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Overview

Milcapy provides two beam element formulations for structural frame analysis:
  • Timoshenko Beam: Accounts for shear deformation (recommended for most cases)
  • Euler-Bernoulli Beam: Classical beam theory without shear deformation
Both elements are 2D frame elements with 3 degrees of freedom (DOF) per node:
  • Translation in x direction
  • Translation in y direction
  • Rotation about z axis
The Timoshenko beam theory is more accurate for deep beams and higher modes of vibration as it includes shear deformation effects.

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Adding Beam Elements

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General Method (Specify Theory)

from milcapy.utils.types import BeamTheoriesType

model.add_member(
    id=1,
    node_i_id=1,
    node_j_id=2,
    section_name='W14x22',
    beam_theory=BeamTheoriesType.TIMOSHENKO
)
Parameters:
  • id (int) → Member ID
  • node_i_id (int) → Initial node ID
  • node_j_id (int) → Final node ID
  • section_name (str) → Section name
  • beam_theory (BeamTheoriesType or str, optional) → Beam theory to use
Available beam theories:
  • 'TIMOSHENKO' or BeamTheoriesType.TIMOSHENKO
  • 'EULER_BERNOULLI' or BeamTheoriesType.EULER_BERNOULLI

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Timoshenko Beam (Direct Method)

model.add_elastic_timoshenko_beam(
    id=1,
    node_i_id=1,
    node_j_id=2,
    section_name='W14x22'
)
The Timoshenko beam element includes:
  • Axial deformation
  • Bending deformation
  • Shear deformation (characterized by the parameter φ)

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Euler-Bernoulli Beam (Direct Method)

model.add_elastic_euler_bernoulli_beam(
    id=1,
    node_i_id=1,
    node_j_id=2,
    section_name='W14x22'
)
The Euler-Bernoulli beam element includes:
  • Axial deformation
  • Bending deformation
  • No shear deformation (φ = 0)

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Degrees of Freedom

Each node has 3 DOF in the global coordinate system: 
Node i: [Uxi, Uyi, Rzi]
Node j: [Uxj, Uyj, Rzj]
Where:
  • Ux = Displacement in x direction
  • Uy = Displacement in y direction
  • Rz = Rotation about z axis

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Advanced Features

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Rigid End Offsets

You can model rigid end zones at beam connections: 
member.la = 0.5  # Rigid offset length at node i
member.lb = 0.3  # Rigid offset length at node j
member.fla = 1.0  # Offset factor at node i (default: 1.0)
member.flb = 1.0  # Offset factor at node j (default: 1.0)

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Member Releases

Release specific DOF at member ends to model pins, rollers, or other connections: 
member.add_releases(
    uxi=False,  # Release axial DOF at node i
    uyi=False,  # Release vertical DOF at node i
    rzi=True,   # Release rotation at node i (pin connection)
    uxj=False,
    uyj=False,
    rzj=False
)
Releasing rzi=True creates a pinned connection at node i, while releasing both rzi=True and rzj=True creates a simply supported beam.

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Distributed Loads

Apply distributed loads along beam members: 
from milcapy.loads.load import DistributedLoad

member.set_current_load_pattern('Dead Load')
member.set_distributed_load(DistributedLoad(
    q_i=-10.0,  # Transverse load at node i (force/length)
    q_j=-10.0,  # Transverse load at node j
    p_i=0.0,    # Axial load at node i
    p_j=0.0     # Axial load at node j
))

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When to Use Each Theory

Timoshenko Beam

Use when:
  • Beam depth/length ratio > 1/10
  • Analyzing deep beams or short spans
  • Shear deformation is significant
  • Higher accuracy is needed
  • Recommended for most cases

Euler-Bernoulli Beam

Use when:
  • Beam is slender (depth/length < 1/10)
  • Long span beams
  • Shear deformation is negligible
  • Classical beam theory is acceptable

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Technical Details

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Shear Deformation Parameter (φ)

For Timoshenko beams, the shear parameter is calculated as: 
φ = (12 * E * I) / (L² * A * k * G)
Where:
  • E = Young’s modulus
  • I = Moment of inertia
  • L = Member length
  • A = Cross-sectional area
  • k = Shear correction factor
  • G = Shear modulus
For Euler-Bernoulli beams: φ = 0

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Transformation Matrix

The local-to-global transformation is computed based on the member orientation angle θ: 
θ = atan2(yj - yi, xj - xi)
T = transformation_matrix(angle=θ)

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Stiffness Matrix

The global stiffness matrix is computed as: 
K_global = T^T * K_local * T
Where K_local is the 6×6 local stiffness matrix including axial, bending, and shear effects (for Timoshenko).

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Example: Simple Frame

import milcapy as mx
from milcapy.utils.types import BeamTheoriesType

# Create model
model = mx.Model()

# Define nodes
model.add_node(1, 0, 0)
model.add_node(2, 5, 0)
model.add_node(3, 5, 3)

# Define material and section
model.add_material('Steel', E=200e9, v=0.3, rho=7850)
model.add_section('W14x22', material='Steel', A=0.0042, I=2.09e-5, k=0.4)

# Add beam members with Timoshenko theory
model.add_member(1, 1, 2, 'W14x22', beam_theory=BeamTheoriesType.TIMOSHENKO)
model.add_member(2, 2, 3, 'W14x22', beam_theory=BeamTheoriesType.TIMOSHENKO)

# Or use direct methods
model.add_elastic_timoshenko_beam(3, 3, 1, 'W14x22')
Always ensure node ordering is correct. The member is oriented from node_i to node_j, which affects the local coordinate system and load direction.

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