Scribd
Upload
356 views3 pages

Concrete Mixture: A Gauss Jordan Elimination Application)

This document discusses concrete mixtures and provides a sample problem to calculate the amounts of cement, sand, gravel, and water needed to produce 50 cubic yards of concrete given weight proportions and a water-cement ratio. A concrete mixture ratio of 1 part cement to 2 parts sand to 3 parts gravel is provided. The sample problem sets up equations based on the given proportions, densities of the materials, and amount of concrete desired. It then uses Gauss-Jordan elimination to solve the system of equations for the amounts of each constituent.

Uploaded by

John Lester Rodriguez
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as DOCX, PDF, TXT or read online on Scribd
356 views3 pages

Concrete Mixture: A Gauss Jordan Elimination Application)

This document discusses concrete mixtures and provides a sample problem to calculate the amounts of cement, sand, gravel, and water needed to produce 50 cubic yards of concrete given weight proportions and a water-cement ratio. A concrete mixture ratio of 1 part cement to 2 parts sand to 3 parts gravel is provided. The sample problem sets up equations based on the given proportions, densities of the materials, and amount of concrete desired. It then uses Gauss-Jordan elimination to solve the system of equations for the amounts of each constituent.

Uploaded by

John Lester Rodriguez
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as DOCX, PDF, TXT or read online on Scribd
You are on page 1/ 3

Concrete Mixture

(A GAUSS JORDAN ELIMINATION APPLICATION)

Concrete - Concrete is the most commonly used man-made material on earth. It is an important construction
materials used extensively in buildings, bridges, roads, and dams. Its uses range from structural applications, to
paviours, kerbs, pipes and drains.

A concrete mixture ratio of 1 part cement, 2 parts sand, and 3 parts aggregate will produce a concrete mix of
approximately 3000 psi. Mixing water with the cement, sand, and stone will form a paste that will bind the
materials together until the mix hardens.

Sample problem:

A concrete mix was designed to have a weight proportions of [1:1.9:2.8] with a water cement ration of 7
gallons of water per sack of cement. How much of each constituent is needed to produce 50 yd^3 of
concrete?

Given:

1 sack of cement = 94 lb

Densities:

Cement = C = 195 lb/ft^3 GRAVEL = G = 165 lb/ft^3

SAND = S = 165 lb lb/ft^3 WATER = W = 62.4 lb/ft^3

Solution:
𝑪 𝑺
= ; 𝟏. 𝟗𝑪 − 𝑺 = 𝟎 (𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟏)
𝟏 𝟏.𝟗
𝑪 𝑮
= ; 𝟐. 𝟖𝑪 − 𝑮 = 𝟎 (𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟐)
𝟏 𝟐.𝟖
𝟏 𝒔𝒂𝒄𝒌 𝟕 𝒈𝒂𝒍𝒍𝒐𝒏𝒔
= ; 𝟓𝟖. 𝟑𝟖𝟕𝟗𝑪 − 𝟗𝟒𝑮 = 𝟎 (𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟑)
𝑪 𝑾
𝟑 𝒇𝒕 𝟑 𝒇𝒕𝟑 𝒇𝒕𝟑 𝒇𝒕𝟑 𝒇𝒕𝟑
𝟓𝟎 𝒚𝒅𝟑 ( ) = 𝑪𝒍𝒃 (𝟏𝟗𝟓𝒍𝒃) + 𝑺𝒍𝒃 (𝟏𝟔𝟓𝒍𝒃) + 𝑮𝒍𝒃 (𝟏𝟗𝟓𝒍𝒃) + 𝑾𝒍𝒃 (𝟏𝟗𝟓𝒍𝒃)
𝟏 𝒚𝒅

𝑪 𝑺 𝑮𝑪 𝑾
𝟏𝟑𝟓𝟎 = + + + (𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟒)
𝟏𝟗𝟓 𝟏𝟔𝟓 𝟏𝟔𝟓 𝟔𝟐.𝟒

Matrix:
C S G W
𝟏. 𝟗 −𝟏 𝟎 𝟎 𝟎
𝟐. 𝟖 𝟎 −𝟏 𝟎 𝟎
𝟓𝟖. 𝟑𝟖𝟕𝟗 𝟎 𝟎 −𝟗𝟒 = [ 𝟎 ]
𝟏 𝟏 𝟏 𝟏
[ 𝟏𝟗𝟓 𝟏𝟔𝟓 𝟏𝟔𝟓 𝟔𝟐.𝟒 ]
𝟏𝟑𝟓𝟎

Make the pivot in the 1st column by dividing the 1st row by 1.9:
𝟏 −𝟎. 𝟓𝟐𝟔𝟑 𝟎 𝟎 𝟎
𝟐. 𝟖 𝟎 −𝟏 𝟎 𝟎
𝟓𝟖. 𝟑𝟖𝟕𝟗 𝟎 𝟎 −𝟗𝟒 = [ 𝟎 ]
𝟏 𝟏 𝟏 𝟏
[ 𝟏𝟗𝟓 𝟏𝟔𝟓 𝟏𝟔𝟓 𝟔𝟐.𝟒 ]
𝟏𝟑𝟓𝟎
Eliminate the 1st column:

𝟏 −𝟎. 𝟓𝟐𝟔𝟑 𝟎 𝟎 𝟎
𝟎 𝟏. 𝟒𝟕𝟑𝟕 −𝟏 𝟎 𝟎
𝟎 𝟑𝟎. 𝟕𝟑𝟎𝟓 𝟎 −𝟗𝟒 = [ 𝟎 ]
𝟏 𝟏
[𝟎 𝟎. 𝟎𝟎𝟖𝟖 𝟏𝟔𝟓 𝟔𝟐.𝟒 ]
𝟏𝟑𝟓𝟎
Make the pivot in the 2nd column by dividing the 2nd row by 1.4737:

𝟏 −𝟎. 𝟓𝟐𝟔𝟑 𝟎 𝟎 𝟎
𝟎 𝟏 −𝟎. 𝟔𝟕𝟖𝟔 𝟎 𝟎
𝟎 𝟑𝟎. 𝟕𝟑𝟎𝟓 𝟎 −𝟗𝟒 = [ 𝟎 ]
𝟏 𝟏
[𝟎 𝟎. 𝟎𝟎𝟖𝟖 𝟏𝟔𝟓 𝟔𝟐.𝟒 ]
𝟏𝟑𝟓𝟎
Eliminate the 2nd column:
𝟏 𝟎 −𝟎. 𝟑𝟓𝟕𝟏 𝟎 𝟎
𝟎 𝟏 −𝟎. 𝟔𝟕𝟖𝟔 𝟎 𝟎
𝟎 𝟎 𝟐𝟎. 𝟖𝟓𝟐𝟖 −𝟗𝟒 = [ 𝟎 ]
𝟏
[𝟎 𝟎 𝟎. 𝟎𝟏𝟐𝟎 ] 𝟏𝟑𝟓𝟎 𝟔𝟐.𝟒
Make the pivot in the 3rd column by dividing the 3rd row by 20.8528:
𝟏 𝟎 −𝟎. 𝟑𝟓𝟕𝟏 𝟎 𝟎
𝟎 𝟏 −𝟎. 𝟔𝟕𝟖𝟔 𝟎 𝟎
𝟎 𝟎 𝟏 −𝟒. 𝟓𝟎𝟕𝟖 = [ 𝟎 ]
𝟏
[𝟎 𝟎 𝟎. 𝟎𝟏𝟐𝟎 ] 𝟏𝟑𝟓𝟎
𝟔𝟐.𝟒

Eliminate the 3rd column:

𝟏 𝟎 𝟎 −𝟏. 𝟔𝟎𝟗𝟗 𝟎
[𝟎 𝟏 𝟎 −𝟑. 𝟎𝟓𝟖𝟗] = [ 𝟎 ]
𝟎 𝟎 𝟏 −𝟒. 𝟓𝟎𝟕𝟖 𝟎
𝟎 𝟎 𝟎 𝟎. 𝟎𝟕𝟎𝟏 𝟏𝟑𝟓𝟎
Make the pivot in the 4th column by dividing the 4th row by 0.0701:

𝟏 𝟎 𝟎 −𝟏. 𝟔𝟎𝟗𝟗 𝟎
[𝟎 𝟏 𝟎 −𝟑. 𝟎𝟓𝟖𝟗] = [ 𝟎 ]
𝟎 𝟎 𝟏 −𝟒. 𝟓𝟎𝟕𝟖 𝟎
𝟎 𝟎 𝟎 𝟏 𝟏𝟗𝟐𝟒𝟕. 𝟐𝟎𝟒𝟓
Eliminate the 4th column:

𝟏 𝟎 𝟎 𝟎 𝟑𝟎𝟗𝟖𝟔. 𝟓𝟎𝟗𝟔
[𝟎 𝟏 𝟎 𝟎] = [𝟓𝟖𝟖𝟕𝟒. 𝟑𝟔𝟖𝟑]
𝟎 𝟎 𝟏 𝟎 𝟖𝟔𝟕𝟔𝟐. 𝟐𝟐𝟕𝟎
𝟎 𝟎 𝟎 𝟏 𝟏𝟗𝟐𝟒𝟕. 𝟐𝟎𝟒𝟓
C = 30,986.5096 lb.
S = 58,874.3683 lb.
G = 86,762.2270 lb.
W = 19,247. 2045 lb.

You might also like