Cracks start at the peak
Fatigue life depends on the local stress at the notch root, not on the average. A Kt of 3 turns 100 MPa nominal into 300 MPa at the fillet.
Stress analysis consultancy · Corporate training · Stress concentration
Find the peak stress before it finds the crack
Most fatigue cracks start at a notch, a fillet or a hole. Our consultants find the peak stress at these features for your design reviews, MRB dispositions and failure investigations, using the classical handbook methods of Peterson's Stress Concentration Factors and validated finite element analysis of the actual part. We also train your engineering teams to do it themselves.
Kt ANALYSIS · HOW IT IS PREPAREDPY-Kt
Industries we supportTypical features we assess and teach in each industry
- AerospaceLugs and fittings, fastener holes, cut-outs in skins and spars
- Rotating equipmentShaft shoulders, grooves, keyseats, splines, crankshafts
- Power · turbinesBlade roots and T-heads, bolt heads, casing ribs
- Pressure equipmentDished heads, wall steps, nozzles and reinforcements
- Automotive · railAxles, springs, gear teeth, stub shafts
- Structures · cranesBeams with web holes, brackets, hooks, pin joints
- Oil and gasDrill-string and pump components, valve bodies
- Manufacturing · MRBOut-of-tolerance radii, blends, reworked holes
Why stress concentration
The nominal stress is rarely the stress that breaks the part
A change of section raises the local stress above the nominal value by the stress concentration factor Kt. A part that is safe on nominal stress can crack at a fillet of the wrong radius. Getting Kt right is the first step of every fatigue and static strength check.
Choose the radius with numbers
Kt against r/d shows how far a larger radius, a relief groove or a shallower step lowers the peak, before the drawing is released.
Disposition of as-found parts
A radius below tolerance or a reworked hole: the as-found Kt and margin decide whether the part can be used as is, reworked or scrapped.
Handbook charts have limits
Many charts come from photoelastic tests on idealised specimens. Outside their range, or for a real part, a finite element model of the geometry gives the defensible value.
Consultancy services
Stress concentration consultancy
Project work for design, manufacturing and operations teams: from a quick handbook check in a design review to a full finite element study of a component, delivered as a clear report with the peak stress and the margin against your allowable. Fixed-price assignments or a retainer for ongoing support.
Kt from charts and equations
The classical solutions for notches, fillets, holes and machine elements, applied within their validity range.
- Tension, bending, torsion and pressure
- Range of every chart checked
- Nominal stress definition stated
- Peak stress and margin
Finite element verification
A model of the real geometry when the part is outside the charts or the result is critical, built with our in-house, customised finite element approach.
- Plane stress, axisymmetric and torsion models
- Validated against commercial FEA tools
- Mesh refined at the notch, convergence shown
- Contact at pins and bearing faces
MRB dispositions
Engineering evidence for parts made outside the drawing tolerance.
- As-found radius, blend or hole size
- Kt and peak stress against the design
- Use as is, rework or scrap
- Traceable report for quality records
Sensitivity study
How much each dimension moves the peak stress.
- Kt at every tolerance limit
- Sensitivity coefficients and tornado chart
- Worst corner of the tolerance box
- Dimensions worth inspecting closely
Lower the peak
Practical changes that reduce stress concentration, each checked by analysis.
- Larger, elliptical or compound fillets
- Relief grooves and relief holes
- Contact moved away from the fillet
- Effect on the nominal stress included
Why did it crack there?
The stress field at the origin, to support root-cause analysis.
- Peak location and stress gradient
- Load case that governs
- Input to fatigue and fracture assessment
- Link to our fitness-for-service work
Corporate training
Training for design, stress and quality teams
Practical courses that build in-house capability, taught by practising consultants with worked examples, ideally from your own products. On site at your office or online; course notes, exercises and a certificate of participation for every attendee.
Stress concentration in practice
Kt, nominal stress definitions and the handbook charts of Peterson's Stress Concentration Factors; the common mistakes and how to avoid them.
Practical FEA for peak stress
Modelling notches, fillets and holes: symmetry, mesh at the notch root, convergence, validation and reading the results with confidence.
Fatigue at notches
From Kt to the fatigue notch factor: notch sensitivity, mean stress, S-N data and design margins for real components.
MRB and non-conformance engineering
Dispositions of out-of-tolerance parts: as-found Kt, peak stress, tolerance sensitivity and the documentation reviewers expect.
Designing for low stress concentration
Fillet shapes, relief grooves and holes, contact positions: practical changes that lower the peak stress, with their effect quantified.
Customised programmes
A course built around your parts, codes and tools, for a team or a whole engineering department, with hands-on sessions in our in-house software.
Plan a courseCase studies
Peak stress across industries
Typical stress concentration problems, each solved with our in-house finite element software. Select a case to watch the analysis from geometry to peak stress.
Aerospace · lifting · Finite element analysis
Pin-loaded lug
A round-ended lug loaded through a pin with 1.5 % diametral clearance. The pin contact is solved in the model; the peak sits on the hole edge just off the net section, where fatigue cracks in lugs start.
- 3.17Kt on the net-section nominal stress
- 475 MPapeak stress at 20 kN
- Pin contactclearance included in the model
Illustrative case computed with PY-Kt (own finite element solver); dimensions and load chosen for illustration.
Case videos · select a category
Pin-loaded lug
A round-ended lug loaded through a pin with 1.5 % diametral clearance. The pin contact is solved in the model; the peak sits on the hole edge just off the net section, where fatigue cracks in lugs start.
T-head blade root
A T-head, as in a blade root or a bolt head, pulled through the shank and supported on its bearing faces. A 2 mm fillet on a 10 mm shank keeps the peak at about three times the shank stress.
Beam with a web hole
A Ø30 mm hole 20 mm below the centre line of a 100 mm deep beam under bending. The governing point is on the compression side of the hole; the tension and compression edges are both checked.
Grooved shaft in torsion
A Ø30 mm shaft with a 1 mm radius U-groove, as for a retaining ring, transmitting 350 N·m. The shear stress peaks at the groove root; the axisymmetric torsion model gives it directly.
Rectangular-wire coil spring
A close-coiled compression spring wound from 4.2 × 3 mm rectangular wire. The shear stress over the wire section is solved as a curved bar in torsion; the peak is on the inside of the coil.
Vessel wall thickness step
A Ø200 mm vessel whose wall changes from 8 to 12 mm with a 3 mm outside fillet, under 20 MPa internal pressure. The meridional stress peaks in the fillet; the axisymmetric model includes the local wall bending at the step.
How we work
How a consultancy project runs
The same route for one fillet or a full design review: a fixed scope, a clear report and a debrief with your team.
- Review the dataDrawings, as-found dimensions, loads, material and allowable stress.
- Handbook valueKt from the applicable solutions, with the range checked.
- Model where neededFinite element analysis of the real geometry, converged.
- Peak and marginPeak stress, margin and the sensitivity to tolerances.
- DecideAccept, modify the design or disposition the part.
Our methods · 1 · Handbook approach (Peterson)
Kt from the chart, step by step
The classical route, as set out in Peterson's Stress Concentration Factors: reduce the geometry to dimensionless ratios, enter the chart for the load case, read Kt and multiply by the nominal stress. It is fast and transparent, and it is the first value in every assessment we do.
- Geometry reduced to ratios such as H/d and r/d
- Kt read from the chart for the load case: tension, bending or torsion
- Nominal stress defined exactly as the chart defines it
- Peak stress σmax = Kt σnom and the margin against the allowable
Stepped flat bar in tension · handbook approach
From the drawing to the peak stress in four steps
Our methods · 2 · Finite element verification
A finite element value behind every chart
For each geometry family we have built a database of converged finite element results over the full range of the dimensions. The handbook value and the finite element value are shown side by side, so a difference is seen before it matters.
- Our own 8-node finite element solver, with the mesh graded to the notch root
- Every model checked against an independent second solver
- Mesh convergence shown for each result
- Out-of-range inputs flagged, never extrapolated silently
- Our in-house, customised FEA approach is well validated against commercial FEA tools
Stepped flat bar in tension · Pythagoras FEA database
Kt rises steeply as the fillet gets sharper
H/d = 2.0H/d = 1.5H/d = 1.2H/d = 1.1
- Geometry and loadDimensions as drawn or as found, load case and allowable stress.
- Handbook valueKt from the applicable charts and equations, with their range checked.
- Finite element modelMesh graded to the notch root, symmetry and contact as in service.
- Peak and marginConverged Kt, peak stress and margin against the allowable.
- DecisionAccept, modify the design or disposition the part.
The tool behind our projects and courses
PY-Kt
Our consultants use our in-house software PY-Kt on every project, and course participants use it in the hands-on sessions: the handbook methods and a validated finite element solver in one application, from the geometry to the peak stress, the margin and the report.
60geometry families
45built-in FEA models
2methods: handbook and FEA
How it makes the calculations easier
- Case library by chapter, part type or load
- Handbook and FEA results side by side
- Inputs checked: out of range is shown, never hidden
- SI and US customary units, switchable
- Mesh, stress, displacement and convergence built in
- Animated stress and displacement, saved as GIF
- Sensitivity study: tolerances, tornado chart, worst case
- Excel and PDF reports
Case library
60 geometry families, 45 with a finite element model
The library covers the design charts of Peterson's Stress Concentration Factors (W. D. Pilkey and D. F. Pilkey), Chapters 2 to 5, with the analysis and design methods of Chapter 6. Every case has the handbook methods; the finite element model can be run for any dimensions, including those outside the charts.
Notches and grooves16 cases · 16 FEA
U, V and flat-bottom notches in bars and plates; circumferential grooves in shafts. Tension, bending, torsion.
Shoulder fillets13 cases · 13 FEA
Stepped flat bars and shafts, lands, tubes, vessel walls; elliptical and compound fillets.
Holes10 cases · 10 FEA
Circular, elliptical, rectangular and triangular holes; rows and patterns; beams with holes; shear.
Machine elements21 cases · 6 FEA
Keyseats, splines, gear teeth, T-heads, lugs, curved bars, springs, crankshafts, U-members, dished heads.
Why work with us
Consultants who also teach
We use the classical solutions where they apply and a validated model of the real part where they do not, and we explain every step, in a report or in a classroom.
01
Experienced consultants
Stress analysis, fatigue and fracture mechanics for design, manufacturing and in-service problems, in one team.
02
Validated in-house FEA
A customised finite element approach built for stress concentration, validated against commercial FEA tools.
03
Clear, traceable reports
Geometry, nominal stress, Kt, peak stress and margin, with every assumption stated for your reviewers.
04
Training that sticks
Courses built on real cases, so your engineers take the methods straight back to their own work.
About us
Pythagoras Engineering Consultancy & Services
We are an engineering consultancy for stress analysis, fitness-for-service and fracture mechanics, and we train engineering teams in the same methods. We help designers, manufacturers and operators find the peak stress in their parts and decide with confidence.
Our stress concentration work ranges from a single fillet radius in a design review to MRB dispositions and finite element studies of complete components, and from a single course to customised training programmes for whole departments. For in-service flaws see our fitness-for-service services.
ConsultancyCorporate trainingStress concentration factorsFinite element analysisMRB dispositionsFatigueDesign modification
Contact us
Tell us about the project, the support or the training you need. For a project or consultancy support: the part, the feature and the load; we reply with the method, the data we need and a time and cost estimate. For training: the team, the topics and the dates.
- Office
- Inspire Workplace, #134 & 135, Karthik Nagar, LRDE Layout, Doddanekundi Post, Bangalore - 560037
- Phone
- +91 70422 43776