Overview The Q6 (Quadrilateral with 6 DOF) membrane element is an enhanced 4-node element featuring:
4 corner nodes 3 DOF per node : translations (x, y) + rotation (z)Drilling rotation to prevent numerical instabilityEnhanced bending representation Converges to Euler-Bernoulli beam solution for beam-like structures
The Q6 element adds rotational DOF (“drilling” degrees of freedom) to improve element compatibility and avoid spurious singular modes in shell-like structures.
Adding Q6 Elements from milcapy.utils.types import ConstitutiveModel model.add_membrane_q6( 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 3 DOF :
Node 1: [Ux1, Uy1, Rz1] Node 2: [Ux2, Uy2, Rz2] Node 3: [Ux3, Uy3, Rz3] Node 4: [Ux4, Uy4, Rz4]
Total: 12 DOF per element
Where:
Ux, Uy = Translations in x and yRz = Rotation about z-axis (“drilling” rotation)
The drilling rotation (Rz) provides enhanced stiffness against in-plane rotation and improves element behavior in shell-like problems.
Shape Functions The Q6 element uses:
For displacement DOF (bilinear): N1 = ( 1 - ξ) * ( 1 - η) / 4 N2 = ( 1 + ξ) * ( 1 - η) / 4 N3 = ( 1 + ξ) * ( 1 + η) / 4 N4 = ( 1 - ξ) * ( 1 + η) / 4
For rotational DOF (enhanced): N5 = ( 1 - ξ²) * ( 1 - η) / 2 # Additional modes N6 = ( 1 - η²) * ( 1 + ξ) / 2 N7 = ( 1 - ξ²) * ( 1 + η) / 2 N8 = ( 1 - η²) * ( 1 - ξ) / 2
Total: 8 shape functions (4 bilinear + 4 quadratic)
Strain-Displacement Relationship The B matrix is partitioned:
Where:
B_disp (3×8): Relates strain to translational DOFB_rot (3×4): Relates strain to rotational DOF
Full B matrix: 3×12
Stiffness Matrix The 12×12 stiffness matrix:
K = Σ (B ^ T @ D @ B * det(J) * t * w_i)
Integrated using 4-point Gauss quadrature.
The DOF ordering after assembly:
[Ux1, Uy1, Rz1, Ux2, Uy2, Rz2, Ux3, Uy3, Rz3, Ux4, Uy4, Rz4]
Key Features Drilling Rotation Benefits
Stability Eliminates spurious zero-energy modes in membrane assemblies.
Compatibility Provides rotational compatibility with beam and shell elements.
Bending Better represents in-plane bending without out-of-plane DOF.
Beam Behavior Converges to Euler-Bernoulli beam theory for slender structures.
Convergence Properties The Q6 element converges to the Euler-Bernoulli beam solution for beam-like structures (without considering shear deformation).
For structures where shear deformation is important, use Q6i instead.
Example: Cantilever Beam import milcapy as mx from milcapy.utils.types import ConstitutiveModel # Model a cantilever beam using Q6 membrane elements model = mx.Model() # Beam dimensions L = 2.0 # Length (m) h = 0.6 # Height (m) # Nodes model.add_node( 1 , 0 , 0 ) model.add_node( 2 , L, 0 ) model.add_node( 3 , L, h) model.add_node( 4 , 0 , h) # Material and section model.add_material( 'Concrete' , E = 25e9 , v = 0.2 , rho = 2400 ) model.add_shell_section( 'Wall' , material = 'Concrete' , t = 0.4 ) # Add Q6 element model.add_membrane_q6( id = 1 , node_ids = [ 1 , 2 , 3 , 4 ], section_name = 'Wall' , state = ConstitutiveModel. PLANE_STRESS ) # Fixed support at left edge model.add_support( node_id = 1 , ux = True , uy = True , rz = True ) model.add_support( node_id = 4 , ux = True , uy = True , rz = True ) # Apply tip load model.add_load_pattern( 'Lateral' ) model.add_nodal_load( node_id = 3 , fx = 0 , fy =- 20000 ) # 20 kN # Solve model.solve( 'Lateral' ) # Get results displacements = model.get_node_displacements( 3 ) print ( f "Tip deflection: { displacements[ 1 ] :.6f} m" )
Example: Plate Under Tension import milcapy as mx from milcapy.utils.types import ConstitutiveModel # Create model model = mx.Model() # Define plate (2m × 1m) def create_q6_mesh ( nx , ny ): """Create structured Q6 mesh""" width, height = 2.0 , 1.0 dx, dy = width / nx, height / ny # Nodes node_id = 1 node_map = {} for j in range (ny + 1 ): for i in range (nx + 1 ): model.add_node(node_id, i * dx, j * dy) node_map[(i,j)] = node_id node_id += 1 # Elements elem_id = 1 for j in range (ny): for i in range (nx): n1 = node_map[(i, j)] n2 = node_map[(i + 1 , j)] n3 = node_map[(i + 1 , j + 1 )] n4 = node_map[(i, j + 1 )] model.add_membrane_q6( id = elem_id, node_ids = [n1, n2, n3, n4], section_name = 'Plate' , state = ConstitutiveModel. PLANE_STRESS ) elem_id += 1 # Material and section model.add_material( 'Steel' , E = 200e9 , v = 0.3 , rho = 7850 ) model.add_shell_section( 'Plate' , material = 'Steel' , t = 0.01 ) # Create mesh (4×2 elements) create_q6_mesh( nx = 4 , ny = 2 ) # Boundary conditions and loads...
Constitutive Models Plane Stress state = ConstitutiveModel. PLANE_STRESS D = (E / ( 1 - v²)) * [[ 1 , v, 0 ], [v, 1 , 0 ], [ 0 , 0 , ( 1 - v) / 2 ]]
Use for thin plates and shells.
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)) ]]
Use for thick structures.
Advantages and Limitations Advantages
Enhanced stability with drilling DOFBetter bending behavior than Q4Compatibility with beam/shell elementsSingle element can represent simple beams wellEliminates spurious mechanisms
Good for shell-like structures
Limitations
More DOF than Q4 (higher computational cost)
Does not account for shear deformation
Converges to Euler-Bernoulli (not Timoshenko)
Still requires multiple elements for complex stress states
Recommendation: The Q6 element is preferred over Q6i and Q6i-Modified for most applications due to its robust formulation.
When to Use Q6 Elements Best for:
Slender beam-like structures
Shell connections to membrane
Structures where drilling stiffness matters
Improved single-element accuracy
Avoiding spurious modes
Consider alternatives:
Q6i : When shear deformation is important (thick beams)Q8 : When highest accuracy is neededQ4 : When minimal DOF is preferred and mesh is refined
Comparison with Other Elements Feature Q4 Q6 Q6i Q8 DOF/node 2 3 2 2 Total DOF 8 12 8 8 Drilling rotation ❌ ✅ ❌ ❌ Shear deformation ❌ ❌ ✅ ✅ Beam theory - Euler-Bernoulli Timoshenko - Single-element accuracy Low Medium Medium High Recommended Refined mesh General use Thick beams Best accuracy
General guideline: Use Q6 as the default membrane element unless you specifically need shear deformation (Q6i) or maximum accuracy (Q8).
