University of the Philippines DilimanCollege of EngineeringINSTITUTE OF CIVIL ENGINEERING

CE 28 ANALYTICAL AND COMPUTATIONAL METHODS IN CIVL ENGINEERING III2nd Semester AY 2014-2015

SHEAR STRESSES OF PRISMATIC SHAFTS WITH CIRCULAR AND TRIANGULAR CROSS-SECTIONS

Submitted by:

Baniqued, Julius Rey2012-

Tuppil, Camille G.2012-24733

CE 28 TXY

Submitted to:

Mr. William MataInstructor

Date Submitted:

March 18, 2015

PROGRAM IMPLEMENTATION

DECLARATION OF LIBRARIES AND VARIABLESThis program makes use of an external library matrix.h composed of matrix functions. In the first part of the program, this library, along with other standard libraries, were called. The value of the constant pi is also defined.

The output of this program is a file named shear.csv. This CSV file will contain a table of values showing the shear stress in regularly-spaced points within the cross-section of a beam. The program creates this file at the start of the program.

INPUTThe program first asks the user if the cross-section of the beam is circular or triangular.

The program then proceeds to ask for other significant values such as Internal Torque (T) in N-m, Length of Shaft (L) in m, Shear Modulus of Elasticity (G) in Pa, radius of cross-section (r) in m for circular cross-sections (cs=1).

For triangular cross-sections (cs=2), the program also asks for Internal Torque (T) in N-m, Length of Shaft (L) in m, Shear Modulus of Elasticity (G) in Pa. The program also asks for the length of one side of the triangle (b) in m. This program assumes that the triangle is equilateral and one side is parallel to the horizontal.

COMPUTING FOR J, POLAR MOMENT OF INERTIAThe program proceeds to compute for the polar moment of inertia J and the angle of twist theta for the cross-section.

For circular cross-sections:

For triangular cross-sections:

SETTING UP THE GRID MATRIX UAfter the user provides the interval h, the program can start setting up the grid matrix U. This matrix represents the beams cross-section and will be the one printed in the output CSV file. The number of grid points is computed and stored in variable k where

For circular cross-sections,

For triangular cross-sections,

K is then set as the number of rows and columns for Matrix U.

SETTING UP THE BOUNDARY CONDITIONSThe program makes sure that only the grid points inside of the designated area are considered. If the grid point is outside the circle or triangle, the program gives it a value of 0. If the grid point is inside, the program gives it a preliminary value or count. This serves as a number for each grid point. The variable count also determines the number of equations needed to solve for the Prandtl Stress Function .

For circular cross-sections:

For triangular cross-sections,

SETTING UP CONSTANT MATRIX ASetting up the constant matrix A is the first step to computing for the values of the Prandtl Stress Function . The number of rows and columns of matrix A is the variable count minus 1. This ensures that the number of rows and columns (equations) corresponds to the number of points inside the cross-section. The program first gives all elements of Matrix A a value of 0.

The program proceeds to set values of Matrix A based on the values of Matrix U. If the element is in the main diagonal of the matrix, a value of -4 is assigned to it. For any other element, the program checks its corresponding element in Matrix U. If the surrounding elements of element A in Matrix B is not zero, the program assigns a value of 1 for the corressponding surrounding element in Matrix A.

For both circular and triangular cross-sections:

SETTING UP MATRIX BMatrix B represents the other side of the matrix equation (Ax = B). The program sets up Matrix B as a column matrix with values equal to -2G for both circular and triangular cross-sections.

SOLVING FOR MATRIX XUsing a Gauss-Jordan function, the program obtains a column matrix X containing the values of for each point in the grid.

TRANSFORMING MATRIX X INTO MATRIX PBecause Matrix X is a column matrix, another square matrix P created to contain the values of in grid form.

GETTING THE DERIVATIVE OF THE PRANDTL STRESS FUNCTION Because solving for the shear stress at each point requires the derivative of the function , a process for getting the derivative of each point in the Matrix P is required. This part of the program uses finite difference formulas (central, forward and backward) to compute for the derivatives.

This part of the program also computes the shear stress of each point in the grid using the given formula for shear stress:

The computed shear stress is stored into its corresponding element in Matrix U.

OUTPUTThe output is printed in a CSV file containing the values of grid Matrix U.

SAMPLE OUTPUT/RESULTS

FOR CIRCULAR CROSS-SECTIONS

*For actual data, see Sample Data Circular.xlsx

FOR TRIANGULAR CROSS-SECTIONS

*For actual data, see Sample Data Circular.xlsx

LIMITATIONS

The contour graphs produced by the program MS Excel show discrepancies at the boundaries. The graphs make it seem like the shear stress becomes lower at the boundaries. However, the proponents believe that this is only because the program interpolates the given data to create a continuous contour.

APPENDIX

FORMULASCircular Cross-SectionsTriangular Cross-Sections

Polar Moment of Inertia, J

Angle of Twist,