Chapter 4, Problem 1P

(a)

To determine

Categorize the given truss is stable or unstable.

Verify the truss is determinate or indeterminate in case the truss is stable.

Find the degree of indeterminacy in case the given truss is statically indeterminate.

Answer to Problem 1P

The plane truss is stable.

The plane truss is statically indeterminate.

The plane truss is indeterminate with indeterminacy of 2.

Explanation of Solution

Use the equations to find the stability of the truss.

$b<2n-3\text{the}\text{truss}\text{is}\text{internally}\text{unstable}$        (1)

$b\ge 2n-3\text{the}\text{truss}\text{is}\text{internally}\text{stable}$        (2)

Here, the b is the number of bars and n is the joints.

Show the conditions for static instability, determinacy, and indeterminacy of the plane truss as follows:

$b+r<2n\text{the truss is statically unstable}$        (3)

$b+r=2n\text{the truss is statically determinate}$        (4)

$b+r>2n\text{the truss is statically indeterminate}$        (5)

Here, r is the support reactions.

Show the expression for the degree of static indeterminacy (i) as follows:

$i=\left(b+r\right)-2n$        (6)

Refer the Figure of the given plane truss.

The number of the bars in the plane truss is $b=11$.

The number of reaction is $r=5$.

The number of the joints in the plane truss is $n=7$.

Substitute the values of b, r, and n with Equation (2).

$\begin{array}{l}11\ge 2\left(7\right)-3\\ 11=11\end{array}$

Thus, the plane truss is stable.

Substitute the values of b, r, and n with Equation (5).

$\begin{array}{l}11+5>2\left(7\right)\\ 16>14\end{array}$

Thus, the plane truss is statically indeterminate.

Find the degree of static indeterminacy of the truss as follows:

$i=\left(b+r\right)-2n$

Substitute the values of b, r, and n with Equation (6).

$\begin{array}{c}i=\left(11+5\right)-2\times 7\\ =16-14\\ =2\end{array}$

Thus, the plane truss is indeterminate with indeterminacy of 2.

(b)

To determine

Categorize the given truss is stable or unstable.

Verify the truss is determinate or indeterminate in case the truss is stable.

Find the degree of indeterminacy in case the given truss is statically indeterminate.

Answer to Problem 1P

The plane truss is stable.

The plane truss is statically indeterminate.

The plane truss is indeterminate with indeterminacy of 2.

Explanation of Solution

Refer the Figure of the given plane truss.

The number of the bars in the plane truss is $b=10$.

The number of reaction is $r=4$.

The number of the joints in the plane truss is $n=6$.

Substitute the values of b, r, and n with Equation (2).

$\begin{array}{l}10\ge 2\left(6\right)-3\\ 10>9\end{array}$

Thus, the plane truss is stable.

Substitute the values of b, r, and n with Equation (5).

$\begin{array}{l}10+4>2\left(6\right)\\ 14>12\end{array}$

Thus, the plane truss is statically indeterminate.

Find the degree of static indeterminacy of the truss as follows:

$i=\left(b+r\right)-2n$

Substitute the values of b, r, and n with Equation (6).

$\begin{array}{c}i=\left(10+4\right)-2\times 6\\ =14-12\\ =2\end{array}$

Thus, the plane truss is indeterminate with indeterminacy of 2.

(c)

To determine

Categorize the given truss is stable or unstable.

Verify the truss is determinate or indeterminate in case the truss is stable.

Find the degree of indeterminacy in case the given truss is statically indeterminate.

Answer to Problem 1P

The plane truss is geometrically unstable.

Explanation of Solution

Refer the Figure of the given plane truss.

The number of the bars in the plane truss is $b=8$.

The number of reaction is $r=4$.

The number of the joints in the plane truss is $n=6$.

Substitute the values of b, r, and n with Equation (1).

$\begin{array}{l}8<2\left(6\right)-3\\ 8=9\end{array}$

Thus, the plane truss is geometrically unstable.

(d)

To determine

Categorize the given truss is stable or unstable.

Verify the truss is determinate or indeterminate in case the truss is stable.

Find the degree of indeterminacy in case the given truss is statically indeterminate.

Answer to Problem 1P

The plane truss is stable.

The plane truss is statically indeterminate.

The plane truss is indeterminate with indeterminacy of 2.

Explanation of Solution

Refer the Figure of the given plane truss.

The number of the bars in the plane truss is $b=10$.

The number of reaction is $r=4$.

The number of the joints in the plane truss is $n=6$.

Substitute the values of b, r, and n with Equation (2).

$\begin{array}{l}10\ge 2\left(6\right)-3\\ 10>9\end{array}$

Thus, the plane truss is stable.

Substitute the values of b, r, and n with Equation (5).

$\begin{array}{l}10+4>2\left(6\right)\\ 14>12\end{array}$

Thus, the plane truss is statically indeterminate.

Find the degree of static indeterminacy of the truss as follows:

$i=\left(b+r\right)-2n$

Substitute the values of b, r, and n with Equation (6).

$\begin{array}{c}i=\left(10+4\right)-2\times 6\\ =14-12\\ =2\end{array}$

Thus, the plane truss is indeterminate with indeterminacy of 2.

(e)

To determine

Categorize the given truss is stable or unstable.

Verify the truss is determinate or indeterminate in case the truss is stable.

Find the degree of indeterminacy in case the given truss is statically indeterminate.

Answer to Problem 1P

The plane truss is geometrically unstable.

Explanation of Solution

Refer the Figure of the given plane truss.

The number of the bars in the plane truss is $b=8$.

The number of reaction is $r=6$.

The number of the joints in the plane truss is $n=7$.

Substitute the values of b, r, and n with Equation (1).

$\begin{array}{l}8<2\left(7\right)-3\\ 8<11\end{array}$

Thus, the plane truss is geometrically unstable.

(f)

To determine

Categorize the given truss is stable or unstable.

Verify the truss is determinate or indeterminate in case the truss is stable.

Find the degree of indeterminacy in case the given truss is statically indeterminate.

Answer to Problem 1P

The plane truss is stable and determinate.

Explanation of Solution

Refer the Figure of the given plane truss.

The number of the bars in the plane truss is $b=38$.

The number of reaction is $r=4$.

The number of the joints in the plane truss is $n=21$.

Substitute the values of b, r, and n with Equation (4).

$\begin{array}{l}38+4=2\left(21\right)\\ 22=22\end{array}$

Thus, the plane truss is stable and determinate.

(g)

To determine

Categorize the given truss is stable or unstable.

Verify the truss is determinate or indeterminate in case the truss is stable.

Find the degree of indeterminacy in case the given truss is statically indeterminate.

Answer to Problem 1P

The plane truss is stable and determinate.

Explanation of Solution

Refer the Figure of the given plane truss.

The number of the bars in the plane truss is $b=10$.

The number of reaction is $r=4$.

The number of the joints in the plane truss is $n=7$.

Substitute the values of b, r, and n with Equation (4).

$\begin{array}{l}10+4=2\left(7\right)\\ 14=14\end{array}$

Thus, the plane truss is stable and determinate.

(h)

To determine

Categorize the given truss is stable or unstable.

Verify the truss is determinate or indeterminate in case the truss is stable.

Find the degree of indeterminacy in case the given truss is statically indeterminate.

Answer to Problem 1P

The plane truss is geometrically unstable.

Explanation of Solution

Refer the Figure of the given plane truss.

The number of the bars in the plane truss is $b=9$.

The number of reaction is $r=3$.

The number of the joints in the plane truss is $n=6$.

The reactions are concurrent.

Refer to the given figure, all the supports are roller. So the structure is geometrically unstable.

Thus, the plane truss is geometrically unstable.