Overview The Q6i (Quadrilateral with 6 Incompatible modes) membrane element is a 4-node quadrilateral featuring:
4 corner nodes 2 DOF per node (translations in x and y)2 internal incompatible modes (condensed out)Prevents shear locking in bending-dominated problemsConverges to Timoshenko beam solution (includes shear deformation)
The Q6i element uses incompatible modes that are statically condensed at the element level, resulting in enhanced bending behavior without adding global DOF.
Adding Q6i Elements from milcapy.utils.types import ConstitutiveModel model.add_membrane_q6i( id = 1 , node_ids = [ 1 , 2 , 3 , 4 ], section_name = 'Shell_10mm' , state = ConstitutiveModel. PLANE_STRESS )
Parameters:
id (int) → Element IDnode_ids (list[int]) → List of 4 node IDssection_name (str) → Section name (shell/membrane section)state (ConstitutiveModel, optional) → Constitutive model
Available constitutive models:
'PLANE_STRESS' or ConstitutiveModel.PLANE_STRESS (default)'PLANE_STRAIN' or ConstitutiveModel.PLANE_STRAIN
Critical: Nodes must be ordered counter-clockwise for correct formulation.
Node Ordering 4 ----------- 3 | | | | | · | | | | | 1 ----------- 2 Order: [1, 2, 3, 4] ✓ (counter-clockwise)
Degrees of Freedom Each node has 2 translational DOF:
Node 1: [Ux1, Uy1] Node 2: [Ux2, Uy2] Node 3: [Ux3, Uy3] Node 4: [Ux4, Uy4]
Total: 8 DOF (same as Q4, but with enhanced formulation)
Internal modes: 2 additional incompatible modes (condensed)The incompatible modes improve element performance but are eliminated through static condensation, so they don’t appear in the global system.
Shape Functions Standard bilinear modes (4): N1 = ( 1 - ξ) * ( 1 - η) / 4 N2 = ( 1 + ξ) * ( 1 - η) / 4 N3 = ( 1 + ξ) * ( 1 + η) / 4 N4 = ( 1 - ξ) * ( 1 - η) / 4
Incompatible modes (2): N5 = ( 1 - ξ²) # Incompatible in ξ direction N6 = ( 1 - η²) # Incompatible in η direction
Total: 6 shape functions (4 compatible + 2 incompatible)
Enhanced B Matrix The strain-displacement matrix includes incompatible modes:
B = [B_compatible | B_incompatible]
Where:
B_compatible (3×8): Standard displacement DOFB_incompatible (3×2): Internal modes
Full B matrix before condensation: 3×10
Static Condensation The incompatible modes are eliminated at the element level:
# 12×12 stiffness with internal modes K_full = [[K_cc, K_cs], [K_sc, K_ss]] # Condensed 8×8 stiffness K_condensed = K_cc - K_cs @ inv(K_ss) @ K_sc
The final element stiffness matrix is 8×8 (same size as Q4).
Static condensation happens automatically. You don’t need to handle internal modes—they’re transparent to the user.
Key Features Shear Locking Prevention
What is Shear Locking? Overly stiff behavior in bending due to spurious shear strain in low-order elements.
How Q6i Helps Incompatible modes allow the element to bend without artificial shear constraints.
Convergence to Timoshenko Theory Unlike Q6 (which converges to Euler-Bernoulli), the Q6i converges to Timoshenko beam theory , accounting for shear deformation effects.
This makes Q6i ideal for:
Moderately thick beams
Short spans
Structures where shear deformation is significant
Example: Deep Beam Analysis import milcapy as mx from milcapy.utils.types import ConstitutiveModel # Model a deep beam where shear deformation matters model = mx.Model() # Deep beam: 3m long × 0.6m deep L = 3.0 h = 0.6 # Nodes model.add_node( 1 , 0.0 , 0.0 ) model.add_node( 2 , L, 0.0 ) model.add_node( 3 , L, h) model.add_node( 4 , 0.0 , h) # Material and section model.add_material( 'Concrete' , E = 25e9 , v = 0.2 , rho = 2400 ) model.add_shell_section( 'Wall' , material = 'Concrete' , t = 0.25 ) # Add Q6i element (single element for demonstration) model.add_membrane_q6i( id = 1 , node_ids = [ 1 , 2 , 3 , 4 ], section_name = 'Wall' , state = ConstitutiveModel. PLANE_STRESS ) # Simply supported model.add_support( node_id = 1 , ux = True , uy = True ) model.add_support( node_id = 2 , ux = False , uy = True ) # Point load at top center model.add_load_pattern( 'Live' ) model.add_nodal_load( node_id = 3 , fx = 0 , fy =- 6000 ) # 6 kN # Solve model.solve( 'Live' ) # Results disp = model.get_node_displacements( 3 ) print ( f "Deflection: { abs (disp[ 1 ]) * 1000 :.3f} mm" )
For this deep beam (L/h = 5), shear deformation is significant. Q6i will give more accurate results than Q6 or standard Q4.
Example: Comparison Study import milcapy as mx from milcapy.utils.types import ConstitutiveModel def analyze_beam ( element_type ): """ Compare Q4, Q6, Q6i, and Q8 elements Args: element_type: 'Q4', 'Q6', 'Q6i', or 'Q8' """ model = mx.Model() # Cantilever beam: 2m × 0.4m model.add_node( 1 , 0.0 , 0.0 ) model.add_node( 2 , 2.0 , 0.0 ) model.add_node( 3 , 2.0 , 0.4 ) model.add_node( 4 , 0.0 , 0.4 ) # Material model.add_material( 'Steel' , E = 200e9 , v = 0.3 , rho = 7850 ) model.add_shell_section( 'Beam' , material = 'Steel' , t = 0.05 ) # Add element based on type if element_type == 'Q4' : model.add_membrane_q4( 1 , [ 1 , 2 , 3 , 4 ], 'Beam' , ConstitutiveModel. PLANE_STRESS ) elif element_type == 'Q6' : model.add_membrane_q6( 1 , [ 1 , 2 , 3 , 4 ], 'Beam' , ConstitutiveModel. PLANE_STRESS ) elif element_type == 'Q6i' : model.add_membrane_q6i( 1 , [ 1 , 2 , 3 , 4 ], 'Beam' , ConstitutiveModel. PLANE_STRESS ) elif element_type == 'Q8' : model.add_membrane_q8( 1 , [ 1 , 2 , 3 , 4 ], 'Beam' , ConstitutiveModel. PLANE_STRESS , IntegrationType. REDUCED ) # Fixed support if element_type == 'Q6' : model.add_support( 1 , ux = True , uy = True , rz = True ) model.add_support( 4 , ux = True , uy = True , rz = True ) else : model.add_support( 1 , ux = True , uy = True ) model.add_support( 4 , ux = True , uy = True ) # Tip load model.add_load_pattern( 'P' ) model.add_nodal_load( 3 , fx = 0 , fy =- 1000 ) model.solve( 'P' ) disp = model.get_node_displacements( 3 ) return abs (disp[ 1 ]) * 1000 # mm # Compare elements for elem in [ 'Q4' , 'Q6' , 'Q6i' , 'Q8' ]: deflection = analyze_beam(elem) print ( f " { elem :4s} : { deflection :.4f} mm" )
Expected behavior:
Q4 : Stiff (shear locking)Q6 : Less stiff (no shear deformation modeled)Q6i : Accounts for shear (most realistic for deep beams)Q8 : Best accuracy overall
Constitutive Models Plane Stress state = ConstitutiveModel. PLANE_STRESS D = (E / ( 1 - v²)) * [[ 1 , v, 0 ], [v, 1 , 0 ], [ 0 , 0 , ( 1 - v) / 2 ]]
Plane Strain state = ConstitutiveModel. PLANE_STRAIN k = E( 1 - v) / (( 1 + v)( 1 - 2v )) D = k * [[ 1 , v / ( 1 - v), 0 ], [v / ( 1 - v), 1 , 0 ], [ 0 , 0 , ( 1 - 2v ) / ( 2 ( 1 - v)) ]]
Advantages and Limitations Advantages
Eliminates shear locking in bending problemsTimoshenko beam convergence (includes shear deformation)Same DOF as Q4 (no extra global unknowns)Better single-element accuracy than Q4
Ideal for moderately thick beams and plates
Good for coarse meshes
Limitations
More complex formulation than Q4
Slightly higher computational cost per element
Not as accurate as Q8 for very coarse meshes
Requires careful implementation of condensation
If static condensation fails (singular K_ss), the element falls back to standard Q4 formulation.
When to Use Q6i Elements Best for:
Thick beams and plates (L/h < 10)Structures where shear deformation is important
Avoiding shear locking in coarse meshes
Single or few elements spanning a member
Deep beam analysis
Consider alternatives:
Q4 : For refined meshes where shear locking isn’t an issueQ6 : For slender beams where shear deformation is negligibleQ8 : For highest accuracy (but more DOF and integration points)
Element Comparison Feature Q4 Q6 Q6i Q8 DOF/node 2 3 2 2 Total DOF 8 12 8 8 Internal modes 0 0 2 (condensed) 4 (condensed) Shear locking Yes No (drilling) No (incompatible) No Beam theory - Euler-Bernoulli Timoshenko - Integration points 4 4 4 4 or 9 Best for Refined mesh General, slender Thick, shear Best accuracy
Choosing between Q6 and Q6i:
Use Q6 for slender structures (shear deformation negligible)
Use Q6i for thick/deep structures (shear deformation important)
Technical Details The incompatible modes satisfy:
Orthogonality to rigid body modes
Zero contribution to constant strain states
Enhanced flexibility in bending
Numerical Integration 4-point Gauss quadrature:
ξ = η = ± 1 / √ 3 weights = [ 1 , 1 , 1 , 1 ]
Patch Test The Q6i element passes the patch test (constant strain reproducing capability) despite having incompatible modes, thanks to proper formulation.
Historical note: Incompatible mode elements were developed by Wilson et al. to overcome shear locking while maintaining simplicity.
