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Overview

Boundary conditions define how the structure is supported and constrained. milcapy provides several types of boundary conditions including fixed restraints, prescribed displacements, elastic supports, and local coordinate systems.

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Restraints (Fixed Supports)

Restraints prevent movement at specific degrees of freedom. Each node has 3 DOFs that can be restrained:
  • ux: Translation in X direction
  • uy: Translation in Y direction
  • rz: Rotation about Z axis

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Adding Restraints

model.add_restraint(
    node_id=1,
    ux=True,   # Restrain X translation
    uy=True,   # Restrain Y translation
    rz=True    # Restrain Z rotation
)

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Common Support Types

Prevents all translations and rotations:
model.add_restraint(
    node_id=1,
    ux=True,
    uy=True,
    rz=True
)
Use cases: Column bases, rigid connections to foundations
Prevents translations but allows rotation:
model.add_restraint(
    node_id=2,
    ux=True,
    uy=True,
    rz=False  # Free to rotate
)
Use cases: Simple beam supports, truss joints
Prevents vertical translation only:
model.add_restraint(
    node_id=3,
    ux=False,  # Free to move horizontally
    uy=True,   # Restrained vertically
    rz=False   # Free to rotate
)
Use cases: Expansion joints, sliding supports
Prevents horizontal translation only:
model.add_restraint(
    node_id=4,
    ux=True,   # Restrained horizontally
    uy=False,  # Free to move vertically
    rz=False
)
Use cases: Vertical sliding supports

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Prescribed Displacements

Prescribed displacements impose specific displacement values at nodes, useful for settlement analysis or imposed deformations: 
model.add_prescribed_dof(
    node_id=1,
    load_pattern_name="SETTLEMENT",
    ux=None,      # No prescribed displacement in X
    uy=-0.010,    # 10mm downward settlement
    rz=None,      # No prescribed rotation
    CSys="GLOBAL" # Coordinate system
)
Use None for DOFs without prescribed displacement. Only specify values for DOFs you want to control.

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Settlement Analysis Example

# Create settlement load pattern
model.add_load_pattern(
    name="SETTLEMENT",
    self_weight_multiplier=0.0
)

# Apply support settlement
model.add_prescribed_dof(
    node_id=1,
    load_pattern_name="SETTLEMENT",
    uy=-0.020  # 20mm settlement
)

model.add_prescribed_dof(
    node_id=4,
    load_pattern_name="SETTLEMENT",
    uy=-0.015  # 15mm settlement (differential settlement)
)
Prescribed displacements are associated with load patterns. Make sure to create the load pattern before adding prescribed DOFs.

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Elastic Supports

Elastic supports provide flexible support using spring stiffness values: 
model.add_elastic_support(
    node_id=1,
    kx=1000000,    # Stiffness in X direction (N/m)
    ky=5000000,    # Stiffness in Y direction (N/m)
    krz=100000,    # Rotational stiffness (N·m/rad)
    CSys="GLOBAL"
)

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Elastic Support Applications

Model soil-structure interaction:
# Soil spring constants (from geotechnical analysis)
k_soil_vertical = 10e6    # N/m
k_soil_horizontal = 5e6   # N/m
k_soil_rotational = 2e6   # N·m/rad

model.add_elastic_support(
    node_id=1,
    kx=k_soil_horizontal,
    ky=k_soil_vertical,
    krz=k_soil_rotational
)
Simulate semi-rigid connections:
# Semi-rigid rotational spring
model.add_elastic_support(
    node_id=5,
    kx=None,  # No spring in X
    ky=None,  # No spring in Y
    krz=5e5   # Rotational spring only
)
Use None for infinite stiffness (rigid restraint in that direction). Use 0 for free movement, or a positive value for elastic support.

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Local Coordinate Systems

Define inclined or rotated coordinate systems at nodes: 
import math

# Define local axis at 45 degrees
model.add_local_axis_for_node(
    node_id=1,
    angle=math.radians(45)  # Angle in radians
)

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Using Local Axes

Local axes affect:
  1. Restraints - Applied in local directions
  2. Loads - When using CSys="LOCAL"
  3. Results - Displacements/forces in local coordinates
import math

# Create inclined support (30° from horizontal)
angle = math.radians(30)
model.add_local_axis_for_node(node_id=1, angle=angle)

# Restrain in local Y direction (perpendicular to incline)
model.add_restraint(
    node_id=1,
    ux=False,
    uy=True,  # Restrained perpendicular to inclined surface
    rz=False
)

# Add load in local coordinates
model.add_point_load(
    node_id=1,
    load_pattern_name="LL",
    fx=0,
    fy=-1000,  # Load perpendicular to inclined surface
    CSys="LOCAL"
)

import math

# 30° incline
angle_30 = math.radians(30)

# 45° incline  
angle_45 = math.radians(45)

# 60° incline
angle_60 = math.radians(60)

model.add_local_axis_for_node(node_id=1, angle=angle_30)

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Combining Boundary Conditions

You can apply multiple boundary condition types to the same node: 
# Node with elastic support and prescribed settlement
model.add_elastic_support(
    node_id=1,
    kx=1e6,
    ky=5e6,
    krz=None
)

model.add_prescribed_dof(
    node_id=1,
    load_pattern_name="SETTLEMENT",
    uy=-0.01  # 10mm settlement on elastic support
)

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Complete Example: Frame with Various Supports

import math
from milcapy import SystemMilcaModel

# Create model and add geometry
model = SystemMilcaModel()

# Add material and sections
model.add_material(name="Steel", modulus_elasticity=200e9,
                   poisson_ratio=0.3, specific_weight=78500)
model.add_rectangular_section(name="COL", material_name="Steel",
                              base=0.30, height=0.30)

# Create nodes
model.add_node(id=1, x=0.0, y=0.0)   # Left column base
model.add_node(id=2, x=0.0, y=4.0)   # Left column top
model.add_node(id=3, x=8.0, y=4.0)   # Right column top
model.add_node(id=4, x=8.0, y=0.0)   # Right column base

# Create columns and beam
model.add_member(id=1, node_i_id=1, node_j_id=2, section_name="COL")
model.add_member(id=2, node_i_id=2, node_j_id=3, section_name="COL")
model.add_member(id=3, node_i_id=4, node_j_id=3, section_name="COL")

# Left support: Fixed
model.add_restraint(
    node_id=1,
    ux=True,
    uy=True,
    rz=True
)

# Right support: Roller (horizontal movement allowed)
model.add_restraint(
    node_id=4,
    ux=False,  # Free horizontal movement
    uy=True,
    rz=True
)

# Add elastic support under roller (foundation spring)
model.add_elastic_support(
    node_id=4,
    ky=10e6  # 10 MN/m vertical spring
)

# Settlement load case
model.add_load_pattern(name="SETTLEMENT", self_weight_multiplier=0.0)
model.add_prescribed_dof(
    node_id=4,
    load_pattern_name="SETTLEMENT",
    uy=-0.025  # 25mm settlement
)

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Boundary Condition Checks

Stability Check: Ensure your structure has sufficient restraints to prevent rigid body motion. A 2D structure typically needs at least 3 restraints to be stable.
Common Minimum Restraints:
  • 1 fixed support (3 restraints)
  • 1 pinned + 1 roller (2 + 1 = 3 restraints)
  • 2 pinned supports + 1 horizontal restraint elsewhere
Over-Constraint: Too many restraints can lead to incompatible constraints. Use elastic supports or releases when needed.

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Best Practices

Coordinate System Consistency: When using local axes, clearly document the angle convention. milcapy uses counter-clockwise positive angles from the global X-axis.
Elastic Support Stiffness:
  • Use realistic spring values from geotechnical reports
  • Very high stiffness (greater than 1e12) approaches rigid restraint
  • Very low stiffness (less than 1) approaches free movement
Units Matter: Ensure consistent units:
  • Translational springs: N/m or kN/m
  • Rotational springs: N·m/rad or kN·m/rad
  • Displacements: meters
  • Angles: radians

Nodes & Elements

Understanding node geometry and connectivity

Load Patterns

Applying loads after defining supports

Running Analysis

Solving models with boundary conditions

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