First-Order: Bending Line

Bending line of a multi-bar frame using the first-order method

This example computes and plots the bending line (deflection curve) of a complex multi-bar frame system subjected to point and line loads. The system contains fixed supports and various load cases.

The FirstOrder solver is used to determine the displacement field, and the resulting bending line is visualized using the plot(kind='bending_line') method.

This serves as a minimal introduction to the basic workflow of sStatics. You can find the example as an executable Python file here.

Define the Structural System

In this section, the static system is modeled. For a detailed step-by-step workflow, refer to the “Getting Started” example.

from sstatics.core.preprocessing import (
    Material, CrossSection, NodePointLoad, Node, Bar, BarLineLoad, System
)
from sstatics.core.calc_methods import FirstOrder
from sstatics.core.postprocessing.graphic_objects import ObjectRenderer, SystemGeo

# 1. Define material and cross-section
mat = Material(young_mod=13700000, poisson=0.1, shear_mod=0.1,
               therm_exp_coeff=0.1)

cs = CrossSection(mom_of_int=58956e-8, area=612e-4, height=612e-4, width=1,
                  shear_cor=0.1)

# 2. Define nodes with supports and point loads
node_1 = Node(x=0, z=4, u='fixed', w='fixed', phi='fixed')
node_2 = Node(x=0, z=0)
node_3 = Node(x=3, z=0, loads=NodePointLoad(z=20))
node_4 = Node(x=11, z=0, loads=NodePointLoad(x=-15))
node_5 = Node(x=11, z=6, u='fixed', w='fixed')

# 3. Define line loads on bars
line_load_1 = BarLineLoad(pi=3.3, pj=2.4, direction='z')
line_load_2 = BarLineLoad(pi=2.4, pj=0, direction='z')
line_load_3 = BarLineLoad(pi=2, pj=2, direction='x')

# 4. Define bars and system
bar_1 = Bar(node_1, node_3, cs, mat)
bar_2 = Bar(node_2, node_3, cs, mat, line_loads=line_load_1)
bar_3 = Bar(node_3, node_4, cs, mat, line_loads=line_load_2)
bar_4 = Bar(node_4, node_5, cs, mat, line_loads=line_load_3)

system_2 = System([bar_1, bar_2, bar_3, bar_4])

# Show system graphic
ObjectRenderer(SystemGeo(system_2), 'mpl').show()

Perform the Structural Analysis

Next, we assemble the bar into a structural system and perform a first-order analysis.

The FirstOrder class differs from the standard Solver in that it accepts the structural system directly, rather than a pre-defined mesh. The computational mesh is generated automatically during the analysis, so you can focus on defining the model and loads.

This approach simplifies the workflow and allows you to quickly compute internal forces, support reactions and other results without manually creating the mesh.

# 7. Perform first-order structural analysis
solution = FirstOrder(system_2)

Plot of the Bending Line

The bending line can be visualized directly using the plot method from the FirstOrder class.

To display the bending line, set the parameter kind='bending_line'

# 6. Plot bending line (deflection curve)
solution.plot(kind='bending_line')