HONGJUN LI Department of Mechanical Engineering University of Strathclyde Glasgow, Scotland, UK

HONGJUN LI Department of Mechanical Engineering University of Strathclyde Glasgow, Scotland, UK

Introduction FEA is an established analysis method used in Pressure Vessel Design Most DBA is still based on elastic analysis Inelastic analysis is becoming more widely used This presentation will describe new approaches to inelastic design developed at the University of Strathclyde by Hongjun Li Martin Muscat Bobby Hamilton Donald Mackenzie Specifically A new criterion of plastic collapse for strain hardening plastic analysis A bounding theorem method for calculating shakedown loads

INELASTIC FAILURE MODES Gross plastic deformation under static load In elastic DBA, prevented by limiting the elastic primary stress in the vessel. In inelastic DBA calculate permissible load through inelastic analysis Incremental plastic collapse (ratchetting) In elastic DBA, shakedown is assured by limiting the elastic primary plus secondary stress to twice yield In inelastic DBA, calculate permissible load through inelastic shakedown analysis

Inelastic Material Models σ PLASTIC σ PLASTIC σ Y YIELD σ Y YIELD E E ELASTIC ELASTIC ε (a) Non-linear stress-strain ε (b) Multi-linear isotropic hardening σ σ Y YIELD PLASTIC E P σ σ Y YIELD PLASTIC E E ELASTIC ELASTIC ε (c) Bilinear kinematic hardening ε (d) Perfect plasticity

Gross Plastic Deformation under Static Load

Types of inelastic analysis Limit analysis Assumes an elastic-perfectly plastic material model and small deformation theory The limit load is the highest load satisfying equilibrium between external and internal forces May be assumed to be the ductile collapse load in DBA. The allowable load is two thirds of limit load Plastic analysis Strain hardening and large deformation effects may be included A criterion of plastic collapse is applied to determine the allowable load The plastic load characterises gross plastic deformation The allowable load is two thirds of plastic load

Limit Analysis Advantages No inelastic stress strain design data needed Load-path independent don t need to know load history Conservative if non-linear geometric weakening is not significant Easier than elastic analysis Disadvantages Not conservative if non-linear geometric weakening is significant Does not account for enhanced strength due to strain hardening Can be difficult to evaluate limit load if solution converges for unrealistic deformations

Example: Multilinear Hardening Plastic Analysis σ ε

ASME Criterion of Plastic Collapse The twice elastic slope criterion (TES) Applied to characteristic loaddeformation curve obtained by plastic analysis Load P P k k/2 Deformation

Deformation Parameter Little guidance on nature and location of deformation parameter given F d P V D. PRESSURISED CLOSED VESSEL Choice can significantly affect the calculated plastic load. A. CANTILEVER BENDING M θ P w Gerdeen proposed the product of the load and deformation parameters should have units of work (Nm) if possible B. MOMENT LOADING P C. PRESSURISED CYLINDER w E. PRESSURE IN HEAD d P F. PRESSURE IN NOZZLE

Problems with the ASME Procedure The method is heuristic or arbitrary Based on experimental experience with specific configurations Choice of suitable load and deformation parameters Especially for non-proportional combined loading In some cases no intersection occurs between the loaddeformation curve and collapse limit line The plastic load is influenced by the elastic response Value of plastic load is dependent on FE modelling assumptions Does not adequately represent the effect material strain hardening has on plastic deformation

A New Plastic Criterion Based on plastic work concepts The evolution of the gross plastic deformation mechanism is characterised by The rate of change of slope of the W p -Q curve, or The curvature at a point on the curve 1 = ρ 1 + d 2 W dq dw dq p 2 p 2 3 2 Plastic Work Load Slope

Curvature and Gross Plastic Collapse Example: bilinear hardening beam in pure bending Post-yield curvature indicates post-yield stress redistribution Plastic deformation The maximum rate of stress redistribution corresponds with maximum curvature Subsequent decrease in curvature indicates decreasing rate of stress redistribution Minimum or zero curvature indicates little or no redistribution Gross plastic deformation M (Nm) curvature Hardening Analysis WP (Nm) Discontinuity

Curvature and Gross Plastic Collapse Plastic Work Curvature or PWC criterion of plastic load The load corresponding to constant or zero curvature after stress redistribution M (Nm) curvature Discontinuity Hardening Analysis WP (Nm)

Pipe Bend Example: ASME Approach Moment-2*kNm 1000 900 800 700 600 500 400 300 200 100 0 Closing & perfectly plastic Closing & 5% bilinear Opening & perfectly plastic Opening & 2% bilinear 0 5 10 15 20 25 30 Rotation -Degree

Pipe Bend Example: Closing Moment Moment knm 637.5 Moment knm 720 500 600 Perfectly Plastic M TES = NA M PWC =637.5 knm Plastic work Plastic work Bilinear Hardening M TES =700 knm M PWC =720 knm

Pipe Bend Example: Opening Moment Moment knm Moment knm 1120 1180 Plastic Work Plastic Work Perfectly Plastic M TES =915 knm M PWC =1120 knm Bilinear Hardening M TES =980 knm M PWC =1180 knm

Shakedown & Ratchetting under Cyclic Load

Ductile Failure under Cyclic Loads Maximum load between first yield and plastic collapse Elastic shakedown Plastic shakedown (alternating plasticity) Ratchetting DBA must ensure shakedown of the vessel

Elastic Shakedown of Thick Cylinder P P S B D A B P Y A C t C D

Preventing Cyclic Failure in PVD High cycle & low cycle fatigue Perform a fatigue analysis Establish the design life of the vessel Ratchetting Ensuring that the structure shakes-down to elastic action

Shakedown Analysis: Plastic DBA Incremental elastic-plastic Finite Element Analysis Simulate the structural response for a given load history Monitor the resulting plastic strain accumulation No plastic strain accumulation & No alternating plasticity: elastic shakedown Alternating plasticity: plastic shakedown Plastic strain increases with each cycle: ratchetting

Elastic Shakedown of thick Vessel Incremental elastic-plastic analysis ε1p P 0 t Py t

Plastic Shakedown of Thick Vessel Incremental elastic-plastic analysis ε1p P 0 t Py

Ratchetting of Thick Vessel Incremental elastic-plastic analysis ε1p P 0 t Py t

Incremental FEA Does not predict the specific value of the shakedown load Demonstrates the structural response for a given load level A number of simulations at different loads needed to identify shakedown load PV design codes simply require that shakedown is demonstrated for a specific load Can be difficult to distinguish between plastic shakedown and ratchetting Large number of cycles may be needed Expensive in computer requirements

Shakedown Bounding Theorems Similar approach to (lower bound) limit load analysis Establishes a lower bound on the shakedown load for design purposes Does not calculate realistic stress and strain values Independent of load history Melan s (lower bound) theorem: For a given time dependent cyclic load set, a structure made of elastic-perfectly plastic material will exhibit shakedown if a constant residual stress field can be found such that the yield condition is not violated for any combination of time dependent cyclic elastic stress and residual stress.

Muscat s Lower Bound Method The lower bound limit load theorem can be coupled with FEA to evaluate elastic shakedown loads Muscat s method uses static FEA to calculate elastic and elastic plus residual stress fields for Melan s theorem Limit analysis is performed with results stored for n load levels up to the limit load Elastic plus residual stress is assumed to be the limit load solution for each load P n Satisfies Melan s theorem Elastic stress is the conventional elastic stress at load P n

Applying Melan s Theorem Residual stress calculated by superposition of Elastic plus residual and Elastic stress fields Residual stress = (Elastic Plus Residual)- (Elastic) Load P n is a lower bound on the shakedown load if Maximum residual stress is less than yield Maximum Elastic plus residual stress is less than yield

Shakedown of a Thick Cylinder Calculate stress fields by limit analysis for load P n Elastic plus Residual Stress: from limit analysis Elastic Stress: From scaled initial elastic response

Calculate residual stress field for load P n - (Elastic plus residual) (Elastic) Stress = Residual Stress

Apply Melan s Theorem - (Elastic plus residual) σy: Yes (Residual) σy: Yes P n is a lower bound shakedown load

Conclusion: Inelastic Analysis in PV DBA Inelastic analysis allows allowable loads to be calculated by simulating inelastic response Limit analysis, Limit load Plastic analysis, Plastic load A criterion for plastic collapse is required for gross plastic deformation design in plastic analysis Problems with TES The Plastic Work Curvature criterion is a consistent and robust criterion which accounts for the effects of geometric non-linearity and strain hardening

Shakedown can be assessed through Performing incremental elastic plastic analysis and monitoring plastic strain accumulation over a number of cycles Applying a lower bound theorem based approach such as Muscat s Method to the results of a (static) limit analysis Requires only a single limit analysis Clearly establishes the elastic shakedown load

I would like thank Dr Mackenzie for his help during the preparation of this presentation! QUESTIONS?