Lecture 36

Lecture 36:

Plotting V, M and Deflection

A comprehensive procedure for the construction of shear, moment, and deflection diagrams for statically determinate beams is given below. Each and every one of these diagrams are plotted from left to the right and positive values are shown above, and negative values below, a reference axis. The procedure is as follows:

loaded beam, FBD and deflection diagram

1. Draw a FBD of the entire structure and find the reactions of the beam.

2. Based on intuition and an educated guess, sketch an approximate deflection curve of the structure. this will help to determine the location of inflection points and thus help to construct the diagrams.

3. Draw a SHEAR DIAGRAM. Start at the left end and plot the external shear values with regard to the following:

The shear diagram is the graphic representation of the shear force at successive points along the beam. Upward acting forces are assumed positive and downward forces negative.
The shear force (V) at any point is equal to the algebraic sum of the external loads and reactions to the left of that point.
Since the entire beam must be in equilibrium (the sum of V = 0), the shear diagram must close to zero at the right end.
Consider the loading for increments along the beam's length in order to determine the shape of the curve.

if there is no change in the load along the incremental length under consideration, the shear curve is a straight horizontal line (or a curve of zero slope). The slope at any point is defined as the tangent to the curve at that point.

if a load exists, but does not change in magnitude over successive increments (uniformly distrib-uted), the slope of the shear curve is constant and non-horizontal.

if a load exists, and increases in magnitude over successive increments, the slope of the shear curve is positive (approaches the vertical); if the magnitude decreases, the slope of the shear curve is negative (approaches the horizontal).
Abrupt changes in loading cause abrupt changes in the slope of the shear curve. Concentrated loads produce vertical lines (a jump) in the shear curve.

4. Draw a MOMENT DIAGRAM. Start at the left end and plot the bending moment values due to the external loads and the reactions with regard to the following:

The moment diagram is the graphical representation of the magnitude of the bending moment at successive points along the beam.
The bending moment for the moment diagram (M) at any point equals the sum of moments of the forces to the left about that point.
Since the entire beam is in equilibrium (Sum of M=0), the bending moment diagram must close to zero at the right side.
Consider the values of successive incremental ordinates along the length of the beam in order to determine the shape of the curve:

if the magnitudes of successive shear ordinates are constant, the moment curve has a constant slope at that increment.

if the magnitudes of successive shear ordinates increase, the slope of the moment curve is positive (it approaches the vertical).

if the magnitudes of successive shear ordinates decreases, the slope of the moment curve is negative (it approaches the horizontal).
Abrupt changes in the shear diagram will produce changes in the shape of the moment curve. Concentrated moments produce vertical lines in the moment curve.

5. Draw a DEFLECTION DIAGRAM.

beam deflection and inflection points

There is no set manner in which the deflection diagram should be drawn. There are, however, a few rules which can be followed:

points of zero moment are inflection points (change in curvature direction) of the deflection curve.
the maximum points of the moment diagram are the points at which the deflection diagram slope is zero (where the tangent of the curve is horizontal).

6. There are a few relationships (based on calculus) between these three diagrams that are helpful to construct them:

The slope of the moment curve at any point is equal to the shear force at the same point.
Points of zero shear are peaks of the moment curve, points of maximum positive or maximum negative moment.
The change in moment value between any two points is equal to the area of the shear diagram between the same two points (provided no other external moment is applied). This is known as the Shear Area Method.
The shapes of the curves from shear to moment always progress in this order; straight horizontal line, straight sloped line, parabolic curve and cubic curve. Therefore, if a shear curve is a horizontal line, the moment curve is a sloped line; if th e shear curve is a sloped line the moment curve is a parabolic curved line; etc. This happens because of the mathematical relationship between loading, shear and moment diagrams.

Example

sample shear, moment and deflection diagrams