BIOE 3200 - Fall 2015

Galileo’s BeamTo read more about the history of beam theory, check this out: https://newtonexcelbach.wordpress.com/2008/02/27/the-history-of-the-theory-of-beam-bending-part-1 /

Normal and Shear Stresses in Bending

Learning objective:◦Determine normal (σx) and shear (τxy) stresses in beams in bending

BIOE 3200 - Fall 2015

Axial loads in long bones create bending

BIOE 3200 - Fall 2015

Normal Stresses in Beams

BIOE 3200 - Fall 2015

From http://www.strucalc.com/engineering-resources/normal-stress-bending-stress-shear-stress/

Normal stress in beam cross-section in bending:

σx = σx = Normal (flexural) stressMz= Bending momenty = vertical distance from the neutral axisIz = Second moment of area

Definition of normal stress:

Shear Stresses in Beams

BIOE 3200 - Fall 2015

Shear stress in beam cross-section in bending:τxy =

τxy = Shear stressQ = first moment of the shaded area with respect to the neutral axis V(x) = Calculated shear at specific sectiony = vertical distance from the neutral axisI = Second moment of areat = Width of beam at depth of specific section

What is the first moment of area?

Area: A = First moment of area

;

◦ Qy=Axc , Qx=A yc ; Q=0 about centroid axes

Centroid: ◦ xc = Qy/A ◦ yc = Qx/A

Second moment of area:

◦ for rectangular cross sections

BIOE 3200 - Fall 2015

Calculating first moment of area Q(y) at a point p in a rectangular cross-section:

: Distance from origin of z-y coordinate (centroid of the rectangle) to centroid of the shaded area

y : Distance from origin of z-y coordinate (centroid of the rectangle) to bottom of the shaded area (where p is located)

: Area value of the shaded area b = width of beam; h = height of

beam

Qx=

BIOE 3200 - Fall 2015