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BeamDeflection

In This Lesson: Basics of Deflection Causes of Deflection Factors that Affect Bending Deflection of Cantilever Beam Deflection of Simply Supported Beam

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

Beam Structure Examples

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

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

Types of Loads

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

Physical Property - Geometry

Beam Supports

Spring Board Deflection Beam Deflections Bridge Deflection

250 lb P 72 in.  Max ? Calculating Deflection on a Spring Diving Board Pine Diving Board Dimensions: Base (B) = 12 in. Height (H) = 2 in. Known: Pine (E) = 1.76 x 106 psi Applied Load (P)= 250 lb

P L  max Deflection of Cantilever Beam with Concentrated Load  max = P x L3 3 xE x I 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 inches4) that is concerned with a surface area and how that area is distributed about a centroidal axis.

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

250 lb P 72 in.  Max Cantilever Beam Load Example Known: Pine (E) = 1.76 x 106 psi Applied Load (P) = 250 lb •  max = P x L3 • 3 x E x I •  max = (250 lb) (72 in.)3 • (3) (1.76 x 106 psi) (8 in.4) •  max = (250 lb) (373248 in.3) • (3) (1.76 x 106 psi) (8 in.4)

max= (9.3312 x 107 lb)(in.3) (5.28 x 106psi)(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 Cantilever Beam Load Example

L P  max Calculating Deflection on a Pine Beam in a Structure Beam Dimensions: Base (B) = 4 in. Height (H) = 6 in. Length (L) = 96 in. Known: Pine (E) = 1.76x106 psi Applied Load (P)= 200 lb

L P  max Deflection of Simply Supported Beam with Concentrated Load  max = P x L3 48 x E x I • 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

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

Simply Supported Beam Example 96 in. Known: Pine (E) = 1.76x106 psi Applied Load (P) = 200 lb P  max •  max = P x L3 • 48 x E x I •  max = (200 lb)(96 in.)3 • (48)(1.76x106psi)(72 in.4) •  max = (200 lb)(884736 in.3) • (48)(1.76x106 psi)(72 in.4)

Simply Supported Beam Example • max= (1.769472 x 108 lb)(in.3) (8.448 x 107psi)(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

References Onouye, B., & Kane, K. (2002). Statics and strength of materials for architecture and building construction. Upper Saddle River, NJ: Prentice Hall. Shaeffer, R.E. (2002). Elementary structures for architects and builders. Upper Saddle River, NJ: Prentice Hall.

Credits Writer: Matt Putman Content Editor: Sam Cox & Wes Terrell Production Work: CJ Amarosa Publisher: CJ Amarosa – Project Lead The Way Virtual Academy for Professional Development – www.pltw.org

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