Truss: Method of Joints

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Question statement

{applet}

Determine the forces in all members of the truss shown when loaded with two forces: $F_B = \var{magB}$ and $F_C = \var{magC}$. For each member, indicate tension or compression.

Give any introductory information the student needs.

			Name	Type	Generated Value

magC	quantity	quantity(350, "kN")
magB	quantity	quantity(300, "kN")
units	string	kN
alpha	integer	-10
debug	boolean	false
applet	ggbapplet	HTML node
params	dict	Dictionary with 4 entries

			Name	Type	Generated Value

vecB	vector	Vector with 3 components
vecC	vector	vector(0,-350,0)
r_ab	vector	vector(1.7320508076,1,0)
r_ac	vector	vector(1.7320508076,0,0)
r_ad	vector	vector(3.4641016151,0,0)
vecD	vector	vector(0,286.3340798453,0)
vecA	vector	Vector with 3 components
check	vector	vector(0,0,0)

			Name	Type	Generated Value

magD	quantity	quantity(286.334079845283838, "kN")
magAx	quantity	quantity(295.442325903662379, "kN")
magAy	quantity	quantity(115.760373454795285, "kN")

			Name	Type	Generated Value

BD	quantity	quantity(572.668159690567676, "kN")
CD	quantity	quantity(495.945174230515465426182645193764, "kN")
BC	quantity	quantity(350, "kN")
AC	quantity	quantity(495.945174230515465426182645193764, "kN")
vecAB	vector	Vector with 3 components
AB	quantity	quantity(231.520746909590542, "kN")
ABy	quantity	quantity(115.760373454795342, "kN")
Abs	quantity	quantity(200.502848326852927, "kN")
vecCD	vector	vector(-495.9451742305,0)
vecBD	vector	vector(701.37,-286.33)

			Name	Type	Generated Value

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
qty(random(100..800#25),units)

JME function reference

Description

Magnitude of force C, always points straight down

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(350, "kN")

← Depends on:

units

→ Used by:

BC
params
vecC

This variable doesn't seem to be used anywhere.

Parts

Reactions Gap-fill
Joint D Gap-fill
Joint C Gap-fill
Joint B Gap-fill
1. Gap Gap 0. Engineering Accuracy with units
  Add
2. Gap Gap 1. Choose one from a list
  Add

Gap-fill

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

Determine the components of the reactions at $A$ and $D$.

$A_x$ = Gap 0. Gap 1. $\qquad A_y$ = Gap 2. Gap 3.

$D_x$ = Gap 4. Gap 5. $\qquad D_y$ = Gap 6. Gap 7.

{precround(vecA,1)} {units} {vecD}

Advice

Find the reactions:

Draw a free body diagram of the entire truss.
Get scalar components of loads $\mathbf{B}$ and $\mathbf{C}$ for later use.
$\begin{align}\\ B_x &= \var{display(qty(vecB[0],units))} & B_y &= \var{display(qty(vecB[1],units))}\\C_x &= \var{display(qty(vecC[0],units))} & C_y &= \var{display(qty(vecC[1],units))}\end{align}$
Take moments at $A$ to find reactions at $D$. Note 1-2-$\sqrt{3}$ triangle.
$\begin{align}\\ \Sigma M_A &= 0\\ 2\sqrt{3} \,D &= \var{if(sign(vecB[0])=-1,'– ','')} 1\, B_x + \var{if(sign(vecB[1])=1,'– ','')} \sqrt{3}\, B_y + \sqrt{3}\,C \\ D &= \dfrac{ \var{if(sign(vecB[0])=-1,'– ','')} B_x}{2\sqrt{3}} + \dfrac{\var{if(sign(vecB[1])=1,'– ','')}\, B_y }{2} + \dfrac{C }{2}\\D&= \dfrac{1}{2}\left(\dfrac{ \var{precround(vecB[0],1)} }{\sqrt{3}} + \var{precround( - vecB[1],1) } + \var{scalar(magC)} \right)&= \var{display(magD)}\text{ up.}\end{align}$
Apply $\Sigma F_x$ and $\Sigma F_y $ equations to get the components of the reaction force at $A$.
$\begin{align} \\\Sigma F_x &= 0\\ A_x &= B_x \\ &= \var{display(qty(abs(vecA[0]),units))}\quad \var{if(sign(vecA[0])=1,'to the right.','to the left.')} \end{align}$

$\begin{align} \\\Sigma F_y &= 0\\ A_y &= C_y - D_y + B_y \\ A_y &= \simplify[!collectNumbers]{{siground(-VecC[1],5)} + {siground(-VecD[1],5)} + {siground(-vecB[1],5)}} \\ &= \var{display(qty(abs(vecA[1]),units))} \quad \var{if(sign(vecA[1])=1,'up.','down.')} \end{align}$

Analyze the joints:

Draw a FBDs of the joints showing your assumed directions for the forces in the members. Members in tension pull away from the joint, compression push towards the joint.
{geogebra_applet('menecfet',[['α',alpha+'°'],['FB','"'+"B = " + string(magB)+'"'],['FC', '"'+"C = " + string(magC)+'"'],['FD', '"'+"D = " + display(magD)+'"'],['FAx', '"'+"A_x = " + display(qty(-vecA[0],units))+'"'],['FAy', '"'+"A_y = " + display(qty(vecA[1],units))+ '"']])}
Pick a joint with two unknowns, joint $D$ for example. Joint $A$ would also work.
Apply equations of equilibrium to joint $D$ to find forces in members $BD$ and $CD$.
$\begin{align}\\D: \Sigma F_y &= 0\\ BD_y &= D\\ BD &= \dfrac{D}{\sin 30°}\\ &= \dfrac{\var{display(magD)}}{0.5}\\ &= \var{display(BD)} \quad \text{Compression.}\end{align}$

$\begin{align}\\D: \Sigma F_x &= 0\\ CD &=BD_x\\ &= BD \cos 30° \\&= \var{display(BD)} \cos 30° \\&= \var{display(CD)} \quad \text{Tension.}\end{align} $
Move to an adjacent joint with two unknowns, joint $C$ in this case. Joint $C$ is a special case.
$\begin{align}\\ C: \Sigma F_y &= 0\\BC &= C\\ &= \var{display(BC)} \quad \text{Tension.}\end{align}$

$\begin{align}\\ C: \Sigma F_x &= 0\\AC &= CD\\ &= \var{display(AC)} \quad \text{Tension.}\end{align}$
Move to joint $B$ and solve for $AB$.

$B: \Sigma F_y=0$

$-AB_y - BC + BD_y \, \mathbf{\var{if(sign(vecB[1])=1,'+','–')}}\, B_y = 0$

$AB_y= \var{vecC[1]} + \var{precround(vecD[1],1)} \, \var{if(sign(vecB[1])=-1,'','+')} \, \var{precround(vecB[1],1)}= \var{display(-ABy)}$

$\begin{align}AB &= \dfrac{AB_y}{\sin 30°}\\ &= \var{display(-2ABy)}\\ &= \var{display(AB)} \var{if(sign(ABy)=-1,' Tension.', ' Compression.')}\end{align}$

Check your work:

At this point you should check your work by verifying that joint $A$ is in equilibrium. If it is not you have made a mistake somewhere above.

$A: \Sigma F_x \stackrel{?}{=} 0$

$-A_x + AB_x + AC = \var{precround(vecA[0],1)}+ \var{precround(-vecAB[0],1) } + \var{precround(-vecCD[0],1) } = \var{vecA[0] -vecAB[0] - vecCD[0]}$

{if((vecA[0] -vecAB[0] - vecCD[0]) = 0,'Check!', if((vecA[0] -vecAB[0] - vecCD[0]) < 0.00001, 'Close Enough', 'Something is wrong.'))}

$A: \Sigma F_y \stackrel{?}{=} 0$

$A_y + AB_y = \var{precround(vecA[1],1)}+ \var{precround(-vecAB[1],1) } = \var{vecA[1] -vecAB[1]}$

{if((vecA[1] -vecAB[1]) = 0,'Check!', if((vecA[1] -vecAB[1]) < 0.00001, 'Close Enough', 'Something is wrong.'))}

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This question is used in the following exams:

Chapter 06 Exercises by Michael Proudman in Engineering Statics.
28. Method of Joints by Michael Proudman in Engineering Statics.
Trusses by Robert Birch in Robert's workspace.
Chapter 6 Exercises by William Haynes in Engineering Statics.
28. Method of Joints by William Haynes in Engineering Statics.

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Truss: Method of Joints

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

William Haynes 2 years, 10 months ago

William Haynes 6 years, 5 months ago

William Haynes 6 years, 5 months ago

William Haynes 6 years, 5 months ago

Error in variable testing condition

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Generated value: quantity

Parts

Gap-fill

Generated value: `quantity`