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This example demonstrates how to model and analyze structures using 2D membrane elements (plane stress elements). You’ll learn how to create a wall-beam portal frame system combining membrane elements with beam members.

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Overview

This example models a portal frame with:
  • Two concrete walls modeled with Q6 membrane elements
  • Three beam members connecting the walls horizontally
  • Fixed supports at the base of both walls
  • Lateral loads applied at wall joints

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What are Membrane Elements?

Membrane elements are 2D finite elements used for plane stress analysis:
  • Model thin structures loaded in their plane
  • Capture in-plane stress distribution
  • No out-of-plane bending behavior
  • Ideal for walls, plates, and thin shells under in-plane loading

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Complete Example

1

Import the library

Import the necessary modules:
from milcapy import SystemModel
2

Define geometric parameters

Set up the dimensional parameters for the structure:
E = 2e6          # Young's modulus (tonf/m²)
v = 0.3          # Poisson's ratio

h = 2.4          # Wall height (m)
b = 1.2          # Wall width (m)
l = 2.4          # Beam length (m)

t = 0.4          # Wall thickness (m)
sec = [0.4, 0.4] # Beam section (m)
F = 100          # Applied load (tonf)
These parameters define:
  • Material properties (E, v)
  • Geometric dimensions (h, b, l)
  • Section properties (t, sec)
  • Loading magnitude (F)
3

Create model and define materials

Initialize the model and create material and section definitions:
wallPortal = SystemModel()

wallPortal.add_material("concreto", E, v)
wallPortal.add_rectangular_section("vig40x40", "concreto", *sec)
wallPortal.add_shell_section("muro", "concreto", t)
  • Material: Concrete with specified E and v
  • vig40x40: Rectangular beam section (0.4m × 0.4m)
  • muro: Shell/membrane section with 0.4m thickness
4

Create the nodal mesh

Define nodes for the two walls:
# Left wall nodes
wallPortal.add_node(1, 0, 0)      # Base
wallPortal.add_node(2, b, 0)      # Base
wallPortal.add_node(3, 0, h)      # Level 1
wallPortal.add_node(4, b, h)      # Level 1
wallPortal.add_node(5, 0, 2*h)    # Level 2
wallPortal.add_node(6, b, 2*h)    # Level 2
wallPortal.add_node(7, 0, 3*h)    # Level 3
wallPortal.add_node(8, b, 3*h)    # Level 3

# Right wall nodes
wallPortal.add_node(9,  b+l, 0)      # Base
wallPortal.add_node(10, b+l+b, 0)    # Base
wallPortal.add_node(11, b+l, h)      # Level 1
wallPortal.add_node(12, b+l+b, h)    # Level 1
wallPortal.add_node(13, b+l, 2*h)    # Level 2
wallPortal.add_node(14, b+l+b, 2*h)  # Level 2
wallPortal.add_node(15, b+l, 3*h)    # Level 3
wallPortal.add_node(16, b+l+b, 3*h)  # Level 3
This creates a 3-level portal frame structure with walls on each side.
5

Add membrane elements

Create Q6 quadrilateral membrane elements for the walls:
# Left wall membrane elements
wallPortal.add_membrane_q6(1, 1, 2, 4, 3, "muro")    # Level 0-1
wallPortal.add_membrane_q6(2, 3, 4, 6, 5, "muro")    # Level 1-2
wallPortal.add_membrane_q6(3, 5, 6, 8, 7, "muro")    # Level 2-3

# Right wall membrane elements
wallPortal.add_membrane_q6(4, 9, 10, 12, 11, "muro") # Level 0-1
wallPortal.add_membrane_q6(5, 11, 12, 14, 13, "muro")# Level 1-2
wallPortal.add_membrane_q6(6, 13, 14, 16, 15, "muro")# Level 2-3
The add_membrane_q6() method creates 6-node quadrilateral elements. Nodes are specified in counter-clockwise order: bottom-left, bottom-right, top-right, top-left.
6

Add beam connectors

Connect the two walls with horizontal beams:
wallPortal.add_elastic_timoshenko_beam(7, 4, 11, "vig40x40")  # Level 1
wallPortal.add_elastic_timoshenko_beam(8, 6, 13, "vig40x40")  # Level 2
wallPortal.add_elastic_timoshenko_beam(9, 8, 15, "vig40x40")  # Level 3
These beams tie the two walls together at each level. Timoshenko beam theory accounts for shear deformation.
Alternatively, you could use Euler-Bernoulli beams with:
wallPortal.add_elastic_euler_bernoulli_beam(7, 4, 11, "vig40x40")
7

Apply boundary conditions

Fix the base nodes of both walls:
wallPortal.add_restraint(1,  *(True, True, True))  # Left wall base
wallPortal.add_restraint(2,  *(True, True, True))  # Left wall base
wallPortal.add_restraint(9,  *(True, True, True))  # Right wall base
wallPortal.add_restraint(10, *(True, True, True))  # Right wall base
All base nodes are fully restrained (x, y, and rotation).
8

Apply lateral loads

Create a load pattern and apply increasing lateral loads up the left wall:
wallPortal.add_load_pattern("carga")

wallPortal.add_point_load(3, "carga", fx=1*F)  # Level 1: 100 tonf
wallPortal.add_point_load(5, "carga", fx=2*F)  # Level 2: 200 tonf
wallPortal.add_point_load(7, "carga", fx=3*F)  # Level 3: 300 tonf
The lateral loads simulate wind or seismic loading, increasing with height.
9

Solve and visualize

Perform the analysis and display results:
wallPortal.solve()
wallPortal.show()
The show() method opens an interactive 3D viewer displaying:
  • Deformed shape
  • Stress contours
  • Displacement magnitudes

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Full Code

from milcapy import SystemModel

# Define parameters
E = 2e6          # Young's modulus (tonf/m²)
v = 0.3          # Poisson's ratio
h = 2.4          # Wall height (m)
b = 1.2          # Wall width (m)
l = 2.4          # Beam length (m)
t = 0.4          # Wall thickness (m)
sec = [0.4, 0.4] # Beam section (m)
F = 100          # Applied load (tonf)

# Create model
wallPortal = SystemModel()

# Define materials and sections
wallPortal.add_material("concreto", E, v)
wallPortal.add_rectangular_section("vig40x40", "concreto", *sec)
wallPortal.add_shell_section("muro", "concreto", t)

# Add nodes - Left wall
wallPortal.add_node(1, 0, 0)
wallPortal.add_node(2, b, 0)
wallPortal.add_node(3, 0, h)
wallPortal.add_node(4, b, h)
wallPortal.add_node(5, 0, 2*h)
wallPortal.add_node(6, b, 2*h)
wallPortal.add_node(7, 0, 3*h)
wallPortal.add_node(8, b, 3*h)

# Add nodes - Right wall
wallPortal.add_node(9,  b+l, 0)
wallPortal.add_node(10, b+l+b, 0)
wallPortal.add_node(11, b+l, h)
wallPortal.add_node(12, b+l+b, h)
wallPortal.add_node(13, b+l, 2*h)
wallPortal.add_node(14, b+l+b, 2*h)
wallPortal.add_node(15, b+l, 3*h)
wallPortal.add_node(16, b+l+b, 3*h)

# Add membrane elements
wallPortal.add_membrane_q6(1, 1, 2, 4, 3, "muro")
wallPortal.add_membrane_q6(2, 3, 4, 6, 5, "muro")
wallPortal.add_membrane_q6(3, 5, 6, 8, 7, "muro")
wallPortal.add_membrane_q6(4, 9, 10, 12, 11, "muro")
wallPortal.add_membrane_q6(5, 11, 12, 14, 13, "muro")
wallPortal.add_membrane_q6(6, 13, 14, 16, 15, "muro")

# Add connecting beams
wallPortal.add_elastic_timoshenko_beam(7, 4, 11, "vig40x40")
wallPortal.add_elastic_timoshenko_beam(8, 6, 13, "vig40x40")
wallPortal.add_elastic_timoshenko_beam(9, 8, 15, "vig40x40")

# Apply boundary conditions
wallPortal.add_restraint(1,  *(True, True, True))
wallPortal.add_restraint(2,  *(True, True, True))
wallPortal.add_restraint(9,  *(True, True, True))
wallPortal.add_restraint(10, *(True, True, True))

# Apply loads
wallPortal.add_load_pattern("carga")
wallPortal.add_point_load(3, "carga", fx=1*F)
wallPortal.add_point_load(5, "carga", fx=2*F)
wallPortal.add_point_load(7, "carga", fx=3*F)

# Solve and visualize
wallPortal.solve()
wallPortal.show()

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Membrane Element Types in milcapy

milcapy supports several membrane element formulations:

Q6 Elements

6-node quadrilateral
  • Standard element with mid-side nodes
  • Good accuracy for most applications
  • Used in this example

Q8 Elements

8-node quadrilateral
  • Higher-order interpolation
  • Better for curved boundaries
  • Use add_membrane_q8()

Q6i Elements

6-node with incompatible modes
  • Enhanced accuracy for bending
  • Reduces shear locking
  • Use add_membrane_q6i()

CST Elements

Constant strain triangle
  • 3-node triangular element
  • Simple but less accurate
  • Use for irregular meshes

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When to Use Membrane Elements

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Appropriate for:

  • Shear walls under lateral loads
  • Deep beams with D/L > 1/4
  • Thin plates under in-plane loading
  • Plane stress/strain analysis
  • Bracket and gusset plate analysis

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Not appropriate for:

  • Out-of-plane bending (use shell elements)
  • Thick sections under pure bending (use beam elements)
  • 3D stress states (use solid elements)
For structures with both in-plane and out-of-plane behavior, consider using shell elements which combine membrane and plate bending behavior.

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Patch Test Validation

The membrane element formulations in milcapy pass standard patch tests, ensuring:
  • Constant strain states are exactly reproduced
  • Element convergence with mesh refinement
  • Proper rank of stiffness matrix
See the patch_test.py example in the source code for validation cases.

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Advanced Topics

Mesh Refinement

Increase element density in high-stress regions for better accuracy

Mixed Formulations

Combine different element types in transition regions

Incompatible Modes

Use enhanced elements (Q6i, MQ6IMod) for improved bending behavior

Material Nonlinearity

Model concrete cracking and steel yielding

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Next Steps

Portal Frame

Learn frame modeling with beam elements

Truss Analysis

Model axial-only structures

Shell Sections

View shell and membrane section API

Material Library

Explore material definitions and properties

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