Beam Column Design Optimization

A rectangular beam column is a structural element that combines both the properties of a beam and a column. It has the ability to resist both bending and axial forces. It is typically made up of reinforced concrete, steel or timber and is used in the construction of buildings, bridges and other structures. Optimize a rectangular beam column to carry an axial (horizontal) load of 25 lb and a transverse (vertical) load of 10 lb.

Design the beam column to be minimum weight while staying within the limits of material yield strength and buckling load. Assume that the beam can only bend in the horizontal (y) and vertical (x) directions and there is no bending in the other horizontal (z) dimension. The beam column is made of steel with a density (specific weight) of 0.3 lb/in³, a Young's modulus of 30x10⁶ psi, and a yield stress of 3x10⁴ psi. The beam must be at least 0.5 inches wide. The width (w) must be less than twice the depth (d).

Optimize the beam column design to minimize the weight. The compressive stress due to P_y is

$$\sigma_y = \frac{P_y}{b\;d}$$

The compressive stress due to P_x is

$$\sigma_x = \frac{P_x \ell d}{2 I_{zz}} = \frac{6 P_x \ell}{b \; d^2}$$

where the length l of the beam is 50 in and `I_{zz}` is the moment of inertia. The load where axial buckling starts is

$$P_{y,crit}=\frac{\pi^2 E I_{zz}}{4 \ell^2} = \frac{\pi^2 E b d^3}{48\ell^2}$$

Determine the optimal solution and show the solution graphically on a contour plot that shows the constraints and weight objective function versus the design variables (beam width and beam depth).

Solution

from gekko import GEKKO
from numpy import pi

m = GEKKO(remote=False)

# Constants
Px = 10 # Transverse load (lb)
Py = 25 # Axial load (lb)
Ln = 50 # Beam length (in)
rho = 0.3 # Material density (lb/in^3)
S_y = 30000 # Yield stress (psi)
E = 30e6 # Young's modulus

# Variables
b = m.Var(lb=0.5) # Beam width (in)
d = m.Var(lb=1e-3) # Beam depth (in)
weight = m.Var(lb=1e-3) # Beam weight (lb)

# Intermediate calculations
I_zz = b*d**3/12 # Moment of Inertia
x_stress = m.Var(ub=S_y) # Compressive stress due to Px
y_stress = m.Var(ub=S_y) # Compressive stress due to Py
# Intermediate: Axial Buckling Load
Py_ab = m.Intermediate(pi**2*E*I_zz/(4*Ln**2))

# Equations
m.Equations([
weight == b*d*Ln*rho,
y_stress*(b*d)==Py,
x_stress*(2*I_zz)==Px*Ln*d,
Py_ab >= Py,
b <= 2*d
])

# Objective
m.Minimize(weight)

# Solve
m.options.SOLVER = 3
m.solve()

print('Weight: ' + str(weight[0]))
print('Depth: ' + str(d[0]))
print('Width: ' + str(b[0]))
print('Vertical stress: ' + str(x_stress[0]))
print('Horizontal stress: ' + str(y_stress[0]))
print('Axial Buckling Load: ' + str(Py_ab[0]))

[$[Get Code]]

The optimal solution is:

  Optimal Weight: 3.354101916
  Optimal Depth: 0.44721359774
  Optimal Width: 0.5
  Optimal Vertical stress: 30000.0
  Optimal Horizontal stress: 111.80340055
  Optimal Axial Buckling Load: 110.34553341

Show solution graphically similar to the Two Bar Truss design optimization.

Generative AI Learning

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

"Quiz me with 4 questions, one at a time, on the beam-column design problem: why a member carrying BOTH axial and transverse load needs stress checks that neither pure-beam nor pure-column analysis covers, how width b and depth d each influence bending stress vs axial buckling differently, what the geometric constraint b less than 2d protects against, and how to tell from the optimizer output whether the design is stress-limited or buckling-limited. Grade my answers and list my misconceptions."

"Before I optimize the beam-column (minimize weight; vertical and horizontal stress below 30,000 psi yield; buckling load above the 25 lb axial load; b at least 0.5 in; b at most 2d): interview me until I commit to which constraints will be active at the optimum and whether the optimal section will be slender or stocky, with the mechanics reasoning. Then critique my logic without revealing numbers - my own GEKKO run will."

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 and your optimal design (b, d, weight) with a check that every constraint is satisfied.
Which constraints are active at the optimum? Compare with your committed prediction and explain any surprise mechanically.
Double the transverse load (10 to 20 lb) and re-solve. Which constraint takes over, and how does the optimal aspect ratio d/b shift?
Include a contour plot of weight over (b, d) with the feasible region shaded and the optimum marked.
From the quiz prompt: one question you missed and the corrected answer.