Influence lines

3 min read • 463 words

Influence lines are constructed for a plane truss.

This example is adapted from a CE221 project  .

An influence line  is a graph that shows a response quantity at a fixed point in a structure as a function of a load’s location. In this study the response quantity we’ll investigate is the axial force in each truss member.

import xara
from xara.load import NodalLoad
import numpy as np
import veux

1) Model  

def create_model():
    model = xara.Model(ndm=2, ndf=2) 
    
    # ------------------------------
    # Geometry parameters
    # ------------------------------
    span = 12.0    # Total length in meters
    num_elem = 11  # Number of elements
    num_node = 7   # Number of node
    node_x_diff= span/(num_node-1)
    
    # ------------------------------
    # Material properties
    # ------------------------------
    # Material & section for W14x38
    E  = 210e9  # Young's modulus in Pascals (steel)
    A1 = 0.00723     # Cross-sectional area in m^2
    Fy = 340e6
    model.uniaxialMaterial("Steel02", 1, Fy, E, 0.02)
    
    # Material & section for HSS10x10x1/2
    A2 = 0.00645  # Cross-sectional area in m^2
    model.uniaxialMaterial("Steel02", 2, Fy, E, 0.02)

    model.section("Truss", 1, area=A1, material=1)
    model.section("Truss", 2, area=A2, material=2)
    
    # ------------------------------
    # Define nodes
    # ------------------------------
    for i in range(1, num_node+1):
        if i <= 4: 
            model.node(i ,((i-1)*node_x_diff*2,0))
        elif i>4:
            model.node(i ,(4*(i-5)+node_x_diff,node_x_diff))

    
    # ------------------------------
    # Define boundary conditions
    # ------------------------------
    model.fix(1, (1, 1))  # Pin at node 1
    model.fix(4, (0, 1))  # Roller at node 4
    
    # ------------------------------
    # Define truss elements
    # ------------------------------
    for i in range(1,num_elem+1):
        if i <=3:
            # Beam: W14x38 between nodes 1 and 2
            model.element("Truss", i, (i, i+1), section=1)
        elif i <=5:
            # Beam: HSS10x10x1/2 between node 5 and 6
            model.element("Truss", i, (i+1, i+2), section=2)
        elif i <=8:
            # Beam: HSS10x10x1/2 between node 1 and 5
            model.element("Truss", i, (i-5, i-1), section=2)
        elif i <=11:
            # Beam: HSS10x10x1/2 between node 2 and 5
            model.element("Truss", i, (i-4, i-7), section=2)
    return model

model = create_model()
veux.render(model)

2) Loading  

def create_load(model, load, loc):
    "Add loading to a the model"

    for node in range(1, 4+1):
        if model.nodeCoord(node)[0] > loc:
            break

    right_node = node
    left_node = node-1
    L = 4
    a = loc - model.nodeCoord(left_node)[0]

    return NodalLoad({
        left_node: (0,  (4-a)/4*load),
        right_node: (0, a/4*load)
    })

 
def analyze(model, load, steps=1):
    # ------------------------------
    # Run analysis 
    # ------------------------------
    xara.solve(model, load)
    
    # ------------------------------
    # Collect outputs
    # ------------------------------
    u, p = [], []
    for i in range(steps):
        # model.analyze(1)
        p.append(model.getTime())
        u.append(model.nodeDisp(6, 2))

    return u, p

3) Analysis  

To construct the influence lines, we’ll begin by initializing a dictionary that holds empty lists for the influence lines corresponding to each member:

forces = {i: {"x": [], "force": []} for i in create_model().getEleTags()}

for x in np.linspace(0, 12, 10):
    model = create_model()
    loads = create_load(model, 1.0, x)
    analyze(model, loads)

    for elem in model.getEleTags():
        forces[elem]["x"].append(x)
        forces[elem]["force"].append(model.eleResponse(elem, "basicForce"))
    

import matplotlib.pyplot as plt
plt.style.use("veux-web")

fig, ax = plt.subplots(figsize=(10, 6))

for elem in range(1, 12):
    ax.plot(forces[elem]["x"], forces[elem]["force"], label=f"Member {elem}")

ax.set_xlabel("Location (m)")
ax.set_ylabel("Axial Response (N/N)")
fig.legend();