Related to curved beam.

1. JAHANGIRABAD INSTIUTE OF TECHNOLOGY
BARABANKI
Department of Mechanical Engineering
Curved Beams & Unsymmetrical Bending
RAVI VISHWAKARMA
10/06/17 Ravi Vishwakarma ,Assistant Professor JIT 1

2. Winkler-Bach theory
In this chapter, we shall study the bending of beams which are initially
curved. We do this by restricting ourselves to the case where the
bending takes place in the plane of curvature. This happens when the
cross section of the beam is symmetrical about the plane of its
curvature and the bending moment acts in this plane. As we did for
straight beams, we first obtain the solution assuming sections that are
initially plane remain plane after bending. The resulting relation
between the stress, moment and the deflection is called as Winkler-
Bach formula.
10/06/17 Ravi Vishwakarma ,Assistant Professor JIT 2

3. 10/06/17 Ravi Vishwakarma ,Assistant Professor JIT 3
As mentioned before, in this section we obtain the
stress field assuming, sections that are plane
before bending remain plane after bending.
Consequently, a transverse section rotates about
an axis called the neutral axis as shown in figure
Let us examine an infinitesimal portion of a
curved beam enclosing an angle Δ . Due to anϕ
applied pure bending moment M, the section AB
rotates through an angle δ(Δ ) about the neutralϕ
axis and occupy the position A′B′. SN denotes the
surface on which the stress is zero and is called
the neutral surface. Since, the stress is zero in this
neutral surface, the length of the material fibers
on this plane and oriented along the axis of the
beam would not have changed.

4. Thus, the linearized strain is given by,
After solving,
The above equation is called Winkler-Bach theory.
10/06/17 Ravi Vishwakarma ,Assistant Professor JIT 4
( )( )
))(( φ
φε ∆−
∆∂
−=
yr
y
n
y
yr
EEyda
M
r
r nn −
=
∫
=





−−
σ
1

5. Unsymmetrical Bending
Load applied in the plane of
symmetry
Load applied at some orientation
10/06/17 Ravi Vishwakarma ,Assistant Professor JIT 5

6. Cross-section is unsymmetrical
10/06/17 Ravi Vishwakarma ,Assistant Professor JIT 6
z P

7. Cross-section is unsymmetrical
10/06/17 Ravi Vishwakarma ,Assistant Professor JIT 6
z P