A short column is made by nailing four 1 × 4-in. planks to a 4 × 4-in. timber. Using an allowable stress of 600 psi, determine the largest compressive load P that can be applied at the center of the top section of the timber column as shown if (a) the column is as described, (b) plank 1 is removed, (c) planks 1 and 2 are removed, (d) planks 1, 2, and 3 are removed, (e) all planks are removed.
Fig. P4.112
(a)
Find the largest compressive load P that can be applied at the center of the top section of the timber column.
Answer to Problem 112P
The largest compressive load P is
Explanation of Solution
Given information:
The compressive load P is
The allowable stress
The width
The depth
The width
The depth
Calculation:
Sketch the centric loading as shown in Figure 1.
Refer to Figure 1.
Find the area of the timber section using the relation:
Substitute
Calculate the largest compressive load P using the relation:
Substitute
Thus, the largest compressive load P is
(b)
Find the largest compressive load P that can be applied at the center of the top section of the timber column without plank 1.
Answer to Problem 112P
The largest compressive load P that can be applied at the center of the top section of the timber column without plank 1 is
Explanation of Solution
Calculation:
Sketch the Eccentric loading as shown in Figure 2.
Find the area of the timber section using the relation:
Substitute
Refer to Figure 2.
Find the centroid
Substitute
Refer to Figure 2.
Find the moment of inertia
Substitute
Find the moment of inertia
Substitute
Find the total moment of inertia as follows:
Substitute
Calculate the largest compressive load P that can be applied at the center of the top section of the timber column without plank 1using the relation:
Here, e is the eccentricity, I is the moment of inertia, A is the area of cross section, and c is the distance between the centroid from extreme fibre.
Substitute
Thus, the largest compressive load P that can be applied at the center of the top section of the timber column without plank 1 is
(c)
Find the largest compressive load P that can be applied at the center of the top section of the timber column without plank 1and 2.
Answer to Problem 112P
The largest compressive load P that can be applied at the center of the top section of the timber column without plank 1 and 2 is
Explanation of Solution
Calculation:
Sketch the centric loading as shown in Figure 3.
Refer to Figure 3.
Find the area of the timber section using the relation:
Substitute
Calculate the largest compressive load P using the relation:
Substitute
Thus, the largest compressive load P is
(d)
Find the largest compressive load P that can be applied at the center of the top section of the timber column without plank , 2, and 3.
Answer to Problem 112P
The largest compressive load P that can be applied at the center of the top section of the timber column without plank 1, 2, and 3 is
Explanation of Solution
Calculation:
Sketch the Eccentric loading as shown in Figure 4.
Refer to Figure 4.
Find the area of the timber section using the relation:
Substitute
Find the centroid
Determine the moment of inertia (I) of eccentric section as follows:
Substitute
Calculate the largest compressive load P that can be applied at the center of the top section of the timber column without plank , 2, and 3 using the relation:
Here, e is the eccentricity, I is the moment of inertia, A is the area of cross section, and c is the distance between the centroid from extreme fibre.
Substitute
The largest compressive load P that can be applied at the center of the top section of the timber column without plank 1, 2, and 3 is
(e)
Find the largest compressive load P that can be applied at the center of the top section of the timber all columns are removed.
Answer to Problem 112P
The largest compressive load P that can be applied at the center of the top section of the timber all columns are removed is
Explanation of Solution
Calculation:
Sketch the centric loading as shown in Figure 5.
Refer to Figure 5.
Find the area of the timber section using the relation:
Substitute
Calculate the largest compressive load P using the relation:
Substitute
Thus, the largest compressive load P that can be applied at the center of the top section of the timber all columns are removed is
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