To decrease the weight and to aid in the hardening process, the concrete sections in Exercise 88

step 1 In this problem, we have a concrete section as shown in the figure. Four cylindrical openings ha

To decrease the weight and to aid in the hardening process,...

Key Concepts

Weight Reduction and Curing Benefits

Reducing the weight of concrete members by incorporating voids not only eases handling and reduces dead loads but can also positively influence the curing process. Lighter sections tend to experience more uniform temperature distribution during curing, potentially leading to more consistent hardening and reduced thermal stresses.

Reinforced Concrete Design

This concept involves designing concrete elements that integrate reinforcement (usually steel) to enhance structural performance under various loads. It encompasses the principles of durability, strength, and stiffness while considering the effects of material properties, which are critical in ensuring that even modified geometries like hollow sections perform adequately.

Geometric Property Analysis

Understanding the geometry of a structural section—including the effects of modifications such as openings—is essential for calculating key properties like cross-sectional area, moment of inertia, and section modulus. These properties directly influence the element’s ability to resist bending, shear, and other stresses under applied loads.

Hollow Structural Sections

This concept focuses on the design and analysis of concrete sections that incorporate voids or openings. Hollow sections reduce self-weight while maintaining sufficient strength and stiffness. They require careful consideration of how the removal of material affects load distribution and overall structural performance.

Transcript

00:01 In this problem, we have a concrete section as shown in the figure.

00:05 Four cylindrical openings have been provided in this concrete section to reduce its weight.

00:11 We are required to find the area of the face of the section.

00:15 After that, we need to find the volume of the concrete in one of the section.

00:19 And finally, we are required to find its weight.

00:23 So let's see how to solve this question.

00:27 Since area of the face is symmetric about y -axis, therefore, the expression to calculate area can be written as 2 into integration 0 to 5 minus 1 upon 3 under root 5 minus x d x plus 2 into 1 into 0 .5 minus and now we are going to reduce the area of these hollow cylindrical sections.

01:03 2 .2.

01:05 Pi by 4 into d squared that means 1 upon 4 to the power 2 minus 2 2 by 5 by 4 into 1 upon 8 to the power 2 and now let's solve this we get area a is equal to 2 in 2 and the integration of 1 d x will be equal to x minus the integration of 1 upon 3 and route 5 minus x d x d x will be equal to minus 2 upon 9 so here we will have plus 2 upon 9 5 minus x to the power 3 upon 2 and the limits are 0 to 5 plus 2 into 1 .5 that means 1 minus and when we calculate these 2 times 2 into pi by 4 into 1 by 4 to the power 2 minus 2 2 into 5 by 4 into 1 by 8 to the power 2 we get 0 .1 to 27.

02:05 Now substitute the limits so we can write 2 into 5 minus 5 minus 2x9 into 5 to the power 3 by 2 plus 1 minus 0 .127.

02:22 When we further solve this, finally we get area a is equal to 5 .908 square meter.

02:34 So this is the final answer for part a and now let's move to part b...

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