MECE5397代写、Python程序设计代写

日期：2023-03-22 08:56

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.

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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.

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