MECE5397: Assignment 1 - design of a beam
Due date: 8:00am on Mar. 21
In this assignment, your task is to optimize the cross sectional geometry of a doubly clamped beam. The
cross section is constructed with two parabolic curves in the form of b(y) that is dependent on bf
, bw and h.
The distributed load applied to the beam is also in a parabolic form p(x) that is dependent on pmax, pmin
and L. The Young’s modulus of the material is E, the yield strength is σy, and the deflection is u(x). Note
that all the question are to be completed in a Jupyter notebook and submitted on blackboard.
L
pmax p(x)
h
bw
bf
pmin
y
x
b(y)
In this problem bw and h are the optimization variables.
Task 1
State the cross sectional equations for beam width - b(y), area - A(y), first moment of area - Q(y), and second
moment of area - I(y). Note that the first moment of area is typically integrated from y to h/2, where as
both A and I are integrated from −h/2 to h/2.
Hint:
A(y) = Z
dA
Q(y) = Z
ydA
I(y) = Z
y
2
dA
Hint: first derive b(y), then substitute dA by b(y)dy.
Task 2
Write the governing ODE equation. Note that p(x) is not a constant, but rather, a function of x. Also note
that pmin ̸= 0.
Task 3
Derive the general solution to the governing ODE.
Task 4
State the boundary conditions.
1
Task 5
Calculate the integration constants.
Task 6
Derive moment, shear, and square of the von mises stress using the following equations.
M(x) = EI d
2u(x)
dx2
V (x) = EI d
3u(x)
dx3
σvm(x, y)
2 = σ
2
xx + 3σ
2
xy
where:
σxx =
M(x)y
I(y)
σxy =
V (x)Q(y)
b(y)I(y)
Task 7
Here is the optimization problem. Please explain in words what the following means
min
h,bw
A(y)Lρ
s.t. σ
2
vm − σ
2
y < 0
umax − ulimit < 0
bmin − bw < 0
hmin − h < 0
(1)
Task 8
Substitute constants into the above equations, so that in the end, the objective and each constraint can be
plotted.
Hint:
Think about at what x coordinates does umax occur.
Think about at what (x, y) coordinates does σmax occur. For this assignment, test four points (x, y) = (0,
h
2
),
(x, y) = ( L
2
, −
h
2
), (x, y) = (0, 0), (x, y) = ( L
4
,
h
4
)
Use constants:
pmin = 0N/m
pmax = −20kN/m
bf = 0.25m
L = 15m
σy = 180MPa
ulimit =
L
250
bmin = 0.01m
bmax = bf
2
hmin = 0.01m
hmax = 0.5m
ρ = 7800kg/m3
E = 200GPa
Substitute constants into the above equations. Note you need a separate sigma function for each of the four
test points above.
Task 9
Lambdify the expressions into Python function definitions, with h and hw as input variables.
Task 10
Use meshgrid to generate a grid of test points for both of the optimization variables h and bw, and generate
values for the mass, the displacement constraint, and the stress constraints.
Task 11
Plot the optimization objective a filled contour plot.
Draw the constraints as not-filled contour plot.
Find the optimal h and bw by looking at this contour plot.