Design of Compression Members

Approach. Box (built-up) column designed to IS 883:1994 cl. 7.6.2: the planks are nailed or bolted to form a hollow square section; the strength depends on $S/\sqrt{d_1^2 + d_2^2}$, where $d_1$ is the overall side and $d_2$ the inside side of the box.

Data and permissible stresses

Length $S = 3.50$ m, axial load $N = 925$ kN, plank thickness $t = 50$ mm. Pin ends, effective length $= S$.

Species: Sal (Shorea robusta), Group A. Location: inside. Grade: Grade I (standard) (Table 1 values are for Grade I).

Permissible stresses (IS 883:1994 Table 1, inside location):

The question gives $E = 12700$ N/mm², $f_{cp} = 10.6$ N/mm² (the Table 1 value for Sal, Grade I, inside), $U = 0.6$ and $q = 1$; these are used.

For plank thickness 50 mm (cl. 7.6.2.5): $U = 0.6$, $q = 1.0$ (values given for 25 mm and 50 mm planks).

Trial section

Square box, overall side $d_1 = 500$ mm, inside side $d_2 = 500 - 2\times50 = 400$ mm.

$$A = d_1^2 - d_2^2 = 500^2 - 400^2 = 90,000\ \text{mm}^2$$ $$\begin{aligned} \frac{S}{\sqrt{d_1^2 + d_2^2}} &= \frac{3500}{\sqrt{500^2 + 400^2}} = \frac{3500}{640.3} = 5.47 \\ K_9 &= \frac{\pi}{2}\sqrt{\frac{UE}{5qf_{cp}}} = \frac{\pi}{2}\sqrt{\frac{0.6\times12700}{5\times1.0\times10.60}} = 18.83 \end{aligned}$$

The ratio 5.47 is less than 8, so it is a short column (cl. 7.6.2.1): $f_c = q f_{cp} = 1.0\times10.60 = 10.60$ N/mm².

Check

$$P = f_cA = 10.60\times90,000/10^3 = 954\ \text{kN} > N = 925\ \text{kN}$$

Safe.

Arrangement of planks (250 mm × 50 mm planks)

   +--------------------------+
   |  plank 50 mm thick     |
   |   +--------------+  |
   |   |   hollow     |  |   outside 500 x 500 mm
   |   |  400 x 400    |  |
   |   +--------------+  |
   +---------------------+

Each 500 mm face is made of two 250 mm planks placed side by side; each 400 mm face is made of one 250 mm and one 150 mm plank (a plank ripped to width), so all planks are 50 mm thick. They are fixed with staggered nails or bolts at a pitch not exceeding about 150 mm to 200 mm so that they act as one unit; the ends are fitted with solid end blocks.

Answer

Provide a box column of overall size 500 mm × 500 mm formed with 50 mm thick Sal planks (inside 400 mm). Capacity 954 kN $>$ 925 kN.

Approach. Working stress design to IS 883:1994 (cl. 7.6 for columns, cl. 7.7 for combined axial compression and bending). Ends are taken as pin-ended, so the effective length equals the unsupported length $S$ (cl. 7.6.1.5).

Data and permissible stresses

Length $S = 4.00$ m = 4000 mm. Axial load $N = 250$ kN.

Species: Deodar (Cedrus deodara), Group C. Location: outside. Grade: Grade I (standard) (Table 1 values are for Grade I).

Permissible stresses (IS 883:1994 Table 1, outside location):

An open shed is an outside location. The grade is not stated, so Grade I (standard) is taken.

Trial section

Try 225 mm × 225 mm (square). Area $A = 50,625$ mm²; least dimension $d = 225$ mm.

Slenderness and permissible axial stress (cl. 7.6.1)

$$\begin{aligned} S/d &= \frac{4000}{225} = 17.8 \\ K_8 &= 0.584\sqrt{\frac{E}{f_{cp}}} = 0.584\sqrt{\frac{9480}{6.90}} = 21.65 \end{aligned}$$

$11 < S/d = 17.8 < K_8 = 21.6$, so it is an intermediate column:

$$f_c = f_{cp}\left[1 - \frac{1}{3}\left(\frac{S/d}{K_8}\right)^4\right] = 6.90\left[1 - \frac{1}{3}\left(\frac{17.8}{21.6}\right)^4\right] = 5.85\ \text{N/mm}^2$$

$S/d = 17.8 < 50$ (cl. 7.6.1.4). OK.

Check

$$P = f_cA = 5.85\times50,625/10^3 = 296.3\ \text{kN} \ge N = 250\ \text{kN}$$

Safe.

Answer

Provide a solid Deodar column of 225 mm × 225 mm cross-section (Grade I (standard), outside location). Capacity $296$ kN $> 250$ kN.

Approach. Box (built-up) column designed to IS 883:1994 cl. 7.6.2: the planks are nailed or bolted to form a hollow square section; the strength depends on $S/\sqrt{d_1^2 + d_2^2}$, where $d_1$ is the overall side and $d_2$ the inside side of the box.

Data and permissible stresses

Length $S = 4.50$ m, axial load $N = 400$ kN, plank thickness $t = 50$ mm. Pin ends, effective length $= S$.

Species: Teak (Tectona grandis), Group B. Location: inside. Grade: Grade I (standard) (Table 1 values are for Grade I).

Permissible stresses (IS 883:1994 Table 1, inside location):

Location is not stated: inside, Grade I assumed.

For plank thickness 50 mm (cl. 7.6.2.5): $U = 0.6$, $q = 1.0$ (values given for 25 mm and 50 mm planks).

Trial section

Square box, overall side $d_1 = 300$ mm, inside side $d_2 = 300 - 2\times50 = 200$ mm.

$$A = d_1^2 - d_2^2 = 300^2 - 200^2 = 50,000\ \text{mm}^2$$ $$\begin{aligned} \frac{S}{\sqrt{d_1^2 + d_2^2}} &= \frac{4500}{\sqrt{300^2 + 200^2}} = \frac{4500}{360.6} = 12.48 \\ K_9 &= \frac{\pi}{2}\sqrt{\frac{UE}{5qf_{cp}}} = \frac{\pi}{2}\sqrt{\frac{0.6\times9970}{5\times1.0\times9.40}} = 17.72 \end{aligned}$$

$8 < 12.48 < K_9 = 17.72$: intermediate column:

$$f_c = q f_{cp}\left[1 - \frac{1}{3}\left(\frac{S}{K_9\sqrt{d_1^2 + d_2^2}}\right)^4\right] = 1.0\times9.40\left[1 - \frac{1}{3}\left(\frac{12.48}{17.72}\right)^4\right] = 8.63\ \text{N/mm}^2$$

Check

$$P = f_cA = 8.63\times50,000/10^3 = 431\ \text{kN} > N = 400\ \text{kN}$$

Safe.

Arrangement

   +--------------------------+
   |  plank 50 mm thick     |
   |   +--------------+  |
   |   |   hollow     |  |   outside 300 x 300 mm
   |   |  200 x 200    |  |
   |   +--------------+  |
   +---------------------+

Planks of 50 mm thickness are fixed with staggered nails or bolts at a pitch not exceeding about 150 mm to 200 mm so that they act as one unit; the ends are fitted with solid end blocks.

Answer

Provide a box column of overall size 300 mm × 300 mm formed with 50 mm thick Teak planks (inside 200 mm). Capacity 431 kN $>$ 400 kN.

Approach. Working stress design to IS 883:1994 (cl. 7.6 for columns, cl. 7.7 for combined axial compression and bending). Ends are taken as pin-ended, so the effective length equals the unsupported length $S$ (cl. 7.6.1.5).

Data and permissible stresses

Length $S = 3.20$ m = 3200 mm. Axial load $N = 200$ kN, bending moment $M = 30$ kN·m.

Species: Sal (Shorea robusta), Group A. Location: inside. Grade: Select grade (Grade I values multiplied by 1.16, IS 883:1994 cl. 6.3).

Permissible stresses (IS 883:1994 Table 1, inside location):

Location is not stated, so inside is taken. The bending moment acts about the major axis (depth $d$); buckling is checked about the minor axis (least dimension $b$).

Trial section

Try 200 mm × 275 mm. Area $A = 55,000$ mm²; least dimension $d = 200$ mm.

Slenderness and permissible axial stress (cl. 7.6.1)

$$\begin{aligned} S/d &= \frac{3200}{200} = 16.0 \\ K_8 &= 0.584\sqrt{\frac{E}{f_{cp}}} = 0.584\sqrt{\frac{12670}{12.30}} = 18.75 \end{aligned}$$

$11 < S/d = 16.0 < K_8 = 18.7$, so it is an intermediate column:

$$f_c = f_{cp}\left[1 - \frac{1}{3}\left(\frac{S/d}{K_8}\right)^4\right] = 12.30\left[1 - \frac{1}{3}\left(\frac{16.0}{18.7}\right)^4\right] = 10.12\ \text{N/mm}^2$$

$S/d = 16.0 < 50$ (cl. 7.6.1.4). OK.

Combined axial compression and bending (cl. 7.7.1)

$$\begin{aligned} f_{ac} &= \frac{N}{A} = \frac{200000}{55,000} = 3.636\ \text{N/mm}^2 \\ f_{ab} &= \frac{6M}{bd^2}\ \text{(bending about the strong axis)} = \frac{6\times30000000}{200\times275^2} = 11.901\ \text{N/mm}^2 \end{aligned}$$

Permissible bending stress $f_b = 19.60$ N/mm²; the form factor is not applied for depths up to 300 mm (cl. 7.5.4).

$$\frac{f_{ac}}{f_c} + \frac{f_{ab}}{f_b} = \frac{3.636}{10.12} + \frac{11.901}{19.60} = 0.966 \le 1.0$$

Safe.

Answer

Provide a solid Sal column of 200 mm × 275 mm cross-section (Select grade, inside location). Combined stress ratio $= 0.97 < 1$.

Approach. A spaced column is two or more equal timber members kept apart by spacer blocks at the ends and at intermediate points and joined by bolts or nails. IS 883:1994 cl. 7.6.3 applies the solid-column formulas with a restraint factor of 2.5 (3 if the end connectors are placed at $S/10$ to $S/20$ from the ends), and the slenderness $S/d$ of each member must not exceed 80 (cl. 7.6.3.3).

Data and permissible stresses

Length $S = 3.5$ m, axial load $N = 120$ kN, ends pin-jointed.

Species: Sal (Shorea robusta), Group A. Location: inside. Grade: Grade I (standard) (Table 1 values are for Grade I).

Permissible stresses (IS 883:1994 Table 1, inside location):

Location and grade not stated: inside, Grade I assumed.

Trial section

Two members, each 100 mm thick × 100 mm wide. Area $A = 2\times100\times100 = 20,000$ mm². The least dimension of one member is $d_1 = 100$ mm.

Slenderness and permissible stress (cl. 7.6.3)

$$\begin{aligned} S/d_1 &= \frac{3500}{100} = 35.0 \ (\le 80,\ \text{OK}) \\ K_{10} &= 0.584\sqrt{\frac{2.5E}{f_{cp}}} = 0.584\sqrt{\frac{2.5\times12670}{10.60}} = 31.9 \end{aligned}$$

$35.0 > K_{10} = 31.9$: long spaced column (cl. 7.6.3.2):

$$f_c = \frac{0.329E\times2.5}{(S/d_1)^2} = \frac{0.329\times12670\times2.5}{35.0^2} = 8.51\ \text{N/mm}^2$$

Check

$$P = f_cA = 8.51\times20,000/10^3 = 170\ \text{kN} > 120\ \text{kN}$$

Safe.

Details of the spaced column

• Clear spacing between the two members taken as $100$ mm (equal to the member thickness; this spacing is a detailing choice), kept by spacer blocks of the same thickness and species at both ends, at mid-height and at the quarter points; the end blocks are placed at $S/20$ from the ends (so the factor 2.5 is used).
• Each spacer block is connected to both members with two bolts (12 mm) or staggered nails.

   |  |   |  |      two members 100 x 100 mm
   |  |===|  |      spacer block
   |  |   |  |
   |  |===|  |      spacer block (mid-height)
   |  |   |  |
   |  |===|  |      end block

Answer

Provide a spaced column of two 100 mm × 100 mm Sal members with 100 mm spacing and spacer blocks as shown. Capacity 170 kN $>$ 120 kN.

Approach. Box (built-up) column designed to IS 883:1994 cl. 7.6.2: the planks are nailed or bolted to form a hollow square section; the strength depends on $S/\sqrt{d_1^2 + d_2^2}$, where $d_1$ is the overall side and $d_2$ the inside side of the box.

Data and permissible stresses

Length $S = 5.00$ m, axial load $N = 400$ kN, plank thickness $t = 50$ mm. Pin ends, effective length $= S$.

Species: Sal (Shorea robusta), Group A. Location: inside. Grade: Grade I (standard) (Table 1 values are for Grade I).

Permissible stresses (IS 883:1994 Table 1, inside location):

Location not stated: inside, Grade I assumed.

For plank thickness 50 mm (cl. 7.6.2.5): $U = 0.6$, $q = 1.0$ (values given for 25 mm and 50 mm planks).

Trial section

Square box, overall side $d_1 = 275$ mm, inside side $d_2 = 275 - 2\times50 = 175$ mm.

$$A = d_1^2 - d_2^2 = 275^2 - 175^2 = 45,000\ \text{mm}^2$$ $$\begin{aligned} \frac{S}{\sqrt{d_1^2 + d_2^2}} &= \frac{5000}{\sqrt{275^2 + 175^2}} = \frac{5000}{326.0} = 15.34 \\ K_9 &= \frac{\pi}{2}\sqrt{\frac{UE}{5qf_{cp}}} = \frac{\pi}{2}\sqrt{\frac{0.6\times12670}{5\times1.0\times10.60}} = 18.81 \end{aligned}$$

$8 < 15.34 < K_9 = 18.81$: intermediate column:

$$f_c = q f_{cp}\left[1 - \frac{1}{3}\left(\frac{S}{K_9\sqrt{d_1^2 + d_2^2}}\right)^4\right] = 1.0\times10.60\left[1 - \frac{1}{3}\left(\frac{15.34}{18.81}\right)^4\right] = 9.04\ \text{N/mm}^2$$

Check

$$P = f_cA = 9.04\times45,000/10^3 = 407\ \text{kN} > N = 400\ \text{kN}$$

Safe.

Arrangement

   +--------------------------+
   |  plank 50 mm thick     |
   |   +--------------+  |
   |   |   hollow     |  |   outside 275 x 275 mm
   |   |  175 x 175    |  |
   |   +--------------+  |
   +---------------------+

Planks of 50 mm thickness are fixed with staggered nails or bolts at a pitch not exceeding about 150 mm to 200 mm so that they act as one unit; the ends are fitted with solid end blocks.

Answer

Provide a box column of overall size 275 mm × 275 mm formed with 50 mm thick Sal planks (inside 175 mm). Capacity 407 kN $>$ 400 kN.

Approach. Working stress design to IS 883:1994 (cl. 7.6 for columns, cl. 7.7 for combined axial compression and bending). Ends are taken as pin-ended, so the effective length equals the unsupported length $S$ (cl. 7.6.1.5).

Data and permissible stresses

Length $S = 4.00$ m = 4000 mm. Axial load $N = 150$ kN, bending moment $M = 15$ kN·m.

Species: Sal (Shorea robusta), Group A. Location: inside. Grade: Grade I (standard) (Table 1 values are for Grade I).

Permissible stresses (IS 883:1994 Table 1, inside location):

Location and grade not stated: inside, Grade I (standard) assumed. Bending about the major axis.

Trial section

Try 175 mm × 250 mm. Area $A = 43,750$ mm²; least dimension $d = 175$ mm.

Slenderness and permissible axial stress (cl. 7.6.1)

$$\begin{aligned} S/d &= \frac{4000}{175} = 22.9 \\ K_8 &= 0.584\sqrt{\frac{E}{f_{cp}}} = 0.584\sqrt{\frac{12670}{10.60}} = 20.19 \end{aligned}$$

$S/d = 22.9 > K_8 = 20.2$, so it is a long column:

$$f_c = \frac{0.329E}{(S/d)^2} = \frac{0.329\times12670}{22.9^2} = 7.98\ \text{N/mm}^2$$

$S/d = 22.9 < 50$ (cl. 7.6.1.4). OK.

Combined axial compression and bending (cl. 7.7.1)

$$\begin{aligned} f_{ac} &= \frac{N}{A} = \frac{150000}{43,750} = 3.429\ \text{N/mm}^2 \\ f_{ab} &= \frac{6M}{bd^2}\ \text{(bending about the strong axis)} = \frac{6\times15000000}{175\times250^2} = 8.229\ \text{N/mm}^2 \end{aligned}$$

Permissible bending stress $f_b = 16.90$ N/mm²; the form factor is not applied for depths up to 300 mm (cl. 7.5.4).

$$\frac{f_{ac}}{f_c} + \frac{f_{ab}}{f_b} = \frac{3.429}{7.98} + \frac{8.229}{16.90} = 0.917 \le 1.0$$

Safe.

Answer

Provide a solid Sal column of 175 mm × 250 mm cross-section (Grade I (standard), inside location). Combined stress ratio $= 0.92 < 1$.

Approach. Combined axial compression and bending, IS 883:1994 cl. 7.7.1: the column is safe if $\dfrac{f_{ac}}{f_c} + \dfrac{f_{ab}}{f_b} \le 1$.

Data and permissible stresses

Section 200 × 200 mm, effective length $S = 2500$ mm, $N = 30$ kN, $M = 1.5$ kN·m.

Species: Deodar (Cedrus deodara), Group C. Location: outside. Grade: Grade I (standard) (Table 1 values are for Grade I).

Permissible stresses (IS 883:1994 Table 1, outside location):

The column is located outside the building, so the outside values are taken; the grade is not stated, so Grade I (standard).

Permissible axial stress

$$\begin{aligned} S/d &= 2500/200 = 12.5 \\ K_8 &= 0.584\sqrt{E/f_{cp}} = 0.584\sqrt{9480/6.9} = 21.65 \end{aligned}$$

$11 < S/d < K_8$, so it is an intermediate column:

$$f_c = f_{cp}\left[1 - \frac{1}{3}\left(\frac{S/d}{K_8}\right)^4\right] = 6.9\left[1 - \frac{1}{3}\left(\frac{12.5}{21.65}\right)^4\right] = 6.644\ \text{N/mm}^2$$

Actual stresses

$$\begin{aligned} f_{ac} &= \frac{30\times10^3}{200\times200} = 0.750\ \text{N/mm}^2 \\ f_{ab} &= \frac{6M}{bd^2} = \frac{6\times1.5\times10^6}{200^3} = 1.125\ \text{N/mm}^2 \end{aligned}$$

Check

$$\frac{f_{ac}}{f_c} + \frac{f_{ab}}{f_b} = \frac{0.750}{6.644} + \frac{1.125}{8.7} = 0.113 + 0.129 = 0.242$$

Answer: the interaction value is 0.24 < 1, so the column is safe. (Depth is 200 mm, so no form factor is applied to the bending stress.)

Approach. Working stress design to IS 883:1994 (cl. 7.6 for columns, cl. 7.7 for combined axial compression and bending). Ends are taken as pin-ended, so the effective length equals the unsupported length $S$ (cl. 7.6.1.5).

Data and permissible stresses

Length $S = 4.00$ m = 4000 mm. Axial load $N = 50$ kN.

Species: Sal (Shorea robusta), Group A. Location: inside. Grade: Grade I (standard) (Table 1 values are for Grade I).

Permissible stresses (IS 883:1994 Table 1, inside location):

Location and grade not stated: inside, Grade I assumed. A square section is chosen.

Trial section

Try 125 mm × 125 mm (square). Area $A = 15,625$ mm²; least dimension $d = 125$ mm.

Slenderness and permissible axial stress (cl. 7.6.1)

$$\begin{aligned} S/d &= \frac{4000}{125} = 32.0 \\ K_8 &= 0.584\sqrt{\frac{E}{f_{cp}}} = 0.584\sqrt{\frac{12670}{10.60}} = 20.19 \end{aligned}$$

$S/d = 32.0 > K_8 = 20.2$, so it is a long column:

$$f_c = \frac{0.329E}{(S/d)^2} = \frac{0.329\times12670}{32.0^2} = 4.07\ \text{N/mm}^2$$

$S/d = 32.0 < 50$ (cl. 7.6.1.4). OK.

Check

$$P = f_cA = 4.07\times15,625/10^3 = 63.6\ \text{kN} \ge N = 50\ \text{kN}$$

Safe.

Answer

Provide a solid Sal column of 125 mm × 125 mm cross-section (Grade I (standard), inside location). Capacity $64$ kN $> 50$ kN.

Slenderness ratio

The slenderness ratio of a timber column is the ratio of its unsupported (effective) length $S$ to its least lateral dimension $d$, $S/d$ (IS 883:1994 cl. 7.6), not to the radius of gyration as in steel.

Types of columns by slenderness

with $K_8 = 0.584\sqrt{E/f_{cp}}$, $K_9 = \dfrac{\pi}{2}\sqrt{\dfrac{UE}{5qf_{cp}}}$ and $K_{10} = 0.584\sqrt{\dfrac{2.5E}{f_{cp}}}$.

Permissible stress in each class

• Short: column crushes; $f_c = f_{cp}$.
• Intermediate: $f_c = f_{cp}\left[1 - \dfrac{1}{3}\left(\dfrac{S/d}{K_8}\right)^4\right]$.
• Long: buckles elastically; $f_c = \dfrac{0.329E}{(S/d)^2}$.

Definition of the slenderness ratio for the three types

1. Solid column: $S/d$, with $d$ the smaller side of the rectangular section (diameter for a circular column; a circular column is treated as a square of equal area). The limit is $S/d \le 50$.
2. Built-up (box) column: $\dfrac{S}{\sqrt{d_1^2 + d_2^2}}$, where $d_1$ is the overall and $d_2$ the inside dimension of the box in the direction of buckling (so the hollow section is judged by an equivalent radius of gyration, $r = \sqrt{(d_1^2 + d_2^2)/12}$). The strength is then multiplied by the constants $q$ and $U$ of the plank thickness (25 mm: $U = 0.8$, $q = 1.0$; 50 mm: $U = 0.6$, $q = 1.0$).
3. Spaced column: $S/d$ of the individual member, with $d$ the least dimension of one member; it must not exceed 80 (cl. 7.6.3.3). The spacer blocks give a restraint factor of 2.5 (3 if the end fasteners are placed at $S/10$ to $S/20$ from the ends), so the long-column stress is $0.329E\times2.5/(S/d)^2$.

Approach. Working stress design to IS 883:1994 (cl. 7.6 for columns, cl. 7.7 for combined axial compression and bending). Ends are taken as pin-ended, so the effective length equals the unsupported length $S$ (cl. 7.6.1.5).

Data and permissible stresses

Length $S = 2.00$ m = 2000 mm. Axial load $N = 75$ kN, bending moment $M = 12$ kN·m.

Species: Sal (Shorea robusta), Group A. Location: inside. Grade: Grade I (standard) (Table 1 values are for Grade I).

Permissible stresses (IS 883:1994 Table 1, inside location):

The loads are used as given (IS 883 is a working-stress code, so they are treated as the design loads); location and grade are not stated: inside, Grade I. Bending about the major axis.

Trial section

Try 125 mm × 225 mm. Area $A = 28,125$ mm²; least dimension $d = 125$ mm.

Slenderness and permissible axial stress (cl. 7.6.1)

$$\begin{aligned} S/d &= \frac{2000}{125} = 16.0 \\ K_8 &= 0.584\sqrt{\frac{E}{f_{cp}}} = 0.584\sqrt{\frac{12670}{10.60}} = 20.19 \end{aligned}$$

$11 < S/d = 16.0 < K_8 = 20.2$, so it is an intermediate column:

$$f_c = f_{cp}\left[1 - \frac{1}{3}\left(\frac{S/d}{K_8}\right)^4\right] = 10.60\left[1 - \frac{1}{3}\left(\frac{16.0}{20.2}\right)^4\right] = 9.21\ \text{N/mm}^2$$

$S/d = 16.0 < 50$ (cl. 7.6.1.4). OK.

Combined axial compression and bending (cl. 7.7.1)

$$\begin{aligned} f_{ac} &= \frac{N}{A} = \frac{75000}{28,125} = 2.667\ \text{N/mm}^2 \\ f_{ab} &= \frac{6M}{bd^2}\ \text{(bending about the strong axis)} = \frac{6\times12000000}{125\times225^2} = 11.378\ \text{N/mm}^2 \end{aligned}$$

Permissible bending stress $f_b = 16.90$ N/mm²; the form factor is not applied for depths up to 300 mm (cl. 7.5.4).

$$\frac{f_{ac}}{f_c} + \frac{f_{ab}}{f_b} = \frac{2.667}{9.21} + \frac{11.378}{16.90} = 0.963 \le 1.0$$

Safe.

Answer

Provide a solid Sal column of 125 mm × 225 mm cross-section (Grade I (standard), inside location). Combined stress ratio $= 0.96 < 1$.

Approach. Working stress design to IS 883:1994 (cl. 7.6 for columns, cl. 7.7 for combined axial compression and bending). Ends are taken as pin-ended, so the effective length equals the unsupported length $S$ (cl. 7.6.1.5).

Data and permissible stresses

Length $S = 2.00$ m = 2000 mm. Axial load $N = 500$ kN, bending moment $M = 50$ kN·m.

Species: Sal (Shorea robusta), Group A. Location: inside. Grade: Select grade (Grade I values multiplied by 1.16, IS 883:1994 cl. 6.3).

Permissible stresses (IS 883:1994 Table 1, inside location):

Location not stated: inside. Bending about the major axis; buckling about the minor axis.

Trial section

Try 225 mm × 375 mm. Area $A = 84,375$ mm²; least dimension $d = 225$ mm.

Slenderness and permissible axial stress (cl. 7.6.1)

$$\begin{aligned} S/d &= \frac{2000}{225} = 8.9 \\ K_8 &= 0.584\sqrt{\frac{E}{f_{cp}}} = 0.584\sqrt{\frac{12670}{12.30}} = 18.75 \end{aligned}$$

$S/d \le 11$, so it is a short column and $f_c = f_{cp} = 12.30$ N/mm².

$S/d = 8.9 < 50$ (cl. 7.6.1.4). OK.

Combined axial compression and bending (cl. 7.7.1)

$$\begin{aligned} f_{ac} &= \frac{N}{A} = \frac{500000}{84,375} = 5.926\ \text{N/mm}^2 \\ f_{ab} &= \frac{6M}{bd^2}\ \text{(bending about the strong axis)} = \frac{6\times50000000}{225\times375^2} = 9.481\ \text{N/mm}^2 \end{aligned}$$

Permissible bending stress $f_b = 19.60$ N/mm²; depth 375 mm $> 300$ mm so the form factor $K = 0.81\,(d^2 + 89400)/(d^2 + 55000) = 0.952$ is applied: $f_b = 18.67$ N/mm².

$$\frac{f_{ac}}{f_c} + \frac{f_{ab}}{f_b} = \frac{5.926}{12.30} + \frac{9.481}{18.67} = 0.990 \le 1.0$$

Safe.

Answer

Provide a solid Sal column of 225 mm × 375 mm cross-section (Select grade, inside location). Combined stress ratio $= 0.99 < 1$.

Approach. Safe axial load of a solid column, IS 883:1994 cl. 7.6.1: $P = f_c\times A$, where $f_c$ depends on the slenderness $S/d$.

Data and permissible stresses

Section 225 × 225 mm, unsupported length $S = 3000$ mm (pin ends), Select grade Sal.

Species: Sal (Shorea robusta), Group A. Location: inside. Grade: Select grade (Grade I values multiplied by 1.16, IS 883:1994 cl. 6.3).

Permissible stresses (IS 883:1994 Table 1, inside location):

Location not stated: inside.

Slenderness

$$\begin{aligned} S/d &= \frac{3000}{225} = 13.33 \\ K_8 &= 0.584\sqrt{\frac{E}{f_{cp}}} = 0.584\sqrt{\frac{12670}{12.30}} = 18.75 \end{aligned}$$

$11 < S/d = 13.33 < K_8 = 18.75$, so it is an intermediate column.

$$f_c = f_{cp}\left[1 - \frac{1}{3}\left(\frac{S/d}{K_8}\right)^4\right] = 12.30\left[1 - \frac{1}{3}\left(\frac{13.33}{18.75}\right)^4\right] = 11.25\ \text{N/mm}^2$$

Safe load

$$P = f_cA = 11.25\times50,625/10^3 = 569\ \text{kN}$$

Answer: the safe axial load is about 569 kN (permissible stress $f_c = 11.25$ N/mm²; $S/d = 13.3 < 50$).

Approach. Box (built-up) column designed to IS 883:1994 cl. 7.6.2: the planks are nailed or bolted to form a hollow square section; the strength depends on $S/\sqrt{d_1^2 + d_2^2}$, where $d_1$ is the overall side and $d_2$ the inside side of the box.

Data and permissible stresses

Length $S = 5.00$ m, axial load $N = 350$ kN, plank thickness $t = 60$ mm. Pin ends, effective length $= S$.

Species: Deodar (Cedrus deodara), Group C. Location: inside. Grade: Grade I (standard) (Table 1 values are for Grade I).

Permissible stresses (IS 883:1994 Table 1, inside location):

Location and grade not stated: inside, Grade I (standard) assumed.

For plank thickness 60 mm (cl. 7.6.2.5): $U = 0.6$, $q = 1.0$ (values given for 25 mm and 50 mm planks; the 50 mm values are used).

Trial section

Square box, overall side $d_1 = 300$ mm, inside side $d_2 = 300 - 2\times60 = 180$ mm.

$$A = d_1^2 - d_2^2 = 300^2 - 180^2 = 57,600\ \text{mm}^2$$ $$\begin{aligned} \frac{S}{\sqrt{d_1^2 + d_2^2}} &= \frac{5000}{\sqrt{300^2 + 180^2}} = \frac{5000}{349.9} = 14.29 \\ K_9 &= \frac{\pi}{2}\sqrt{\frac{UE}{5qf_{cp}}} = \frac{\pi}{2}\sqrt{\frac{0.6\times9480}{5\times1.0\times7.80}} = 18.97 \end{aligned}$$

$8 < 14.29 < K_9 = 18.97$: intermediate column:

$$f_c = q f_{cp}\left[1 - \frac{1}{3}\left(\frac{S}{K_9\sqrt{d_1^2 + d_2^2}}\right)^4\right] = 1.0\times7.80\left[1 - \frac{1}{3}\left(\frac{14.29}{18.97}\right)^4\right] = 6.96\ \text{N/mm}^2$$

Check

$$P = f_cA = 6.96\times57,600/10^3 = 401\ \text{kN} > N = 350\ \text{kN}$$

Safe.

Arrangement

   +--------------------------+
   |  plank 60 mm thick     |
   |   +--------------+  |
   |   |   hollow     |  |   outside 300 x 300 mm
   |   |  180 x 180    |  |
   |   +--------------+  |
   +---------------------+

Planks of 50 mm thickness are fixed with staggered nails or bolts at a pitch not exceeding about 150 mm to 200 mm so that they act as one unit; the ends are fitted with solid end blocks.

Answer

Provide a box column of overall size 300 mm × 300 mm formed with 60 mm thick Deodar planks (inside 180 mm). Capacity 401 kN $>$ 350 kN.

Approach. Working stress design to IS 883:1994 (cl. 7.6 for columns, cl. 7.7 for combined axial compression and bending). Ends are taken as pin-ended, so the effective length equals the unsupported length $S$ (cl. 7.6.1.5).

Data and permissible stresses

Length $S = 4.00$ m = 4000 mm. Axial load $N = 350$ kN.

Species: Deodar (Cedrus deodara), Group C. Location: outside. Grade: Select grade (Grade I values multiplied by 1.16, IS 883:1994 cl. 6.3).

Permissible stresses (IS 883:1994 Table 1, outside location):

The question states outside location and Select grade.

Trial section

Try 250 mm × 250 mm (square). Area $A = 62,500$ mm²; least dimension $d = 250$ mm.

Slenderness and permissible axial stress (cl. 7.6.1)

$$\begin{aligned} S/d &= \frac{4000}{250} = 16.0 \\ K_8 &= 0.584\sqrt{\frac{E}{f_{cp}}} = 0.584\sqrt{\frac{9480}{8.00}} = 20.10 \end{aligned}$$

$11 < S/d = 16.0 < K_8 = 20.1$, so it is an intermediate column:

$$f_c = f_{cp}\left[1 - \frac{1}{3}\left(\frac{S/d}{K_8}\right)^4\right] = 8.00\left[1 - \frac{1}{3}\left(\frac{16.0}{20.1}\right)^4\right] = 6.93\ \text{N/mm}^2$$

$S/d = 16.0 < 50$ (cl. 7.6.1.4). OK.

Check

$$P = f_cA = 6.93\times62,500/10^3 = 433.3\ \text{kN} \ge N = 350\ \text{kN}$$

Safe.

Answer

Provide a solid Deodar column of 250 mm × 250 mm cross-section (Select grade, outside location). Capacity $433$ kN $> 350$ kN.

Approach. IS 883:1994 cl. 7.6.1.6: the permissible load on a circular column must not exceed that of a square column of equal cross-sectional area. So the circular column is designed through its equivalent square side $a$, with $a^2 = \pi D^2/4$, i.e. $a = 0.886D$.

Data and permissible stresses

Height (unsupported length) $S = 3.5$ m, axial load $N = 200$ kN, pin ends.

Species: Sal (Shorea robusta), Group A. Location: outside. Grade: Grade I (standard) (Table 1 values are for Grade I).

Permissible stresses (IS 883:1994 Table 1, outside location):

An open shed is an outside location; grade not stated: Grade I.

Trial section

Try diameter $D = 180$ mm. Area $A = \pi D^2/4 = 25,447$ mm². Equivalent square side:

$$a = 0.886D = 159.5\ \text{mm}$$

Slenderness and permissible stress

$$\begin{aligned} S/a &= \frac{3500}{159.5} = 21.94 \\ K_8 &= 0.584\sqrt{\frac{E}{f_{cp}}} = 0.584\sqrt{\frac{12670}{9.40}} = 21.44 \end{aligned}$$

Class: long; $f_c = 8.66$ N/mm² (long column: $0.329E/(S/a)^2$).

Check

$$P = f_cA = 8.66\times25,447/10^3 = 220\ \text{kN} > 200\ \text{kN}$$

Safe.

Answer: provide a circular Sal column of diameter 180 mm. (The permissible load equals that of a 160 mm square column of the same area.)

Approach. Working stress design to IS 883:1994 (cl. 7.6 for columns, cl. 7.7 for combined axial compression and bending). Ends are taken as pin-ended, so the effective length equals the unsupported length $S$ (cl. 7.6.1.5).

Data and permissible stresses

Length $S = 4.00$ m = 4000 mm. Axial load $N = 300$ kN.

Species: Deodar (Cedrus deodara), Group C. Location: inside. Grade: Grade I (standard) (Table 1 values are for Grade I).

Permissible stresses (IS 883:1994 Table 1, inside location):

Location and grade not stated: inside, Grade I (standard) assumed; a square section is chosen.

Trial section

Try 225 mm × 225 mm (square). Area $A = 50,625$ mm²; least dimension $d = 225$ mm.

Slenderness and permissible axial stress (cl. 7.6.1)

$$\begin{aligned} S/d &= \frac{4000}{225} = 17.8 \\ K_8 &= 0.584\sqrt{\frac{E}{f_{cp}}} = 0.584\sqrt{\frac{9480}{7.80}} = 20.36 \end{aligned}$$

$11 < S/d = 17.8 < K_8 = 20.4$, so it is an intermediate column:

$$f_c = f_{cp}\left[1 - \frac{1}{3}\left(\frac{S/d}{K_8}\right)^4\right] = 7.80\left[1 - \frac{1}{3}\left(\frac{17.8}{20.4}\right)^4\right] = 6.29\ \text{N/mm}^2$$

$S/d = 17.8 < 50$ (cl. 7.6.1.4). OK.

Check

$$P = f_cA = 6.29\times50,625/10^3 = 318.4\ \text{kN} \ge N = 300\ \text{kN}$$

Safe.

Answer

Provide a solid Deodar column of 225 mm × 225 mm cross-section (Grade I (standard), inside location). Capacity $318$ kN $> 300$ kN.

Approach. Combined axial compression and bending, IS 883:1994 cl. 7.7.1: safe if $\dfrac{f_{ac}}{f_c} + \dfrac{f_{ab}}{f_b} \le 1$.

Data and permissible stresses

Section 250 × 250 mm, $S = 4000$ mm (pin ends), $N = 200$ kN, $M = 20$ kN·m.

Species: Sal (Shorea robusta), Group A. Location: inside. Grade: Grade I (standard) (Table 1 values are for Grade I).

Permissible stresses (IS 883:1994 Table 1, inside location):

Location and grade are not stated: inside, Grade I (standard) assumed.

Permissible axial stress

$$\begin{aligned} S/d &= 4000/250 = 16.0 \\ K_8 &= 0.584\sqrt{12670/10.6} = 20.19 \end{aligned}$$

$11 < S/d < K_8$: intermediate column.

$$f_c = 10.6\left[1 - \frac{1}{3}\left(\frac{16.0}{20.19}\right)^4\right] = 9.207\ \text{N/mm}^2$$

Actual stresses

$$\begin{aligned} f_{ac} &= \frac{200\times10^3}{250\times250} = 3.200\ \text{N/mm}^2 \\ f_{ab} &= \frac{6M}{bd^2} = \frac{6\times20\times10^6}{250^3} = 7.680\ \text{N/mm}^2 \end{aligned}$$

(The depth is 250 mm, which is less than 300 mm, so no form factor: $f_b = 16.9$ N/mm².)

Check

$$\frac{3.200}{9.207} + \frac{7.680}{16.9} = 0.348 + 0.454 = 0.802$$

Answer: the value is 0.80, which is less than 1: the column can carry the 200 kN load and 20 kN·m moment safely.