Structural Member PropertiesMoment of Inertia (I) is a mathematical property of a cross section (measured in inches4) that gives important information about how that cross-sectional area is distributed about a centroidal axis.

In general, a higher Moment of Inertia produces a greater resistance to deformation.

Stiffness of an object related to its shape

©iStockphoto.com ©iStockphoto.com

Calculating Moment of Inertia - Rectangles

Why did beam B have greater deformation than beam A?

Moment of Inertia Principles

Difference in Moment of Inertia due to the orientation of the beam

Calculating Moment of Inertia

31.5 in. 5.5 in.

= 12

31.5 in. 166.375 in.=

12

4249.5625 in.=

12

4= 20.8 in.

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

Roof Materials, Structure Dead Load

Walls, Floors,Materials, StructureDead Load

Occupants, MovableFixtures, Furniture Live Load

LoadingSnow Live Load

Roof Materials, Structure Dead Load

Walls, Floors,Materials, StructureDead Load

Occupants, MovableFixtures, Furniture Live Load

Types of Loads

Factors that Affect Bending

– Material Property– Physical Property– Supports

Physical Property - Geometry

Beam Supports

Beam Deflections

Spring Board DeflectionBridge Deflection

Calculating Deflection on a Spring Diving Board

Known:Pine (E) = 1.76 x 106 psiApplied Load (P)= 250 lb

Pine Diving Board Dimensions: Base (B) = 12 in. Height (H) = 2 in.

72 in.P

Max ?

250 lb

Deflection of Cantilever Beam with Concentrated Load

max = P x L3

3 x E x I

Where: max is the maximum deflection

P is the applied loadL is the lengthE is the elastic modulusI is the cross section moment of

inertia

PL

max

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.

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

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)

Known:Pine (E) = 1.76 x 106 psiApplied Load (P) = 250 lb 72 in. P

Max

250 lb

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

Known:Pine (E) = 1.76x106 psiApplied Load (P)= 200 lb

Beam Dimensions: Base (B) = 4 in. Height (H) = 6 in.Length (L) = 96 in. P

L

max

Deflection of Simply Supported Beam with Concentrated Load

max = P x L3

48 x E x INote 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

PL

max

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

max = P x L3 48 x E x I

max = (200 lb)(96 in.)3 (48)(1.76x106 psi)(72 in.4)

max = (200 lb)(884736 in.3) (48)(1.76x106 psi)(72 in.4)

Known:Pine (E) = 1.76x106 psiApplied Load (P) = 200 lb

P96 in.

max

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