SME
Mechanics of Materials (Solid Mechanics I) · Problem type
Shear and moment of a uniform load
Use this for “A 6 m simply supported beam carries 10 kN/m. Find V and M 2 m from A, and the largest M.”
Diagram
ΣF = 0: RA + RB = 30 + 30 = 60 kN, the total load.
|M| is largest at x = 3 m: 45 kN·m.
The area under V between two points is the change in M.
This is an example. Type your own values to start a new problem.
w
Uniform load
example
L
Span
example
x
Section from A
example
R
Each reaction
calculated
V
Shear at x
calculated
M
Moment at x
calculated
Mₘₐₓ
Largest moment
calculated
Formulas
R
=
w
×
L
2
30
=
10
×
6
2
V
=
w
×
(
L
2
−
x
)
10
=
10
×
(
6
2
−
2)
M
=
w
×
x
×
L
−
x
2
40
=
10
×
2
×
6 − 2
2
Mₘₐₓ
=
w
×
L
2
8
45
=
10
×
6
2
8
Assumptions
•
A pin at A and a roller at B; w covers the whole span, the beam’s own weight included.
•
Smile convention: M > 0 sags the beam; V > 0 when the left part is pushed up.
•
Linear elastic with small deflections, so the loads act on the straight beam.
Step-by-step
WE KNOW
w = 10 kN/m, L = 6 m, x = 2 m
FIND
each reaction (R), shear at x (V), moment at x (M), largest moment (Mₘₐₓ)
Step 1 · Find each reaction (R)
Use R = w × L ÷ 2
The whole load wL is shared equally by the two supports.
R
=
10
×
6
2
R
=
60
2
R
=
30
kN
Step 2 · Find shear at x (V)
Use V = w × (L ÷ 2 − x)
Left
of
the
cut,
R
=
wL
2
pushes
up
and
wx
pushes
down,
so
V
=
wL
2
−
wx.
V
=
10
×
(
6
2
−
2)
V
=
10
×
(3
−
2)
V
=
10
kN
Step 3 · Find moment at x (M)
Use M = w × x × (L − x) ÷ 2
Moments
of
the
left
part
about
the
cut:
Rx
−
wx
×
x
2,
which
is
wx
(
L
−
x
)
2
.
M
(
x
)
is
the
integral
of
V
(
x
).
M
=
10
×
2
×
6 − 2
2
M
=
10
×
2
×
4
2
M
=
20
×
4
2
M
=
80
2
M
=
40
kN·m
Step 4 · Find largest moment (Mₘₐₓ)
Use Mₘₐₓ = w × L² ÷ 8
M
is
largest
where
V
=
0,
at
midspan:
x
=
L
2
in
wx
(
L
−
x
)
2
gives
wL
²
8
.
Mₘₐₓ
=
10
×
6
2
8
Mₘₐₓ
=
10
×
36
8
Mₘₐₓ
=
360
8
Mₘₐₓ = 45 kN·m
Check
Put the numbers back in. Both sides should match.
30
=
10
×
6
2
✓
10
=
10
×
(
6
2
−
2)
✓
40
=
10
×
2
×
6 − 2
2
✓
45
=
10
×
6
2
8
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.