Moment of inertia: built-up beam with angles

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Name	Status	Author	Last Modified
Moment of inertia: built-up beam with angles	Ready to use	William Haynes	22/03/2021 14:13
Moment of inertia: 2 angles forming box beam	Ready to use	William Haynes	22/03/2021 14:13

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Question statement

{geogebra_applet('vvcydmsb',['A': A,'d': scalar(depth),'C':[ definition: C, fixed: true]])}

A built-up beam is made up of two {name} angles welded to a {b} $\times$ {h} plate, as shown in cross section below.

Use this table of properties for the angle sections.

Give any introductory information the student needs.

			Name	Type	Generated Value

name	string	L9 $\times$ 4 $\times$ 1/2
depth	quantity	quantity(9, "in")
A_L	quantity	quantity(6.25, "in^2")
Ixx	quantity	quantity(53.2, "in^4")
ybar	quantity	quantity(3.31, "in")
Iyy	quantity	quantity(6.92, "in^4")
xbar	quantity	quantity(0.81, "in")

			Name	Type	Generated Value

A_R	quantity	quantity(40.5, "in^2")
A_T	quantity	quantity(53, "in^2")
ybar_R	quantity	quantity(0.75, "in")
ybar_L	quantity	quantity(2.31, "in")
xbar_L	quantity	quantity(10.19, "in")
Qx	quantity	quantity(59.25, "in^3")
ybar_T	quantity	quantity(1.117924528301886792452830188679245283019, "in")

			Name	Type	Generated Value

Ix_R	quantity	quantity(30.375, "in^4")
Ix_L	quantity	quantity(40.270625, "in^4")
Ix_T	quantity	quantity(110.91625, "in^4")
Iy_R	quantity	quantity(2460.375, "in^4")
Iy_L	quantity	quantity(702.175625, "in^4")
Iy_T	quantity	quantity(3864.72625, "in^4")
Ibarx'	quantity	quantity(44.67922169811320754716981132075471698113, "in^4")
kx'	quantity	quantity(0.9181525924285505533616555494537353198838, "in")

			Name	Type	Generated Value

index	integer	0
b	quantity	quantity(27, "in")
h	quantity	quantity(1.5, "in")
A	vector	vector(13.5,1.5)
C	vector	vector(0,1.1179245283)
debug	boolean	false
applet	ggbapplet	HTML node

			Name	Type	Generated Value

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
["L9  $\\times$  4  $\\times$  1/2","L8  $\\times$  6  $\\times$  5/8","L8  $\\times$  6  $\\times$  1/2","L8  $\\times$  4  $\\times$  3/4","L8  $\\times$  4  $\\times$  1/2","L7  $\\times$  4  $\\times$  3/4","L7  $\\times$  4  $\\times$  5/8","L7  $\\times$  4  $\\times$  1/2","L7  $\\times$  4  $\\times$  3/8","L6  $\\times$  4  $\\times$  7/8","L6  $\\times$  4  $\\times$  3/4","L6  $\\times$  4  $\\times$  1/2","L6  $\\times$  4  $\\times$  3/8","L5  $\\times$  3  $\\times$  5/8","L5  $\\times$  3  $\\times$  1/2"][index]

JME function reference

Description

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

Can an exam override the value of this variable?

Generated value: `string`

L9 $\times$ 4 $\times$ 1/2

← Depends on:

index

→ Used by:

Statement

Parts

Gap-fill

Gap-fill

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

Find the area of the composite shape.

$A_T = $ $A$ {A_T}

Find the distance between the $x$ axis at the bottom of the plate and the parallel $x'$ axis passing through the centroid of the composite shape.

$\bar{y} = $ $\bar{y}$ {ybar_T}

Find the moment of inertia of the composite shape about the centroidal $y$ axis.

$\bar{I}_y = $ $\bar{I}_y$ {Iy_T}

Find the moment of inertia of the composite shape about the $x$-axis

$I_x=$ $\bar{I}_x$ {Ix_T}

Use the parallel axis theorem to find the moment of inertia of the composite shape about the centroidal $x'$-axis.

$\bar{I}_{x'}= $ $\bar{I}_{x'}$ {Ibarx'}

Find the corresponding radius of gyration

$k_{x'} = $ $k_{x'}$ {kx'}

Advice

Let the rectangle be designated by subscript $R$, the rectangle by $L$, and the total composite shape by $T$.

Find rectangle properties

$b = \var{b}, h = \var{h}$

$A_R = b \cdot h = \var{A_R}$

$\bar{y}_R = h/2 = \var{ybar_R}$

Find the angle properties

From the table:

$A_L= \var{A_L}$,

$\bar{x}=\var{xbar}, \bar{y} = \var{ybar}$,

$I_{xx} = \var{Ixx}, I_{yy} = \var{Iyy}$

Find the centroid of the angle shapes

Note that the angle shape is rotated 90° from the table diagram.

Measuring from the bottom center of the rectangle:

$\bar{y}_L = h + \bar{x} = \var{ybar_L}$

$\bar{x}_L = \pm (b/2 - \bar{y}) = \pm \var{xbar_L}$

Find the centroid of composite shape

Measuring from the bottom center of the rectangle:

$\begin{align}A_T &= A_R + 2\, A_L \\&= \var{A_R} + 2 \,(\var{A_L})\\ &=\var{A_T}\end{align}\\$

$\bar{x}_T = 0$, by symmetry.

$\\\begin{align} \bar{y}_T &= \dfrac{\Sigma A_i \bar{y}_i}{\Sigma A_i}\\ &=\dfrac{A_R \bar{y}_R + 2\, A_L \bar{y}_L}{A_T}\\&= \dfrac{(\var{A_R}) (\var{ybar_R}) + 2\, (\var{A_L}) (\var{ybar_L})}{\var{A_T} }\\&=\var{disp(ybar_T)} = \bar{y}\end{align}$

Find the moment of inertia about the centroidal y- axis

$\begin{align} \bar{I}_y &= (I_y)_R + 2\, (I_y)_L\\&= \frac{h b^3}{12} + 2\, [\bar{I} + A d^2]_L\\ &=\frac{h b^3}{12} + 2\, [I_{xx} + A_L (\bar{x}_L)^2]\\ &=\frac{(\var{h})(\var{b})^3}{12} + 2\, [\var{Ixx} + (\var{A_L}) (\var{xbar_L})^2]\\&=\var{disp(Iy_R)} + 2\,[\var{disp(Iy_L)}] \\&=\var{disp(Iy_T)}\end{align}$

Find the moment of inertia about the x-axis

$\begin{align} I_x &= (I_x)_R + 2\, (I_x)_L\\&= \frac{b h^3}{3} + 2\, [\bar{I} + A d^2]_L\\ &=\frac{b h^3}{3} + 2\, [I_{yy} + A_L (\bar{y}_L)^2]\\ &=\frac{(\var{b})(\var{h})^3}{3} + 2\, [\var{Iyy} + (\var{A_L}) (\var{ybar_L})^2]\\&= \var{disp(Ix_r)} + 2\, [\var{disp(Ix_L)}]\\&=\var{disp(Ix_T)}\end{align}$

Use the parallel axis theorem to find the moment of inertia about the centroidal x'-axis

$\begin{align} I &= \bar{I} + A d^2 \\ \bar{I}_{x'} &= I_x - A d^2\\ &=I_x - A_T (\bar{y}_T)^2\\ &=\var{disp(Ix_T)} - (\var{A_T}) (\var{disp(ybar_T)})^2\\&=\var{disp(Ibarx')}\end{align}$

Find the radius of gyration with respect to the $x'$- axis.

$\begin{align} k_{x'} &= \sqrt{\dfrac{\bar{I}_{x'}}{A_T}} \\&=\sqrt{\dfrac{\var{disp(Ibarx')}}{\var{A_T}}} \\&=\var{disp(kx')}\end{align}$

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This question is used in the following exams:

Chapter 10 Exercises by Michael Proudman in Engineering Statics.
42. Standard Sections by Michael Proudman in Engineering Statics.
41. Standard Sections by William Haynes in Engineering Statics.
Chapter 10 Exercises by William Haynes in Engineering Statics.

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Moment of inertia: built-up beam with angles

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William Haynes 4 years, 2 months ago

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William Haynes 4 years, 7 months ago

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William Haynes 5 years, 10 months ago

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Moment of inertia: built-up beam with angles

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

William Haynes 4 years, 2 months ago

William Haynes 4 years, 7 months ago

William Haynes 4 years, 7 months ago

William Haynes 5 years, 10 months ago

William Haynes 5 years, 10 months ago

William Haynes 5 years, 10 months ago

Error in variable testing condition

Error

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Generated value: string

Parts

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Generated value: `string`