How to Calculate the Shear Force and Bending Moment?
Shear Force Calculation
Shear force (V) is the force acting on a beam segment’s point perpendicular to that segment’s longitudinal plane. They occur from the unsymmetrical sharing of external load and reactions that act on a specific part of the structure. When determining shear force, it is possible to use three approaches: the direct, Graphical, and Numerical. In this case, we only discuss the direct method, which is much simpler and relevant for most applications.
Definition of Sign Convention
When determining the shear force, there must be a convention of its sign, which must be positive or negative.
The most common convention is as follows:
-It should also be mentioned that a positive sign is assigned to any vertical forces acting downwards.
-It should be noted that the vertical forces are hostile specifically.
-Left shear forces are positive, and right shear forces are hostile.
Step-by-step Calculation
1. Determine all point loads, distributed loads, and reactions located at the beam.
2. Generally, we select an origin (slightly to the left of the beam) and assign the intrinsic + ve and – ve signs to the former.
3. Subdivide the beam into a small elemental length (Δx, which is a differential length).
4. Find out the resultant force applied on each part of it (with point loads as well as the distributed loads and the reactions).
5. Force that stays at a specific point of the beam is the sum of shear forces accumulated in the narrowed segment of the beam located on the left side of this point only.
Example: A simply supported beam having a point load (P) at mid-span and a uniformly distributed load (w) covering the whole span of the beam of length (L).
1. Point loads: P (at the center of the beam)
2. Distributed load: w (which is evenly distributed throughout the span of the beam).
3. Reactions: R1 and R2 (at the supports)
4. Calculate the reactions: R1 = R2 = wL/2
5. Break down the beam. The shear force at any point to the left of the center is given by the expression:
for ( x < L / 2 ) ;
x : V ( x ) = w ( 0 ) – w ( L ) x + w L / 2.
V(x) = w(x) – P + wL/2 for L/2 < x ≤ L
Bending Moment Calculation
Force, what was felt at a beam segment, due to load transfer along the length of the beam is called bending moment (M). It is the outcome of an external loads reaction and shear forces which come into play in any fixed segment of structure. It is also possible to calculate the bending moment through a direct method, which deserves our attention below.
Definition of Sign Convention
Similar to shear force, a sign convention is necessary for the calculation of the bending moment:
-Clockwise moments are good ones.
-Moments that are counterclockwise are negative.
-Moments of the applied loads occurring at the left of the point are taken as positive, and at the right of the end, are taken as unfavorable.
Step-by-step Calculation
1. All the point loads, distributed loads, and reactions that are on a beam should be determined.
2. Pick a reference axis (typically the left end of the beam), and then assign plus (+) and minus (-) signs.
3. Subdivide the beam into small segments of equal dimensions required to be considered (Δx).
4. Point load, distributed load, reaction, or shear force stretches are determined, then, the force or net moment through segments of a beam for point loads, distributed loads, reactions, or shear forces.
5. The algebraic sum of moments of the forces applied to the beam up to a given point that is acting on the beam segment.
Using the same example as before, the bending moment at any point x to the left of the center is given by the expression:
M(x) = w(x^2)/2 – wLx/2 For 0 ≤ x ≤ L/2
Shear force and Bending moment diagrams
These are to be sized after getting the shear force and/or bending moment values of a segment, and the respective diagrams may be plotted. They show how shear force and bending moment change across the beam (and give a graphical description of how a beam behaves under load).
The steps for plotting these diagrams are as follows:
Shear Force Diagram
1. Shear force values are graphed and mapped to the respective points on the beam.
2. Connect the points together in perfectly lit, straight, and beautiful elegant lines.
3. If necessary, find the slope of the distributed loads that will be useful in extending the lines between the data points.
Bending Moment Diagram
1. Plotting the variation of bending moment as a function of a specific coordination point on the beam.
2. Make a straight line between the points.
3. If necessary, use the slope of the distributed loads and the shear force diagram in order to guide you in drawing lines between the data points.
4. Find points of inflection which marks the change of direction of bending moment from tension to compression or vice versa.
Through these diagrams, engineers are well positioned in identifying areas that need more reinforcements within beams. They can also calculate deflection of the beam, stress to be experienced by the beam, structural capability of the beam and load to be borne by it to ensure it can safely exert under the given loading conditions.
Thus, simple shear force and bending moment calculation and creation of the corresponding diagrams are considered a crucial set of competencies for structural engineers. The engineers can follow numerous steps accordingly and calculate different conditions of beams under load and guarantee that the corresponding structures will be safe and efficient.
