Centroide e inercia en vigas

t r ic me es Geo erti rop p
15 Geometric properties Copyright © G G Schierle, 2001-05 press Esc to end, ↓ for next, ↑ for previous slide 1

Geometric properties
Type of property:
1. 2. 3. 4. 5.6. Cross section area A Centroid C Moment of Inertia I Polar Moment of Inertia J Section Modulus S Radius of Gyration r

Defines:
Axial stress fa and shear stress fv Center of mass (Neutral Axis)Bending stress fb and deflection Δ Torsion stress τ Max. bending stress fb (S = I/c) Column slenderness r = (I/A)1/2

Today’s topics:

Centroid Parallel Axis Theorem
15 Geometric propertiesCopyright © G G Schierle, 2001-05

Centroidal Moment of Inertia, etc. Moment of Inertia of composite sections

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Centroid
Centroid is thecenter of mass of a body or surface area. Beam centroid is the Neutral Axis of zero bending stress. Centroid also defines distributed load center of mass, etc. 1 Centroid C of freeform body 2 CentroidC of composite cross section (with centroid outside cross section area) Centroid is a point where the moment of all partial areas is zero, i. e., the area is balanced at the centroid. Defining thetotal area A =Σda with lever arms x’ and y’ from an arbitrary origin fo partial areas da with lever arms x and y to that origin, yields: Σ Mx = 0 → x’A - Σ x da = 0 A = Σda → x’Σda = Σx da x’= Σx da /Σda y’= Σy da / Σda

15 Geometric properties

Copyright © G G Schierle, 2001-05

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X’=8/2=4”

Centroid
Beam centroid example Assume: A1= 8”x2” A2 = 2 x 2” x 6” Y1 = 6 + 1 Y2 = 6/2 Due to symmetry: X’ = 8/2 A1 = 16 in2 A2 = 24 in2 Y1 = 7” Y2 = 3” X’ = 4”

Y’= ΣAY / ΣA = (A1 Y1+A2 Y2) / (A1+A2) Part 1 2 Σ Y’ = 184 / 40
15 Geometricproperties Copyright © G G Schierle, 2001-05 press Esc to end, ↓ for next, ↑ for previous slide

A (in2) 16 24 40

Y (in) 7 3

A Y (in3) 112 72 184 Y’ = 4.6”
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1. T-beam centroid...