Introduction to Structural Member Properties Moment of Inertia Principles of EngineeringTM Unit 4 – Lesson 4.1 - Statics Introduction to Structural Member Properties

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

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.

Moment of Inertia Principles Principles of EngineeringTM Unit 4 – Lesson 4.1 - 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

Moment of Inertia Principles Principles of EngineeringTM Unit 4 – Lesson 4.1 - 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?

Calculating Moment of Inertia – Rectangles Moment of Inertia Principles Moment of Inertia Principles of EngineeringTM Unit 4 – Lesson 4.1 - 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

Calculating Moment of Inertia Principles of EngineeringTM Unit 4 – Lesson 4.1 - Statics Calculate beam A moment of inertia

Calculating Moment of Inertia Principles of EngineeringTM Unit 4 – Lesson 4.1 - Statics Calculate beam B moment of inertia

Moment of Inertia 14Times Stiffer Beam A Beam B Moment of Inertia Principles of EngineeringTM Unit 4 – Lesson 4.1 - Statics 14Times Stiffer Beam A Beam B

Moment of Inertia – Composite Shapes Principles of EngineeringTM Unit 4 – Lesson 4.1 - Statics Moment of Inertia – Composite Shapes Why are composite shapes used in structural design?

Non-Composite vs. Composite Beams Moment of Inertia Principles of EngineeringTM Unit 4 – Lesson 4.1 - 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 10.67 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

Structural Member Properties Moment of Inertia Principles of EngineeringTM Unit 4 – Lesson 4.1 - 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.

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

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)

Modulus of Elasticity Principles Moment of Inertia Principles of EngineeringTM Unit 4 – Lesson 4.1 - Statics Characteristics of objects that affect deflection (Δ) Applied force or load Length of span between supports Modulus of elasticity Moment of inertia

Calculating Beam Deflection Moment of Inertia Calculating Beam Deflection Principles of EngineeringTM Unit 4 – Lesson 4.1 - 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

Calculating Beam Deflection Moment of Inertia Calculating Beam Deflection Principles of EngineeringTM Unit 4 – Lesson 4.1 - 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

Calculating Beam Deflection Moment of Inertia Calculating Beam Deflection Principles of EngineeringTM Unit 4 – Lesson 4.1 - 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

Douglas Fir vs. ABS Plastic Moment of Inertia Douglas Fir vs. ABS Plastic Principles of EngineeringTM Unit 4 – Lesson 4.1 - Statics 4.24 times less deflection