Chapter 14, Problem 14.1P

Calculate the section modulus for:
(a) a $\text{6}$ -in-by- $\text{1}0$ -in. (S4S) timber cross section, with respect to the strong axis (check with Appendix E).
(b) a pipe cross section having an inner diameter of $\text{1}00$ mm and an outer diameter of $\text{114}$ mm.
(c) the cross section of Example 14.16, with respect to the weak axis.

To determine

(a)

The section modulus of a 6-in.-by-10in. (S4S) timber cross section should be determined.

Answer to Problem 14.1P

   $82.7i{n}^3$.

Explanation of Solution

Given:

The nominal size of timber is 6in. x 16 in.

We can get the dressed dimension of the timber from the table of properties of solid rectangular timber as $5\frac{1}{2}in.\times 15\frac{1}{2}in.$

Concept Used:

The cross-section diagram of S4S timber is as below:

  

Calculation:

We can calculate the section of modulus for given rectangular beam with the strong axis from formula,

   $\begin{array}{l}{s}_x=\frac{I}{c}\\ s_x=\frac{(b{h}^3/12)}{(h/2)}\\ s_x=\frac{b{h}^2}{6}\\ s_x=\frac{5.5\times {(9.5)}^2}{6}\\ s_x=82.7i{n}^3\end{array}$

Conclusion:

The section modulus of a 6-in.-by-10in. (S4S) timber cross section with respect to the strong axis is $82.7i{n}^3$.

To determine

(b)

The section modulus of a pipe cross section having diameter of 100 mm and 114 mm inner and outer respectively should be determined. $59332m{m}^3$

Answer to Problem 14.1P

   $59332m{m}^3$

Explanation of Solution

Given:

The cross section of a pipe with the measures of diameter as,

   $d_i=100mm$ and $d_o=114mm$

Concept Used:

The cross section of the given pipe with inner diameter of 100mm and outer diameter of 114mm is as below:

  

Calculation:

We can calculate the section of modulus for given pipe from formula, 9136124030

   $\begin{array}{l}{s}_x=\frac{I}{c}\\ s_x=\frac{(\pi (d_o{}^4-d_i^4/64)}{(d_o/2)}\\ s_x=\frac{(\frac{\pi ({114}^4-{110}^4)}{64}}{(114/2)}\\ s_x=59332m{m}^3\end{array}$

Conclusion:

The section modulus is $59332m{m}^3$ for pipe cross section having diameter of 100 mm and 114 mm inner and outer respectively.

To determine

(c)

The section modulus of the cross section of example 14.16, with respect to the weak axis should be determined.

Answer to Problem 14.1P

   $s_x=72i{n}^3$.

Explanation of Solution

Given:

The data which we can get from problem 14.16 is b =12 in. and h =6 in.

Concept Used:

The cross section of the given beam as per problem 14.16 is as below:

  

Calculation:

We can calculate the section of modulus for given rectangular beam with the weakest axis from formula,

   $\begin{array}{l}{s}_x=\frac{I}{c}\\ s_x=\frac{(b{h}^3/12)}{(h/2)}\\ s_x=\frac{b{h}^2}{6}\\ s_x=\frac{12\times {(6)}^2}{6}\\ s_x=72i{n}^3\end{array}$

Conclusion:

The section modulus is $72i{n}^3$ for the cross section of rectangular beam as per example 14.16, with respect to the weak axis.