1. 7.2 Shear and Moment Equations and Diagrams
Beams – structural members designed to support loadings perpendicular to their axes
Beams – straight long bars with constant cross-sectional areas
A simply supported beam is pinned at one end
and roller supported at
the other
A cantilevered beam is
fixed at one end and free
at the other

2. 7.2 Shear and Moment Equations and Diagrams
For actual design of a beam, apply
- Internal shear force V and the bending moment M analysis
- Theory of mechanics of materials
Appropriate engineering code to determine beam’s required cross-sectional area
Variations of V and M obtained by the method of sections
Graphical variations of V and M are termed as shear diagram and bending moment diagram

3. 7.2 Shear and Moment Equations and Diagrams
Internal shear and bending moment functions generally discontinuous, or their slopes will be discontinuous at points where a distributed load changes or where concentrated forces or couple moments are applied
Functions must be applied for each segment of the beam located between any two discontinuities of loadings
Internal normal force will not be considered

4. 7.2 Shear and Moment Equations and Diagrams
Load applied to a beam act perpendicular to the beam’s axis and hence produce only an internal shear force and bending moment
For design purpose, the beam’s resistance to shear, and particularly to bending, is more important than its ability to resist a normal force

5. 7.2 Shear and Moment Equations and Diagrams
Sign Convention
To define a positive and negative shear force and bending moment acting on the beam
Positive directions are denoted by an internal shear force that causes clockwise rotation of the member on which it acts and by an internal moment that causes compression or pushing on the upper part of the member

6. 7.2 Shear and Moment Equations and Diagrams
Sign Convention
A positive moment would tend to bend the member if it were elastic, concave upwards
Loadings opposite to the above are considered negative

7. 7.2 Shear and Moment Equations and Diagrams
Procedure for Analysis
Support Reactions
Determine all the reactive forces and couple moments acting on the beam’
Resolve them into components acting perpendicular or parallel to the beam’s axis

8. 7.2 Shear and Moment Equations and Diagrams
Procedure for Analysis
Shear and Moment Reactions
Specify separate coordinates x having an origin at the beam’s left end and extending to regions of the beams between concentrated force and/or couple moments or where there is no continuity of distributed loadings
Section the beam perpendicular to its axis at each distance x and draw the FBD of one of the segments

9. 7.2 Shear and Moment Equations and Diagrams
Procedure for Analysis
Shear and Moment Reactions
V and M are shown acting in their positive sense
The shear V is obtained by summing the forces perpendicular to the beam’s axis
The moment M is obtained by summing moments about the sectioned end of the segment

10. 7.2 Shear and Moment Equations and Diagrams
Procedure for Analysis
Shear and Moment Diagrams
Plot the shear diagram (V versus x) and the moment diagram (M versus x)
If computed values of the functions describing V and M are positive, the values are plotted above the x axis, whereas negative values are plotted below the x axis
Convenient to plot the shear and the bending moment diagrams below the FBD of the beam

11. 7.2 Shear and Moment Equations and Diagrams
Example 7.7
Draw the shear and bending moments
diagrams for the shaft. The support at A is a
thrust bearing and the support at C is a
journal bearing.

12. 7.2 Shear and Moment Equations and Diagrams
Solution
Support Reactions
FBD of the shaft

13. 7.2 Shear and Moment Equations and Diagrams
Solution

14. 7.2 Shear and Moment Equations and Diagrams
Solution

15. 7.2 Shear and Moment Equations and Diagrams
Solution
Shear diagram
internal shear force is always positive within the shaft AB
Just to the right of B, the shear force changes sign and remains at constant value for segment BC
Moment diagram
Starts at zero, increases linearly to B and therefore decreases to zero

16. 7.2 Shear and Moment Equations and Diagrams
Solution
Graph of shear and moment diagrams is discontinuous at points of concentrated force ie, A, B, C
All loading discontinuous are mathematical, arising from the idealization of a concentrated force and couple moment

17. 7.2 Shear and Moment Equations and Diagrams
Example 7.8
Draw the shear and bending diagrams for
the beam.

18. 7.2 Shear and Moment Equations and Diagrams
Solution
Support Reactions
FBD of the beam
View Free Body Diagram

19. 7.2 Shear and Moment Equations and Diagrams
Solution
Distributed loading acting on this segment has an intensity of 2/3 x at its end and is replaced by a resultant force after the segment is isolated as a FBD
For magnitude of the
resultant force,
½ (x)(2/3 x) = 1/3 x2

20. 7.2 Shear and Moment Equations and Diagrams
Solution
Resultant force acts through the centroid of the distributed loading area, 1/3 x from the right

21. 7.2 Shear and Moment Equations and Diagrams
Solution
For point of zero shear,
For maximum moment,