Internal force: overhanging beam

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William Haynes

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Question statement

{applet(500,215,false)}

The beam shown supports a uniformly distributed load of $w$ = {Wa}. load that varies uniformly from {Wa} at the left end to {Wb} at the right end.

The lengths of the beam segments are $d_1$ = {display(d_1)}, $d_2$ = {display(d_2)}, and $d_3$ = {display(d_3)}.

Give any introductory information the student needs.

			Name	Type	Generated Value

L	quantity	quantity(12, "m")
Ax	quantity	quantity(0, "m")
Bx	quantity	quantity(12, "m")
Cx	quantity	quantity(2.000000000000004, "m")
Dx	quantity	quantity(9.999999999999996, "m")
d_1	quantity	quantity(2.000000000000004, "m")
d_2	quantity	quantity(8, "m")
d_3	quantity	quantity(1.999999999999996, "m")
Wmax	quantity	quantity(200, "N/m")
Wa	quantity	quantity(400, "N/m")
Wd	quantity	quantity(233.3333333333332666666666666666666666667, "N/m")
Wb	quantity	quantity(200, "N/m")
h1	quantity	quantity(200, "N/m")
h2	quantity	quantity(200, "N/m")
dw	quantity	quantity(5.333333333333333333333333333333333333333, "m")
W_t	quantity	quantity(3600, "N")
FC	quantity	quantity(9599.9999999999808000000000000384, "N")
FA	quantity	quantity(-5999.9999999999808000000000000384, "N")
dw'	quantity	quantity(4.561403508771927778393351800554081936142, "m")
W_p	quantity	quantity(3166.6666666666650666666666666668, "N")
FV	quantity	quantity(433.3333333333349333333333333332, "N")
M_D	quantity	quantity(-422.2222222221449333333333334905777777748, "N*m")

			Name	Type	Generated Value

F_C	vector	vector(0,9600)
F_Wt	vector	vector(0,-3600)
F_A	vector	vector(0,-6000)
F_Wp	vector	vector(0,-3166.6666666667)
F_V	vector	vector(0,433.3333333333)

			Name	Type	Generated Value

A	vector	vector(0,0)
B	vector	vector(12,0)
C	vector	vector(2,0)
D	vector	vector(10,0)
w_A	number	2
w_B	number	1
ggb_points	list	Nested 5×2 list
units	list	[ "N", "m" ]
debug	boolean	false

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
qty(random(6,8,12,16,20,24),units[1])

JME function reference

Description

overall length of the beam.

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

Can an exam override the value of this variable?

Generated value: `quantity`

quantity(12, "m")

← Depends on:

units

→ Used by:

This variable doesn't seem to be used anywhere.

Parts

Reactions Gap-fill
1. Gap $F_A$ Engineering Accuracy with units
  Add
2. Gap $F_C$ Engineering Accuracy with units
  Add
Internal Load Gap-fill
1. Gap $V$ Engineering Accuracy with units
  Add
2. Gap $M_D$ Engineering Accuracy with units
  Add

Gap-fill

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

Determine the reactions at pin $A$ and roller $C$. Let positive values indicate upward forces.

$A$ = $F_A$ {display(FA)}

$C$ = $F_C$ {display(FC)}

Advice

1. Draw a FBD of the entire beam and find reactions at A and C.

{applet(500,566,true)}

a. Replace distributed load with an equivalent concentrated load.

$W$ = {W_t}, $d$ = {display(dw)}

b. Apply equilibrium equations.

$\begin{align} \Sigma M_A &= 0 & \Sigma F_y &=0 \\C \cdot\var{if(C[0]<D[0],latex('d_1'),latex('(d_1+d_2)'))} &= W \cdot d & \simplify{{if(scalar(FA)<0,-1,1)} A} + C &= W = \var{W_t}\\ C &= \var{display(FC)} \uparrow & A &= \var{display(abs(FA))} \var{if(F_A[1]>0,latex('\\uparrow'),latex('\\downarrow'))}\end{align}$

2. Draw a FBD of the portion of the beam between A and D and find the shear and bending moment.

a. Determine the distributed load above point D.

Let $w_A$, $w_B$, and $w_D$ be the magnitude of the distributed load at points $A$, $B$, and $D$ respectively, and $L$ the length of the beam.

Given: $w_A = \var{wA},\qquad w_B = \var{wB}\qquad L = (d_1 + d_2 + d_3) = \var{L}$

$w_D = \var{display(wD)}$

Using similar triangles: $w_D = w_A + \dfrac{\var{if(C[0]>D[0],latex('d_1'),latex('(d_1+d_2)'))}}{L} \left( w_B - w_A \right) = \var{display(wD)}$

b. Replace distributed load with an equivalent concentrated load.

$W'$ = {display(W_p)}, $d'$ = {display(dw')}

c. Apply equilibrium equations.

Let $\ell = \var{if(C[0]>D[0],latex('d_1'),latex('d_1+d_2'))} = \var{display(Dx)}\text{, and} \qquad d_\perp = (\ell - d') = \var{display(Dx - dw')}$

$\begin{align} \Sigma M_D &= 0 & \Sigma F_y &=0 \\ \simplify{{if(scalar(FA)<0,1,-1)} A} \ell + W' d_\perp \var{if(Cx < Dx, latex('+C d_2'),'')} + M_D&= 0 & \simplify{{if(scalar(FA)<0,-1,1)} A} \var{if(Cx < Dx, latex('+ C'),'')} - W' - V &=0 \\ M_D &= \simplify{{if(scalar(FA)>0,1,-1)} A} \ell - W' d_\perp \var{if(Cx < Dx, latex('C d_2'),'')} & V &= \simplify{{if(scalar(FA)<0,-1,1) }A} \var{if(Cx < Dx, latex('+ C'),'')} - W' \\ &= \var{display(M_D)} & V &= \var{display(FV)} \end{align}$

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This question is used in the following exams:

Chapter 08 Exercises by Michael Proudman in Engineering Statics.
35. Internal Forces by Michael Proudman in Engineering Statics.
35. Internal Forces by William Haynes in Engineering Statics.
Chapter 8 Exercises by William Haynes in Engineering Statics.
Lars's copy of 35. Internal Forces by Lars Hansson in Lars's workspace.

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Taxonomy: Context

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William Haynes 4 years, 3 months ago

William Haynes 4 years, 7 months ago

William Haynes 6 years, 4 months ago

William Haynes 6 years, 4 months ago

William Haynes 6 years, 4 months ago

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Generated value: `quantity`

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Internal force: overhanging beam

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

William Haynes 4 years, 3 months ago

William Haynes 4 years, 7 months ago

William Haynes 6 years, 4 months ago

William Haynes 6 years, 4 months ago

William Haynes 6 years, 4 months ago

Error in variable testing condition

Error

Warning

Generated value: quantity

Parts

Gap-fill

Generated value: `quantity`