Influence Line: Internal Force (Mechanism)

This example demonstrates how to compute the influence line for the internal force \(f_m\) at midspan of the first bar in a simple bar system when the modified structure becomes a mechanism.

Influence lines for internal forces are typically obtained by:

• releasing the bar at the point where the force is to be evaluated, and

• applying a unit deformation corresponding to the desired internal force quantity.

However, in this example, releasing the bar at midspan causes the structure to lose stiffness, turning it into a kinematically admissible mechanism. As a result, the system cannot be solved using the displacement method, since the stiffness matrix no longer has full rank.

In such cases, the influence line no longer corresponds to a bending line of a modified elastic system. Instead, it matches the displacement figure of the resulting rigid-body motion.

To obtain this displacement figure, a pole plan (Polplan) is constructed first. From this geometric construction, the corresponding displacement shape can be derived, which represents the influence line for the force quantity \(f_m\).

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

Import Modules

We start by importing the required classes needed to set up the structure and define the

InfluenceLine instance:

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

Create System

We first define the material, cross-section, nodes, and bars for the simple two-span bar system:

# 1. Define material and cross-section
mat = Material(21000, 0.1, 8100, 0.1)          # S235
cs = CrossSection(2769, 76.84, 20, 10, 0.1)   # HEA-240

# 2. Define nodes with supports
node_1 = Node(0, 0, u='fixed', w='fixed')
node_2 = Node(300, 0, w='fixed')
node_3 = Node(600, 0)

# 3. Define bars and system
bar_1 = Bar(node_1, node_2, cs, mat)
bar_2 = Bar(node_2, node_3, cs, mat)
system = System([bar_1, bar_2])

# Visualize system
ObjectRenderer(SystemGeo(system, show_bar_text=True), 'mpl').show(show_axis=False)

Create Influence-Line

We now create an InfluenceLine object that performs the computation:

# 4. Define Influence line
il = InfluenceLine(system)

Compute Influence Line for Internal Force \(f_m\)

To compute the influence line of the internal force \(f_m\) at the midpoint of the first bar, we specify:

• force type: 'fm'

• object: bar_1

• local position = 0.5 (midspan)

il.force(kind='fm', obj=bar_1, position=0.5)