Structural Member Properties Moment of Inertia I is

Structural Member Properties Moment of Inertia (I) is a mathematical property of a cross section (measured in inches 4) that gives important information about how that crosssectional area is distributed about a centroidal axis. Stiffness of an object related to its shape In general, a higher Moment of Inertia produces a greater resistance to deformation. ©i. Stockphoto. com

Moment of Inertia Principles 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

Calculating Moment of Inertia Calculate beam A Moment of Inertia

Moment of Inertia – Composite Shapes Why are composite shapes used in structural design?

Beam Deflection Measurement of deformation – Importance of stiffness – Change in vertical position – Scalar value – Deflection formulas –

Beam Structure Examples

What Causes Deflection? Snow Live Load Walls, Floors, Materials, Structure Dead Load Roof Materials, Structure Dead Load Occupants, Movable Fixtures, Furniture Live Load

Loading Snow Live Load Walls, Floors, Materials, Structure Dead Load Roof Materials, Structure Dead Load Occupants, Movable Fixtures, Furniture Live Load

Types of Loads

Factors that Affect Bending Material Property – Physical Property – Supports –

Physical Property - Geometry

Beam Supports

Beam Deflections Spring Board Deflection Bridge Deflection

Calculating Deflection on a Spring Diving Board Pine Diving Board Dimensions: Base (B) = 12 in. Height (H) = 2 in. 72 in. P 250 lb Max ? Known: Pine (E) = 1. 76 x 106 psi Applied Load (P)= 250 lb

Deflection of Cantilever Beam with Concentrated Load max = P x L 3 3 x. Ex. I L P max Where: max is the maximum deflection P is the applied load L is the length E is the elastic modulus I is the cross section moment of inertia

Moment of Inertia (MOI) Moment of Inertia (I) is a mathematical property of a cross section (measured in inches 4) that is concerned with a surface area and how that area is distributed about a centroidal axis.

Calculating Moment of Inertia (I) I = (12 in. )(2 in. )3 12 I = (12 in. )(8 in. 3) 12 I = 96 in. 4 12 I = 8 in. 4

Cantilever Beam Load Example Known: Pine (E) = 1. 76 x 106 psi Applied Load (P) = 250 lb 72 in. max = P x L 3 3 x. Ex. I max = (250 lb) (72 in. )3 (3) (1. 76 x 106 psi) (8 in. 4) max = (250 lb) (373248 in. 3) (1. 76 x 106 psi) (8 in. 4) P 250 lb Max

Cantilever Beam Load Example max = (9. 3312 x 107 lb)(in. 3) (5. 28 x 106 psi)(8 in. 4) max = (9. 3312 x 107 lb)(in. 3) (4. 224 x 107 psi)(in. 4) max = (9. 3312 x 107) (4. 224 x 107 in. ) max = 2. 21 inches

Calculating Deflection on a Pine Beam in a Structure Beam Dimensions: Base (B) = 4 in. Height (H) = 6 in. Length (L) = 96 in. L P max Known: Pine (E) = 1. 76 x 106 psi Applied Load (P)= 200 lb

Deflection of Simply Supported Beam with Concentrated Load max = L 3 Px 48 x E x I P L max Note that the simply supported beam is pinned at one end. A roller support is provided at the other end. Where: max is the maximum deflection P is the applied load L is the length E is the elastic modulus I is the cross section moment of inertia

Calculating Moment of Inertia (I) I = (4 in. )(6 in. )3 12 I = (4 in. )(216 in. 3) 12 I = 864 in. 4 12 I = 72 in. 4

Simply Supported Beam Example Known: Pine (E) = 1. 76 x 106 psi Applied Load (P) = 200 lb 96 in. P max = P x L 3 48 x E x I max = (200 lb)(96 in. )3 (48)(1. 76 x 106 psi)(72 in. 4) max = (200 lb)(884736 in. 3) (48)(1. 76 x 106 psi)(72 in. 4)

Simply Supported Beam Example max = (1. 769472 x 108 lb)(in. 3) (8. 448 x 107 psi)(72 in. 4) max = (1. 769472 x 108 lb)(in. 3) (6. 08256 x 109 psi)(in. 4) max = (1. 769472 x 108) (6. 08256 x 109 in. ) max = 0. 029 inches