Structural Analysis Workflow

Critical Parameters:

• Young's Modulus (E)
• Poisson's Ratio (ν)
• Yield Strength (σ_y)
• Ultimate Tensile Strength (σ_u)
• Density (ρ)
• Thermal expansion coefficient (α)

2. Load Application

Pressure Loads:

• Apply internal pressure to casing wetted surfaces
• Use static pressure at design point or maximum operating pressure
• For impellers, apply pressure distribution from CFD analysis
• Consider pressure pulsations for fatigue analysis

Centrifugal Forces (Impellers):

• Apply rotational velocity to impeller body
• Centrifugal force: F_c = mω²r
• Ensure correct direction and magnitude
• Consider overspeed conditions (typically 120% of rated speed)

Hydraulic Forces:

• Radial thrust on impeller
• Axial thrust from pressure difference
• Tangential forces from fluid momentum change
• Unbalanced forces at off-design conditions

Thermal Loads:

• Temperature distribution from thermal analysis
• Differential expansion between components
• Thermal gradients during startup/shutdown

3. Boundary Conditions (Constraints)

Casing Constraints:

• Fixed supports at mounting feet
• Symmetry planes if applicable
• Pipe connections (nozzles) - typically allow thermal expansion
• Anchor points: fully constrained in all directions

Impeller Constraints:

• Hub constrained to shaft (cylindrical constraint)
• Contact between impeller and shaft (may use bonded or frictional contact)
• Symmetry conditions if analyzing only a sector

Connection Types:

• Bonded: No relative motion (welded joints)
• Frictional: Coulomb friction model (shrink fits)
• No separation: Contact that can slide but not separate
• Frictionless: Free sliding contact

4. Mesh Generation

Element Types:

• Tetrahedral (Tet10): General purpose, automatic meshing
• Hexahedral (Hex20): Higher accuracy, structured regions
• Wedge (Prism): Transition between tet and hex meshes
• Shell elements: Thin-walled sections of casings

Mesh Quality Criteria:

• Element aspect ratio < 5:1 (ideally < 3:1)
• Skewness < 0.85
• Jacobian ratio > 0.6
• Minimum of 3 elements through thickness
• Finer mesh at stress concentrations (fillets, holes, welds)

Mesh Refinement:

• Use sphere of influence around critical features
• Create local mesh controls at:
  • Fillet radii
  • Nozzle-to-shell junctions
  • Impeller eye (inlet)
  • Impeller blade-to-hub junction
  • Wear ring gaps
• Perform mesh convergence study (verify results converge as mesh density increases)

Recommended Element Sizes:

• Impeller blade thickness: minimum 4 elements
• Casing wall thickness: minimum 3 elements
• Fillet radius: minimum 6 elements around arc
• Global element size: 5-10% of characteristic dimension

5. Solver Selection

Linear Static Analysis:

• Use when: Deformations are small, material is elastic, no time-dependent effects
• Suitable for: Initial design verification, factor of safety calculations
• Fast computation, good for parametric studies

Nonlinear Analysis:

• Use when: Large deformations, material plasticity, contact nonlinearity
• Required for: Plastic collapse analysis, contact pressure distribution
• Slower computation, requires incremental loading

Modal Analysis:

• Determine natural frequencies and mode shapes
• Avoid resonance with pump operating speed and harmonics
• Check: 1st critical frequency > 1.5 × operating frequency

Harmonic/Transient Analysis:

• Pressure pulsations at blade passing frequency
• Startup and shutdown transients
• Water hammer events

Fatigue Analysis:

• High-cycle fatigue (> 10^4 cycles): Impellers, shafts
• Low-cycle fatigue: Thermal cycling, startup/shutdown
• Use S-N curves or strain-life methods

Solver Settings:

• Convergence criteria: typically 0.1% force/moment convergence
• Large deflection: ON for impellers with thin blades
• Contact settings: program-controlled for initial runs
• Stabilization: may be needed for very thin features

6. Results Interpretation

Primary Results:

Von Mises Stress (σ_vm):

• Equivalent stress for ductile materials
• Compare against material yield strength
• σ_vm = √[(σ₁-σ₂)² + (σ₂-σ₃)² + (σ₃-σ₁)²] / √2
• Most commonly used failure criterion

Principal Stresses (σ₁, σ₂, σ₃):

• Maximum and minimum normal stresses
• Critical for brittle materials
• Check maximum principal stress against ultimate strength

Deformation:

• Total deformation: overall displacement magnitude
• Directional deformation: check clearances (impeller-to-casing)
• Critical gaps: wear ring clearances, axial thrust bearing clearances

Safety Factor:

• Factor of Safety (FOS) = σ_allowable / σ_actual
• Color-coded: FOS < 1 (red/failure), FOS > 1 (safe)

Secondary Results:

Shear Stress:

• Critical at welds, keyways, shaft connections
• Maximum shear stress theory (Tresca criterion)

Strain:

• Elastic strain vs. plastic strain
• Strain energy density
• Equivalent plastic strain for ductile failure

Contact Pressure:

• Shrink fit interfaces
• Bearing surfaces
• Sealing faces

Critical Review Checklist:

1. Do stress concentrations align with geometric features?
2. Are boundary conditions realistic (no rigid body motion)?
3. Is the deformation pattern physically reasonable?
4. Are peak stresses at expected locations?
5. Does the solution satisfy equilibrium (sum of reactions = applied loads)?

7. Safety Factor Evaluation

Design Codes and Standards:

• ASME Section VIII Division 1: Pressure vessels (casings)
• ASME Section VIII Division 2: Higher pressure, FEA-based design
• API 610: Centrifugal pumps for petroleum industry
• ISO 13709: Centrifugal pumps for petroleum, petrochemical and natural gas industries

Required Safety Factors:

Static Loads:

• Against yield: FOS ≥ 1.5 (general machinery)
• Against yield: FOS ≥ 2.0 (pressure vessels per ASME)
• Against ultimate: FOS ≥ 2.5 (static loads)
• Against ultimate: FOS ≥ 4.0 (dynamic/impact loads)

Fatigue Loads:

• Infinite life design: Stress < endurance limit
• Finite life: Use cumulative damage theory (Miner's rule)
• FOS on fatigue life: typically 2.0 or higher

Safety Factor Calculation:

For ductile materials:

FOS_yield = σ_y / σ_vm
FOS_ultimate = σ_u / σ_vm
Design FOS = min(FOS_yield, FOS_ultimate)

For brittle materials:

FOS = σ_u / σ₁ (based on maximum principal stress)

Material-Specific Adjustments:

• Cast iron: Reduce allowable stress by 25%
• Welded joints: Apply joint efficiency factor (0.7-1.0)
• High temperature: Apply creep reduction factors
• Corrosive environment: Add corrosion allowance

Applications to Pumps

Pump Casing Pressure Containment

Analysis Objectives:

• Verify casing can withstand maximum allowable working pressure (MAWP)
• Check stress at nozzle-to-shell junctions
• Evaluate flange sealing capability
• Assess casing distortion effects on wear ring clearances

Critical Regions:

• Volute cutwater (point of maximum pressure gradient)
• Nozzle reinforcement pads
• Casing split lines (horizontal or vertical split)
• Mounting foot attachments

Load Cases:

• Maximum operating pressure
• Hydrostatic test pressure (typically 1.5 × design pressure)
• Pressure + thermal expansion
• Pressure + piping loads (nozzle forces and moments)

Acceptance Criteria:

• Von Mises stress < allowable stress
• Flange face distortion < gasket capability
• No yielding at bolt holes
• Maximum deformation < 0.5% of casing diameter

Common Failure Modes:

• Casing rupture at cutwater
• Flange leakage due to uneven loading
• Cracking at weld toes
• Bolt failure under pressure cycling

Impeller Centrifugal Stress

Analysis Objectives:

• Calculate maximum stress in blades, shrouds, and hub
• Evaluate stress at blade-to-shroud junctions
• Check deflection to ensure clearance maintenance
• Assess fatigue life at blade passing frequency

Critical Regions:

• Blade trailing edge (maximum bending moment)
• Blade-to-hub fillet (stress concentration)
• Blade-to-shroud junction (closed impellers)
• Hub bore (stress concentration from keyway)

Load Cases:

• Centrifugal force at rated speed
• Centrifugal force at overspeed (120% of rated)
• Combined centrifugal + hydraulic pressure
• Thermal loads from fluid temperature

Stress Components:

Centrifugal Stress (Hoop Stress):

σ_hoop = ρω²r² (for thin disk)