Tresca s or Mises Yield Condition in Pressure Vessel Design Franz Rauscher Institute for Pressure Vessels and Plant Technology Vienna University of Technology Austria
Why using Tresca s yield condition in a Design Check? Simulation Evaluation of real behaviour of a structure Design Check Show sufficient safety margin against relevant failure mode Conservative approach
EN13445-3: Unfired Pressure Vessels Part 3 Design: DBF (Design by Formulas) Fatigue Annex B: Direct Route for DBA with reference to Fatigue (DBF section) Annex C: Stress Categorisation for DBA
Direct Route Gross Plastic Deformation Design Check (GPD-DC) Progressive Plastic Deformation Design Check (PD-DC) Instability Design Check (I-DC) Fatigue Design Check (F-DC) Static Equilibrium Design Check (SE-DC)
Gross Plastic Deformation (GPD) Principle: Design actions have to be carried by a model with: First order theory A linear-elastic ideal-plastic constitutive law Tresca s yield condition and associated flow rule A maximum absolute value of the principal structural strains of 5% Yield strength = RM RM d = ( design Material strength RM ( material strength parameter) γ ( partial safety factor) R d parameter)
Tresca s vs. Mises yield condition Mises' yield surface with reduced yield strength Mises' yield surface σ 2 Tresca's yield surface σ 1 RM d _ M = 3 2 RM d
One dimensional Stress σ 2 σ σ σ 1 Rod in tension Beam in bending Limit pressure difference: 15.5%
Closed Cylinder with internal pressure σ σ 2 σ Closed cylinder σ 1 Limit pressure difference: 0%
Sphere with internal pressure σ σ σ 2 Sphere with internal pressure σ 1 Limit pressure difference: 15.5%
Example: Thin unwelded flat end Case Limit pressure [bar] Max. abs. value of total principal strain Iterations CPU time [s] Pure Tresca 71.4 5% 1011 init 23.5 x (Time for Mises) Pure Mises 69 4.5% 30 full Mises equivalent stress at Tresca limit Limit pressure difference: 3.4%
Example: Storage tank Max. principal strains at Tresca limit Internal pressure Hydrostatic pressure
Results for storage tank Case Limit pressure [bar] Max. abs. value of principal strain Iterations Limit pressure difference [%] Internal pressure pure Tresca 1.262 4.58% 2279 init 0.5% Internal pressure pure Mises 1.2 4.3% 15 full Hydrostatic pressure pure Tresca 3.175 0.34% 260 init 11% Hydrostatic pressure Tresca-95% Mises 3.08 0.34% Hydrostatic pressure pure Mises 2.828 3.86% 72 full
Example dished end Max. principal strain at Tresca limit Case Limit pressure [bar] Max. abs. value of total principal strain Iterations CPU time [s] Pure Tresca 17.54 2% 991 23.5 x (Time for Mises) Pure Mises 16.36 2.3% 46 Limit pressure difference: 6.7%
Example: Sphere with nozzles Case Limit pressure [bar] Max. abs. value of total principal strain Pure Mises 351.2 4.5 Tresca with 95% Mises 368.2 3.5% Pure Tresca (CAD-FEM) 371 5% plastic equivalent strain 371 Max principal PS max = = 309bar 1.2 strain at Tresca limit ( DBF :273.9bar) Experiment :750bar (2.5%)
Hydrogen Reactor M F p, M, F Pressure Limit Pressure p Moment M, Force F Time t p Material: Shell: 10CrMo9 10 Nozzle: 11CrMo 9 10 NT M = 285 knm F = 60 knm
Evaluation of Limits for Hydrogen Reactor 20 021 Elements, Solid 45: 8-node isoparametric 3Dsolid, reduced integration, hourglass control Case Constant Moment Constant Force Limit pressure [bar] Max. abs. value of total principal strain Iterations Pure Tresca 229 1.15 1694 init 7775 Pure Mises 216.4 5.0 46 full 1697 Limit pressure difference: 5.5% CPU time [s] PIV 2.5GHz, 1GB
Results: Case Tresca limit (pressure) / Mises limit (pressure) CPU time for Tresca limit / CPU-time for Mises limit Beam in tension and 1.155 analytical results bending Closed Cylinder with 1 internal pressure Open cylinder with internal 1.155 pressure Sphere with internal pressure 1.155 Flat end 1.0348 23.5 Storage tank with internal 1.0517 129 pressure Storage tank with 1.1227 28.5 hydrostatic pressure Dished end 1.0721 37 Sphere with nozzles 1.0564 not noted Hydrogen Reactor 1.58 5
Conclusions: In general: Tresca limit may be up to 15.5% higher than the Mises limit When undisturbed cylinder or sphere fails under internal pressure: Closed cylinder fails: no difference Open cylinder fails: Tresca limit 15.5% higher Sphere fails: Tresca limit 15.5% higher In the considered examples: Tresca limit is between 3.5 and 12.3% higher Procedure for Tresca s yield condition would be needed in commercial FE packages.