SME
Advanced Solid Mechanics (Solid Mechanics II)
Plasticity and failure criteria
Use this for “A 60 × 120 mm steel bar yields at 300 MPa. Find MY, Mₚ and the shape factor.”
Diagram
Fully plastic: σY everywhere, split where the areas above and below are equal; first yield dashed.
Elastic section modulus S = 144,000 mm³
Plastic section modulus Z = 216,000 mm³
Yield moment MY = 43.2 kN·m
Shape factor f = 1.5
This is an example. Type your own values to start a new problem.
S (mm³)
=
b (mm)
×
(
h (mm)
)
2
6
Z (mm³)
=
b (mm)
×
(
h (mm)
)
2
4
MY (kN·m)
=
σY (MPa)
×
S (mm³)
,
Mₚ (kN·m)
=
σY (MPa)
×
Z (mm³)
f
Shape factor
worked out (not typed)
Formulas
S
=
b
×
h
2
6
144,000
=
60
×
120
2
6
Z
=
b
×
h
2
4
216,000
=
60
×
120
2
4
MY
=
σY
×
S
43,200,000 = 300 × 144,000
Mₚ
=
σY
×
Z
64,800,000 = 300 × 216,000
f
=
Mₚ
MY
1.5
=
64,800,000
43,200,000
Formulas are worked in N·mm. The step-by-step shows the conversions.
Assumptions
•
Elastic–perfectly plastic: σ = Eε up to σY, then stays at σY.
•
At MY only the edges reach σY; at Mₚ every fiber has yielded.
•
A rectangle bent about its horizontal axis, worked in N, mm and MPa.
Step-by-step
WE KNOW
b = 60 mm, h = 120 mm, σY = 300 MPa
FIND
elastic section modulus (S), plastic section modulus (Z), yield moment (MY), plastic moment (Mₚ), shape factor (f)
Step 1 · Find elastic section modulus (S)
Use S = b × h² ÷ 6
The
elastic
modulus
I
c
of
a
rectangle:
bh
³
12
over
h
2
.
S
=
60
×
120
2
6
S
=
60
×
14,400
6
S
=
864,000
6
S
=
144,000
mm³
Step 2 · Find plastic section modulus (Z)
Use Z = b × h² ÷ 4
Each
half,
area
bh
2,
pulls
or
pushes
at
its
own
center;
the
two
centers
are
h
2
apart.
Z
=
60
×
120
2
4
Z
=
60
×
14,400
4
Z
=
864,000
4
Z
=
216,000
mm³
Step 3 · Find yield moment (MY)
Use MY = σY × S
The flexure formula with the edge stress just at σY.
MY = 300 × 144,000
MY = 43,200,000 N·mm
Step 4 · Find plastic moment (Mₚ)
Use Mₚ = σY × Z
Every fiber carries σY, tension on one side of the axis and compression on the other.
Mₚ = 300 × 216,000
Mₚ = 64,800,000 N·mm
Step 5 · Find shape factor (f)
Use f = Mₚ ÷ MY
How
much
more
moment
full
yielding
carries
than
first
yield;
Z
S
is
1.5
for
any
rectangle.
f
=
64,800,000
43,200,000
f
=
1.5
Step 6 · Convert the answers
Change each answer back to the unit you chose.
MY = 43,200,000 N·mm = 43.2 kN·m (1 kN·m = 10⁶ N·mm)
Mₚ = 64,800,000 N·mm = 64.8 kN·m (1 kN·m = 10⁶ N·mm)
Check
Put the numbers back in. Both sides should match.
144,000
=
60
×
120
2
6
✓
216,000
=
60
×
120
2
4
✓
43,200,000 = 300 × 144,000 ✓
64,800,000 = 300 × 216,000 ✓
1.5
=
64,800,000
43,200,000
✓
← Previous lesson
Impact of a dropped weight on a bar
Next lesson →
Von Mises and Tresca from three principal stresses
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.