(a)
The distance
Answer to Problem 5.2.2P
Explanation of Solution
Given:
An unsymmetrical flexural member consists of a
Concept Used:
As the given flexural member is unsymmetrical, therefore, the plastic neutral axis of the section won’t lie on the center of the member.
We will use the concept of equilibrium of forces, to calculate the distance
Calculation:
Now, as we have the following equation
Where,
As the force is equal, we have
Now, calculating the area of the top flange, as follows:
Substituting the values, we have
Calculating the area of the web above the neutral axis, as follows:
Substituting the values, we have
Now, calculating the area of component above the plastic neutral axis as follows:
Now, calculating the area of the bottom flange, we have
Substitute the
Calculating the area of the web below the neutral axis, as follows:
Substituting the values, we have
Calculating the area of the component below the plastic neutral axis as follows:
Now for computing the neutral axis, use
Substitute the values, we have
Conclusion:
Therefore, the distance
(b)
Plastic moment MPfor the horizontal plastic neutral axis.
Answer to Problem 5.2.2P
Explanation of Solution
Given:
An unsymmetrical flexural member consists of a
Calculation:
We have the following formula for the plastic moment of the section
Where,
And we have following formula for calculating the plastic section modulus
We have,
As we have calculated, the distance
We now have the following diagram to consider:
Calculate the area of the top flange as follows:
Now, the area of the web portion that is left is as follows:
Now, calculating the centroidal distance of the web as follows:
Now, calculating the centroidal distance of the top flange as follows:
Following figure shows the centroidal distances that were found in the above steps.
Calculating
Member Component | |||
Web | |||
Top Flange | |||
Total |
Now calculating the value of
Now, similarly find for the lower half section, we have the following figure
Calculate the area of the bottom flange as follows:
Now, the area of the web portion that is left is as follows:
Now, calculating the centroidal distance of the web as follows:
Now, calculating the centroidal distance of the bottom flange as follows:
Calculating
Member Component | |||
Web | |||
Top Flange | |||
Total |
Now calculating the value of
Now, calculating the plastic section modulus as follows:
Substituting the values, we have
And
Now, for the plastic moment of the section, we have
Substituting the values
Conclusion:
Therefore, the plastic moment MP for the horizontal plastic neutral axis is
(c)
The plastic section modulus Z with respect to the minor principal axis.
Answer to Problem 5.2.2P
Explanation of Solution
Given:
An unsymmetrical flexural member consists of a
Calculation:
To find thePlastic section modulus Z with respect to the minor principal axis, we need to know that the vertical line or axis is considered to be the minor principal axis, therefore as we know that the member is symmetrical along the y-axis, and the same axis is the minor principal axis which confirms that the plastic neutral axis is passing through its center.
We have the following formula for the plastic section modulus:
Where,
and A is the total area of cross section.
Now, as we know that the plastic neutral axis is passing through the center of the minor principal axis, we conclude
And know we need to find any one of them and let that be equal to
Thus, we have
We can find the
Member Component | |||
Top Flange | |||
Web | |||
Bottom Flange | |||
Total |
Calculating the centroidal distance as follows:
Calculation of plastic section modulus is as follows:
Substituting the values, we have
Conclusion:
Therefore, the value of plastic section modulus Z with respect to the minor principal axis is
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Chapter 5 Solutions
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