Moment of inertia of a composite shape: U or T

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William Haynes

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William Haynes 4 years, 7 months ago

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William Haynes 4 years, 7 months ago

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William Haynes 6 years, 5 months ago

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William Haynes 6 years, 5 months ago

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William Haynes 6 years, 5 months ago

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William Haynes 6 years, 5 months ago

Created this as a copy of Centroid of composite area: rectangles (not finished).

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Question statement

A composite shape is drawn to scale below. Grid units are [{units}].

{geogebra_applet('ez5xyd5w',[['A',A],['B',B]])}

Dimensions:

Rectangle 1: $b_1 = \var{b1} \qquad h_1 = \var{h1}$

Rectangle 2: $b_2 = \var{b2}\qquad h_2 = \var{h2}$ {if(B[1]<A[1],'Removed','')}

Area:

$A_1 = b_1 h_1 = \var{A1}$

$A_2 = b_2 h_2 = \var{A2}$ {if(B[1]<A[1],'Removed','')}

$A_t = A_1 + A_2 = \var{A1} + \var{if(B[1]<A[1],'–','')} \var{A2} = \var{A_t}$

For the x-axis:

$I_{1x} = \frac{1}{3}b_1 h_1^3 = \var{siground(I1x,4)}$

$I_{2x} = \bar{I} + A d^2 = \frac{1}{12} (\var{B2}) (\var{H2})^3 +(\var{A2}) (\var{dx})^2 = \var{siground(I2x,4)}$ {if(B[1]<A[1],'Removed','')}

$I_x = I_{1x} + I_{2x} = \var{siground(I1x,4)} + \var{if(B[1]<A[1],'–','')} \var{siground(I2x,4)} = \var{siground(I_tx,4)}$

$k_x = \sqrt{\frac{I_x}{A}} = \var{siground(kx,4)}$

For the centroidal y'-axis:

$I_{1y'} = \frac{1}{12}h_1 b_1^3 = \var{siground(I1y,4)}$

$I_{2y'} = \frac{1}{12}h_2 b_2^3 = \var{siground(I2y,4)}$ {if(B[1]<A[1],'Removed','')}

$\bar{I}_{y'} = I_{1y'} + I_{2y'} = \var{siground(I1y,4)} + \var{if(B[1]<A[1],'–','')} \var{siground(I2y,4)} = \var{siground(Ibar_y',4)}$

For the y-axis, using the parallel axis theorem:

$I_y = \bar{I}_{y'}+ A d^2 = \var{siground(Ibar_y',4)} + (\var{A_t}) (\var{dy})^2 = \var{siground(I_ty,4)}$

$k_y = \sqrt{\frac{I_y}{A}} = \var{siground(ky,4)}$

Give any introductory information the student needs.

			Name	Type	Generated Value

A	vector	vector(1.5,3)
B	vector	vector(4.5,5)
units	string	in
debug	boolean	false
axis	string	y

			Name	Type	Generated Value

b1	quantity	quantity(3, "in")
h1	quantity	quantity(3, "in")
b2	quantity	quantity(9, "in")
h2	quantity	quantity(2, "in")
I1x	quantity	quantity(27, "in^4")
I1y	quantity	quantity(6.75, "in^4")
I2x	quantity	quantity(294, "in^4")
I2y	quantity	quantity(121.5, "in^4")
I_tx	quantity	quantity(321, "in^4")
I_ty	quantity	quantity(675, "in^4")
dx	quantity	quantity(4, "in")
dy	quantity	quantity(4.5, "in")
A1	quantity	quantity(9, "in^2")
A2	quantity	quantity(18, "in^2")
A_t	quantity	quantity(27, "in^2")
kx	quantity	quantity(3.448026810929533489912866480875665789434, "in")
ky	quantity	quantity(5, "in")
Ibar_y'	quantity	quantity(128.25, "in^4")

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
vector(random(0.5..6#0.5),random(1..9))

JME function reference

Description

This point is at the half width of rectangle 1

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

Can an exam override the value of this variable?

Generated value: `vector`

vector(1.5,3)

→ Used by:

A_t
I_tx
I_ty
Ibar_y'
b1
dx
dy
h1
h2

Statement
Advice
Variable testing condition

Parts

Area Gap-fill
1. Gap A Engineering Accuracy with units
  Add
Moment of Inertia Gap-fill
1. Gap $I_\var{axis} $ Engineering Accuracy with units
  Add
Radius of gyration Gap-fill
1. Gap $k_\var{axis}$ 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 total area of the shape.

$A$ = A {A_t}

Advice

Dimensions:

$b_1 = \var{b1} \qquad h_1 = \var{h1}\\b_2 = \var{b2}\qquad h_2 = \var{h2}$

Area:

$A_1 = b_1 h_1 = \var{A1}$

$A_2 = b_2 h_2 = \var{A2}$ {if(B[1]<A[1],'Removed','')}

$A_t = A_1 + A_2 = \var{A1} + \var{if(B[1]<A[1],'–','')} \var{A2} = \var{A_t}$

For the x-axis:

$I_{1x} = \frac{1}{3}b_1 h_1^3 = \var{siground(I1x,4)}$

$I_{2x} = \bar{I} + A d^2 = \frac{1}{12} (\var{B2}) (\var{H2})^3 +(\var{A2}) (\var{dx})^2 = \var{siground(I2x,4)} $ {if(B[1]<A[1],'Removed','')}

$I_x = I_{1x} + I_{2x} = \var{siground(I1x,4)} + \var{if(B[1]<A[1],'–','')} \var{siground(I2x,4)} = \var{siground(I_tx,4)}$

$k_x = \sqrt{\frac{I_x}{A}} = \var{siground(kx,4)}$

For the centroidal y'-axis:

$I_{1y'} = \frac{1}{12}h_1 b_1^3 = \var{siground(I1y,4)}$

$I_{2y'} = \frac{1}{12}h_2 b_2^3 = \var{siground(I2y,4)} $ {if(B[1]<A[1],'Removed','')}

$\bar{I}_{y'} = I_{1y'} + I_{2y'} = \var{siground(I1y,4)} + \var{if(B[1]<A[1],'–','')} \var{siground(I2y,4)} = \var{siground(Ibar_y',4)}$

For the y-axis, using the parallel axis theorem:

$I_y = \bar{I}_{y'}+ A d^2 = \var{siground(Ibar_y',4)} + (\var{A_t}) (\var{dy})^2 = \var{siground(I_ty,4)}$

$k_y = \sqrt{\frac{I_y}{A}} = \var{siground(ky,4)}$

Give a worked solution to the whole question.

Use this tab to check that this question works as expected.

There was an error which means the tests can't run:

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

This question is used in the following exams:

Chapter 10 Exercises by Michael Proudman in Engineering Statics.
24. Parallel Axis Theorem by Michael Proudman in Engineering Statics.
Chapter 10 Exercises by William Haynes in Engineering Statics.
24. Parallel Axis Theorem by William Haynes in Engineering Statics.

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Moment of inertia of a composite shape: U or T

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Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

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Checkpoint description

William Haynes 4 years, 7 months ago

William Haynes 4 years, 7 months ago

William Haynes 6 years, 5 months ago

William Haynes 6 years, 5 months ago

William Haynes 6 years, 5 months ago

William Haynes 6 years, 5 months ago

Error in variable testing condition

Error

Warning

Generated value: `vector`

Parts

Gap-fill

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Error

Moment of inertia of a composite shape: U or T

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, 5 months ago

William Haynes 6 years, 5 months ago

William Haynes 6 years, 5 months ago

William Haynes 6 years, 5 months ago

Error in variable testing condition

Error

Warning

Generated value: vector

Parts

Gap-fill

Generated value: `vector`