V&M: Triangular loading

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Question statement

A cantilever beam supports a uniformly varying distributed load as shown.

Give any introductory information the student needs.

			Name	Type	Generated Value

unitsF	string	lb
unitsL	string	ft
unitsW	string	lb/ft
L	integer	18
w	integer	90
Vmp	quantity	quantity(-202.5, "lbs")
Mmp	quantity	quantity(-607.5, "lbs*ft")

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
'lb'

JME function reference

Description

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

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Generated value: `string`

→ Used by:

Mmp
Vmp
unitsW

This variable doesn't seem to be used anywhere.

Parts

Functions Gap-fill
1. Gap V(x) Mathematical expression
  Add
2. Gap M(x) Mathematical expression
  Add
Values at midpoint. Gap-fill
1. Gap V({L/2}) Engineering Accuracy with units
  Add
2. Gap M({L/2}) Engineering Accuracy with units
  Add

Gap-fill

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

Derive expressions for shear and bending moments as a function of $x$. Use the standard sign convention for shear and bending moments.

$V(x) =$ V(x) $\qquad M(x) = $ M(x)

Advice

Draw a free body diagram.

For a general solution, take an imaginary cut at $x$ and draw a free body diagram of the portion of the beam to the left of that point. $V(x)$ and $M(x)$ are drawn in the positive directions for shear and bending moment.

At location $x$ the downward loading is proportionally less than the maximum loading at the wall, $w$.

$w(x) = \left(\frac{x}{L}\right) w$

Determine equivalent load, $W$, by finding the area and centroid of the triangular load.

$W = \dfrac{(w x/L) (x)}{2} = \dfrac {w x^2}{2 L}$ acting at a distance of $\dfrac{2 x}{3}$ from the left, or $\dfrac{x}{3}$ from the cut.

Apply equations of equilibrium to the free body diagram.

$\begin{align} \Sigma F_y &= 0 & \Sigma M_{cut} &= 0\\ -W - V(x) &=0 & (W)(x/3) + M(x) &=0\\ V(x) &=-\dfrac {w x^2}{2 L}& M(x) &= -\dfrac {w x^3}{6 L} \end{align}$

Evaluating for given values, $w = \var{qty(w, unitsW)}$ and $L = \var{qty(L, unitsL)}$.

$\begin{align} V(x) &=-\simplify[all]{{w}/2/{L}} x^2 & M(x) &= -\simplify[all]{{w}/6/{L}} x^3 \end{align}$

Evaluate functions at $x = \var{qty(L/2,unitsL)}$.

$\begin{align} V(\var{L/2}) &=\var{Vmp} & M(\var{L/2}) &= \var{Mmp}\end{align}$

Give a worked solution to the whole question.

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William Haynes 4 years, 2 months ago

William Haynes 4 years, 3 months ago

William Haynes 4 years, 3 months ago

Error in variable testing condition

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Generated value: `string`

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V&M: Triangular loading

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, 3 months ago

William Haynes 4 years, 3 months ago

Error in variable testing condition

Error

Warning

Generated value: string

Parts

Gap-fill

Generated value: `string`