V-M 5a: Arbitrary parabolic loading

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William Haynes

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William Haynes 1 week, 6 days ago

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replaced geogebra_file() with geogebra_applet()

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William Haynes 2 years, 9 months ago

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Made significant revisions. Now student is shown shear and bending moment curves, but must integrate the loading to determine the shear and bending moment functions themselves.

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Starting modifications.

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Should work now after bug fix. https://github.com/numbas/numbas-extension-geogebra/commit/12c803d3b95c4bf60b440a1043be5db1b4f0def6

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State prior to testing. Uses a variable callling a javascript function.

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William Haynes 4 years, 2 months ago

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William Haynes 4 years, 4 months ago

Created this as a copy of V-M 5: Symmetric parabolic loading.

Name	Status	Author	Last Modified
V-M 1: Concentrated forces	Ready to use	William Haynes	16/08/2024 19:05
V-M 5: Symmetric parabolic loading	Ready to use	William Haynes	08/03/2025 17:57
V-M 4: Combined loads	Ready to use	William Haynes	16/08/2024 19:13
V-M 2: Concentrated moments	Ready to use	William Haynes	04/05/2022 23:54
V-M 5a: Arbitrary parabolic loading	Ready to use	William Haynes	08/03/2025 18:07
V-M 1a: Concentrated forces, no diagrams	Ready to use	William Haynes	16/08/2024 19:08
V-M 2a: Concentrated moments, no diagrams	Ready to use	William Haynes	05/05/2022 00:14
V-M 4a: Combined loads, no diagrams	Ready to use	William Haynes	27/11/2024 15:38

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Question statement

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

$w\!(x) = \simplify[fractionNumbers,canonicalOrder]{{w}}$ [{units[1]} / {units[0]}]

The shear and bending moment diagrams are also shown.

{applet}

Started working on maximum shear and moment, but did not finish. Need to solve a cubic equation by writing equation solver. I would like to turn the visibility of the curves on as a hint, but need a signal for that.

Give any introductory information the student needs.

			Name	Type	Generated Value

A	vector	vector(0,0)
B	vector	vector(10,180)
C	vector	vector(5,0)
scale	integer	60
beamLen	integer	10
units	list	[ "m", "kN" ]

			Name	Type	Generated Value

applet	ggbapplet	HTML node
params	dict	Dictionary with 11 entries
matrix	matrix	Matrix of size 3×4
w	polynomial	polynomial(3.6x^2 - 18x)
v	polynomial	polynomial(-1.2x^3 + 9x^2)
m	polynomial	polynomial(-0.3x^4 + 3x^3)

			Name	Type	Generated Value

load	number	300
dQy	polynomial	polynomial(3.6x^3 - 18x^2)
intdQy	polynomial	polynomial(0.9x^4 - 6x^3)
Qy	number	3000
xBar	number	10
intW	polynomial	polynomial(1.2x^3 - 9x^2)
R_B	number	300
R_A	number	0
check	boolean	true

			Name	Type	Generated Value

roots	list	[ 0, 0, 5, 10 ]
v_extremes	list	List of 4 items
v_max	vector	vector(10,-300)

			Name	Type	Generated Value

			d	number	324
			debug	boolean	false

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
vector(0,random(-4..3) scale )

JME function reference

Description

Leftmost point on loading curve (x, w(x))

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(0,0)

← Depends on:

This variable doesn't seem to be used anywhere.

Parts

Reactions Gap-fill
1. Gap A Engineering Accuracy with units
  Add
2. Gap B Engineering Accuracy with units
  Add
Shear Function Gap-fill
1. Gap $V\!(x)$ Mathematical expression
  Add
Bending Moment Gap-fill
1. Gap $M\!(x)$ Mathematical expression
  Add

Gap-fill

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

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

$R_A = $ A $\qquad R_B=$ B

Advice

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

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

$\begin{align} W & = \int w\!(x) \text{d}x \\ & = \simplify[fractionNumbers,canonicalOrder]{defint({w}, x, 0, {beamlen})} \\ &= \left[\simplify[fractionNumbers,canonicalOrder]{{intw}}+C_1 \right]_0^\var{beamlen}\\ &= \var{siground(qty(load,units[1]),5)} = \var{siground(qty(abs(load),units[1]),5)} \var{arrow(-load)}\end{align}$

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

$\begin{align} Q_y &= \int_0^L x\ w\!(x) \text{d}x\\&= \simplify[fractionNumbers,canonicalOrder]{defint({dqy},x,0,L)}\\& = \left[\simplify[fractionNumbers,canonicalOrder]{{intdqy}}+C_2 \right]_0^\var{beamlen}\\ &= \var{siground(qty(Qy,units[0] + " " + units[1]),4)} \end{align}$

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.

$\begin{align}\sum M_B &= 0 & \sum M_A &=0 \\ R_A L &= \simplify{{sign(load)} W }(L-\bar{x}) & + R_B L &= \simplify{{sign(load)} W }\bar{x} & \\ R_A&= \var{show(qty(abs(R_A),units[1]))} \var{arrow(R_A)} & R_B &= \var{show(qty(abs(R_B),units[1]))} \var{arrow(R_B)} \end{align}$

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 $-w\!(x)$.

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

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

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

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}

$\simplify{{show(v_extremes[1])}}$

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

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

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V-M 5a: Arbitrary parabolic loading

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

William Haynes 1 week, 6 days ago

William Haynes 2 years, 9 months ago

William Haynes 2 years, 9 months ago

William Haynes 2 years, 9 months ago

William Haynes 2 years, 9 months ago

William Haynes 2 years, 10 months ago

William Haynes 2 years, 10 months ago

William Haynes 2 years, 10 months ago

William Haynes 4 years, 2 months ago

William Haynes 4 years, 4 months ago

William Haynes 4 years, 4 months ago

Error in variable testing condition

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Generated value: vector

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Generated value: `vector`