Question

Define Mohr’s theorem

Answer 1

2.2 Mohr’s First Theorem (Mohr I)
Development
Noting that the angles are always measured in radians, we have:
ds R d
ds R
d
θ
θ
= ⋅
∴ =
From the Euler-Bernoulli Theory of Bending, we know:
1 M
R EI =
Hence:
M d ds
EI
θ = ⋅
But for small deflections, the chord and arc length are similar, i.e. ds dx ≈ , giving:
M d dx
EI
θ = ⋅
The total change in rotation between A and B is thus:
B B
A A
M d dx
EI

The term M EI is the curvature and the diagram of this terms as it changes along a
beam is the curvature diagram (or more simply the M EI diagram). Thus we have:
B
BA B A
A
M d dx
EI
θ θθ =−= ∫
This is interpreted as:
[ ] Change in slope Area of diagram AB
AB
M
EI
⎡ ⎤ = ⎢ ⎥ ⎣ ⎦
This is Mohr’s First Theorem (Mohr I):
The change in slope over any length of a member subjected to bending is equal
to the area of the curvature diagram over that length.
Usually the beam is prismatic and so E and I do not change over the length AB,
whereas the bending moment M will change. Thus:
1 B
AB
A
M dx
EI
θ = ∫
[ ] [Area of diagram] Change in slope AB

2.3 Mohr’s Second Theorem (Mohr II)
Development
From the main diagram, we can see that:
d xd ∆ = ⋅ θ
But, as we know from previous,
M d dx
EI
θ = ⋅
Thus:
M d x dx
EI
∆ = ⋅⋅
And so for the portion AB, we have:
First moment of diagram about
B B
A A
B
BA
A
M d x dx
EI
M dx x
EI
M B
EI
∆= ⋅ ⋅
⎡ ⎤ ∆= ⋅ ⎢ ⎥ ⎣ ⎦
=
∫ ∫
∫
This is easily interpreted as:
Vertical
Intercept diagram of diagram BA
BA BA
B
M M
EI EI
⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = × ⎛ ⎞ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎜ ⎟ ⎣ ⎦ ⎢ ⎝ ⎠ ⎥ ⎣ ⎦
This is Mohr’s Second Theorem (Mohr II):
For an originally straight beam, subject to bending moment, the vertical
intercept between one terminal and the tangent to the curve of another
terminal is the first moment of the curvature diagram about the terminal where
the intercept is measured.
There are two crucial things to note from this definition:
• Vertical intercept is not deflection; look again at the fundamental diagram – it
is the distance from the deformed position of the beam to the tangent of the
deformed shape of the beam at another location. That is:
∆ ≠ δ
• The moment of the curvature diagram must be taken about the point where the
vertical intercept is required. That is:
∆BA AB











This is Mohr's Second Theorem(Mohr II): For an originally straight beam, subject to bending moment, the vertical intercept between one terminal and the tangent to the curve of another terminal is the first moment of the curvature diagram about the terminal where the intercept is measured.

The moment-areatheorem is an engineering tool to derive the slope, rotation and deflection of beams and frames. This theorem was developed byMohr and later stated namely by Charles Ezra Greene in 1873.