structural member properties moment of inertia (i) is a mathematical property of a cross section...
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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
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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
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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
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Moment of Inertia – Composite Shapes
Why are composite shapes used in structural design?
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Beam Deflection
– Measurement of deformation– Importance of stiffness– Change in vertical position– Scalar value– Deflection formulas
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Beam Structure Examples
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What Causes Deflection?Snow Live Load
Roof Materials, Structure Dead Load
Walls, Floors,Materials, StructureDead Load
Occupants, MovableFixtures, Furniture Live Load
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LoadingSnow Live Load
Roof Materials, Structure Dead Load
Walls, Floors,Materials, StructureDead Load
Occupants, MovableFixtures, Furniture Live Load
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Types of Loads
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Factors that Affect Bending
– Material Property– Physical Property– Supports
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Physical Property - Geometry
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Beam Supports
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Beam Deflections
Spring Board DeflectionBridge Deflection
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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
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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
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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.
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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
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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
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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
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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
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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
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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
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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
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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