Chapter 6.2, Problem 6.74P

6.70 through 6.74 classify as determinate or indeterminate. (All members act both in tension and in compression.)

Fig. P6.74

To determine

Classify the given structure as completely, partially, or improperly constrained and if completely constrained, further classify as determinate or indeterminate.

Answer to Problem 6.74P

The given structure is completely constrained and determinate.

Explanation of Solution

The structure is shown in Fig. P6.74 (a). The arrangement can be redrawn by labeling different points in the truss and is given in Figure 1.

The free-body diagram of the section $EH$ and other portions of the truss is given separately in Figure 2.

Write the condition for a truss to be completely constrained.

$m+r=2n$ (I)

Here, $m$ is the number of members of the truss, $n$ is the number of joints and $r$ is the number of reaction components at its supports.

Write the condition for a truss to be partially constrained.

$m+r<2n$ (II)

Write the expression for a truss to be indeterminate.

$m+r>2n$ (III)

Write the expression for the condition for equilibrium.

$\Sigma M=0$ (IV)

$\Sigma F=0$ (V)

Here, $M$ is the moments of force and $F$ is the force.

Conclusion:

For the given truss,

$m=12$

$r=4$

$n=8$

Compute $m+r$ by substituting $12$ for $m$ and $4$ for $r$.

$\begin{array}{c}m+r=12+4\\ =16\end{array}$

Compute $2n$ by substituting $8$ for $n$.

$\begin{array}{c}2n=2\left(8\right)\\ =16\end{array}$

So, for the given system, $m+r=16=2n$ and hence the condition in equation (I) is satisfied.

Apply the condition for equilibrium in equation (IV) about point $H$ of the structure in the free-body diagrams given in Figure 2.

$\Sigma M_H=0\Rightarrow F_{DE}$

Apply the condition for equilibrium in equation (V) about various points of the structure in the free-body diagrams given in Figure 2.

$\begin{array}{l}\Sigma F_x=0\Rightarrow F_{GH}\\ \Sigma F_y=0\Rightarrow H_y\end{array}$

So, the truss $ABCDGF$ is a simple truss and all forces can be determined.

Therefore, the given structure is completely constrained and determinate.

To determine

Classify the given structure as completely, partially, or improperly constrained and if completely constrained, further classify as determinate or indeterminate.

Answer to Problem 6.74P

The given structure is partially constrained.

Explanation of Solution

The structure is shown in Fig. P6.74 (b). The arrangement can be redrawn by labeling different points in the truss and is given in Figure 3.

From equations (I), (II), and (III), the conditions for the state of constrain of the structure is as follows,

$\begin{array}{l}m+r=2n\left(\text{completely constarined}\right)\\ m+r<2n\left(\text{partially constrained}\right)\\ m+r>2n\left(\text{constrained and indeterminate}\right)\end{array}$

Conclusion:

For the given truss,

$m=12$

$r=3$

$n=8$

Compute $m+r$ by substituting $12$ for $m$ and $3$ for $r$.

$\begin{array}{c}m+r=12+3\\ =15\end{array}$

Compute $2n$ by substituting $8$ for $n$.

$\begin{array}{c}2n=2\left(8\right)\\ =16\end{array}$

So, for the given system, $m+r=15<2n=16$ and hence the condition in equation (II) is satisfied.

Therefore, the given structure is partially constrained.

To determine

Classify the given structure as completely, partially, or improperly constrained and if completely constrained, further classify as determinate or indeterminate.

Answer to Problem 6.74P

The given structure is completely constrained and indeterminate.

Explanation of Solution

The structure is shown in Fig. P6.74 (c). The arrangement can be redrawn by labeling different points in the truss and is given in Figure 4.

From equations (I), (II), and (III), the conditions for the state of constrain of the structure is as follows,

$\begin{array}{l}m+r=2n\left(\text{completely constarined}\right)\\ m+r<2n\left(\text{partially constrained}\right)\\ m+r>2n\left(\text{constrained and indeterminate}\right)\end{array}$

Conclusion:

For the given truss,

$m=13$

$r=4$

$n=8$

Compute $m+r$ by substituting $13$ for $m$ and $4$ for $r$.

$\begin{array}{c}m+r=13+4\\ =17\end{array}$

Compute $2n$ by substituting $8$ for $n$.

$\begin{array}{c}2n=2\left(8\right)\\ =16\end{array}$

So, for the given system, $m+r=17>2n=16$ and hence the condition in equation (III) is satisfied.

Therefore, the given structure is completely constrained and indeterminate.