// Numbas version: finer_feedback_settings {"name": "37. Relationships between Shear and Bending Moment", "metadata": {"description": "

Homeworks 36, 37, 38 and 39 don't resume properly in Numbas LTI, so broken into individual problems.

", "licence": "Creative Commons Attribution-ShareAlike 4.0 International"}, "duration": 0, "percentPass": 0, "showQuestionGroupNames": false, "shuffleQuestionGroups": false, "showstudentname": true, "question_groups": [{"name": "Group", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", ""], "variable_overrides": [[], []], "questions": [{"name": "VM7: Symmetric parabolic loading", "extensions": ["jsxgraph", "polynomials", "quantities"], "custom_part_types": [{"source": {"pk": 19, "author": {"name": "William Haynes", "pk": 2530}, "edit_page": "/part_type/19/edit"}, "name": "Engineering Accuracy with units", "short_name": "engineering-answer", "description": "

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\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)"}, {"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.

", "definition": "switch(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"}, {"name": "percent_error", "description": "", "definition": "try(\nscalar(abs((correct_quantity - student_quantity)/correct_quantity))*100 \n,msg,\nif(student_quantity=correct_quantity,0,100))\n "}, {"name": "right", "description": "", "definition": "percent_error <= settings['right']\n"}, {"name": "close", "description": "

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/newVM_cA1Sig0.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": ["gets answer from ggb", "Mechanics", "mechanics", "shear and bending moment", "Statics", "statics"], "metadata": {"description": "

Students must derive the shear and bending moment functions for beam loaded with a parabolic distributed load described with a loading function. Diagrams are given, but the students must derive the equations by integration.

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

A {qty(2b, units[0])} long, simply-supported beam is subjected to a symmetrical parabolic loading, where

\\[w(x) =\\simplify[fractionNumbers]{{w}} \\quad [\\var{units_string(qty(1,units[3]))}]\\].

$V(x) =\\simplify[fractionNumbers]{{v}}$ $[\\var{units_string(qty(1,units[1]))}]$.

$M(x) =\\simplify[fractionNumbers]{{m}}$ $[\\var{units_string(qty(1,units[2]))}]$.

{loadBoard}

{shearBoard}

{momentBoard}

", "advice": "

Given: The beam is $\\var{qty(L, units[0])}$ long. The loading function is \\[w(x) = \\simplify[fractionNumbers]{{w}}.\\]

The area under the loading function is the downward force on the beam. Here, because positive values of $w$ point down, we replace $w(x)$ with $-w(x)$ to agree with the standard convention that negative forces point down.

1. The total load acting on the beam is the integral loading function from $0$ to $L$.

\\[\\begin{align}W &= -\\int_0^L w(x)\\; dx \\\\
&= \\int_0^L \\simplify[fractionNumbers]{{-w}}\\; dx\\\\
&= \\left. \\simplify[fractionNumbers]{{polynomial(x,[ 0,0,-h/b, h/(b^2)/3])}}\\;\\right |_0^L \\\\
&= \\var{force(load_integral)}
\\end{align}\\]

By symmetry, the reactions at $x=0$ and $x= L$ are each half this load, pointing up, so:

\\[\\begin{align}A &= B=W/2 \\\\
&=\\var{-force(load_integral/2)}\\end{align}\\]

2. The shear function $V(x)$ is the integral of the loading function $w(x)$.

\\[\\begin{align} V(x) &= -\\int w(x)\\; dx \\\\
&= \\int \\simplify[fractionNumbers]{{-w}}\\;dx\\\\
&= \\simplify[fractionNumbers]{{polynomial(x,[ 0,0, -h/b, h/(b^2)/3])}} + C_0 \\end{align}\\]

The constant of integration can be found by evaluating the shear function at $x=0$, where $V(0) = A$ to get

\\[V(0) = C_0 = \\var{force(-load_integral/2)} \\]

5. The moment function is the integral of the shear function

\\[\\begin{align} M(x) &= \\int V(x)\\; dx \\\\
&= \\int \\simplify[fractionNumbers]{{polynomial(x,[ 0,0, -h/b, h/(b^2)/3])}} + C_0 \\;dx\\\\
& = \\simplify[fractionNumbers]{{m}} + C_1\\\\
\\end{align}\\]

Since the moment $M(0) = 0$, $C_1 = 0$

6. The Maximum shear occurs at the ends, where $dV /dx = -w(x) = 0$

\\[\\begin{align}V_{max} &= V(0) = V(L)\\\\
&= \\var{force(maximums[1])}\\end{align}
\\]

7. The Maximum moment occurs at midpoint, $x = \\var{qty(L/2,units[0])}$, where $dM/dx = 0$

\\[\\begin{align}M_{max} &= M(\\var{b})\\\\
&= \\var{moment(maximums[2])}\\end{align}\\]

", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"units": {"name": "units", "group": "Unnamed group", "definition": "random(['ft','lb', 'ft*lb', 'lb/ft'],['m','kN', 'kN*m', 'kN/m'])", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "0.5 L", "description": "

half width of beam \"base\"

", "templateType": "anything", "can_override": false}, "h": {"name": "h", "group": "Ungrouped variables", "definition": "random([1,2,2.5,5,7.5,10])random([10,100])", "description": "

distributed load at the center of the beam. The \"height\" of the parabola

", "templateType": "anything", "can_override": false}, "w": {"name": "w", "group": "Ungrouped variables", "definition": "polynomial(x,[0,2 h/b,-h/(b^2)])", "description": "

loading function w(x)

", "templateType": "anything", "can_override": false}, "F_A": {"name": "F_A", "group": "Unnamed group", "definition": "rational(2/3 b h)", "description": "

Reaction at A = half the downward force ( 2/3 bh)

Reaction at B is the same.

", "templateType": "anything", "can_override": false}, "v": {"name": "v", "group": "Ungrouped variables", "definition": "polynomial(x,[ F_A,0, -h/b, h/(b^2)/3])", "description": "

the shear function. result of integrating -w(x)

", "templateType": "anything", "can_override": false}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "polynomial(x,[0, F_A,0, -h/b/3, h/(b^2)/3/4])", "description": "

The moment function m(x), result of integrating v(x)

", "templateType": "anything", "can_override": false}, "w_t": {"name": "w_t", "group": "Unnamed group", "definition": "2 F_A", "description": "

Total Weight supported by the beam

", "templateType": "anything", "can_override": false}, "debug": {"name": "debug", "group": "Ungrouped variables", "definition": "false", "description": "", "templateType": "anything", "can_override": false}, "loadBoard": {"name": "loadBoard", "group": "Ungrouped variables", "definition": "let(max, maximums[0],\n jsxgraph(\n 480,200,[-2, 1.2 max, 2 b+2, - max/2],\n merge(\n [ \"$w(x)$\": ['functiongraph', [load_function, 0, L], [name: \"a\", withLabel: true, strokeWidth: 3, strokeColor: \"#CC0000\", fillColor: \"#CC0000\", \n fillOpacity: 0.1,label:[fixed: false, fontSize: 14]]]],\narrows,\n [\"beam\": ['segment', [[0,0],[L,0]], [strokeWidth: 5, color: \"black\"]] ],\n [ \"$A$\": ['segment', [[0,-max 0.4], [0,0]], merge(reactionoptions, [label: [fixed: false, color: \"#CC0000\", fontSize: 16]])]],\n [ \"$B$\": ['segment', [[L,-max 0.4],[L,0],], merge(reactionoptions, [label: [fixed: false, color: \"#CC0000\", fontSize: 16]])]]\n )\n)\n)", "description": "", "templateType": "anything", "can_override": false}, "load_function": {"name": "load_function", "group": "Ungrouped variables", "definition": "expr(polynomials[0])", "description": "", "templateType": "anything", "can_override": false}, "maximums": {"name": "maximums", "group": "Ungrouped variables", "definition": "let(\n w, polynomials[0],\n v, polynomials[1],\n m, polynomials[2],\n [eval(w,b),eval(v,0),eval(m,b)])", "description": "

load function at peak

", "templateType": "anything", "can_override": false}, "L": {"name": "L", "group": "Ungrouped variables", "definition": "20", "description": "

beamLength

", "templateType": "anything", "can_override": false}, "shear_function": {"name": "shear_function", "group": "Ungrouped variables", "definition": "expr(polynomials[1])", "description": "", "templateType": "anything", "can_override": false}, "moment_function": {"name": "moment_function", "group": "Ungrouped variables", "definition": "expr(polynomials[2])", "description": "", "templateType": "anything", "can_override": false}, "polynomials": {"name": "polynomials", "group": "Ungrouped variables", "definition": "[\npolynomial(x,[0,2 h/b,-h/(b^2)]),\npolynomial(x,[ F_A,0, -h/b, h/(b^2)/3]),\npolynomial(x,[0, F_A,0, -h/b/3, h/(b^2)/3/4])\n]", "description": "", "templateType": "anything", "can_override": false}, "shearBoard": {"name": "shearBoard", "group": "Ungrouped variables", "definition": "let(max, 1.2*maximums[1],\n jsxgraph(\n 480,200,[-2, 1.1 max, 2 b+2, -max],\n [ \n \"$V(x)$\": ['functiongraph', [shear_function, 0, L], [withLabel: true, strokeWidth: 3, strokeColor: \"#CC0000\" ,label:[fixed: false, fontSize: 14]]],\n\n ]\n))", "description": "", "templateType": "anything", "can_override": false}, "momentBoard": {"name": "momentBoard", "group": "Ungrouped variables", "definition": " let(max, maximums[2], color, \"#3070AD\", \n jsxgraph(\n 480,200,[-2, 1.2 max, 2 b+2, -max/3],\n [\"$M(x)$\": ['functiongraph', [moment_function, 0, L], [withLabel: true, strokeWidth: 3, strokeColor: color, label:[fixed: false, fontSize: 14]]],\n ],\n ))", "description": "", "templateType": "anything", "can_override": false}, "shearload": {"name": "shearload", "group": "Ungrouped variables", "definition": "eval(-w,L)", "description": "", "templateType": "anything", "can_override": false}, "force": {"name": "force", "group": "Ungrouped variables", "definition": "f -> siground(qty(f,units[1]),4)", "description": "", "templateType": "anything", "can_override": false}, "moment": {"name": "moment", "group": "Ungrouped variables", "definition": "m -> siground(qty(m,units[2]),4)", "description": "", "templateType": "anything", "can_override": false}, "load_integral": {"name": "load_integral", "group": "Ungrouped variables", "definition": "eval(polynomial(x,[0,0, -h/b, h/(b^2)/3]),L)", "description": "", "templateType": "anything", "can_override": false}, "Arrow": {"name": "Arrow", "group": "Ungrouped variables", "definition": "let( w , polynomials[0],\n (x) -> ['segment', [[x, eval(w,x)],[x,0]],[ fixed: true, color: \"black\",strokewidth: 1.5,highlight: false, lastArrow: [ type: 4 ]]])", "description": "", "templateType": "anything", "can_override": false}, "reactionOptions": {"name": "reactionOptions", "group": "Ungrouped variables", "definition": "[withLabel: true, \"fixed\": true, \"color\": \"#CC0000\", \"strokewidth\": 4, \"highlight\": false, \"lastArrow\": [ \"type\": 4 ] ]", "description": "", "templateType": "anything", "can_override": false}, "Arrows": {"name": "Arrows", "group": "Ungrouped variables", "definition": "dict(map((i)->([(\"A\"+i),Arrow(i)]), 1..L))", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["debug", "L", "b", "h", "w", "v", "m", "maximums", "load_function", "shear_function", "moment_function", "polynomials", "loadBoard", "shearBoard", "momentBoard", "shearload", "force", "moment", "load_integral", "Arrow", "reactionOptions", "Arrows"], "variable_groups": [{"name": "Unnamed group", "variables": ["w_t", "F_A", "units"]}], "functions": {"show_curvesxxx": {"parameters": [["app", "?"]], "type": "anything", "language": "javascript", "definition": "app.promise.then(function(d) {\n d.app.setVisible(\"showV\", true,false);\n d.app.setVisible(\"showM\", true,false);\n d.app.setValue(\"showV\",true);\n d.app.setValue(\"showM\",true);\n});\nreturn \"\"//new Numbas.jme.types.ggbapplet(app);\n"}}, "preamble": {"js": "\n", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": true, "customName": "Load and Reactions", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Determine the total weight supported by the beam and the reactions at $A$ and $B$.

$W = $ [[0]]

$R_A =$ [[1]]

$R_B = $ [[2]]

$R_A = R_B =\\simplify[fractionNumbers]{{F_A}}$ = {force(F_A)}

", "stepsPenalty": "10", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

The total weight is found by integrating the loading function $w(x)$. Positive values indicate a downward load.

$\\begin{align}W &= \\simplify{defint(w,x,0,L)}\\\\ &= \\simplify[fractionNumbers,canonicalOrder]{defint({w},x,0,{2b})}\\end{align}$

By symmetry, the reactions are half the weight:

$F_A = F_B = \\dfrac{W}{2}$. These point up.

"}], "gaps": [{"type": "engineering-answer", "useCustomName": true, "customName": "$W$", "marks": "10", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "settings": {"correctAnswer": "qty(2 F_A,units[1])", "right": "0.2", "close": "1.0", "C1": "75", "C2": "50", "C3": "25"}}, {"type": "engineering-answer", "useCustomName": true, "customName": "$R_A$", "marks": "5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "settings": {"correctAnswer": "force(F_A)", "right": "0.2", "close": "1.0", "C1": "75", "C2": "50", "C3": "25"}}, {"type": "engineering-answer", "useCustomName": true, "customName": "$R_B$", "marks": "5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "settings": {"correctAnswer": "force(F_A)", "right": "0.2", "close": "1.0", "C1": "75", "C2": "50", "C3": "25"}}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": true, "customName": "Shear and Bending Moment Functions", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Determine expressions for:

$V\\!(x) =$ [[0]]

$M\\!(x) =$ [[1]]

and then plot the shear and bending moment diagrams.

$V(x) = \\simplify[canonicalOrder,fractionNumbers]{{v}}$

$M(x) = \\simplify[canonicalOrder,fractionNumbers]{{m}}$

", "stepsPenalty": "10", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {"validate": {"script": "{show_curves(applet)}", "order": "after"}}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

The shear function is found by integrating the loading function. The negative sign is because positive values of w(x) represent a load which points down.

$\\begin{align}V(x) &= \\simplify{-defint(w,x,0,x)} \\\\&= \\simplify[fractionNumbers,canonicalOrder]{defint({-w},x,0,L)} \\\\&= \\simplify[fractionNumbers,canonicalOrder]{{v}}\\end{align}$

The bending moment function is found by integrating the shear function.

$\\begin{align}M(x) &= \\simplify{defint(V,x,0,x)}\\\\&= \\simplify[fractionNumbers,canonicalOrder]{defint({v},x,0,L)} \\\\&= \\simplify[fractionNumbers,canonicalOrder]{{M}}\\end{align}$

In both cases, the resulting constant of integration is determined by considering the values at $x=0$.

$V(0) = F_A, \\text{ and } M(0) = 0$

"}], "gaps": [{"type": "jme", "useCustomName": true, "customName": "V(x)", "marks": "10", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{expr(v)}", "answerSimplification": "fractionNumbers", "showPreview": true, "checkingType": "sigfig", "checkingAccuracy": "2", "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}, {"type": "jme", "useCustomName": true, "customName": "M(x)", "marks": "10", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{expr(m)}", "answerSimplification": "fractionNumbers", "showPreview": true, "checkingType": "sigfig", "checkingAccuracy": "2", "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": true, "customName": "Maximum internal loads", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Give the (signed) values of the largest shear and bending moment occuring anywhere within the beam. Use the standard sign convention for shear and bending moments.

$V_{\\text{max}} = $ [[0]] $\\qquad M_{\\text{max}} = $ [[1]]

$V_{max} = \\var{siground(qty(eval(v,0),units[1]),4)}$

$M_{max} = \\var{siground(qty(eval(m,b),units[0]+\" \" + units[1]),4)}$

The maximum shear occurs at the ends, where $\\simplify{diff(V,x,1)} = -w(x) = 0$ i.e. at $x=0$, or $x=\\var{qty(2b,units[0])}$.

$\\begin{align}V_{max} &= V(0) \\\\ &= \\var{siground(qty(F_A,units[1]),4)}\\end{align}$

Maximum moment occurs at midpoint, where $\\simplify{diff(M,x,1)} = V = 0$

$\\begin{align}M_{max} &= M(\\var{b}) \\\\&= \\var{siground(qty(eval(m,b),units[0] + ' ' + units[1]),4)}\\end{align}$

"}], "gaps": [{"type": "engineering-answer", "useCustomName": true, "customName": "$V_{max}$", "marks": "10", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "settings": {"correctAnswer": "force(F_A)", "right": "0.5", "close": "1", "C1": "75", "C2": "50", "C3": "25"}}, {"type": "engineering-answer", "useCustomName": true, "customName": "$M_{max}$", "marks": "10", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "settings": {"correctAnswer": "moment(eval(m,b))", "right": "0.5", "close": "1", "C1": "75", "C2": "50", "C3": "25"}}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "VM8: Arbitrary loading function", "extensions": ["jsxgraph", "linear-algebra", "polynomials", "quantities"], "custom_part_types": [{"source": {"pk": 19, "author": {"name": "William Haynes", "pk": 2530}, "edit_page": "/part_type/19/edit"}, "name": "Engineering Accuracy with units", "short_name": "engineering-answer", "description": "

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.

", "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": [], "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", "shear and bending moment", "Statics", "statics"], "metadata": {"description": "

Shear and bending moment diagram for a beam loaded with an arbitrary parabolic distributed load with a loading function given. Students must integrate to find reactions and equations for shear and bending moment. Problem sometimes fails when simplifying up the equations.

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

A simply-supported beam of length $L = \\var {qty(L, units[0])}$ is subjected to a parabolic loading $w(x)$ in {units[3]} given by the function:

\\[w(x) = \\simplify[fractionNumbers,canonicalOrder]{{w}}\\]

The load, shear and bending moment diagrams are shown below.

{loadBoard}

{shearBoard}

{momentBoard}

", "advice": "

Given $w(x)$ and beam length $L$,

\\[ w(x) = \\simplify[fractionNumbers,canonicalOrder]{{w}} \\]

1. Find load $W= \\int_0^L w(x)\\; dx$

\\[\\begin{align} W & = \\int w(x) \\text{d}x \\\\ & = \\simplify[fractionNumbers,canonicalOrder]{defint({w}, x, 0, {L})} \\\\ &= \\left[\\simplify[fractionNumbers,canonicalOrder]{{intw}}\\right]_0^\\var{L}\\\\ &= \\var{siground(qty(load,units[1]),5)} = \\var{siground(qty(abs(load),units[1]),5)} \\var{arrow(-load)}\\end{align}\\]

2. Find $Q_y = \\int_0^L x w(x)\\;dx$

\\[\\begin{align} Q_y &= \\int_0^L x\\ w(x) \\text{d}x\\\\&= \\simplify[fractionNumbers,canonicalOrder]{defint({dqy},x,0,L)}\\\\& = \\left[\\simplify[fractionNumbers,canonicalOrder]{{intdqy}} \\right]_0^\\var{L}\\\\ &= \\var{siground(qty(Qy,units[0] + \" \" + units[1]),4)} \\end{align}\\]

3. Find centroid $\\bar{x}=\\frac{Q_y}{W}$

\\[\\bar{x} = \\dfrac{Q_y}{W} = \\dfrac{\\var{siground(Qy,4)}}{\\var{siground(load,4)}} = \\var{siground(qty(xbar,units[0]),4)}\\]

4. Take moments about $A$ and $B$ to find reactions.

\\[\\begin{align}
\\sum M_B &= 0 & \\sum M_A &=0 \\\\
R_A L &= \\simplify{{sign(load)} W }(L-\\bar{x}) & + R_B L &= \\simplify{{sign(load)} W }\\bar{x} & \\\\
R_A&= \\var{force(abs(R_A))} \\var{arrow(R_A)} & R_B &= \\var{force(abs(R_B))} \\var{arrow(R_B)}
\\end{align}\\]

5. Integrate $V(x) = \\int w(x)\\; dx$ to get the shear function.

Note: Positive values of $w(x)$ indicate downward forces, so the integrations is over $-w(x)$.

\\[\\begin{align} V(x) &= \\int - w(x) dx \\\\
&= \\simplify[fractionNumbers, canonicalOrder]{int({-w},x)} \\\\
&= \\simplify[fractionNumbers,canonicalOrder]{{v -v[0]}}+C_3 && v(0) = R_A \\therefore C_3 = \\var{force(R_A)}
\\\\ &= \\simplify[fractionNumbers,canonicalOrder]{{v}} && [\\var{units[1]}] &\\end{align}\\]

6. Integrate $M(x) = \\int V(x)\\; dx$ to get moment function.

\\[\\begin{align}
M(x) &= \\int V(x) dx \\\\
&= \\simplify[fractionNumbers, canonicalOrder]{int({v},x)}\\\\
&= \\simplify[fractionNumbers,canonicalOrder]{{m}}+C_4 && M(0) =0 \\therefore C_4 = 0 \\\\
&= \\simplify[fractionNumbers,canonicalOrder]{{m}} && [\\var{units[1]}\\text{-}\\var{units[0]} ]&
\\end{align}\\]

Maximum moment occurs where $dM/dx = V = 0$

Evaluate $M(x)$ at point of maximum bending moment.

Maximum shear occurs where $d V /d x = -w(x) = 0$

Evaluate $V(x)$ at point of maximum shear.

To find maximum shear, determine where $\\simplify{diff(v,x,1)} = -w(x) = 0$ using the quadratic equation.

$x_{max} = \\frac{- b \\pm \\sqrt{b^2 - 4 a c}}{2a}$ where, $a= \\var[fractionNumbers]{-w[2]}, b=\\var[fractionNumbers]{-w[1]}$ and $c=\\var[fractionNumbers]{-w[0]}$.

Any real roots of $w(x)$ between the beam's endpoints plus the endpoints themselves are locations of local maximums or minimums.

{v_extremes}

The maximum of these local maximums is the the one we want

$V_{max} = \\var{show(qty(v_max, units[1]))}$

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loading function w(x)

", "templateType": "anything", "can_override": false}, "v": {"name": "v", "group": "functions", "definition": "-polynomial(x, [-R_A, w[0], w[1]/2, w[2]/3, w[3]/4])", "description": "

This is the shear function V(x) = - \\int w(x) dx

Constant C is reaction at x=0

", "templateType": "anything", "can_override": false}, "m": {"name": "m", "group": "functions", "definition": "polynomial(x,moment_coefficients)", "description": "", "templateType": "anything", "can_override": false}, "load": {"name": "load", "group": "Reactions", "definition": "eval(intW,L) - eval(intW,0)", "description": "

downward force on beam = $\\simplify{defint(w,x,0,{beamlen})$

Positive = down

", "templateType": "anything", "can_override": false}, "dQy": {"name": "dQy", "group": "Reactions", "definition": "polynomial(x) w", "description": "", "templateType": "anything", "can_override": false}, "intdQy": {"name": "intdQy", "group": "Reactions", "definition": "polynomial(x, [0, dQy[0], dQy[1]/2, dQy[2]/3, dQy[3]/4, dQy[4]/5])", "description": "

integral of dQy

", "templateType": "anything", "can_override": false}, "Qy": {"name": "Qy", "group": "Reactions", "definition": "eval(intdqy,L)", "description": "

definite integral of x dw from 0 to L

", "templateType": "anything", "can_override": false}, "xBar": {"name": "xBar", "group": "Reactions", "definition": "Qy/load", "description": "", "templateType": "anything", "can_override": false}, "intW": {"name": "intW", "group": "Reactions", "definition": "polynomial(x, [0, w[0], w[1]/2, w[2]/3, w[3]/4])", "description": "", "templateType": "anything", "can_override": false}, "R_B": {"name": "R_B", "group": "Reactions", "definition": "xbar load / L", "description": "

reaction at B Positive is up

", "templateType": "anything", "can_override": false}, "R_A": {"name": "R_A", "group": "Reactions", "definition": "(L-xbar) load/ L", "description": "", "templateType": "anything", "can_override": false}, "check": {"name": "check", "group": "Reactions", "definition": "withintolerance(R_A+R_B,load,10^10)", "description": "

check sometimes has roundoff error.

", "templateType": "anything", "can_override": false}, "roots": {"name": "roots", "group": "maxshear", "definition": "sort(\n if(d>=0, \n filter(x>=0 and x <=L, x,\n let([a: w[2], b: w[1], c: w[0]], \n [(-b + sqrt(d))/(2 a), (-b - sqrt(d))/(2 a)]\n)),[]) + [0,L])", "description": "

use quadratic equation to find where w(x) is zero, filter out any roots not on interval, or imaginary.

The maximum shear will occur at one of these points

", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "w[1]^2 - 4 w[2] * w[0]", "description": "

descriminant for quadratic equation w(x)

", "templateType": "anything", "can_override": false}, "v_extremes": {"name": "v_extremes", "group": "maxshear", "definition": "map(vector(x,eval(v,x)),x,roots)", "description": "

extreme points

list of shear force v(x) at ends and at roots of w(x) -- potential maximums or minimums

", "templateType": "anything", "can_override": false}, "v_max": {"name": "v_max", "group": "functions", "definition": "max([v_maxmin[0],-v_maxmin[1]])", "description": "

filters the list of v_extremes to find the one with the greatest absolute value of shear

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hack to scale load diagram

", "templateType": "anything", "can_override": false}, "w_maxmin": {"name": "w_maxmin", "group": "functions", "definition": "[max(map(x->eval(w,x),0..l)) |> round(), round(min(map(x->eval(w,x),0..l)))]", "description": "", "templateType": "anything", "can_override": false}, "shearBoard": {"name": "shearBoard", "group": "Ungrouped variables", "definition": "\n jsxgraph(\n 500,250,[-2, 1.2 v_max, L+2, -v_max],\n [ \n \"$V(x)$\": ['functiongraph', [shear_function, 0, L], [withLabel: true, strokeWidth: 3, strokeColor: \"#CC0000\", label:[fixed: false, fontSize: 14]]],\n\n ]\n)", "description": "", "templateType": "anything", "can_override": false}, "shear_function": {"name": "shear_function", "group": "functions", "definition": "expr(v)", "description": "", "templateType": "anything", "can_override": false}, "v_maxmin": {"name": "v_maxmin", "group": "functions", "definition": "[max(map(x->eval(v,x),0..l)) |> round(), round(min(map(x->eval(v,x),0..l)))]", "description": "", "templateType": "anything", "can_override": false}, "moment_function": {"name": "moment_function", "group": "functions", "definition": "expr(m)", "description": "", "templateType": "anything", "can_override": false}, "m_maxmin": {"name": "m_maxmin", "group": "functions", "definition": "[max(map(x->eval(m,x),0..l)) |> round(), round(min(map(x->eval(m,x),0..l)))]", "description": "", "templateType": "anything", "can_override": false}, "m_max": {"name": "m_max", "group": "functions", "definition": "max([m_maxmin[0],-m_maxmin[1]])", "description": "", "templateType": "anything", "can_override": false}, "momentBoard": {"name": "momentBoard", "group": "Ungrouped variables", "definition": " let(max, m_maxmin[0], min, m_maxmin[1],\n jsxgraph(\n 500,250,[-2, 1.2 max(1.2 max, -min/2), L+2, min(1.2 min, -max/2)],\n [\"$M(x)$\": ['functiongraph', [moment_function, 0, L], [withLabel: true, strokeWidth: 3, strokeColor: \"#3070AD\", label:[fixed: false, fontSize: 14] ]],\n ]\n ) )", "description": "

max(1.2 max[0],-max[1]/2)

", "templateType": "anything", "can_override": false}, "force": {"name": "force", "group": "Ungrouped variables", "definition": "(f)->siground(qty(f,units[1]),4)", "description": "", "templateType": "anything", "can_override": false}, "load_coefficients": {"name": "load_coefficients", "group": "functions", "definition": "random(coefficients)", "description": "

get coefficients puts curve through three points, A,B,C siground tries to make the coefficients simpler fractions

", "templateType": "anything", "can_override": false}, "moment_coefficients": {"name": "moment_coefficients", "group": "functions", "definition": "[0,v[0],v[1]/2,v[2]/3,v[3]/4,v[4]/5]", "description": "", "templateType": "anything", "can_override": false}, "shear_coefficients": {"name": "shear_coefficients", "group": "functions", "definition": "[-R_A, w[0], w[1]/2, w[2]/3, w[3]/4]", "description": "", "templateType": "anything", "can_override": false}, "random_frac": {"name": "random_frac", "group": "Ungrouped variables", "definition": "() -> random(-9..9 except 0)/random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "coefficients": {"name": "coefficients", "group": "functions", "definition": "[[ 0, -10, 3, -0.1 ],\n[ 0, -10, 4, -0.1 ],\n[ 0, -10, 4, -0.1 ],\n[ 0, -10, 4, -0.2 ],\n[ 0, -20, 4, -0.2 ],\n[ 0, -20, 5, -0.2 ],\n[ 0, -3, 0.3, -0.006 ],\n[ 0, -4, 0.04, 0.009 ],\n[ 0, -4, 0.07, 0.006 ],\n[ 0, -4, 0.3, -0.007 ],\n[ 0, -4, 0.4, -0.008 ],\n[ 0, -5, 0.5, -0.01 ],\n[ 0, -5, 0.6, -0.02 ],\n[ 0, -5, 0.7, -0.02 ],\n[ 0, -6, 0.5, -0.01 ],\n[ 0, -6, 0.8, -0.03 ],\n[ 0, -7, 0.9, -0.03 ],\n[ 0, -8, 1, -0.03 ],\n[ 0, -8, 1, -0.04 ],\n[ 0, -9, 1, -0.04 ],\n[ 0, 2, -0.04, -0.004 ],\n[ 0, 2, -0.05, -0.004 ],\n[ 0, 2, -0.1, 0 ],\n[ 0, 20, -5, 0.2 ],\n[ 0, 3, -0.03, -0.007 ],\n[ 0, 3, -0.05, -0.005 ],\n[ 0, 3, -0.1, 0 ],\n[ 0, 3, -0.3, 0.006 ],\n[ 0, 4, -0.03, -0.007 ],\n[ 0, 4, -0.04, -0.008 ],\n[ 0, 4, -0.04, -0.009 ],\n[ 0, 4, -0.07, -0.006 ],\n[ 0, 4, -0.2, 0 ],\n[ 0, 5, -0.05, -0.01 ],\n[ 0, 5, -0.7, 0.02 ],\n[ 0, 6, -0.05, -0.01 ],\n[ 0, 6, -0.06, -0.01 ],\n[ 0, 6, -0.8, 0.03 ],\n[ 0, 8, -1, 0.03 ],\n[ 0, -3, 0.3, -0.007 ]]", "description": "

These coefficients do not cause \"Simplifier is stuck in a loop\" errors. The were generated by giving three random points to getCoefficients() extension function

", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["d", "debug", "loadBoard", "shearBoard", "momentBoard", "reactionoptions", "drawArrow", "arrows", "force", "random_frac"], "variable_groups": [{"name": "Unnamed group", "variables": ["L", "units"]}, {"name": "functions", "variables": ["w", "v", "m", "coefficients", "load_coefficients", "shear_coefficients", "moment_coefficients", "load_function", "shear_function", "moment_function", "w_maxmin", "v_maxmin", "m_maxmin", "w_max", "v_max", "m_max"]}, {"name": "Reactions", "variables": ["load", "dQy", "intdQy", "Qy", "xBar", "intW", "R_B", "R_A", "check"]}, {"name": "maxshear", "variables": ["roots", "v_extremes"]}], "functions": {"arrow": {"parameters": [["F", "?"]], "type": "anything", "language": "jme", "definition": "if(sign(F) >= 0 ,latex(\"\\\\uparrow\"), latex(\"\\\\downarrow\"))"}, "getCoefficients": {"parameters": [["A", "list of integer"], ["B", "list of integer"], ["C", "list of integer"]], "type": "list", "language": "javascript", "definition": " const jme = Numbas.jme;\n const extension = Numbas.extensions.polynomial\n const [x0, y1] = A;\n const [x2, y2] = B;\n const [x3, y3] = C;\n\n // Sanity check: first point must be at x=0\n if (x0 !== 0) {\n throw new Error(\"First point A must have x = 0\");\n }\n\n const d = y1;\n\n // Right-hand side: subtract y1 to isolate a,b,c terms\n const r1 = y2 - y1;\n const r2 = y3 - y1;\n\n // Matrix coefficients\n const a11 = Math.pow(x2, 3);\n const a12 = Math.pow(x2, 2);\n const a13 = x2;\n\n const a21 = Math.pow(x3, 3);\n const a22 = Math.pow(x3, 2);\n const a23 = x3;\n\n // Solve the system using Cramer's Rule or elimination\n // We'll eliminate c first\n\n // Multiply row 1 by x3 and row 2 by x2 to align c terms\n const m1 = x3, m2 = x2;\n\n const A1 = m1 * a11 - m2 * a21;\n const B1 = m1 * a12 - m2 * a22;\n const R1 = m1 * r1 - m2 * r2;\n\n // Now solve A1 * a + B1 * b = R1\n // We'll solve for b in terms of a, then back-substitute\n\n // Solve for a and b\n // System: A1 * a + B1 * b = R1\n // a13 * a + a12 * b + c = r1\n\n // Choose to solve directly (2x2 linear system)\n const det = A1 * (-a12) - B1 * (-a13);\n if (Math.abs(det) < 1e-12) {\n throw new Error(\"Points are degenerate or lead to singular system.\");\n }\n\n // Use Cramer's Rule\n const detA = R1 * (-a12) - B1 * (r1 - a13 * 0); // assuming a = 0 temporarily\n const detB = A1 * (r1 - a13 * 0) - R1 * (-a13);\n\n const a = detA / det;\n const b = detB / det;\n\n // Solve for c using first equation\n const c = (r1 - a * a11 - b * a12) / a13;\n\n // Return function a * x ** 3 + b * x ** 2 + c * x + d;\n return [d,c,b,a]\n \n"}, "show": {"parameters": [["A", "number"]], "type": "anything", "language": "jme", "definition": "if(abs(if(type(A)=type(qty(1,'m')),scalar(A),A))<1,\n precround(A,4),siground(A,4))"}}, "preamble": {"js": "question.signals.on('adviceDisplayed',function() {\n try{\n }\n catch(err){} \n})\n\n\n", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": true, "customName": "Reactions", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Find the reactions at the end suppports. Use negative values to indicate downward forces.

$R_A = $ [[0]]

$ R_B= $ [[1]]

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Use integration to determine the equation of the shear function $V (x) = \\int -w(x)\\ dx$

$V\\!(x) = \\simplify[fractionNumbers,canonicalOrder]{{v}} = \\simplify[canonicalOrder]{{v}} $

$V\\!(x) = $[[0]]

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Use integration to find the bending moment function $M(x) =\\int V(x)\\ dx$.

$M(x) = \\simplify[fractionNumbers,canonicalOrder]{{m}}$

$=\\simplify[canonicalOrder]{{m}}$

$M(x) = $ [[0]]

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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.

", "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/newVM_cA1Sig0.ggb", "/srv/numbas/media/question-resources/newVM_cA1Sig0.ggb"]]}