Chapter 3, Problem 1P

(a)

To determine

Categorize the given structure as unstable, determinate or indeterminate.

Find the degree of indeterminacy if the structure is statically indeterminate.

Answer to Problem 1P

The given structure is geometrically unstable.

Explanation of Solution

Given information:

The structure is given in the Figure.

Type of supports and reactions:

Let the number of reactions be r.

• The roller support has only one vertical reaction. Hence, the number of reactions at roller support is 1.
• The hinged support has one vertical and one horizontal reaction. Hence, the number of reactions at hinged support is 2.
• The fixed support has one horizontal reaction, one vertical reaction, and a couple of moment. Hence, the number of reactions at fixed support is 3.

Refer the given Figure:

• The beam has three roller supports.
• The beam is supported by three vertical reactions. Hence, the total reactions are $r=3$.
• The three reactions are parallel to each other. So the structure is geometrically unstable.

Therefore, the given structure is geometrically unstable.

(b)

To determine

Categorize the given structure as unstable, determinate or indeterminate.

Find the degree of indeterminacy if the structure is statically indeterminate.

Answer to Problem 1P

The given structure is statically determinate.

Explanation of Solution

Given information:

The structure is given in the Figure.

Calculation:

Refer the give Figure:

• The beam is supported by two vertical reactions and one horizontal reaction. Hence, the total reactions are $r=3$.
• The internally stable structure is known as statically determinate if the total reactions of the supports can be determined by solving the equations of equilibrium.
• The structure is internally stable so the structure is statically determinate.

Therefore, the given structure is statically determinate.

(c)

To determine

Categorize the given structure as unstable, determinate or indeterminate.

Find the degree of indeterminacy if the structure is statically indeterminate.

Answer to Problem 1P

The given structure is statically indeterminate.

The degree of external indeterminacy is 2.

Explanation of Solution

Given information:

The structure is given in the Figure.

Calculation:

Refer the give Figure:

• The beam is supported by three vertical reactions, a horizontal reaction, and a couple moment. Hence, the total reactions are $r=5$.
• If the structure is supported more than three reactions, it cannot be determined by the three equations of equilibrium. So, the structure is called as statically indeterminate.
• The value of r is greater than 3 $\left(r>3\right)$. So the structure is called as statically indeterminate.

Therefore, the given structure is statically indeterminate.

Find the degree of external indeterminacy using the relation:

$i_e=r-3$

Substitute 5 for r.

$\begin{array}{c}{i}_e=5-3\\ =2\end{array}$

Therefore, the degree of external indeterminacy is 2.

(d)

To determine

Categorize the given structure as unstable, determinate or indeterminate.

Find the degree of indeterminacy if the structure is statically indeterminate.

Answer to Problem 1P

The given structure is statically indeterminate.

The degree of external indeterminacy is 1.

Explanation of Solution

Given information:

The structure is given in the Figure.

Calculation:

Refer the give Figure:

• The structure has two fixed supports and two internal hinges. Hence, the total reactions is $r=6$.
• The structure is connected by 2 internal hinge so the value of $e_c$ is 2.
• The value of r is greater than 3. So, the structure is internally unstable and It is considered as statically indeterminate.

Therefore, the given structure is statically indeterminate.

Find the degree of external indeterminacy using the relation:

$i_e=r-\left(3+e_c\right)$

Substitute 6 for r and 2 for $e_c$.

$\begin{array}{c}{i}_e=6-\left(3+2\right)\\ =1\end{array}$

Therefore, the degree of external indeterminacy is 1.