SME
Mechanics of Materials (Solid Mechanics I)
Bending and shear
Use this for “A 100 × 200 mm beam carries a 20 kN·m bending moment. Find the largest bending stress.”
Diagram
I = 66,670,000 mm⁴ about the centroidal axis
σ
=
Mc
I
:
zero
at
the
neutral
axis
through
the
centroid,
largest
at
the
far
edge.
This is an example. Type your own values to start a new problem.
I (mm⁴)
=
b (mm)
×
(
h (mm)
)
3
12
σₘₐₓ (MPa)
=
M (kN·m)
×
c (mm)
I (mm⁴)
Formulas
I
=
b
×
h
3
12
6.6667 ×
10
7
=
100
×
200
3
12
c
=
h
2
100
=
200
2
σₘₐₓ
=
M
×
c
I
30
=
20,000,000
×
100
6.6667 ×
10
7
Formulas are worked in N·mm. The step-by-step shows the conversions.
Assumptions
•
Plane sections stay plane and the beam stays linear elastic, so σ grows in a straight line from 0 at the neutral axis.
•
M bends the beam about the horizontal axis through the centroid, at mid-depth of the rectangle.
•
Worked in N, mm and MPa: N·mm × mm ÷ mm⁴ is N/mm², which is MPa.
Step-by-step
WE KNOW
M = 20 kN·m, b = 100 mm, h = 200 mm
FIND
second moment of area (I), neutral axis to the far edge (c), largest bending stress (σₘₐₓ)
Step 1 · Convert to the formula’s units
The formulas don’t work directly in the units you chose, so convert first (to N·mm).
M = 20 kN·m = 20,000,000 N·mm (1 kN·m = 10⁶ N·mm)
Step 2 · Find second moment of area (I)
Use I = b × h³ ÷ 12
A
rectangle
about
its
own
centroidal
axis:
I
=
∫
y
2
dA
over
the
area,
which
is
bh
³
12
.
I
=
100
×
200
3
12
I
=
100
×
8,000,000
12
I
=
800,000,000
12
I
=
6.6667 ×
10
7
mm⁴
Step 3 · Find neutral axis to the far edge (c)
Use c = h ÷ 2
The neutral axis is at mid-depth, so the top and bottom edges are half the depth from it.
c
=
200
2
c
=
100
mm
Step 4 · Find largest bending stress (σₘₐₓ)
Use σₘₐₓ = M × c ÷ I
The flexure formula: the stress is largest at the edge farthest from the neutral axis.
σₘₐₓ
=
20,000,000
×
100
6.6667 ×
10
7
σₘₐₓ
=
2 ×
10
9
6.6667 ×
10
7
σₘₐₓ = 30 MPa
Check
Put the numbers back in. Both sides should match.
6.6667 ×
10
7
=
100
×
200
3
12
✓
100
=
200
2
✓
30
=
20,000,000
×
100
6.6667 ×
10
7
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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