Shear Force and Bending Moment

>>Types of supports:

Movable Hinged support (or) Simple support (or) Roller support:

Support ‘B’ of the fig. Shown

• No bending moment
• Free to rotate
• Horizontal Displacement possible
• Only vertical reaction or reaction plane of rolling
• Number of independent reaction  components 1

Immovable hinged support:

Support ‘A’ of the fig. Shown.

• Free to rotate — No translation
• No bending moment
• Number  of independent reaction 2

Fixed support:

Support ‘C’ of the fig. Shown.

• Neither rotation nor translation
• Vertical and horizontal reactions and B.M.
• Number of independent reaction  components 3

>> Shear force (F): Algebraic sum of all transverse forces either to the left or right hand side of a section

Bending moment (M):  Algebraic Sum of moments of all transverse forces either to the left or right of a section.

>> Relation between load, S.F. and B.M. :

For maximum B .M., dM/dx = 0

Conclusions:

• Rate of change of S.F. (and hence slope of S.F.D.) is equal to intensity of loading
• The rate of change of B.M (and hence slope of B.M.D) is equal to shear force.
• The change of B.M. from ‘O’ to ‘x’ is proportional to the area of S.F.D. from ‘O’ to ‘X’
• B.M is maximum; when S.F., is zero or changes sign
•  Point of contraflexure
• Point where B.M. changes sign and is equal to zero.

>> Variation of S.F and B.M for different loading on spans of beams:

>> Inclined loading : If the external load is not at right angles to the axis of the beam, the loading can be resolved axially and transversely to the beam

Transverse: Components (sinØ) produces B.M. and S.F.

Axial: Component (cosØ) produces pull or push

>> Horizontal thrust diagram:

Diagram showing the variation of axial Thrust over a span

>> Internal hinge: Bending moment is zero

>> Link: Only vertical force. Can not resist horizontal force ‘H’ and B.M.