These diagrams can provide information about the shear and moment variations along the axis of a member. This information can help determine where reinforcements need to be placed.
The method of sections can be used to find internal loadings at a specific point. This is a "cut" made through the member where internal loadings are to be determined. A free body diagram of one of the 2 parts is drawn and solved for internal forces. This process is used throughout the member to create shear & moment diagrams.
In general, the functions are discontinuous for these diagrams.
Sign convention: always show positive direction. internal moment will cause compression in the top fibers of the material. The internal shear will cause clockwise rotation on the section of member.
Bending Deformation of a Straight Member
When a straight member is subjected to a moment the longitudinal lines will become curved while the vertical (transverse) lines will undergo rotation but the cross section lines will stay straight.
Depending on the moment either the top or bottom of the member will be in compression while the other end will be tensile. There is a neutral surface in the longitudinal fibers where there is no change in length. Compression = shorter length. Tension = longer length.
Any cross section the longitudinal normal strain will vary linearly with the position y from the neutral surface.
Flexure Formula
This formula relates the longitudinal stress distribution in a member to the internal bending moment.
σ (normal stress) will vary from zero, at the neutral axis, to a maximum at a distance furthest from the neutral surface.
σmax = Md / I -- Maximum normal stress.
- M is the resultant internal moment about the neutral axis.
- I is the moment of inertia of the cross section.
- d is the perpendicular distance from the neutral axis to the furthest point.
The Flexure formula can also be applied to cross section of any shape with a resultant internal moment that acts in any direction.
Moment arbitarily applied...
- Resolve the moment in to components.
- Flexure formula to find normal stress in each component. σ= - (Mz y / Iz) - (My z / Iy)
- Superposition: resultant normal stress at the point can be found.
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