Introduction to Beam
Theory
Area Moments of Inertia, Deflection, and Volumes of
Beams
What is a Beam?
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Horizontal structural member
used to support horizontal
loads such as floors, roofs,
and decks.
Types of beam loads
ƒ Uniform
ƒ Varied by length
ƒ Single point
ƒ Combination
Common Beam Shapes
p
Hollow
Box
I Beam
H Beam
Solid
Box
T Beam
Beam Terminology
gy
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The parallel portions on an I-beam or H-beam are referred to as the
flanges. The portion that connects the flanges is referred to as the
web.
Web
Web
Flanges
Flanges
Support
pp
Configurations
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Source: Statics (Fifth Edition), Meriam and Kraige, Wiley
Load and Force Configurations
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Concentrated Load
Source: Statics (Fifth Edition), Meriam and Kraige, Wiley
Distributed Load
Beam Geometry
y
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Consider a simply supported beam of length, L.
The cross section is rectangular
rectangular, with width
width, b
b, and height
height, h
h.
h
L
b
Beam Centroid
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An area has a centroid, which is similar to a center of gravity of a
solid body.
The centroid
of a symmetric cross section can be easily found by
inspection. X and Y axes intersect at the centroid of a symmetric
cross section, as shown on the rectangular cross section.
Y - Axis
Centroid
h/2
X - Axis
h/2
b/2
b/2
Area Moment of Inertia (I)
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Inertia is a measure of a body’s ability to resist movement,
bending, or rotation
Moment of inertia (I) is a measure of a beam’s
ƒ Stiffness with respect to its cross section
ƒ Ability
y to resist bending
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As I increases, bending decreases
As I decreases, bending increases
Units of I are (length)4, e.g.
e g in4, ft4, or cm4
I for Common Cross
Cross-Sections
Sections
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I can be derived for any common area using calculus. However,
moment of inertia equations for common cross sections (e.g.,
rectangular,
t
l circular,
i l ttriangular)
i
l ) are readily
dil available
il bl iin math
th and
d
engineering textbooks.
For a solid rectangular cross section,
bh 3
Ix =
12
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X-axis (passing
through centroid)
b is the dimension parallel to the bending axis
h is the dimension perpendicular to the bending axis
h
b
Which Beam Will Bend (or Deflect) the
Most About the X-Axis?
P
P
h = 1.00”
X-Axis
X-Axis
h = 0.25”
YA i
Y-Axis
Y-Axis
b = 0.25”
b = 1.00”
Solid Rectangular
g
Beam #1
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Calculate the moment of inertia about the X-axis
x
I =
(
)(
2
1
I =
x
X-Axis
I x = 0.02083 in 4
b = 0.25”
)
3
h = 1.00”
n
i
0
0
.
1
n
i
5
2
3 2
.
h
2 0
b 11
Y-Axis
Solid Rectangular
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Beam #2
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Calculate the moment of inertia about the X-axis
Y Axis
Y-Axis
X-Axis
h = 0.25”
bh 3
IX =
12
(1.00 in )(0.25 in )
3
b = 1.00”
IX =
12
I X = 0.00130
0 00130 in
4
Compare
p
Values of Ix
Y-Axis
Y-Axis
X-Axis
h = 1.00”
h = 0.25”
X-Axis
b = 1.00”
b = 0.25”
I x = 0.02083 in
4
I X = 0.00130 in 4
Which beam will bend or deflect the most? Why?
Concentrated ((“Point”)) Load
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Suppose a concentrated load, P (lbf), is applied to the center of the
simply supported beam
P
L
Deflection
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The beam will bend or deflect downward as a result of the load P
(lbf).
P
Deflection (Δ)
( )
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Δ is a measure of the vertical displacement of the beam as a result
of the load P (lbf).
P
L
Deflection,
e ec o , Δ
Deflection (Δ)
( )
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Δ of a simply supported, center loaded beam can be calculated from
the following formula:
P
Deflection, Δ
3
PL
Δ =
48EI
L
P = concentrated load (lbf)
L = span length of beam (in)
E = modulus of elasticity (psi or lbf/in2)
I = moment of inertia of axis perpendicular to load P (in4)
Deflection (Δ)
( )
3
PL
Δ =
48EI
I, the Moment of Inertia, is a significant variable
in the determination of beam deflection
But….What is E?
Modulus of Elasticity
y ((E))
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Material property that indicates stiffness and rigidity
Values of E for many
y materials are readilyy available in tables in
textbooks.
Some common values are
Material
E (psi)
Steel
30 x 106
Aluminum
10 x 106
Wood
~ 2 x 106
Consider…
If the cross-sectional area of a solid wood beam is enlarged, how
does the Modulus of Elasticity, E, change?
PL3
Δ =
48EI
bh 3
Ix =
12
Material
E (psi)
Steel
30 x 106
Aluminum
10 x 106
Wood
~ 2 x 106
Consider…
Assuming the same rectangular cross-sectional area, which will have
the larger Moment of Inertia, I, steel or wood?
3
PL
Δ =
48EI
bh 3
Ix =
12
Material
E (psi)
Steel
30 x 106
Aluminum
10 x 106
Wood
~ 2 x 106
Consider…
Assuming beams with the same cross-sectional area and length,
which will have the larger deflection, Δ, steel or wood?
PL3
Δ =
48EI
bh 3
Ix =
12
Material
E (psi)
Steel
30 x 106
Aluminum
10 x 106
Wood
~ 2 x 106
More Complex
p
Designs
g
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The calculations for Moment of Inertia are very simple for a solid,
symmetric cross section.
C l l ti th
Calculating
the momentt off iinertia
ti ffor more complex
l cross-sectional
ti
l
areas takes a little more effort.
Consider a hollow box beam as shown below:
0.25 in.
6 in.
4 in.
Hollow Box Beams
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The same equation for moment of inertia, I = bh3/12, can be used
but is used in a different way.
Treat the outer dimensions as a positive area and the inner
dimensions as a negative area, as the centroids of both are about
the same X-axis.
X axis.
Negative Area
X-axis
Positive Area
Hollow Box Beams
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Calculate the moment of inertia about the X-axis for the positive
area and the negative area using I = bh3/12.
The outer dimensions will be denoted with subscript “o” and the
inner dimensions will be denoted with subscript “i”.
ho = 6 in.
X axis
X-axis
hi = 5.5
5 5 in.
in
bo = 4 in.
bi = 3.5 in.
Hollow Box Beams
ho = 6 in.
X-axis
hi = 5.5 in.
bi = 3.5 in.
bo = 4 in.
I pos
bo h o
=
12
3
I neg
(4 in )(6 in )
3
I pos =
12
I neg =
b h
= i i
12
3
(3.5 in )(5.5 in )3
12
Hollow Box Beams
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Simply subtract Ineg from Ipos to calculate the moment of inertia of the
box beam, Ibox
I box = I pos - I neg
0.25 in.
3
6 in.
I box
b h
b h
= o o − i i
12
12
3
I box =
3
(4 in )(6 in )3 − (3.5
3 5 in )(5.5
5 5 in )
I box =
(4 in )(216 in 3 ) − (3.5 in )(166.4 in 3 )
4 in.
12
12
I box = 23.5 in 4
12
12
Important
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In order to use the “positive-negative area” approach, the centroids
of both the positive and negative areas must be on the same axis!
ho = 6 in.
X-axis
hi = 5.5 in.
bo = 4 in.
bi = 3.5 in.
I Beams
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The moment of inertia about the X-axis of an I-beam can be
calculated in a similar manner.
I Beams
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Identify the positive and negative areas…
Centroids of the
positive area and
both negative areas
are aligned on the xaxis!
X-axis
a s
2 Negative Areas
Positive Area
I pos
bo h o
=
12
3
I neg
bi h i
=
12
3
I Beams
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…and calculate the moment of inertia about the X-axis similar to the
box beam
R
Remember
b th
there are ttwo negative
ti areas!!
Itotal = Ipos – 2 * Ineg
ho
X-Axis
hi
bo
3
I I − beam
bi
b h
2 bi h i
= o o −
12
12
3
bi
H Beams
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Can we use the “positive-negative area” approach to calculate the
Moment of Inertia about the X-axis (Ix) on an H-Beam?
X-Axis
H Beams
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Where are the centroids located?
X-Axis
They don’t align on the X-axis. Therefore, we can’t use the
“positive-negative approach” to calculate Ix!
We could use it to calculate Iy…but that’s beyond the scope of this
class.
H Beams
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We need to use a different approach.
Divide the H-beam into three positive areas.
Notice the centroids for all three areas are aligned on the X-axis.
X-Axis
h2
h1
h1
b2
b1
I H -beam =
3
1
b1
3
b1 h
b h
b h
+ 2 2 + 1
12
12
12
3
1
3
OR
I H -beam
2 b1 h1 b 2 h 2
=
+
12
12
3
Assignment
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Requirements
q
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Individual
ƒ Sketches of 3 beam
alternatives
ƒ Engineering calculations
ƒ Decision matrix
ƒ Final recommendation to
team
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Team
ƒ Evaluate designs proposed
by all members
ƒ Choose the top 3 designs
proposed by all members
ƒ Evaluate the top 3 designs
ƒ Select the best design
ƒ Submit a Test Data Sheet
ƒ Sketch of final design
ƒ Engineering
calculations
ƒ Decision matrix
ƒ Materials receipt
Test Data Sheet
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Problem statement
Sketch of final design
Calculations
Decision Matrix
Bill of materials and receipts
Performance data
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Design load
Volume
Weight
Moment of Inertia
Deflection
Engineering
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g Presentation
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Agenda
Problem definition
ƒ Design Requirements
ƒ Constraints
ƒ Assumptions
Project Plan
ƒ Work Breakdown Structure
ƒ Schedule
S h d l
ƒ Resources
Research Results
ƒ Benchmark Investigation
ƒ Literature Search
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Proposed Design Alternatives
Alternatives Assessment
(Decision Matrix)
Final Design
Benefits and Costs of the Final
Design
Expected vs. Actual Costs
Expected vs
vs. Actual
Performance
Project Plan Results
Conclusion and Summary
Project
j
Plan
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Start with the 5-step design process
Develop a work breakdown structure
ƒ List all tasks/activities
ƒ Determine priority and order
ƒ Identify milestone and critical path activities
Allocate resources
Create a Gantt chart
ƒ MS Project
P j t
ƒ Excel
ƒ Word
For Next Class…
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Read Chapter 8, Introduction to Engineering, pages 227 through
273