Problem 003-mj | Method of Joints

At joint C
$\Sigma F_V = 0$

$\frac{5}{\sqrt{29}}F_{CD} = 80$

$F_{CD} = 86.16 \, \text{ kN tension}$
 

$\Sigma F_H = 0$

$F_{BC} = \frac{2}{\sqrt{29}}F_{CD}$

$F_{BC} = \frac{2}{\sqrt{29}}(86.16)$

$F_{BC} = 32 \, \text{ kN compression}$
 

At joint D
$\Sigma F_V = 0$

$\frac{5}{\sqrt{29}}F_{BD} = \frac{5}{\sqrt{29}}F_{CD}$

$F_{BD} = F_{CD}$

$F_{BD} = 86.16 \, \text{ kN compression}$
 

$\Sigma F_H = 0$

$F_{DE} = \frac{2}{\sqrt{29}}F_{CD} + \frac{2}{\sqrt{29}}F_{BD}$

$F_{DE} = \frac{2}{\sqrt{29}}(86.16) + \frac{2}{\sqrt{29}}(86.16)$

$F_{DE} = 64 \, \text{ kN tension}$
 

At joint B
$\Sigma F_V = 0$

$\frac{5}{\sqrt{29}}F_{BE} = \frac{5}{\sqrt{29}}F_{BD} + 60$

$\frac{5}{\sqrt{29}}F_{BE} = \frac{5}{\sqrt{29}}(86.16) + 60$

$F_{BE} = 150.78 \, \text{ kN tension}$
 

$\Sigma F_H = 0$

$F_{AB} = F_{BC} + \frac{2}{\sqrt{29}}F_{BD} + \frac{2}{\sqrt{29}}F_{BE}$

$F_{AB} = 32 + \frac{2}{\sqrt{29}}(86.16) + \frac{2}{\sqrt{29}}(150.78)$

$F_{AB} = 120 \, \text{ kN compression}$
 

At joint E
$\Sigma F_V = 0$

$\frac{5}{\sqrt{29}}F_{AE} = \frac{5}{\sqrt{29}}F_{BE}$

$F_{AE} = F_{BE}$

$F_{AE} = 150.78 \, \text{ kN compression}$
 

$\Sigma F_H = 0$

$F_{EF} = F_{DE} + \frac{2}{\sqrt{29}}F_{BE} + \frac{2}{\sqrt{29}}F_{AE}$

$F_{EF} = 64 + \frac{2}{\sqrt{29}}(150.78) + \frac{2}{\sqrt{29}}(150.78)$

$F_{EF} = 176 \, \text{ kN tension}$
 

At joint A
$\Sigma F_V = 0$

$F_{AF} = \frac{5}{\sqrt{29}}F_{AE}$

$F_{AF} = \frac{5}{\sqrt{29}}(150.78)$

$F_{AF} = 140 \, \text{ kN tension}$
 

$\Sigma F_H = 0$

$R_A = F_{AB} + \frac{2}{\sqrt{29}}F_{AE}$

$R_A = 120 + \frac{2}{\sqrt{29}}(150.78)$

$R_A = 176 \, \text{ kN to the left}$
 

At joint F
$\Sigma F_H = 0$

$F_H = F_{EF}$

$F_H = 176 \, \text{ kN to the right}$
 

$\Sigma F_V = 0$

$F_V = F_{AF}$

$F_V = 140 \, \text{ kN upward}$
 

Checking
$\Sigma M_F = 0$

$5R_A = 8(80) + 4(60)$

$R_A = 176 \, \text{ kN to the left}$           check
 

 

$\Sigma F_H = 0$

$F_H = R_A$

$F_H = 176 \, \text{ kN to the right}$           check
 

$\Sigma F_V = 0$

$F_V = 80 + 60$

$F_V = 140 \, \text{ kN upward}$           check
 

Summary
 

 

Top chords
F_DE = 64 kN tension
F_EF = 176 kN tension
 

Bottom chords
F_AB = 120 kN compression
F_BC = 32 kN compression
 

Web members
F_AF = 140 kN tension
F_AE = 150.78 kN compression
F_BE = 150.78 kN tension
F_BD = 86.16 kN compression
F_CD = 86.16 kN tension