Theory of Bending Moment
Theory of simple bending (assumptions)
Material of beam is homogenous and isotropic => constant E in all direction Young’s modulus is constant in compression and tension => to simplify analysis Transverse section which are plane before bending before bending remain plain after bending. => Eliminate effects of strains in other direction (next slide) Beam is initially straight and all longitudinal filaments bend in circular arcs => simplify calculations Radius of curvature is large compared with dimension of cross sections => simplify calculations Each layer of the beam is free to expand or contract => Otherwise they will generate additional internal stresses.
Bending in beams
Key Points: 1. Internal bending moment causes beam to deform. 2. For this case, top fibers in compression, bottom in tension.
Bending in beams
Key Points: 1. Neutral surface – no change in length. 2. Neutral Axis – Line of intersection of neutral surface with the transverse section. 3. All cross-sections remain plane and perpendicular to longitudinal axis.
Bending in beams
Key Points: 1. Bending moment causes beam to deform. 2. X = longitudinal axis 3. Y = axis of symmetry 4. Neutral surface – does not undergo a change in length
Consider the simply supported beam below:
Bending Stress in beams
Radius of Curvature, R
P
A Neutral Surface B
Deflected Shape
RA
M M
M
M
RB
What stresses are generated within, due to bending?
Axial Stress Due to Bending:
M=Bending Moment σx (Compression)
M
Neutral Surface
M
Beam
σx=0 σx (Tension)
stress generated due to bending:
σx is NOT UNIFORM through the section depth
σx DEPENDS ON:
(i) Bending Moment, M (ii) Geometry of Cross-section
Bending Stress in beams
Bending Stress in beams
Stresses due to bending
R N’ E B’ A’ C’ N’ F D’ Strain in layer EF
y = R
Stress _ in _ the _ layer _ EF E= Strain _ in _ the _ layer _ EF σ E=
Material of beam is homogenous and isotropic => constant E in all direction Young’s modulus is constant in compression and tension => to simplify analysis Transverse section which are plane before bending before bending remain plain after bending. => Eliminate effects of strains in other direction (next slide) Beam is initially straight and all longitudinal filaments bend in circular arcs => simplify calculations Radius of curvature is large compared with dimension of cross sections => simplify calculations Each layer of the beam is free to expand or contract => Otherwise they will generate additional internal stresses.
Bending in beams
Key Points: 1. Internal bending moment causes beam to deform. 2. For this case, top fibers in compression, bottom in tension.
Bending in beams
Key Points: 1. Neutral surface – no change in length. 2. Neutral Axis – Line of intersection of neutral surface with the transverse section. 3. All cross-sections remain plane and perpendicular to longitudinal axis.
Bending in beams
Key Points: 1. Bending moment causes beam to deform. 2. X = longitudinal axis 3. Y = axis of symmetry 4. Neutral surface – does not undergo a change in length
Consider the simply supported beam below:
Bending Stress in beams
Radius of Curvature, R
P
A Neutral Surface B
Deflected Shape
RA
M M
M
M
RB
What stresses are generated within, due to bending?
Axial Stress Due to Bending:
M=Bending Moment σx (Compression)
M
Neutral Surface
M
Beam
σx=0 σx (Tension)
stress generated due to bending:
σx is NOT UNIFORM through the section depth
σx DEPENDS ON:
(i) Bending Moment, M (ii) Geometry of Cross-section
Bending Stress in beams
Bending Stress in beams
Stresses due to bending
R N’ E B’ A’ C’ N’ F D’ Strain in layer EF
y = R
Stress _ in _ the _ layer _ EF E= Strain _ in _ the _ layer _ EF σ E=