Problem 2 (10 marks)

An I-beam is shown in the figure to the right.

Given the following equations and constraints,

develop a mathematical model and find the

dimensions of a beam with a minimal cross

sectional area.

Cross sectional area:

Section modulus:

Bending moment:

Axial force:

Bending stress:

Axial stress:

Stress constraint:

Buckling constraint:

X₂

↓

X4

TH

A = x₁x₂ + 2x3x4 - 2x₂x4

S = X₁ (X3X4+*1X2)

M = 350

P = 140

Og =

Op

-142*

1000M

S

OB + Op-

10P

A

180 ≤0

(1+%²

1+171

SO

And subject to the following constraints on plate thickness and width:

x₁:38,40,41

x₂: 1.9, 2.2, 2.4

x3: 28,30,33

X4: 1.1, 1.3, 1.5

·X₂

cm²

cm³

kNm

kN

MPa

MPa

MPa