3.3 Normal Stress
Using the proportional correlation of stress and strain:
max
x,
x
σ
c
y
σ
=
(3.13) in (3.11):
∑
=
0
F
x
:
0
dA
y
c
σ
dA
σ
c
y
dA
σ
max
max
x
∫
∫
∫
=
⋅
=
⋅
=
⋅
∫
dA
y
:
first moment of cross section (statical moment)
→
about the neutral axis =0
→
neutral axis = centroidal axis
∑
=
0
M
z
:
z
2
max
x
M
dA
y
c
σ
dA
σ
y
=
⋅
=
⋅
⋅
∫
∫
z
2
max
M
dA
y
c
σ
=
⋅
∫
dA
y
I
2
⋅
=
∫
:
second moment of cross section (moment of inertia)
Transformation of (3.14):
z
z
max
I
c
M
σ
=
(3.13) in (3.15):
z
z
x
I
y
M
σ
=
Introducing:
c
I
S
=
elastic section modulus
(3.15) becomes:
z
z
max
S
M
σ
=
since
I
y
M
ε
E
σ
⋅
=
⋅
=
→
I
E
y
M
⋅
⋅
=
ε
recalling (3.12):
ρ
y
ε
=
in (3.18):
κ
=
⋅
=
ρ
I
E
M
1
(continued in chapter 7, deflection of beams)
(3.13)
(3.14)
(3.15)
(3.16)
flexual stress (linear elastic)
elastic flecture formulas
(3.17)
(3.18)
(3.19)
curvature of neutral axis
EI
= bending or flexual
stiffness
fig 3.22: stress distribution along section of beam
c
x
y
neutral axis

σ
max
+
σ
max
M
z
24