Centroids and Distributed Loads

1
Centroids and Distributed Loads

• ENGR 221
• February 10, 2003

2
Lecture Goals

• 5.4 Centroids of Composite Bodies
• 5.6 Distributed Loads on Beams
• 5.7 Forces on Submerged Surfaces

3
Centroids Composite Bodies
The calculation of centroid uses the following
equation where AT is the total area and x and
y bar are the centroid of the body.
4
Centroids Composite Bodies
The equation can be broken into integrals of
smaller areas.
5
Centroids Composite Bodies
If each integral is replaced with its centroid
and area, the centroid of the entire body can be
computed using
6
Centroids for Volume -Composite Bodies
The same technique can be applied to finding the
centroid of the volume of a body using
components
7
Centroids Composite Bodies
Each shape has a centroid in the x and y
directions.
8
Centroids Composite Bodies
Each shape has a centroid in the x and y
directions. This figure 5.1 out of your text for
2-D figures.
9
Centroids Composite Bodies
Each volume has a centroid in the x, y, and z
directions. This figure 5.2 out of your text for
3-D figures.
10
Centroids Simple Example for a Composite Body
Find the centroid of the given body
11
Centroids Simple Example for a Composite Body
To find the centroid,
Determine the area of the components
12
Centroids Simple Example for a Composite Body
The total area is
13
Centroids Simple Example for a Composite Body
To centroid of each component Compute the x
centroid
14
Centroids Simple Example for a Composite Body
To centroid of each component Compute the y
centroid
15
Centroids Simple Example for a Composite Body
The problem can be done using a table to
represent the composite body.
16
Centroids Simple Example for a Composite Body
17
Centroids Simple Example for a Composite Body
The problem can be done using a table to
represent the composite body.
18
Centroids Example for a Composite Body
Find the centroid of the given body
19
Centroids Example for a Composite Body
Determine the area of the components
20
Centroids Example for a Composite Body
The total area is
21
Centroids Example for a Composite Body
22
Centroids Example for a Composite Body
An Alternative Method would be to subtract to
areas
23
Centroids Class Problem
Find the centroid of the body
24
Centroids Class Problem
Find the centroid of the body
25
Distributed Loads
How do you determine the equivalent load
(magnitude) and its location?
26
Distributed Loads
Treat the load as a body and determine its area
(magnitude) and its centroid (location of the
resultant)
27
Distributed Loads
So that
28
Distributed Loads Example
A beam supports a distributed load, determine the
equivalent concentrated load.
29
Distributed Loads Example
The load can be broken up into two triangular
loads where the magnitude of the load can be
determined
30
Distributed Loads Example
The center of the loads are Total load is
31
Distributed Loads Example
The location of the resultant load is
32
Distributed Loads Example
One can use the table method to find the loading
acting on the beam.
33
Distributed Loads Example
An alternative loading would be to use a
distributed load of (1.5 kN/m) and ramp load of 3
kN/m /m.
34
Distributed Loads Example
An alternative loading would be to use a
distributed load of (1.5 kN/m) and ramp load of 3
kN/m /m.
35
Distributed Loads Example Problem
Determine the resultant R of the system of
distributed loads and locate its line of action
with respect to the left of the support for
36
Distributed Loads Example Problem
Break the problem into three parts
37
Distributed Loads Example Problem
Break the problem into three parts
38
Distributed Loads Example Problem
Put it in a table format
39
Distributed Loads Class Problem
Determine the resultant R of the system of
distributed loads and locate its line of action
with respect to the left of the support for
40
Force on Submerged Surfaces
In a fluid at rest, the weight of the liquid will
create a pressure on the surface of a body. This
pressure is defined as the hydrostatic pressure.
where PA is pressure absolute, P0 is the
initial pressure and g is the specific weight of
the fluid in F/L3 and d is the depth.
41
Force on Submerged Surfaces
The density of fluid, r is multiplied by the g to
get the specific weight of the fluid and PG
(gauge pressure) is defined as.
42
Force on a Submerged Surface
The pressure acts as a function of depth.
R
The resultant force, R
gd
43
Force on a Submerged Surface- Example
A 3- by 3 ft gate is placed in a wall below water
level as shown. Determine the magnitude and
location of the resultant of the forces exerted
by the water on the gate. (g 62.4 lb/ft3)
44
Force on a Submerged Surface- Example
The pressure distribution on the wall is
45
Force on a Submerged Surface- Example
The pressure distribution on the wall is
How does one obtain the distribution force?
MULTIPLY by the width (3 ft)
46
Force on a Submerged Surface- Example
The equivalent load on wall is
47
Force on a Submerged Surface- Example
The equivalent force is
48
Force on a Submerged Surface- Example
Use a table to find the location
Total force is 1965.6 lb at 3.71 ft from the
surface.
49
Force on a Submerged Surface
How does on find the forces on a submerge surface
at an angle?
Draw the the free-body diagram.
50
Force on a Submerged Surface
The free-body diagram would have
The pressure and the weight of the fluid.
51
Force on a Submerged Surface
The resulting force distribution without the
weight of the water would look like,
52
Force on a Submerged Surface
If we were take a look at the distribution on a
non-linear surface the results would
The force can be represented as
53
Force on a Submerged Surface Class Problem
The quick action gate AB is 1.75 ft wide and is
held in it closed position by a vertical cable
and by hinges located along its top edge B. For
a depth of water d 6-ft determine the force
acting on the gate and location of the force.
54
Homework (Due 2/17/03)
Problems
5-2, 5-4, 5-6, 5-12, 5-14
55
Bonus Slides
An automatic value consists of a square plate 225
by 225 mm, which pivoted about a horizontal axis
through A located at a distance h100 mm above
the lower edge. Determine the depth of the water
d for which the valve will open.
56
Bonus Slides
The bent plate ABCD is 2 m wide and is hinged at
A. Determine the reactions on A and D for the
water level.