Pre-stressed Concrete Structures (04)

Dear Viewers,
Hello & Good-Day to everyday. Hope all of you are well with a very very happy & fresh mind.
Today I am going to explain you about some Mathematical Problem based of Three (03) concepts of Pre-stressed Concrete Structures.

At my this Blog, I have already explained about the theoretical term of Three (03) Concepts of Pre-stressed Concrete Structures which you can review by

http://seasoft022.blogspot.com/2013/05/pre-stressed-concrete-structures-02.html

Now I will explain you these Three (03) Concepts of Pre-stressed Concrete Structures with Mathematical Problem & Solution.
So, Lets start,

Before starting the mathematical problem & solution at first lets review the three (03) concepts of Pre-stressed Concrete Structures again shortly.

There are Three (03) Concepts Pre-stressed Concrete Structures, these are as follows,

1^st concept: Pre-stressing to transform concrete into elastic material,

2^nd concept: Pre-stressing for combination of high strength steel to high strength concrete,

3^rd concrete: Pre-stressing to achieve load balancing.

Now we have considered a Mathematical Problem. This Mathematical Problem which will be solved by using Three (03) Concepts of Pre-stressed Concrete Structures.
So, the Question of our desire Mathematical Problem is as follows,

Question: A Pre-stressed Concrete Beam of 12" x 36" has a simple span of 30' and is loaded by a uniform load of 3.27 kip/ft including its own weight. The Pre-stressing Tension is located at an eccentrically of e=8" and produce an effective prestress of 350 kip.

Compute fiber stresses in the concrete at mid-span section.

Solution: Now lets go to solve this mathematical problem at first by using 1st concept of Pre-stressed Concrete Structures,
Picture-01 & Picture-02 indicates the mathematical solution by using the 1st Concept of Pre-stressed Concrete Structures,

1st Concept of Pre-stressed Concrete Structures, Page 01 of 02
By Engr. Snehashish Bhattacharjee (Tushar), seasoft022.blogspot.com

Picture-01: Mathematical Solution by using the 1st Concept of Pre-stressed Concrete Structures (Page 01 of 02)

1st Concept of Pre-stressed Concrete Structures, Page 02 of 02
By Engr. Snehashish Bhattacharjee (Tushar), seasoft022.blogspot.com

Picture-02: Mathematical Solution by using the 1st Concept of Pre-stressed Concrete Structures (Page 02 of 02)

Now, solve the same mathematical problem by using 2nd Concept of Pre-stressed Concrete Structures,
Picture-03, Picture-04 & Picture-05 indicates the mathematical solution by using the 2nd Concept of Pre-stressed Concrete Structures,

2nd Concept of Pre-stressed Concrete Structures, Page 01 of 03
By Engr. Snehashish Bhattacharjee (Tushar), seasoft022.blogspot.com

Picture-03: Mathematical Solution by using the 2nd Concept of Pre-stressed Concrete Structures (Page 01 of 03)

2nd Concept of Pre-stressed Concrete Structures, Page 02 of 03
By Engr. Snehashish Bhattacharjee (Tushar), seasoft022.blogspot.com

Picture-04: Mathematical Solution by using the 2nd Concept of Pre-stressed Concrete Structures (Page 02 of 03)

2nd Concept of Pre-stressed Concrete Structures, Page 03 of 03
By Engr. Snehashish Bhattacharjee (Tushar), seasoft022.blogspot.com

Picture-05: Mathematical Solution by using the 2nd Concept of Pre-stressed Concrete Structures (Page 03 of 03)

At Last, Solve the same mathematical problem by using 3rd Concept of Pre-stressed Concrete Structures,
Picture-06 & Picture-07 indicates the mathematical solution by using the 3rd Concept of Pre-stressed Concrete Structures,

3rd Concept of Pre-stressed Concrete Structures, Page 01 of 02
By Engr. Snehashish Bhattacharjee (Tushar), seasoft022.blogspot.com

Picture-06: Mathematical Solution by using the 3rd Concept of Pre-stressed Concrete Structures (Page 01 of 02)

3rd Concept of Pre-stressed Concrete Structures, Page 02 of 02
By Engr. Snehashish Bhattacharjee (Tushar), seasoft022.blogspot.com

Picture-07: Mathematical Solution by using the 3rd Concept of Pre-stressed Concrete Structures (Page 02 of 02)

Now lets analyze these Three (03) types of solutions of a same mathematical problem.

Analyze:
*** At first process (1st Concept), there is no extra method to solve this problem.
*** At second process (2nd Concept), there is consideration about compression & tension force,
*** At the third & last process (3rd concept), moment will be calculated by using unbalanced load (Upward Uniform Load from Pre-stressing Force-Download Uniform Load), so that this moment (133.425 kip-ft) will be called unbalanced moment.
*** We can understand that the result of all of these solutions will be more or less same.