Centroidal moment of inertia: box beam

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William Haynes

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William Haynes 4 years, 7 months ago

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William Haynes 6 years, 4 months ago

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Question statement

{geogebra_applet('pv5edgwm',[["A",A],["B",B]])}

The cross-section of a hollow box beam is shown. Grid units are [{units}].

Give any introductory information the student needs.

			Name	Type	Generated Value

b1	quantity	quantity(14, "cm")
h1	quantity	quantity(12, "cm")
b2	quantity	quantity(10, "cm")
h2	quantity	quantity(6, "cm")
A1	quantity	quantity(168, "cm^2")
A2	quantity	quantity(60, "cm^2")
ybar	quantity	quantity(5.444444444444444444444444444444444444444, "cm")
ybar1	quantity	quantity(6, "cm")
ybar2	quantity	quantity(7, "cm")
Ix1	quantity	quantity(8064, "cm^4")
Ix2	quantity	quantity(3120, "cm^4")
Ix	quantity	quantity(4944, "cm^4")
Area	quantity	quantity(108, "cm^2")
Ibar	quantity	quantity(1742.666666666666666666666666666666666667, "cm^4")

			Name	Type	Generated Value

t	integer	2
A	vector	vector(14,12)
B	vector	vector(2,4)
units	string	cm
debug	boolean	false

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
qty(A[0],units)

JME function reference

Description

base for large rectangle

Describe what this variable represents, and list any assumptions made about its value.

Can an exam override the value of this variable?

Generated value: `quantity`

quantity(14, "cm")

← Depends on:

This variable doesn't seem to be used anywhere.

Parts

Gap-fill
1. Gap Gap 0. Engineering Accuracy with units
  Add
2. Gap Gap 1. Engineering Accuracy with units
  Add

Gap-fill

Ask the student a question, and give any hints about how they should answer this part.

Determine the vertical location of the centroid of the cross section.

$\bar{y} =$ Gap 0. {display(ybar)}

Determine the moment of inertia of the shape about a horizontal ${x'}$ axis (the neutral axis) passing through the centroid.

${I_{x'} = }$ Gap 1. {display(Ibar)}

Advice

Part 1: Big Rectangle:

$b_1$ = {b1}, $h_1$ = {h1}, $A_1$ = {A1}, $\bar{y}_1$ = {ybar1}

Part 2: Small rectangle (removed):

$b_2$ = {b2}, $h_2$ = {h2}, $A_2$ = {A2}, $\bar{y}_2$ = {ybar2}

Total Area:

$A = \Sigma A_i = A_1 - A_2 =$ {display(Area)}

Centroid:

$\bar{y} = \dfrac{\Sigma A_i \bar{y}_i}{\Sigma A_i} = \dfrac{A_1 \bar{y}_1 - A_2 \bar{y}_2}{ A_1 - A_2}$ = {display(ybar)}

Moments of inerta about the x axis:

$I_{x_1} = \dfrac{b_1 (h_1)^3}{3} = $ {display(Ix1)}

$I_{x_2}= \bar{I} + A d^2 = \dfrac{b_2 (h_2)^3}{12} + A_2 (\bar{y}_2)^2 = $ {display(Ix2)}

$I_x = I_{x_1} - I_{x_2}= $ {display(Ix)}

Centroidal Moment of inertia:

Using the parallel axis theorem,

$I = \bar{I} + A d^2 \implies \bar{I}_x= I_x - A (\bar{y})^2$ = {display(Ibar)}

Give a worked solution to the whole question.

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Part	Test	Passed?
Gap-fill
Gap-fill		Hasn't run yet
Engineering Accuracy with units
Engineering Accuracy with units		Hasn't run yet
Engineering Accuracy with units
Engineering Accuracy with units		Hasn't run yet

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Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

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William Haynes 4 years, 7 months ago

William Haynes 4 years, 7 months ago

William Haynes 6 years, 4 months ago

William Haynes 6 years, 4 months ago

William Haynes 6 years, 4 months ago

Error in variable testing condition

Error

Warning

Generated value: `quantity`

Parts

Gap-fill

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Centroidal moment of inertia: box beam

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

William Haynes 4 years, 7 months ago

William Haynes 4 years, 7 months ago

William Haynes 6 years, 4 months ago

William Haynes 6 years, 4 months ago

William Haynes 6 years, 4 months ago

Error in variable testing condition

Error

Warning

Generated value: quantity

Parts

Gap-fill

Generated value: `quantity`