// Numbas version: finer_feedback_settings {"name": "41. Standard Sections", "metadata": {"description": "

Homework set. Calculate Moment of Inertia for I beams, channels, angles, etc.

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A value with units marked right if within an adjustable % error of the correct value. Marked close if within a wider margin of error.

", "help_url": "", "input_widget": "string", "input_options": {"correctAnswer": "siground(settings['correctAnswer'],4)", "hint": {"static": true, "value": ""}, "allowEmpty": {"static": true, "value": true}}, "can_be_gap": true, "can_be_step": true, "marking_script": "mark:\nswitch( \n right and good_units and right_sign, add_credit(1.0,'Correct.'),\n right and good_units and not right_sign, add_credit(settings['C2'],'Wrong sign.'),\n right and right_sign and not good_units, add_credit(settings['C2'],'Correct value, but wrong or missing units.'),\n close and good_units, add_credit(settings['C1'],'Close.'),\n close and not good_units, add_credit(settings['C3'],'Answer is close, but wrong or missing units.'),\n incorrect('Wrong answer.')\n)\n\n\ninterpreted_answer:\nqty(student_scalar, student_units)\n\n\n\ncorrect_quantity:\nsettings[\"correctAnswer\"]\n\n\n\ncorrect_units:\nunits(correct_quantity)\n\n\nallowed_notation_styles:\n[\"plain\",\"en\"]\n\nmatch_student_number:\nmatchnumber(studentAnswer,allowed_notation_styles)\n\nstudent_scalar:\nmatch_student_number[1]\n\nstudent_units:\nreplace_regex('ohms','ohm',\n replace_regex('\u00b0', ' deg',\n replace_regex('-', ' ' ,\n studentAnswer[len(match_student_number[0])..len(studentAnswer)])),\"i\")\n\ngood_units:\ntry(\ncompatible(quantity(1, student_units),correct_units),\nmsg,\nfeedback(msg);false)\n\n\nstudent_quantity:\nswitch(not good_units, \n student_scalar * correct_units, \n not right_sign,\n -quantity(student_scalar, student_units),\n quantity(student_scalar,student_units)\n)\n \n\n\npercent_error:\ntry(\nscalar(abs((correct_quantity - student_quantity)/correct_quantity))*100 \n,msg,\nif(student_quantity=correct_quantity,0,100))\n \n\nright:\npercent_error <= settings['right']\n\n\nclose:\nright_sign and percent_error <= settings['close']\n\nright_sign:\nsign(student_scalar) = sign(correct_quantity)", "marking_notes": [{"name": "mark", "description": "This is the main marking note. It should award credit and provide feedback based on the student's answer.", "definition": "switch( \n right and good_units and right_sign, add_credit(1.0,'Correct.'),\n right and good_units and not right_sign, add_credit(settings['C2'],'Wrong sign.'),\n right and right_sign and not good_units, add_credit(settings['C2'],'Correct value, but wrong or missing units.'),\n close and good_units, add_credit(settings['C1'],'Close.'),\n close and not good_units, add_credit(settings['C3'],'Answer is close, but wrong or missing units.'),\n incorrect('Wrong answer.')\n)\n"}, {"name": "interpreted_answer", "description": "A value representing the student's answer to this part.", "definition": "qty(student_scalar, student_units)\n\n"}, {"name": "correct_quantity", "description": "", "definition": "settings[\"correctAnswer\"]\n\n"}, {"name": "correct_units", "description": "", "definition": "units(correct_quantity)\n"}, {"name": "allowed_notation_styles", "description": "", "definition": "[\"plain\",\"en\"]"}, {"name": "match_student_number", "description": "", "definition": "matchnumber(studentAnswer,allowed_notation_styles)"}, {"name": "student_scalar", "description": "", "definition": "match_student_number[1]"}, {"name": "student_units", "description": "

Modify the unit portion of the student's answer by

1. replacing \"ohms\" with \"ohm\" case insensitive

2. replacing '-' with ' '

3. replacing '°' with ' deg'

to allow answers like 10 ft-lb and 30°

", "definition": "replace_regex('ohms','ohm',\n replace_regex('\u00b0', ' deg',\n replace_regex('-', ' ' ,\n studentAnswer[len(match_student_number[0])..len(studentAnswer)])),\"i\")"}, {"name": "good_units", "description": "", "definition": "try(\ncompatible(quantity(1, student_units),correct_units),\nmsg,\nfeedback(msg);false)\n"}, {"name": "student_quantity", "description": "

This fixes the student answer for two common errors.

If student_units are wrong - replace with correct units

If student_scalar has the wrong sign - replace with right sign

If student makes both errors, only one gets fixed.

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Only marked close if the student actually has the right sign.

", "definition": "right_sign and percent_error <= settings['close']"}, {"name": "right_sign", "description": "", "definition": "sign(student_scalar) = sign(correct_quantity) "}], "settings": [{"name": "correctAnswer", "label": "Correct Quantity.", "help_url": "", "hint": "The correct answer given as a JME quantity.", "input_type": "code", "default_value": "", "evaluate": true}, {"name": "right", "label": "% Accuracy for right.", "help_url": "", "hint": "Question will be considered correct if the scalar part of the student's answer is within this % of correct value.", "input_type": "code", "default_value": "0.2", "evaluate": true}, {"name": "close", "label": "% Accuracy for close.", "help_url": "", "hint": "Question will be considered close if the scalar part of the student's answer is within this % of correct value.", "input_type": "code", "default_value": "1.0", "evaluate": true}, {"name": "C1", "label": "Close with units.", "help_url": "", "hint": "Partial Credit for close value with appropriate units. if correct answer is 100 N and close is ±1%,
99 N is accepted.", "input_type": "percent", "default_value": "75"}, {"name": "C2", "label": "No units or wrong sign", "help_url": "", "hint": "Partial credit for forgetting units or using wrong sign.
If the correct answer is 100 N, both 100 and -100 N are accepted.", "input_type": "percent", "default_value": "50"}, {"name": "C3", "label": "Close, no units.", "help_url": "", "hint": "Partial Credit for close value but forgotten units.
This value would be close if the expected units were provided. If the correct answer is 100 N, and close is ±1%,
99 is accepted.", "input_type": "percent", "default_value": "25"}], "public_availability": "always", "published": true, "extensions": ["quantities"]}], "resources": [["question-resources/channel_EfJjxiR.png", "/srv/numbas/media/question-resources/channel_EfJjxiR.png"], ["question-resources/builtupbeam.ggb", "/srv/numbas/media/question-resources/builtupbeam.ggb"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "William Haynes", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2530/"}], "tags": ["built-up beam", "Mechanics", "mechanics", "moment of inertia", "radius of gyration", "standard sections", "Statics", "statics"], "metadata": {"description": "

Find moment of inertia and radius of gyration for a built-up beam made of two channels and two plates.

", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "

A built-up beam is constructed by welding two channels separated by a distance $b = \\var{b}$ to two ({w} $\\times$ {h}) plates.

\n\n\n\n\n\n\n\n

{geogebra_applet('rgft4vzy',[\"b\":ggb(b),'bf': ggb(bf),'w': ggb(w) ,'d': ggb(d),'t': ggb(t),'h': ggb(h)])}

Channel properties:

$\\begin{align}\\var{desc}\\\\\\text{Area}&= \\var{A_C} \\\\D &= \\var{d} \\\\b_f &= \\var{bf}\\\\\\bar{x} &= \\var{xbar}\\\\\\bar{I}_{xx} &= \\var{Ibarx_C} \\\\\\bar{I}_{yy} &= \\var{Ibary_C}\\end{align}$

", "advice": "

Determine moment of inertia with respect to the x-axis

For the channel, since the x-axis passes through its centroid:

$(I_x)_C = \\bar{I}_{xx} = \\var{Ibarx_C} $

For the plate, the x-axis does not pass through its centroid so the parallel axis theorem must be used.

$(I_x)_R = [\\bar{I} + A d^2] $ where,

$\\bar{I} = \\dfrac{b h^3}{12} = \\dfrac{(\\var{w})(\\var{h})^3}{12} = \\var{siground(Ibarx_R,4)}$

$A = b h = (\\var{w})\\,(\\var{h}) = \\var{A_r}$

$d = D/2 + h/2 = \\dfrac{\\var{d}+ \\var{h}}{2} = \\var{d_y} $

$(I_x)_R = \\var{disp(Ix_R)}$

For the composite shape,

$A = 2 [A_C + A_R] = \\var{disp(A_T)}$

$I_x = 2 [ (I_x)_C + (I_x)_R ] = \\var{disp(Ix)}$, and $k_x = \\sqrt{\\dfrac{I_x}{A}} = \\var{disp(kx)}$

Determine moment of inertia with respect to the y-axis

For the rectangle, since the y-axis passes through its centroid:

$(I_y)_R = \\dfrac{h b^3}{12} = \\dfrac{(\\var{h}) (\\var{w})^3}{12} = \\var{disp(Ibary_r)}$

For the channel, the y-axis does not pass through its centroid so the parallel axis theorem must be used.

$(I_y)_C = [\\bar{I} + A d^2] $ where,

$\\bar{I} = \\bar{I}_{yy} = \\var{Ibary_C}$

$A = A_C = \\var{A_C}$

$d = b/2 + \\bar{x} = \\dfrac{\\var{b}}{2} + \\var{xbar}= \\var{d_x} $

$(I_y)_C = \\var{disp(Iy_C)}$

For the composite shape,

$A = 2 [A_C + A_R] = \\var{disp(A_T)}$

$I_y = 2 [ (I_y)_C + (I_y)_R ] = \\var{disp(Iy)}$, and $k_y = \\sqrt{\\dfrac{I_y}{A}} = \\var{disp(ky)}$

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horizontal distance to centroid from flange

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random channel index

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hash of data for selected channel

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area of channel

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0 = inches, 1 = mm

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flange width

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width of plate a little longer than necessary

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depth of channel

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centroidal momentof inertia

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designation

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width of plate

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AISC data for channels

", "templateType": "json"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "round( random(0.4.. 1.2) d, if(unit=0,'0.25 in', '10 mm'))", "description": "

distance between channels

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Determine the moment of inertia and radii of gyration of the composite beam with respect to the $x$- and $y$-axes.

$I_x = $[[0]] $\\qquad k_x = $[[1]]

$I_y = $[[2]] $\\qquad k_y = $[[3]]

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A value with units marked right if within an adjustable % error of the correct value. Marked close if within a wider margin of error.

Modify the unit portion of the student's answer by

1. replacing \"ohms\" with \"ohm\" case insensitive

2. replacing '-' with ' '

3. replacing '°' with ' deg'

to allow answers like 10 ft-lb and 30°

This fixes the student answer for two common errors.

If student_units are wrong - replace with correct units

If student_scalar has the wrong sign - replace with right sign

If student makes both errors, only one gets fixed.

Only marked close if the student actually has the right sign.

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99 is accepted.", "input_type": "percent", "default_value": "25"}], "public_availability": "always", "published": true, "extensions": ["quantities"]}], "resources": [["question-resources/2boxbeams_aMoCPFM.ggb", "/srv/numbas/media/question-resources/2boxbeams_aMoCPFM.ggb"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "William Haynes", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2530/"}], "tags": ["Mechanics", "mechanics", "moment of inertia", "radius of gyration", "standard sections", "Statics", "statics"], "metadata": {"description": "

Find centroidal moments of inertia and radius of gyration for a beam composed of two angle sections forming a box beam.

", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "

{geogebra_applet('u9qngrny',[['h',h],['w',w],['t',t],['P',centroid]])}

A built-up beam is made up of two {name} angles welded together to form a box beam.

Use this table of properties for the angle sections.

", "advice": "

Channel properties from table

$\\begin{align}\\text{Area:}& &A&= \\var{A_L}\\\\\\text{Centroid:} & &\\bar{x} &= \\var{xbar} & \\bar{y} &= \\var{ybar}\\\\\\text{MOI:} && \\bar{I}_{xx} &= \\var{ixx} & \\bar{I}_{yy} &= \\var{Iyy}\\end{align}$

These properties are for angle sections oriented with the long side vertical.

Find moment of inertia

$\\bar{I}_y = 2 \\, [ \\bar{I} + A d^2]$ where,

$\\bar{I} = \\var{latex(if(rotated,safe('\\\\bar{I}_{xx}'),safe('\\\\bar{I}_{yy}')))} =\\var{Ibar_y'}$

$A =\\var{A_L}$, and

$d = \\left(\\dfrac{w+t}{2}-\\var{latex(if(rotated,safe('\\\\bar{y}'),safe('\\\\bar{x}')))} \\right)= \\var{d_x}$, so

$\\bar{I}_y = \\var{disp(Iy)}\\\\$

$\\bar{I}_x = 2 \\, [ \\bar{I} + A d^2]$ where,

$\\bar{I} = \\var{latex(if(rotated,safe('\\\\bar{I}_{yy}'),safe('\\\\bar{I}_{xx}')))} =\\var{Ibar_x'}$

$A =\\var{A_L}$, and

$d =\\left( \\dfrac{h}{2}-\\var{latex(if(rotated,safe('\\\\bar{x}'),safe('\\\\bar{y}')))}\\right) = \\var{d_y}$, so

$\\bar{I}_x = \\var{disp(Ix)}\\\\$

Find Radii of gyration

$k_x = \\sqrt{\\dfrac{\\bar{I}_x} {2 A}} = \\var{disp(kx)} \\qquad k_y = \\sqrt{\\dfrac{\\bar{I}_y} {2 A}} = \\var{disp(ky)}$

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true indicates long side is horizontal

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", "templateType": "anything"}, "ybar": {"name": "ybar", "group": "angle properties", "definition": "qty([3.31,2.52,2.47,2.95,2.86,2.51,2.46,2.42,2.37,2.12,2.08,1.99,1.94,1.8,1.75][index],'in')", "description": "", "templateType": "anything"}, "ky": {"name": "ky", "group": "MOI", "definition": "qty(sqrt(scalar(Iy)/(2 scalar(A_L))),'in')", "description": "", "templateType": "anything"}, "A_L": {"name": "A_L", "group": "angle properties", "definition": "qty([6.25,8.36,6.75,8.44,5.75,7.69,6.48,5.25,3.98,7.98,6.94,4.75,3.61,4.61,3.75][index],'in^2')", "description": "", "templateType": "anything"}, "xbar": {"name": "xbar", "group": "angle properties", "definition": "qty([0.81,1.52,1.47,0.953,0.859,1.01,0.963,0.917,0.87,1.12,1.08,0.987,0.941,0.796,0.75][index],'in')", "description": "", "templateType": "anything"}, "debug": {"name": "debug", "group": "Setup values", "definition": "false", "description": "", "templateType": "anything"}, "Ixx": {"name": "Ixx", "group": "angle properties", "definition": "qty([53.2,54.1,44.3,54.9,38.5,37.8,32.4,26.7,20.6,27.7,24.5,17.4,13.5,11.4,9.45][index],'in^4')", "description": "", "templateType": "anything"}, "Ix": {"name": "Ix", "group": "MOI", "definition": "2 (Ibar_x'+A_L d_y d_y)", "description": "

MOI of box around x axis

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MOI of box around y axis

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horizontal distance from centroid of angle to y axis

", "templateType": "anything"}, "Ibar_x'": {"name": "Ibar_x'", "group": "MOI", "definition": "if(rotated,Iyy,Ixx)", "description": "

centroidal MOI for angle about its centroid

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centroidal MOI for angle about its centroid

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", "templateType": "anything"}, "d_y": {"name": "d_y", "group": "MOI", "definition": "qty(h/2,'in')- if(rotated,xbar,ybar)", "description": "

vertical distance from centroid of angle to x axis

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Find the moment of inertia and radius of gyration with respect to the x axis.

$\\bar{I}_x = $[[0]] $\\quad k_x = $[[1]]

Find the moment of inertial and radius of gyration with respect to the y axis.

$\\bar{I}_y = $[[2]] $\\quad k_y = $[[3]]

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A value with units marked right if within an adjustable % error of the correct value. Marked close if within a wider margin of error.

Modify the unit portion of the student's answer by

1. replacing \"ohms\" with \"ohm\" case insensitive

2. replacing '-' with ' '

3. replacing '°' with ' deg'

to allow answers like 10 ft-lb and 30°

This fixes the student answer for two common errors.

If student_units are wrong - replace with correct units

If student_scalar has the wrong sign - replace with right sign

If student makes both errors, only one gets fixed.

Only marked close if the student actually has the right sign.

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99 N is accepted.", "input_type": "percent", "default_value": "75"}, {"name": "C2", "label": "No units or wrong sign", "help_url": "", "hint": "Partial credit for forgetting units or using wrong sign.
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This value would be close if the expected units were provided. If the correct answer is 100 N, and close is ±1%,
99 is accepted.", "input_type": "percent", "default_value": "25"}], "public_availability": "always", "published": true, "extensions": ["quantities"]}], "resources": [["question-resources/plate-angles.ggb", "/srv/numbas/media/question-resources/plate-angles.ggb"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "William Haynes", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2530/"}], "tags": ["beam", "Mechanics", "mechanics", "moment of inertia", "neutral axis", "parallel axis theorem", "standard sections", "Statics", "statics"], "metadata": {"description": "

Find the centroid and the centroidal moments of inertia for a beam composed of a flat plate and two angle sections.

", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "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.

", "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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coordinates of top right corner of rectangle

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height of rectangle

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centroid of composite shape as a point for geogebra

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base width of rectangle

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Find the area of the composite shape.

$A_T = $ [[0]] {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} = $ [[1]] {ybar_T}

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

$\\bar{I}_y = $ [[2]] {Iy_T}

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

$I_x=$ [[3]] {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'}= $ [[4]] {Ibarx'}

Find the corresponding radius of gyration

$k_{x'} = $ [[5]] {kx'}

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A value with units marked right if within an adjustable % error of the correct value. Marked close if within a wider margin of error.

Modify the unit portion of the student's answer by

1. replacing \"ohms\" with \"ohm\" case insensitive

2. replacing '-' with ' '

3. replacing '°' with ' deg'

to allow answers like 10 ft-lb and 30°

This fixes the student answer for two common errors.

If student_units are wrong - replace with correct units

If student_scalar has the wrong sign - replace with right sign

If student makes both errors, only one gets fixed.

Only marked close if the student actually has the right sign.

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