Lecture Notes of Resultant of a Cunccurent, Non-Coplanar Force System

University of Babylon
College of Engineering
Department of Environmental Engineering
Engineering Analysis I (ENAN 103)







Resultant of a Concurrent Non-coplanar Force System

Undergraduate Level, 1st Stage



Mr. Waleed Ali Tameemi
College of Engineering/ Babylon University
M.Sc. Civil Engineering/ the University of Kansas/ USA



2016-2017

Course Outline
Introduction
Parallelogram Law
Forces Resolution
Resultant of a Cunccurent, Coplanar Force System
Moment
Couples
Resultant of a Non-concurrent, Coplanar Force System
Resultant of a Concurrent Non-coplanar Force System
Equilibrium
Fraction
Truss
Method of Joints
Method of Sections








Resultant of a Concurrent Non-coplanar Force System
The resultant of this system is single force pass through the concurrence point. In order to calculate the resultant magnitude and direction, the following steps are required.




Resolve each single force to its perpendicular components.
f_x=f×cos???_x ?
f_y=f×cos???_y ?
f_z=f×cos???_z ?

p_x=p×cos???_x ?
p_y=p×cos???_y ?
p_z=p×cos???_z ?

Calculate the resultant’s components in x direction, y and z direction.
R_x=??F_x =f_x+p_x
R_y=??F_y =f_y+p_y
R_z=??F_z =f_z+p_z

Calculate the magnitude of the resultant.
R=?(R_x^2+R_y^2+R_z^2 )

Calculate the resultant’s slope in x direction, y direction, and z direction.
?_x=cos^(-1)??(R_x/R?)
?_y=cos^(-1)??(R_y/R?)
?_z=cos^(-1)??(R_z/R?)

Applications
Problem 1:
For the given force system, calculate the magnitude and the slope of its resultant.





Force ID Magnitude (N) ?_x ?_y ?_z
F 100 30 60 30
P 150 45 45 60

Solution
Resolve each single force to its perpendicular components.
f_x=f×cos???_x ?=100×cos?(30)=86.6 N ?
f_y=f×sin???_y ?=100×cos?(60)=50 N ?
f_z=f×sin???_z ?=100×cos?(30)=86.6 N

p_x=p×cos???_x ?=150×cos?(45)=106.1 N ?
p_y=p×sin???_y ?=150×cos?(45)=-106.1 N=106.1?
p_z=p×sin???_z ?=150×cos?(60)=75 N
Calculate the resultant’s components in x direction, y direction, and z direction.
R_x=??F_x =f_x+p_x=86.6+106.1=192.7 N ?
R_y=??F_y =f_y+p_y=50-106.1=-56.1 N=56.1 N ?
R_z=??F_z =f_z+p_z=86.6+75=161.6 N

Calculate the magnitude of the resultant.
R=?(R_x^2+R_y^2+R_z^2 )=?(?192.7?^2+?56.1?^2+?161.6?^2 )=257.7 N

Calculate the resultant’s slope x direction, y and z direction.
?_x=cos^(-1)??(R_x/R?)=cos^(-1)??(192.7/257.7?)=41.6°
?_y=cos^(-1)??(R_y/R?)=cos^(-1)??((-56.1)/257.7?)=102.6°
?_z=cos^(-1)??(R_z/R?)=cos^(-1)??(16.1.6/257.7?)=51.2°