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
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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
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