example showing how Hashin criteria identify failure modes and how stiffness degradation drives progressive failure in a cross-ply laminate.Fracture Energy Damage Evolution ExampleA worked example showing how fracture energy and characteristic length determine the failure displacement and ensure mesh-objective energy dissipation.Cohesive-Zone Traction–Separation ExampleA worked example showing how a cohesive-zone traction-separation law represents interface behaviour — initial stiffness, peak traction, softening and fracture energy.Mixed-Mode Delamination ExampleA worked example showing how mixed-mode energy release rates are combined using a power law criterion to predict delamination propagation.Low-Velocity Composite Impact ExampleA worked example showing how impact energy, absorbed energy and damage energy relate for a low-velocity impact on a composite laminate.Compression-After-Impact ExampleA worked example showing how delamination size and sublaminate thickness affect the CAI strength through sublaminate buckling.Bonded Composite Joint ExampleA worked example showing how adhesive shear and peel stresses distribute along a single-lap bonded composite joint overlap.