Chapter 6.2, Problem 6.70P

Classify each of the structures shown as completely, partially, or improperly constrained; if completely constrained, further classify as determinate or indeterminate. (All members can act both in tension and in compression.)

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 6.70P

The given structure is completely constrained and determinate.

Explanation of Solution

The structure is shown in Fig. P6.70 (a). The free-body diagram of the truss is given in Figure 1.

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,

$r=4$

$m=16$

$n=10$

Compute $m+r$ by substituting $16$ for $m$ and $4$ for $r$.

$\begin{array}{c}m+r=16+4\\ =20\end{array}$

Compute $2n$ by substituting $10$ for $n$.

$\begin{array}{c}2n=2\left(10\right)\\ =20\end{array}$

So, for the given system, $m+r=20=2n$ and hence the condition in equation (I) is satisfied.

Apply the condition for equilibrium in equation (IV) about various points of the structure in the free-body diagrams $\text{I}$ and $\text{II}$ given in Figure-1.

$\begin{array}{l}\text{FBD}\text{I:}\Sigma M_A=0\Rightarrow T_1\\ \text{FBD}\text{II:}\Sigma M_E=0\Rightarrow C_y\end{array}$

Apply the condition for equilibrium in equation (V) about various points of the structure in the free-body diagrams $\text{I}$ and $\text{II}$ given in Figure-1.

$\begin{array}{l}\text{FBD}\text{II:}\Sigma F_x=0\Rightarrow T_2\\ \text{FBD}\text{I}\text{:}\Sigma F_x=0\Rightarrow A_x\\ \text{FBD}\text{I}\text{:}\Sigma F_y=0\Rightarrow A_y\\ \text{FBD}\text{II:}\Sigma F_y=0\Rightarrow E_y\end{array}$

Since each section is a simple truss with reactions determined, the given structure is completely constrained and determinate.

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 6.70P

The given structure is partially constrained.

Explanation of Solution

The structure is shown in Fig. P6.70 (b). It is a non-simple truss.

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

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

Conclusion:

For the given truss,

$r=3$

$m=16$

$n=10$

Compute $m+r$ by substituting $16$ for $m$ and $3$ for $r$.

$\begin{array}{c}m+r=16+3\\ =19\end{array}$

Compute $2n$ by substituting $10$ for $n$.

$\begin{array}{c}2n=2\left(10\right)\\ =20\end{array}$

So, for the given system, $m+r=19<2n=20$ 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 6.70P

The given structure is improperly constrained and indeterminate.

Explanation of Solution

The structure is shown in Fig. P6.70 (c). The free-body diagram of the truss is given in Figure 2.

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

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

Conclusion:

For the given truss,

$r=3$

$m=17$

$n=10$

Compute $m+r$ by substituting $17$ for $m$ and $3$ for $r$.

$\begin{array}{c}m+r=17+3\\ =20\end{array}$

Compute $2n$ by substituting $10$ for $n$.

$\begin{array}{c}2n=2\left(10\right)\\ =20\end{array}$

It is clear that for the given system, the condition in equation (I) is satisfied, but the horizontal reaction forces $A_x$ and $E_x$ are collinear and no equilibrium equation will resolve them, so the given structure is improperly constrained and indeterminate.

Therefore, the given structure is improperly constrained and indeterminate.