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Question statement

{geogebra_applet('wrp4gmsn',var_1)}

The frame shown supports a {F} load at point $F$ . Determine the $x$ - and $y$ - components of the forces acting on member $ADC$ at pins $A$ , $D$ and $C$ .

Give any introductory information the student needs.

			Name	Type	Generated Value

AD	quantity	quantity(1.125, "ft")
DE	quantity	quantity(6.75, "ft")
EF	quantity	quantity(0.5, "ft")
b_y	quantity	quantity(335, "lbs")
a_y	quantity	quantity(25, "lbs")
d_y	quantity	quantity(26.66666666666666666666666666666666666667, "lbs")
d_x	quantity	quantity(-27, "lbs")
e_y	quantity	quantity(386.6666666666666666666666666666666666667, "lbs")
e_x	quantity	quantity(-27, "lbs")
c_y	quantity	quantity(51.66666666666666666666666666666666666667, "lbs")
c_x	quantity	quantity(-27, "lbs")
check	decimal	dec("0")
a_x	quantity	quantity(0, "lbs")

			Name	Type	Generated Value

b	integer	9
h	integer	10
L	number	7.25
d	rational	5/2
unit_d	string	ft
var_1	list	Nested 5×2 list
F	quantity	quantity(360, "lbs")
unit_F	string	lb
var_2	list	Nested 5×2 list
debug	boolean	false

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
qty(b/2 d/h,unit_d)

JME function reference

Description

sum of moment about A for entire frame

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(1.125, "ft")

← Depends on:

b
d
h
unit_d

→ Used by:

Advice
Variable testing condition

Parts

Pin A Gap-fill
1. Gap Ax Engineering Accuracy with units
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2. Gap L-R Choose one from a list
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3. Gap Ay Engineering Accuracy with units
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4. Gap U-D Choose one from a list
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Pin D Gap-fill
1. Gap Dx Engineering Accuracy with units
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2. Gap L-R Choose one from a list
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3. Gap Dy Engineering Accuracy with units
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4. Gap U-D Choose one from a list
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Pin C Gap-fill
1. Gap Cx Engineering Accuracy with units
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2. Gap L-R Choose one from a list
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3. Gap Cy Engineering Accuracy with units
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4. Gap U-D Choose one from a list
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Gap-fill

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

$A_x =$ Ax L-R {a_x} $A_y =$ Ay U-D {d(a_y)}

Advice

$L = \var{L}$ $b = \var{b}$ $h =\var{h}$ $d=\var{d}$

$DE = \var{DE}$

$EF = \var{EF}$

Start by drawing neat, labeled, consistent free body diagrams of the whole frame and each of its parts. Exposed internal forces must occur in equal and opposite pairs. In the diagram below, the external forces are drawn in red.

{geogebra_applet('x3npnvsb',var_2)}

Count the unknowns on each diagram. If any diagram has three unknowns, consider starting there. In this case FBD I has three unknowns ($A_x, A_y$, and $B$).

FBD I

\[\begin{align} \text{I: }\Sigma M_A &= 0\\ B ( \simplify[]{{AD}+{DE} + {AD} }) &= F( \simplify[!collectNumbers]{{AD}+{DE} + {EF} })\\B&=\var{F} \left( \dfrac{\var{AD+qty(L,unit_d)}}{\var{qty(b,unit_d)}}\right) \\ &= \var{b_y}\uparrow \\\\\text{I: }\Sigma F_y &=0\\A_y+B &= F\\A_y &= \var{F} - \var{b_y}\\&=\var{(a_y)}\uparrow\\\\\text{I: }\Sigma F_x &= 0\\A_x&=0\end{align}\]

FBD II

With $A_x, A_y$, and $B$ known, the remaining free body diagrams all have four unknowns but, luckily in this problem, the lines of action of forces $E_x, E_y$, and $D_x$ all intersect at point $E$ on FBD II. By taking moments there, we can find $D_y$. If this was not the case, you would need to select two free body diagrams and take moments appropriately to yield two equations with two unknowns which can be solved simultaneously.

\[\begin{align} \text{II: }\Sigma M_E &=0\\ D_y(\var{DE}) &= F(\var{EF})\\D_y &= \var{F}\left(\dfrac{\var{EF}}{\var{DE}}\right)\\ &=\var{d(D_y)}\downarrow\end{align}\]

FBD III

With $A_x, A_y$ and $D_y$ found, only three unknowns remain on FBD III, so it can be solved to satisfy the problem statement. Remember that negative results indicate that the force acts in the direction opposite to the one assumed on the free body diagram.

\[\begin{align} \text{III: }\Sigma M_C &= 0\\ - A_y ( \var{b/2}) - D_x(\var{h-d}) - D_y(\var{scalar(DE/2)}) &= 0 \\D_x &=\dfrac{-\var{b/2} A_y - \var{scalar(DE/2)}D_y}{\var{h-d}} \\ &=-\dfrac{\var{b/2} (\var{d(A_y)}) +\var{scalar(DE/2)}(\var{d(D_y)})}{\var{h-d}} \\&= \var{d(d_x)} \leftarrow\\\\\text{III: }\Sigma F_y &=0\\A_y+D_y - C_y &= 0\\C_y &= \var{d(A_y)} +\var{d(D_y)}\\&=\var{d(C_y) }\downarrow\\\\\text{III: }\Sigma F_x &= 0\\D_x-C_x&=0\\C_x &= \var{d(C_x)} \rightarrow \end{align}\]

FBD II

With $D_x$ determined, the remainin unknown on FBD II can be found.

\[\begin{align} \text{II: }\Sigma F_x &= 0\\D_x-E_x&=0\\E_x &= \var{d(E_x)} \leftarrow \end{align}\]

FBD IV

All the unknown reactions are now known. You can check your work by verifying that these force put member $CEB$ in equilibrium. If they don't, you have made an error.

\[\begin{align} \text{IV: }\Sigma M_c &\stackrel{?}{=} 0\\ 0&\stackrel{?}{=} B_y (\var{qty(b/2,unit_d)}) - E_y (\var{DE/2}) + E_x (\var{qty(h-d,unit_d)})\\0&\stackrel{?}{=} (\var{d(B_y)})(\var{qty(b/2,unit_d)}) - (\var{d(E_y)})(\var{DE/2}) + (\var{d(E_x)}) (\var{qty(h-d,unit_d)})\\0&\stackrel{?}{=} (\var{d(B_y qty(b/2,unit_d))}) - (\var{d(E_y DE/2)}) + (\var{d(E_x qty(h-d,unit_d))})\\0&= \var{check} \, \checkmark\end{align}\]

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This question is used in the following exams:

Chapter 06 Exercises by Michael Proudman in Engineering Statics.
31. Frames and Machines 1 by Michael Proudman in Engineering Statics.
Homework 18: Frames by Sarah Sojka in Statics at Randolph College.
31. Frames and Machines 1 by William Haynes in Engineering Statics.
Chapter 6 Exercises by William Haynes in Engineering Statics.

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William Haynes 5 years, 10 months ago

William Haynes 5 years, 10 months ago

William Haynes 5 years, 10 months ago

Error in variable testing condition

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William Haynes 5 years, 10 months ago

William Haynes 5 years, 10 months ago

William Haynes 5 years, 10 months ago

Error in variable testing condition

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Generated value: quantity

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Generated value: `quantity`