1. Introduction to Structural Member Properties
Moment of Inertia
Principles of EngineeringTM
Unit 4 – Lesson Statics
Introduction to Structural Member Properties

2. What is a Beam?
Horizontal structural member used to support horizontal loads such as floors, roofs, and decks.
Types of beam loads
Uniform
Varied by length
Single point
Combination

3. Area Moment of Inertia (I)
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
Stiffness with respect to its cross section
Ability to resist bending
As I increases, bending decreases
As I decreases, bending increases
Units of I are (length)4, e.g. in4, ft4, or cm4
In general, a higher moment of inertia produces a greater resistance to deformation.

4. Moment of Inertia Principles
Principles of EngineeringTM
Unit 4 – Lesson Statics
Joist
Plank
Because of the orientation, the joist has a greater moment of inertia. The joist is 9 times as stiff as the plank in this example.
Beam
Material
Length
Width
Height
Area A
Douglas Fir
8 ft
1 ½ in.
5 ½ in.
8 ¼ in.2 B

5. Moment of Inertia Principles
Principles of EngineeringTM
Unit 4 – Lesson Statics
What distinguishes beam A from beam B?
Because of the orientation, the joist has a greater moment of inertia. The joist is 13.4 times as stiff as the plank in this example.
Will beam A or beam B have a greater resistance to bending, resulting in the least amount of deformation, if an identical load is applied to both beams at the same location?

6. Calculating Moment of Inertia – Rectangles
Moment of Inertia Principles
Moment of Inertia
Principles of EngineeringTM
Unit 4 – Lesson Statics
Why did beam B have greater deformation than beam A?
Difference in moment of inertia due to the orientation of the beam
Calculating Moment of Inertia – Rectangles
b is the dimension parallel to the bending axis
h is the dimension perpendicular to the bending axis

7. Calculating Moment of Inertia
Principles of EngineeringTM
Unit 4 – Lesson Statics
Calculate beam A moment of inertia

8. Calculating Moment of Inertia
Principles of EngineeringTM
Unit 4 – Lesson Statics
Calculate beam B moment of inertia

9. Moment of Inertia 14Times Stiffer Beam A Beam B Moment of Inertia
Principles of EngineeringTM
Unit 4 – Lesson Statics
14Times Stiffer
Beam A
Beam B

10. Moment of Inertia – Composite Shapes
Principles of EngineeringTM
Unit 4 – Lesson Statics
Moment of Inertia – Composite Shapes
Why are composite shapes used in structural design?

11. Non-Composite vs. Composite Beams
Moment of Inertia
Principles of EngineeringTM
Unit 4 – Lesson Statics
Doing more with less
Calculating the moment of inertia for a composite shape, such as an I-beam, is beyond the scope of this presentation. The value is given for purposes of comparison.
Both of these shapes are 2 in. wide x 4 in. tall, and both beams are comprised of the same material. The I-beam’s flanges and web are 0.38 in. thick.
The moment of inertia for the rectangular beam is in.4 Its area is 8 in.2.
The moment of inertia for the I-Beam is 6.08 in.4. Its area is 2.75 in.2.
The I-beam may be 43% less stiff than the rectangular beam, BUT it uses 66% less material.
Increasing the height of the I-beam by about 1 inch will make the moment of inertia for both of the shapes equal, but the I-beam will still use less material (61% less).
Area = 8.00in.2
Area = 2.70in.2

12. Structural Member Properties
Moment of Inertia
Principles of EngineeringTM
Unit 4 – Lesson Statics
Structural Member Properties
Chemical Makeup
Modulus of Elasticity (E) The ratio of some specified form of stress to some specified form of strain. E defines the stiffness of an object related to material chemical properties.
In general, a higher modulus of elasticity produces a greater resistance to deformation.

13. Deflection
The beam will bend or deflect downward as a result of the load P (lbf).
The ability of a material to deform and return to its original shape.
Deflection, Δ L P

14. Deflection (Δ)
Δ of a simply supported, center loaded beam can be calculated from the following formula: L P
Deflection, Δ
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)

15. Modulus of Elasticity Principles
Moment of Inertia
Principles of EngineeringTM
Unit 4 – Lesson Statics
Characteristics of objects that affect deflection (Δ)
Applied force or load
Length of span between supports
Modulus of elasticity
Moment of inertia

16. Calculating Beam Deflection
Moment of Inertia
Calculating Beam Deflection
Principles of EngineeringTM
Unit 4 – Lesson Statics
Beam
Material
Length
(L)
Moment of Inertia
(I)
Modulus of Elasticity
(E)
Force (F) A
Douglas Fir
8.0 ft
20.80 in.4
1,800,000 psi
250 lbf B
ABS Plastic
419,000 psi

17. Calculating Beam Deflection
Moment of Inertia
Calculating Beam Deflection
Principles of EngineeringTM
Unit 4 – Lesson Statics
Calculate beam deflection for beam A
Beam
Material
Length I E
Load A
Douglas Fir
8.0 ft
20.80 in.4
1,800,000 psi
250 lbf

18. Calculating Beam Deflection
Moment of Inertia
Calculating Beam Deflection
Principles of EngineeringTM
Unit 4 – Lesson Statics
Calculate beam deflection for beam B
Beam
Material
Length I E
Load B
ABS Plastic
8.0 ft
20.80 in.4
419,000 psi
250 lbf

19. Douglas Fir vs. ABS Plastic
Moment of Inertia
Douglas Fir vs. ABS Plastic
Principles of EngineeringTM
Unit 4 – Lesson Statics
4.24 times less deflection