Unsymmetrical Bending

 Page 1


Unsymmetrical Bending
Page 2


Unsymmetrical Bending
PURE BENDING
? Bending is a very severe form of stressing a structure
? The simple bending theory applies when bending takes place about an 
axis which is perpendicular to a plane of symmetry.
? Bending moments acts along the axis of the member. 
Assumptions made in pure bending
1. The normal planes remain normal even after bending.
2. There is no net internal axial force.
3. Stress varies linearly over cross section.
4. Zero stress exists at the centroid and the line of centroid is the
Neutral Axis (N.A)
Page 3


Unsymmetrical Bending
PURE BENDING
? Bending is a very severe form of stressing a structure
? The simple bending theory applies when bending takes place about an 
axis which is perpendicular to a plane of symmetry.
? Bending moments acts along the axis of the member. 
Assumptions made in pure bending
1. The normal planes remain normal even after bending.
2. There is no net internal axial force.
3. Stress varies linearly over cross section.
4. Zero stress exists at the centroid and the line of centroid is the
Neutral Axis (N.A)
•Bending stress and strain at any point may be computed as 
Page 4


Unsymmetrical Bending
PURE BENDING
? Bending is a very severe form of stressing a structure
? The simple bending theory applies when bending takes place about an 
axis which is perpendicular to a plane of symmetry.
? Bending moments acts along the axis of the member. 
Assumptions made in pure bending
1. The normal planes remain normal even after bending.
2. There is no net internal axial force.
3. Stress varies linearly over cross section.
4. Zero stress exists at the centroid and the line of centroid is the
Neutral Axis (N.A)
•Bending stress and strain at any point may be computed as 
? Symmetrical bending : The plane of loading or the plane of
bending is co-incident with or parallel to, a plane containing
principal centroidal axes of inertia of the cross-section of the
beam.
? Bending stress is given by
s
z
=
M
x
I
x
y+
M
y
I
y
x
? Bending stress along N.A iss
z
= 0
Page 5


Unsymmetrical Bending
PURE BENDING
? Bending is a very severe form of stressing a structure
? The simple bending theory applies when bending takes place about an 
axis which is perpendicular to a plane of symmetry.
? Bending moments acts along the axis of the member. 
Assumptions made in pure bending
1. The normal planes remain normal even after bending.
2. There is no net internal axial force.
3. Stress varies linearly over cross section.
4. Zero stress exists at the centroid and the line of centroid is the
Neutral Axis (N.A)
•Bending stress and strain at any point may be computed as 
? Symmetrical bending : The plane of loading or the plane of
bending is co-incident with or parallel to, a plane containing
principal centroidal axes of inertia of the cross-section of the
beam.
? Bending stress is given by
s
z
=
M
x
I
x
y+
M
y
I
y
x
? Bending stress along N.A iss
z
= 0
UNSYMMETRICAL BENDING
Assumptions
1. The plane sections of the beam remain plane after bending 
2. The material of the beam is homogeneous and linearly elastic.
3. There is no net internal axial force.
Sign conventions and notation
? u, v and w are the
displacement components of
any point within beam parallel to
x, y, z axes.
? P = axial load and T = torque
??? ?? (z) and ?? ?? (z)are
distributed loads
??? ?? and?? ?? are applied bending
moments