Equivalent Force Systems | Additional Documents & Tests for Civil Engineering (CE) PDF Download

 Page 1


EQUIVALENT FORCE SYSTEM 
• Equivalent Vectors – Two vectors are equivalent in a certain capacity if each 
produces the very same effect in this capacity. Effect of forces on a rigid 
body is manifested in motion (or lack of motion) of body induced by force.  
• Two forces are equivalent if they can initiate same motion of the rigid body 
• Equal Vectors – Two vectors are equal if they have same dimensions, 
magnitude, and direction 
TRANSLATION OF A FORCE TO PARALLEL POSITION 
• If a force is moved to a line of action ‘d’ distance away from original line of 
action, then equivalent force system consists of a force of same magnitude 
and a couple moment with magnitude ‘Fd’ (equal to moment of original 
force about a point on new line of action) 
• In vector form, couple moment is given by ?? ?? =??¯ × ?? ¯
, ??¯ is the position vector 
from a point on original line of action to a point on new line of action 
• Conversely, a force and a couple in the same plane can be reduced to a 
single equivalent force by introducing a force of equal magnitude at a 
parallel line of action as that of the original force 
 
 
 
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EQUIVALENT FORCE SYSTEM 
• Equivalent Vectors – Two vectors are equivalent in a certain capacity if each 
produces the very same effect in this capacity. Effect of forces on a rigid 
body is manifested in motion (or lack of motion) of body induced by force.  
• Two forces are equivalent if they can initiate same motion of the rigid body 
• Equal Vectors – Two vectors are equal if they have same dimensions, 
magnitude, and direction 
TRANSLATION OF A FORCE TO PARALLEL POSITION 
• If a force is moved to a line of action ‘d’ distance away from original line of 
action, then equivalent force system consists of a force of same magnitude 
and a couple moment with magnitude ‘Fd’ (equal to moment of original 
force about a point on new line of action) 
• In vector form, couple moment is given by ?? ?? =??¯ × ?? ¯
, ??¯ is the position vector 
from a point on original line of action to a point on new line of action 
• Conversely, a force and a couple in the same plane can be reduced to a 
single equivalent force by introducing a force of equal magnitude at a 
parallel line of action as that of the original force 
 
 
 
SIMPLEST RESULTANT OF SPECIAL FORCE SYSTEMS 
1. Coplanar Force System 
• If there are several forces and couple moments in a plane, their 
equivalent force system will be a single force with a specific line of 
action or a couple moment (if sum of forces is zero), or a zero vector 
• As an example, consider a coplanar force system on xy plane, then 
Forces are of the form F=F
x
?? ^ +F
y
?? ^ and couples of the form C=C
z
?? ^
. Then, 
by moving all forces to origin, then we get a resultant force,  
F
R
=?(?? ?? ?? ^ + ?? ?? ?? ^ ), 
and a resultant couple,   
C
R
=?(?? ?? ?? ?? )?? ^
+ ?(?? ?? )?? ^
, 
where, ?? ?? is the perpendicular distance from the origin to ?? ?? . This 
system can further be reduced to a single force (if resultant force in 
preceding step is not a zero vector) such that the moment   of this 
force about origin is equal to resultant couple moment obtained in 
the preceding step. If resultant force is zero, then the coplanar 
system has a couple moment or a zero vector 
 
 
 
 
 
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EQUIVALENT FORCE SYSTEM 
• Equivalent Vectors – Two vectors are equivalent in a certain capacity if each 
produces the very same effect in this capacity. Effect of forces on a rigid 
body is manifested in motion (or lack of motion) of body induced by force.  
• Two forces are equivalent if they can initiate same motion of the rigid body 
• Equal Vectors – Two vectors are equal if they have same dimensions, 
magnitude, and direction 
TRANSLATION OF A FORCE TO PARALLEL POSITION 
• If a force is moved to a line of action ‘d’ distance away from original line of 
action, then equivalent force system consists of a force of same magnitude 
and a couple moment with magnitude ‘Fd’ (equal to moment of original 
force about a point on new line of action) 
• In vector form, couple moment is given by ?? ?? =??¯ × ?? ¯
, ??¯ is the position vector 
from a point on original line of action to a point on new line of action 
• Conversely, a force and a couple in the same plane can be reduced to a 
single equivalent force by introducing a force of equal magnitude at a 
parallel line of action as that of the original force 
 
 
 
SIMPLEST RESULTANT OF SPECIAL FORCE SYSTEMS 
1. Coplanar Force System 
• If there are several forces and couple moments in a plane, their 
equivalent force system will be a single force with a specific line of 
action or a couple moment (if sum of forces is zero), or a zero vector 
• As an example, consider a coplanar force system on xy plane, then 
Forces are of the form F=F
x
?? ^ +F
y
?? ^ and couples of the form C=C
z
?? ^
. Then, 
by moving all forces to origin, then we get a resultant force,  
F
R
=?(?? ?? ?? ^ + ?? ?? ?? ^ ), 
and a resultant couple,   
C
R
=?(?? ?? ?? ?? )?? ^
+ ?(?? ?? )?? ^
, 
where, ?? ?? is the perpendicular distance from the origin to ?? ?? . This 
system can further be reduced to a single force (if resultant force in 
preceding step is not a zero vector) such that the moment   of this 
force about origin is equal to resultant couple moment obtained in 
the preceding step. If resultant force is zero, then the coplanar 
system has a couple moment or a zero vector 
 
 
 
 
 
2. Parallel Force System: 
• Simplest resultant of a parallel force system is either a force with a 
specific line of action, or a single couple moment or zero vector 
• For a parallel system of forces, parallel to z-axis, forces will be of the 
form F=F
x
?? ^
 and couples of the form C
p
=C
p
?? ^ + C
p
?? ^ . Then,  
F
R
=?(?? ?? ?? ^ ) and,  C
R
=?(?? × ?? ?? )+?(?? ????
?? ^ + ?? ????
?? ^ ) 
If F
r
 is not equal to zero, then resultant couple can be negated by 
moving the force to a specific line of action. If F
r
 is a zero vector, then 
there can be a couple moment or zero vector (if the resultant couple 
is zero) 
 
 
 
3. General force system: 
• Simplest resultant is a force and a couple moment which are 
collinear/parallel. Simplest resultant can also be a single force oe a 
single couple moment or zero vector. 
• Simplest force system comprising of a force and a couple moment 
collinear to force is called wrench 
 
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EQUIVALENT FORCE SYSTEM 
• Equivalent Vectors – Two vectors are equivalent in a certain capacity if each 
produces the very same effect in this capacity. Effect of forces on a rigid 
body is manifested in motion (or lack of motion) of body induced by force.  
• Two forces are equivalent if they can initiate same motion of the rigid body 
• Equal Vectors – Two vectors are equal if they have same dimensions, 
magnitude, and direction 
TRANSLATION OF A FORCE TO PARALLEL POSITION 
• If a force is moved to a line of action ‘d’ distance away from original line of 
action, then equivalent force system consists of a force of same magnitude 
and a couple moment with magnitude ‘Fd’ (equal to moment of original 
force about a point on new line of action) 
• In vector form, couple moment is given by ?? ?? =??¯ × ?? ¯
, ??¯ is the position vector 
from a point on original line of action to a point on new line of action 
• Conversely, a force and a couple in the same plane can be reduced to a 
single equivalent force by introducing a force of equal magnitude at a 
parallel line of action as that of the original force 
 
 
 
SIMPLEST RESULTANT OF SPECIAL FORCE SYSTEMS 
1. Coplanar Force System 
• If there are several forces and couple moments in a plane, their 
equivalent force system will be a single force with a specific line of 
action or a couple moment (if sum of forces is zero), or a zero vector 
• As an example, consider a coplanar force system on xy plane, then 
Forces are of the form F=F
x
?? ^ +F
y
?? ^ and couples of the form C=C
z
?? ^
. Then, 
by moving all forces to origin, then we get a resultant force,  
F
R
=?(?? ?? ?? ^ + ?? ?? ?? ^ ), 
and a resultant couple,   
C
R
=?(?? ?? ?? ?? )?? ^
+ ?(?? ?? )?? ^
, 
where, ?? ?? is the perpendicular distance from the origin to ?? ?? . This 
system can further be reduced to a single force (if resultant force in 
preceding step is not a zero vector) such that the moment   of this 
force about origin is equal to resultant couple moment obtained in 
the preceding step. If resultant force is zero, then the coplanar 
system has a couple moment or a zero vector 
 
 
 
 
 
2. Parallel Force System: 
• Simplest resultant of a parallel force system is either a force with a 
specific line of action, or a single couple moment or zero vector 
• For a parallel system of forces, parallel to z-axis, forces will be of the 
form F=F
x
?? ^
 and couples of the form C
p
=C
p
?? ^ + C
p
?? ^ . Then,  
F
R
=?(?? ?? ?? ^ ) and,  C
R
=?(?? × ?? ?? )+?(?? ????
?? ^ + ?? ????
?? ^ ) 
If F
r
 is not equal to zero, then resultant couple can be negated by 
moving the force to a specific line of action. If F
r
 is a zero vector, then 
there can be a couple moment or zero vector (if the resultant couple 
is zero) 
 
 
 
3. General force system: 
• Simplest resultant is a force and a couple moment which are 
collinear/parallel. Simplest resultant can also be a single force oe a 
single couple moment or zero vector. 
• Simplest force system comprising of a force and a couple moment 
collinear to force is called wrench 
 
DISTRIBUTED FORCE SYSTEMS AND THEIR RESULTANTS 
• Vectors or scalars may be continuously distributed throughout a finite 
volume called vector or scalar fields. For example, temperature distribution 
is scalar field and gravitational force field of earth is a vector field. Vector 
field are represented as F(x,y,z,t) = F
x
(x,y,z,t)?? ^ + F
y
(x,y,z,t)?? ^ + F
z
(x,y,z,t)?? ^
. 
• Force distributions such as gravitational force, that exert influence directly 
on elements of mass distributed throughout the body are termed as body 
force distribution 
• Force distributions over a surface are called surface force distribution or 
surface tractions 
• Intensity of loading – In case of a continuous load on a beam, the load can 
be replaced by an equivalent coplanar distribution that acts at the central 
plane denoted as ‘w’, and is called the intensity of loading, which is a force 
system distributed over a line 
 
1. Parallel Body Force System and Centre of Gravity 
• Consider a rigid body with density ?(x,y,z) subjected to gravitational 
body force given by B(x,y,z)= -g?? ^
 per unit mass 
• Gravitational force on a differential mass element is dm= -g(?dV)?? ^
, 
where, dV is the volume of element 
• Resultant force, F
R
=? -?? (?????? )?? ^
?? = -Mg ?? ^
, where, M is the entire mass 
• Resultant act at the point (??¯, ??¯, ??¯) known as centre of gravity where 
entire weight of the body is assumed to be concentrated, and is 
computed as: ??¯ =
? ???????? ?? , ??¯ =
? ???????? ?? , ??¯ =
? ???????? ?? 
 
2. Parallel Force distribution over a plane surface and Centre of pressure 
• For pressure distribution given as p(x,y), simplest resultant force is 
F
R
=- ?
(?????? )
?? ?? ^
 acting at the point (??¯ =
? ???????? ? ??????
, ??¯ =
? ????????
? ??????
)  
 
 
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EQUIVALENT FORCE SYSTEM 
• Equivalent Vectors – Two vectors are equivalent in a certain capacity if each 
produces the very same effect in this capacity. Effect of forces on a rigid 
body is manifested in motion (or lack of motion) of body induced by force.  
• Two forces are equivalent if they can initiate same motion of the rigid body 
• Equal Vectors – Two vectors are equal if they have same dimensions, 
magnitude, and direction 
TRANSLATION OF A FORCE TO PARALLEL POSITION 
• If a force is moved to a line of action ‘d’ distance away from original line of 
action, then equivalent force system consists of a force of same magnitude 
and a couple moment with magnitude ‘Fd’ (equal to moment of original 
force about a point on new line of action) 
• In vector form, couple moment is given by ?? ?? =??¯ × ?? ¯
, ??¯ is the position vector 
from a point on original line of action to a point on new line of action 
• Conversely, a force and a couple in the same plane can be reduced to a 
single equivalent force by introducing a force of equal magnitude at a 
parallel line of action as that of the original force 
 
 
 
SIMPLEST RESULTANT OF SPECIAL FORCE SYSTEMS 
1. Coplanar Force System 
• If there are several forces and couple moments in a plane, their 
equivalent force system will be a single force with a specific line of 
action or a couple moment (if sum of forces is zero), or a zero vector 
• As an example, consider a coplanar force system on xy plane, then 
Forces are of the form F=F
x
?? ^ +F
y
?? ^ and couples of the form C=C
z
?? ^
. Then, 
by moving all forces to origin, then we get a resultant force,  
F
R
=?(?? ?? ?? ^ + ?? ?? ?? ^ ), 
and a resultant couple,   
C
R
=?(?? ?? ?? ?? )?? ^
+ ?(?? ?? )?? ^
, 
where, ?? ?? is the perpendicular distance from the origin to ?? ?? . This 
system can further be reduced to a single force (if resultant force in 
preceding step is not a zero vector) such that the moment   of this 
force about origin is equal to resultant couple moment obtained in 
the preceding step. If resultant force is zero, then the coplanar 
system has a couple moment or a zero vector 
 
 
 
 
 
2. Parallel Force System: 
• Simplest resultant of a parallel force system is either a force with a 
specific line of action, or a single couple moment or zero vector 
• For a parallel system of forces, parallel to z-axis, forces will be of the 
form F=F
x
?? ^
 and couples of the form C
p
=C
p
?? ^ + C
p
?? ^ . Then,  
F
R
=?(?? ?? ?? ^ ) and,  C
R
=?(?? × ?? ?? )+?(?? ????
?? ^ + ?? ????
?? ^ ) 
If F
r
 is not equal to zero, then resultant couple can be negated by 
moving the force to a specific line of action. If F
r
 is a zero vector, then 
there can be a couple moment or zero vector (if the resultant couple 
is zero) 
 
 
 
3. General force system: 
• Simplest resultant is a force and a couple moment which are 
collinear/parallel. Simplest resultant can also be a single force oe a 
single couple moment or zero vector. 
• Simplest force system comprising of a force and a couple moment 
collinear to force is called wrench 
 
DISTRIBUTED FORCE SYSTEMS AND THEIR RESULTANTS 
• Vectors or scalars may be continuously distributed throughout a finite 
volume called vector or scalar fields. For example, temperature distribution 
is scalar field and gravitational force field of earth is a vector field. Vector 
field are represented as F(x,y,z,t) = F
x
(x,y,z,t)?? ^ + F
y
(x,y,z,t)?? ^ + F
z
(x,y,z,t)?? ^
. 
• Force distributions such as gravitational force, that exert influence directly 
on elements of mass distributed throughout the body are termed as body 
force distribution 
• Force distributions over a surface are called surface force distribution or 
surface tractions 
• Intensity of loading – In case of a continuous load on a beam, the load can 
be replaced by an equivalent coplanar distribution that acts at the central 
plane denoted as ‘w’, and is called the intensity of loading, which is a force 
system distributed over a line 
 
1. Parallel Body Force System and Centre of Gravity 
• Consider a rigid body with density ?(x,y,z) subjected to gravitational 
body force given by B(x,y,z)= -g?? ^
 per unit mass 
• Gravitational force on a differential mass element is dm= -g(?dV)?? ^
, 
where, dV is the volume of element 
• Resultant force, F
R
=? -?? (?????? )?? ^
?? = -Mg ?? ^
, where, M is the entire mass 
• Resultant act at the point (??¯, ??¯, ??¯) known as centre of gravity where 
entire weight of the body is assumed to be concentrated, and is 
computed as: ??¯ =
? ???????? ?? , ??¯ =
? ???????? ?? , ??¯ =
? ???????? ?? 
 
2. Parallel Force distribution over a plane surface and Centre of pressure 
• For pressure distribution given as p(x,y), simplest resultant force is 
F
R
=- ?
(?????? )
?? ?? ^
 acting at the point (??¯ =
? ???????? ? ??????
, ??¯ =
? ????????
? ??????
)  
 
 
3. Coplanar Parallel Force distribution 
• Type of loading where loading is symmetrical over the longitudinal 
midplane of a body can be considered as a coplanar parallel force 
distribution. For example, load on beams 
• Load in such cases can be represented by an intensity function w(x) 
• Resultant force is given as F
R
=- ? ?? (?? )???? ?? ^ , which act at the point 
??¯ =
? ???? (?? )???? ?? ^
? ?? (?? )???? ?? ^