Stress: Stress Distribution in a Bar

This example demonstrates how to compute normal, shear, and bending stresses along a beam using the BarStressDistribution class.

We consider a simply supported beam with:

• One fixed node

• One vertically restrained node

The beam is subjected to a constant line load, and we evaluate the stress distribution along the bar at 10 discrete points.

The example prints:

• Normal stresses \(\sigma(x)\)

• Shear stresses \(\tau(x)\)

• Bending stresses \(\sigma_{b,\text{top}}(x)\) and \(\sigma_{b,\text{bottom}}(x)\)

Additionally, stresses are evaluated at a specific cross-section height \(z\).

You can find the example as an executable Python file here.

Import Modules

We start by importing the required classes for defining the Stress object.

from sstatics.core.preprocessing import (
    Bar, System, Material, Node, NodePointLoad, CrossSection, BarLineLoad
)
from sstatics.core.preprocessing.geometry import Polygon
from sstatics.core.calc_methods import FirstOrder

Create First-Order Analysis

We create an instance of FirstOrder, to perform a linear structural analysis:

# 1. Define Material
mat = Material(210_000_000, 81_000_000, 0.1, 0.1)

# 2. Define Cross-section (rectangle 10 × 40)
rect = Polygon(points=[(0, 0), (10, 0), (10, 40), (0, 40), (0, 0)])
cs = CrossSection(geometry=[rect])

# 3. Define System: single-span beam (4 m)
n1 = Node(0, 0, u='fixed', w='fixed')
n2 = Node(4, 0, w='fixed', loads=(NodePointLoad(x=1)))

line_load = BarLineLoad(1, 1)
b1 = Bar(n1, n2, cs, mat, line_loads=line_load)
system = System([b1])

# 4. Solve system
solution = FirstOrder(system)

Stress Distribution Along the Beam

We evaluate the stress at 10 points along the bar:

n_disc = 10
stress_list = solution.stress_distribution(n_disc=n_disc)

The output includes:

• Normal stress: \(\sigma(x)\)

• Shear stress: \(\tau(x)\)

• Bending stress at bottom fiber: \(\sigma_{b,\text{bottom}}(x)\)

• Bending stress at top fiber: \(\sigma_{b,\text{top}}(x)\)

for i, bar_stress in enumerate(stress_list):
    sigma = bar_stress.stress_disc
    print(f"\nBar {i}: Stress distribution at {n_disc+1} points")
    print("  σ_normal(x): \n", sigma[:, 0])
    print("  τ_shear(x): \n", sigma[:, 1])
    print("  σ_bending_bottom(x): \n", sigma[:, 2])
    print("  σ_bending_top(x): \n", sigma[:, 3])

Bar 0: Stress distribution at 11 points
  σ_normal(x):
 [0.0025 0.0025 0.0025 0.0025 0.0025 0.0025 0.0025 0.0025 0.0025 0.0025
 0.0025]
  τ_shear(x):
 [ 7.50000000e-03  6.00000000e-03  4.50000000e-03  3.00000000e-03
  1.50000000e-03 -8.32667268e-19 -1.50000000e-03 -3.00000000e-03
 -4.50000000e-03 -6.00000000e-03 -7.50000000e-03]
  σ_bending_bottom(x):
 [ 8.32667268e-20 -2.70000000e-04 -4.80000000e-04 -6.30000000e-04
 -7.20000000e-04 -7.50000000e-04 -7.20000000e-04 -6.30000000e-04
 -4.80000000e-04 -2.70000000e-04  3.33066907e-19]
  σ_bending_top(x):
 [-8.32667268e-20  2.70000000e-04  4.80000000e-04  6.30000000e-04
  7.20000000e-04  7.50000000e-04  7.20000000e-04  6.30000000e-04
  4.80000000e-04  2.70000000e-04 -3.33066907e-19]

Stress at a Specific Cross-Section Height

We can also evaluate stresses at a specific height \(z\) of the cross-section (e.g., \(z=0\)):

z = 0

print(f"\nStress evaluation at z = {z}:")

for i, bar_stress in enumerate(stress_list):
    sigma_z = bar_stress.stress_at_z(z)
    print(f"\nBar {i}: Stress distribution at {n_disc+1} points at height "
          f"z = {z}")
    print("  σ_normal(x): \n", sigma_z[:, 0])
    print("  τ_shear(x): \n", sigma_z[:, 1])
    print("  σ_bending(x): \n", sigma_z[:, 2])

Stress evaluation at z = 0:

Bar 0: Stress distribution at 11 points at height z = 0
  σ_normal(x):
 [0.0025 0.0025 0.0025 0.0025 0.0025 0.0025 0.0025 0.0025 0.0025 0.0025
 0.0025]
  τ_shear(x):
 [0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]
  σ_bending(x):
 [ 8.32667268e-20 -2.70000000e-04 -4.80000000e-04 -6.30000000e-04
 -7.20000000e-04 -7.50000000e-04 -7.20000000e-04 -6.30000000e-04
 -4.80000000e-04 -2.70000000e-04  3.33066907e-19]

Plot Normal Stress

Finally, we can plot the normal stress distribution along the beam. This provides a visual representation of the stress along the beam, making it easy to interpret the effect of the line load on the structure.

solution.plot_stress('normal')