Two Bar Truss Design

A two-bar truss is a type of truss structure consisting of two bars connected at the ends by either pins or joints. This type of truss is commonly used in construction and engineering applications.

A design of the truss is specified by a unique set of values for the analysis variables: height (H), diameter, (d), thickness (t), separation distance (B), modulus of elasticity (E), and material density (rho). Suppose we are interested in designing a truss that has a minimum weight, will not yield, will not buckle, and does not deflect "excessively,” and so we decide our model should calculate weight, stress, buckling stress and deflection.

Two Bar Assignment

This introductory assignment is designed as a means of demonstrating the optimization capabilities of a number of software packages. Below are tutorials for solving this problem with a number of software tools. Below are a few step-by-step tutorials.

Python (GEKKO) Solution

GEKKO is optimization software for mixed-integer and differential algebraic equations. It is coupled with large-scale solvers for linear, quadratic, nonlinear, and mixed integer programming (LP, QP, NLP, MILP, MINLP).

import numpy as np

# import gekko, pip install if needed
from gekko import GEKKO

# create new model
m = GEKKO()

# declare model parameters
width = m.Param(value=60)
thickness = m.Param(value=0.15)
density = m.Param(value=0.3)
modulus = m.Param(value=30000)
load = m.Param(value=66)

# declare variables and initial guesses
height = m.Var(value=30.00,lb=10.0,ub=50.0)
diameter = m.Var(value=3.00,lb=1.0,ub=4.0)
weight = m.Var()

# intermediate variables with explicit equations
leng = m.Intermediate(m.sqrt((width/2)**2 + height**2))
area = m.Intermediate(np.pi * diameter * thickness)
iovera = m.Intermediate((diameter**2 + thickness**2)/8)
stress = m.Intermediate(load * leng / (2*area*height))
buckling = m.Intermediate(np.pi**2 * modulus \
* iovera / (leng**2))
deflection = m.Intermediate(load * leng**3 \
/ (2 * modulus * area * height**2))

# implicit equations
m.Equation(weight==2*density*area*leng)
m.Equation(weight < 24)
m.Equation(stress < 100)
m.Equation(stress < buckling)
m.Equation(deflection < 0.25)

# minimize weight
m.Minimize(weight)

# solve optimization
m.solve() # remote=False for local solve

print ('')
print ('--- Results of the Optimization Problem ---')
print ('Height: ' + str(height.value))
print ('Diameter: ' + str(diameter.value))
print ('Weight: ' + str(weight.value))

[$[Get Code]]

Solution Contour Plot (Python)

## Generate a contour plot
# Import some other libraries that we'll need
# matplotlib and numpy packages must also be installed
import matplotlib
import numpy as np
import matplotlib.pyplot as plt

# Constants
pi = 3.14159
dens = 0.3
modu = 30000.0
load = 66.0

# Analysis variables
wdth = 60.0
thik = 0.15

# Design variables at mesh points
x = np.arange(10.0, 30.0, 2.0)
y = np.arange(1.0, 3.0, 0.3)
hght, diam = np.meshgrid(x, y)

# Equations and Constraints
leng = ((wdth/2.0)**2.0 + hght**2)**0.5
area = pi * diam * thik
iovera = (diam**2.0 + thik**2.0)/8.0
wght = 2.0 * dens * leng * area
strs = load * leng / (2.0 * area * hght)
buck = pi**2.0 * modu * iovera / (leng**2.0)
defl = load * leng**3.0 / (2.0*modu * area * hght**2.0)

# Create a contour plot
# Visit https://matplotlib.org/examples/pylab_examples/contour_demo.html
# for more examples and options for contour plots
plt.figure()
# Weight contours
CS = plt.contour(hght, diam, wght)
plt.clabel(CS, inline=1, fontsize=10)
# Stress<100
CS = plt.contour(hght, diam, strs,[100.0],colors='k',linewidths=[4.0])
plt.clabel(CS, inline=1, fontsize=10)
# Deflection<0.25
CS = plt.contour(hght, diam, defl,[0.25],colors='b',linewidths=[4.0])
plt.clabel(CS, inline=1, fontsize=10)
# Stress-Buckling<0
CS = plt.contour(hght, diam, strs-buck,[0.0],colors='r',linewidths=[4.0])
plt.clabel(CS, inline=1, fontsize=10)
# Add some labels
plt.title('Two Bar Optimization Problem')
plt.xlabel('Height')
plt.ylabel('Diameter')
# Save the figure as a PNG
plt.savefig('contour1.png')

# Create a new figure to see more detail
plt.figure()
# Weight contours
CS = plt.contour(hght, diam, wght)
plt.clabel(CS, inline=1, fontsize=10)
# Stress<100
CS = plt.contour(hght, diam, strs,[90.0,100.0],colors='k',linewidths=[0.5, 4.0])
plt.clabel(CS, inline=1, fontsize=10)
# Deflection<0.25
CS = plt.contour(hght, diam, defl,[0.22,0.25],colors='b',linewidths=[0.5, 4.0])
plt.clabel(CS, inline=1, fontsize=10)
# Stress-Buckling<0
CS = plt.contour(hght, diam, strs-buck,[-5.0,0.0],colors='r',linewidths=[0.5, 4.0])
plt.clabel(CS, inline=1, fontsize=10)
# Add some labels
plt.title('Two Bar Optimization Problem')
plt.xlabel('Height')
plt.ylabel('Diameter')
# Save the figure as a PNG
plt.savefig('contour2.png')

# Show the plots
plt.show()

[$[Get Code]]

APM Python Tutorial

APM Python Two Bar Files

APM MATLAB Tutorial

APM MATLAB Two Bar Files

Objective Function Plot

One part of the assignment asks you to select width and load as variables for a 3d optimal surface plot and plot the solution of the optimization problem to minimize deflection at each of the width / load combinations. This tutorial example shows how to do this same activity but for the alternative problem of minimizing weight.

Two Bar Objective Plot Files

This assignment can be completed in collaboration with others. Additional guidelines on individual, collaborative, and group assignments are provided under the Expectations link.

Generative AI Learning

Use these prompts to test your understanding after completing the exercise. Direct the AI - it may critique and question, but the formulation and the optimization runs are yours.

"Quiz me with 5 questions, one at a time, on the two-bar truss design problem: which quantities are design variables (height, diameter) vs analysis variables (stress, weight, deflection) and why the distinction matters, why the stress and buckling constraints compete as height changes, what it means for a constraint to be active at the optimum, what happens to the optimal weight if the stress limit is relaxed from 100 to 110, and why bounds on the design variables are part of an honest formulation. Grade my answers and list my misconceptions."

"Before I run the optimizer on the two-bar truss (minimize weight subject to stress, buckling, and deflection constraints, design variables height and diameter): interview me, one question at a time, until I commit to a prediction of (a) which constraints will be active at the optimum and (b) whether the optimal height is nearer 10 or 50 inches, with physical reasoning about the stress-vs-buckling tradeoff. Only after I commit, evaluate my reasoning - do not reveal the actual optimum, I will get that from my own GEKKO run."

Tip: After solving, re-run with the stress limit at 90, 100, and 110 and plot optimal weight vs the limit. The slope is the constraint's shadow price - you will meet it formally as a Lagrange multiplier in the KKT unit. The contour plot on this page shows why: the optimum slides along the active constraint boundary.

What to Turn In

Submit a short report (PDF, 1-2 pages) that curates your results into a demonstration of what you learned. You may use Generative AI to help write the report, but you must guide it to the correct results, justifications, and assumptions. Answer these questions:

Present the formulation in standard form (objective, design variables with bounds, constraints with units) and the optimal design from your own run (height, diameter, weight).
Which constraints are active at the optimum? Compare with the prediction you committed to in the AI interview - where was your physical reasoning right or wrong?
Show the optimal weight for stress limits of 90, 100, and 110. What is the approximate shadow price (weight change per unit of stress limit), and what would you tell a design lead who asks whether buying stronger material is worth it?
Include the contour plot with the feasible region shaded and your optimum marked. Why does the unconstrained minimum of weight lie outside the feasible region?
From the quiz prompt: one question you missed and the corrected answer.

Design Optimization