Normal and shear stresses in beams in bending

1. BIOE 3200 - Fall 2015
Galileo’s Beam
To 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

2. Learning objective:
◦ Determine normal (σx) and shear (τxy)
stresses in beams in bending
BIOE 3200 - Fall 2015

3. Axial loads in long bones create
bending
BIOE 3200 - Fall 2015

4. 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 =
𝑀 𝑧
𝐼
𝑦 𝑑𝐴 =
𝑀 𝑧
𝑦
𝐼 𝑧
σx = Normal (flexural) stress
Mz= Bending moment
y = vertical distance from the neutral axis
Iz = Second moment of area
Definition of
normal stress:

5. Shear Stresses in Beams
BIOE 3200 - Fall 2015
Shear stress in beam cross-section in bending:
τxy =
𝑉 𝑥 𝑑𝑥
𝐼
𝑦 𝑑𝐴 =
𝑉(𝑥)𝑄
𝑡𝐼 𝑧
τxy = Shear stress
Q = first moment of the shaded area with respect to the neutral axis
V(x) = Calculated shear at specific section
y = vertical distance from the neutral axis
I = Second moment of area
t = Width of beam at depth of specific section

6. What is the first moment of area?
 Area: A = 𝐴
𝑑𝐴
 First moment of area
𝑄 𝑦 = 𝐴
𝑥 𝑑𝐴 ; 𝑄 𝑥 = 𝐴
𝑦 𝑑𝐴
◦ Qy=Axc , Qx=A yc ; Q=0 about centroid
axes
 Centroid:
◦ xc = Qy/A
◦ yc = Qx/A
 Second moment of area:
 𝐼x = 𝑦2
𝑑𝐴
◦ 𝐼 = 𝑏ℎ3
12
for rectangular cross sections
BIOE 3200 - Fall 2015

7. Calculating first moment of area Q(y) at
a point p in a rectangular cross-section:
 𝑦𝑝 : 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)
 𝐴 𝑝: Area value of the shaded area
 b = width of beam; h = height of beam
 Qx=𝐴 𝑝 𝑦𝑝
BIOE 3200 - Fall 2015