Determine the resultant force and specify where it acts on the beam measured from A. Assume p = 230 lb/ft.
1.Determine the magnitude of the resultant force.
2.Determine the distance between A and the resultant’s line of action.

Answer
General guidance


Expression to find theresultant force on beam carrying uniformly distributed load
…… (1)
Here,
is the resultant force on the beam acting in downward direction,
is the load on beam per unit length of the beam and
is the length of the portion of the beam on which uniformly distributed load is applied.
Expression to find theresultant force on beam carrying linearly varying load
…… (2)
Here,
is the resultant force on the beam acting in downward direction,
is the load acting on beam per unit length and
is the length of the portion of the beam on which linearly varying load is applied.
General sign convection for moment:
The moment is considered positive in counter-clockwise direction and negative in clockwise direction
Expression for the equilibrium of the rigid body:
Net force acting on a body is equal to zero. Also, the sum of the moment of all the forces at any point is equal to zero.
In
plane,
And
Here,
is the force acting on the rigid body in
direction,
is the force acting on the rigid body in
direction and
is the moment of forces acting at point
.
Point of application of resultant force for uniformly distributed load
Resultant force
lies at
distance from left hand side.
Point of application of resultant force for linearly varying load:
Resultant force
lies at
distance from left hand side or low end side of the triangle.
Step-by-step
Step 1 of 2
(1)
Free body diagram of the beam,
Here,
is the distance between point
and point
,
is the distance between point
and point
and
is load per unit length acting on the beam.
From the above figure, the beam is carrying linearly varying load from point
to point
and the beam is carrying uniformly distributed load from point
to point
.
Calculate the magnitude of the resultant force.
Add equation (1) and equation (2) to calculate total resultant of force
acting on the beam.
Substitute
for
,
for
and
for
.
The magnitude of the resultant force is
.
The magnitude of the resultant force is .
Determine the expression for total resultant force by adding the expression of resultant force due to uniformly distributed load and uniformly varying load. Substitute the values of ,
and
.Then calculate the total resultant force on the beam.
Step 2 of 2
(2)
Calculate the distance between point A and the resultant’s line of action.
Here,
is the resultant force acting due to uniformly varying load on member
of the beam,
is the resultant force acting due to uniformly distributed load on member
of the beam,
is the net resultant force acting at
from point
.
is the distance between point
and point
and
is the distance between point
and point
.
Distance between
and line of action of
For uniformly varying load at member
,
acts at
from point
.
Distance between
and line of action of
For uniformly distributed load at member
,
acts at
from point
.
Calculation for resultant force
acting on member
.
Here,
is load acting per unit length and
is the length of member
.
Substitute
for
and
for
Calculate resultant force
acting on member
.
Here,
is load acting per unit length and
is the length of member
.
Substitute
for
and
for
Use moment equilibrium equation at point
.
Thus,
Substitute
for
,
for
,
for
,
for
and
for
The distance between point A and the resultant line of action is .
Calculate the values of and
. Determine the point of application of
and
. Apply moment equilibrium equation at point
. Substitute the values of
and
in moment equilibrium equation. Solve for the distance between point
and resultant’s line of action.
Answer
The magnitude of the resultant force is .
The distance between point A and the resultant line of action is .
Answer only
The magnitude of the resultant force is .
The distance between point A and the resultant line of action is .