Introduction to Beams

1
Introduction to Beams

• A beam is a horizontal structural member used to
  support loads
• Beams are used to support the roof and floors in
  buildings

2
Introduction to Beams

• Common shapes are
• I Angle
  Channel
• Common materials are steel and wood

Source Load Resistance Factor Design (First
Edition), AISC
3
Introduction to Beams

• 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
Flanges
Web
Flanges
4
Introduction to Beams

• Beams are supported in structures via different
  configurations

Source Statics (Fifth Edition), Meriam and
Kraige, Wiley
5
Introduction to Beams

• Beams are designed to support various types of
  loads and forces

Concentrated Load
Distributed Load
Source Statics (Fifth Edition), Meriam and
Kraige, Wiley
6
Beam Theory

• Consider a simply supported beam of length, L.
  The cross section is rectangular, with width, b,
  and depth, h.

h
b
L
7
Beam Theory

• 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
h/2
X - Axis
h/2
b/2
b/2
8
Beam Theory

• An important variable in beam design is the
  moment of inertia of the cross section, denoted
  by I.
• Inertia is a measure of a bodys ability to
  resist rotation.
• Moment of inertia is a measure of the stiffness
  of the beam with respect to the cross section and
  the ability of the beam to resist bending.
• As I increases, bending and deflection will
  decrease.
• Units are (LENGTH)4, e.g. in4, ft4, cm4

9
Beam Theory

• I can be derived for any common area using
  calculus. However, moment of inertia equations
  for common cross sections (e.g., rectangular,
  circular, triangular) are readily available in
  math and engineering textbooks.
• For a rectangular cross section,
• b is the dimension parallel to the bending axis.
  h is the dimension perpendicular to the bending
  axis.

X-axis (passing through centroid)
h
b
10
Beam Theory

• Example Calculate the moment of inertia about
  the X-axis for a yardstick that is 1 high and ¼
  thick.

Y-Axis
h 1.00
X-Axis
b 0.25
11
Beam Theory

• Example Calculate the moment of inertia about
  the Y-axis for a yardstick that is 1 high and ¼
  thick.

X-Axis
h 0.25
Y-Axis
b 1.00
12
Beam Theory

• Suppose a concentrated load, P, is applied to the
  center of the simply supported beam.

P
L
13
Beam Theory

• The beam will bend downward as a result of the
  load P.

P
14
Beam Theory

• The deflection (?) is the vertical displacement
  of the of the beam as a result of the load P.

Deflection, ?
L
15
Beam Theory

• The deflection (?) of a simply supported, center
  loaded beam can be calculated from the following
  formula

where, P concentrated load (lbs.) L span
length of beam (in.) E modulus of elasticity
(lbs./in.2) I moment of inertia of axis
perpendicular to load P (in.4)
P
L
16
Beam Theory

• Modulus of elasticity, E, is a property that
  indicates the stiffness and rigidity of the beam
  material. For example, steel has a much larger
  modulus of elasticity than wood. Values of E for
  many materials are readily available in tables in
  textbooks. Some common values are

17
Beam Theory

• Example Calculate the deflection in the steel
  beam supporting a 500 lb load shown below.

P 500 lb
h 2
b 3
L 36
18
Beam Theory

• Step 1 Calculate the moment of inertia, I.

19
Beam Theory

• Step 2 Calculate the deflection, ?.

20
Beam Theory

• These calculations are very simple for a solid,
  symmetric cross section.
• Now consider slightly more complex symmetric
  cross sections, e.g. hollow box beams.
  Calculating the moment of inertia takes a little
  more effort.
• Consider a hollow box beam as shown below

0.25 in.
6 in.
4 in.
21
Beam Theory

• The same equation for moment of inertia, I
  bh3/12, can be used.
• 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
X-axis
Negative Area
Positive Area
22
Beam Theory

• 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
hi 5.5 in.
bi 3.5 in.
bo 4 in.
23
Beam Theory
ho 6 in.
X-axis
hi 5.5 in.
bi 3.5 in.
bo 4 in.
24
Beam Theory

• Simply subtract Ineg from Ipos to calculate the
  moment of inertia of the box beam, Ibox

0.25 in.
6 in.
4 in.
25
Beam Theory

• The moment of inertia of an I-beam can be
  calculated in a similar manner.

26
Beam Theory

• Identify the positive and negative areas

Positive Area
Negative Area
27
Beam Theory

• and calculate the moment of inertia similar to
  the box beam (note the negative area dimensions
  and that it is multiplied by 2).

ho
hi
bi
bi
bo
28
Beam Theory

• The moment of inertia of an H-beam can be
  calculated in a similar manner

29
Beam Theory

• The moment of inertia of an H-beam can be
  calculated in a similar manner

30
Beam Theory

• however, the H-beam is divided into three
  positive areas.

h2
h1
h1
b2
b1
b1
31
Beam Theory

• Example Calculate the deflection in the I-beam
  shown below. The I-beam is composed of three ½
  x 4 steel plates welded together.

P 5000 lbf
L 8 ft
½ x 4 steel plate (typ.)
32
Beam Theory

• First, calculate the moment of inertia for an
  I-beam as previously shown, i.e. divide the cross
  section of the beam into positive and negative
  areas.

ho 5 in.
hi 4 in.
bi
bi
bo 4 in.
33
Beam Theory

• First, calculate the moment of inertia for an
  I-beam as previously shown, i.e. divide the cross
  section of the beam into positive and negative
  areas.

ho 5 in.
hi 4 in.
bi 1.75in
bi
bo 4 in.
34
Beam Theory

• Next, calculate the deflection (Esteel 30 x 106
  psi).

P 5000 lbf
L 8 ft
35
Beam Theory

• Calculate the deflection, ?.

36
Beam Theory

• Example Calculate the volume and mass of the
  beam if the density of steel is 490 lbm/ft3.

L 8 ft
½ x 4 steel plate (typ.)
37
Beam Theory

• Volume (Area) x (Length)

38
Beam Theory

• Convert to cubic feet

39
Beam Theory

• Calculate mass of the beam
• Mass Density x Volume

40
Materials

• Basswood can be purchased from hobby or craft
  stores. Hobby Lobby carries many common sizes of
  basswood. DO NOT purchase balsa wood.
• 1201 teams must submit a receipt for the
  basswood.
• The piece of basswood in the Discovery Box WILL
  NOT be used for Project 2.
• Clamps and glue are provided in the Discovery
  Box. Use only the glue provided.

41
Assembly

• I-beams and H-beams Begin by marking the
  flanges along the center where the web will be
  glued.
• Box beams No marking is necessary.

• I-beams and H-beams Apply a small amount of
  glue along the length of the web and also to the
  flange.
• Box beams Apply a small amount of glue two one
  side and the bottom to form an L shaped section.

42
Assembly

• I-beams and H-beams Press the two pieces
  together and hold for a couple of minutes.
• Box beams Press the two pieces together into an
  L shape and hold for a couple of minutes.

• I-beams and H-beams Clamp the pieces and allow
  the glue to cure as instructed on the bottle.
• Box beams Clamp the pieces and allow the glue to
  cure as instructed on the bottle.
• YOU MUST ALLOW EACH GLUE JOINT TO CURE COMPLETELY
  BEFORE CONTINUING ON TO ADDITIONAL GLUE JOINTS!

43
Assembly

• Stiffeners may be applied to the web of an I or H
  beam or between sides of a box beam.
• Stiffeners are small pieces of wood that aid in
  gluing and clamping the beam together.
• Stiffeners may be necessary if the pieces of wood
  that the team has chosen are thin (less than
  3/16). Thin pieces of wood may collapse when
  clamped together.
• Stiffeners keep the flanges stable during
  clamping and glue curing.