Shear force and Bending Moment in beams
Concept of Shear Force and Bending moment in beams:
When the beam is loaded in some arbitrarily manner, the internal forces and moments are developed and the terms shear force and bending moments come into pictures which are helpful to analyze the beams further. Let us define these terms
Now let us consider the beam as shown in fig 1(a) which is supporting the loads P1, P2, P3 and is simply supported at two points creating the reactions R1 and R2respectively. Now let us assume that the beam is to divided into or imagined to be cut into two portions at a section AA. Now let us assume that the resultant of loads and reactions to the left of AA is „F' vertically upwards, and
since the entire beam is to remain in equilibrium, thus the resultant of forces to the right of AA must also be F, acting downwards. This forces „F' is as a shear force. The shearing force at any x-
section of a beam represents the tendency for the portion of the beam to one side of the section to slide or shear laterally relative to the other portion.
Therefore, now we are in a position to define the shear force „F' to as follows:
At any x-section of a beam, the shear force „F' is the algebraic sum of all the lateral components of the forces acting on either side of the x-section.
Sign Convention for Shear Force:
The usual sign conventions to be followed for the shear forces have been illustrated in figures 2 and 3.
Let us again consider the beam which is simply supported at the two prints, carrying loads P1, P2 and P3 and having the reactions R1 and R2 at the supports Fig 4. Now, let us imagine that the beam is cut into two potions at the x-section AA. In a similar manner, as done for the case of shear force, if we say that the resultant moment about the section AA of all the loads and
reactions to the left of the x-section at AA is M in C.W direction, then moment of forces to the right of x-section AA must be „M' in C.C.W. Then „M' is called as the Bending moment and is
abbreviated as B.M. Now one can define the bending moment to be simply as the algebraic sum of the moments about an x-section of all the forces acting on either side of the section
Sign Conventions for the Bending Moment:
For the bending moment, following sign conventions may be adopted as indicated in Fig 5 and Fig 6.
Some times, the terms „Sagging' and Hogging are generally used for the positive and negative bending moments respectively.
Bending Moment and Shear Force Diagrams:
The diagrams which illustrate the variations in B.M and S.F values along the length of the beam for any fixed loading conditions would be helpful to analyze the beam further.
Thus, a shear force diagram is a graphical plot, which depicts how the internal shear force „F' varies along the length of beam. If x dentotes the length of the beam, then F is function x i.e. F(x).
Similarly a bending moment diagram is a graphical plot which depicts how the internal bending moment „M' varies along the length of the beam. Again M is a function x i.e. M(x).
Basic Relationship Between The Rate of Loading, Shear Force and Bending Moment:
The construction of the shear force diagram and bending moment diagrams is greatly simplified if the relationship among load, shear force and bending moment is established.
Let us consider a simply supported beam AB carrying a uniformly distributed load w/length. Let us imagine to cut a short slice of length dx cut out from this loaded beam at distance „x' from the origin „0'.
Let us detach this portion of the beam and draw its free body diagram.
The forces acting on the free body diagram of the detached portion of this loaded beam are the following
• The shearing force F and F+ dF at the section x and x + dx respectively.
• The bending moment at the sections x and x + dx be M and M + dM respectively.
• Force due to external loading, if „w' is the mean rate of loading per unit length then the total loading on this slice of length dx is w. dx, which is approximately acting through the centre „c'.
If the loading is assumed to be uniformly distributed then it would pass exactly through the centre „c'.
This small element must be in equilibrium under the action of these forces and couples.
Now let us take the moments at the point „c'. Such that
Conclusions: From the above relations,the following important conclusions may be drawn
• From Equation (1), the area of the shear force diagram between any two points, from the basic calculus is the bending moment diagram
• The slope of bending moment diagram is the shear force,thus
Thus, if F=0; the slope of the bending moment diagram is zero and the bending moment is therefore constant.'
• The maximum or minimum Bending moment occurs where
The slope of the shear force diagram is equal to the magnitude of the intensity of the distributed loading at any position along the beam. The -ve sign is as a consequence of our particular choice of sign conventions
Procedure for drawing shear force and bending moment diagram:
The advantage of plotting a variation of shear force F and bending moment M in a beam as a function of „x' measured from one end of the beam is that it becomes easier to determine the
maximum absolute value of shear force and bending moment.
Further, the determination of value of M as a function of „x' becomes of paramount importance so as to determine the value of deflection of beam subjected to a given loading.
Construction of shear force and bending moment diagrams:
A shear force diagram can be constructed from the loading diagram of the beam. In order to draw this, first the reactions must be determined always. Then the vertical components of forces and reactions are successively summed from the left end of the beam to preserve the mathematical sign conventions adopted. The shear at a section is simply equal to the sum of all the vertical forces to the left of the section.
When the successive summation process is used, the shear force diagram should end up with the previously calculated shear (reaction at right end of the beam. No shear force acts through the beam just beyond the last vertical force or reaction. If the shear force diagram closes in this fashion, then it gives an important check on mathematical calculations.
The bending moment diagram is obtained by proceeding continuously along the length of beam from the left hand end and summing up the areas of shear force diagrams giving due regard to sign. The process of obtaining the moment diagram from the shear force diagram by summation is exactly the same as that for drawing shear force diagram from load diagram.
It may also be observed that a constant shear force produces a uniform change in the bending moment, resulting in straight line in the moment diagram. If no shear force exists along a certain portion of a beam, then it indicates that there is no change in moment takes place. It may also further observe that dm/dx= F therefore, from the fundamental theorem of calculus the maximum or minimum moment occurs where the shear is zero. In order to check the validity of the bending moment diagram, the terminal conditions for the moment must be satisfied. If the end is free or pinned, the computed sum must be equal to zero. If the end is built in, the moment computed by the summation must be equal to the one calculated initially for the reaction. These conditions must always be satisfied.
Cantilever beams - problems
Cantilever with a point load at the free end:
Mx = - w.x
A cantilever of length carries a concentrated load 'W' at its free end.
Draw shear force and bending moment.
At a section a distance x from free end consider the forces to the left, then F = -W (for all values of x) -ve sign means the shear force to the left of the x-section are in downward direction and therefore negative
Taking moments about the section gives (obviously to the left of the section)
M = -Wx (-ve sign means that the moment on the left hand side of the portion is in the anticlockwise direction and is therefore taken as -ve according to the sign convention)
so that the maximum bending moment occurs at the fixed end i.e. M = -W l
Simplysupported beam -problems
Simply supported beam subjected to a central load (i.e. load acting at the mid-way)
By symmetry the reactions at the two supports would be W/2 and W/2. now consider any section X-X from the left end then, the beam is under the action of following forces.
.So the shear force at any X-section would be = W/2 [Which is constant upto x < l/2]
If we consider another section Y-Y which is beyond l/2 then
for all values greater = l/2
SSB with central point load:
Overhanging beams - problems
In the problem given below, the intensity of loading varies from q1 kN/m at one end to the q2 kN/m at the other end.This problem can be treated by considering a U.d.i of intensity q1 kN/m over the entire span and a uniformly varying load of 0 to ( q2- q1)kN/m over the entire span and then super impose teh two loadings.
Point of Contraflexure:
Consider the loaded beam a shown below along with the shear force and Bending moment diagrams for It may be observed that this case, the bending moment diagram is completely positive so that the curvature of the beam varies along its length, but it is always concave upwards or sagging.However if we consider a again a loaded beam as shown below along with the S.F and B.M diagrams, then
It may be noticed that for the beam loaded as in this case,
The bending moment diagram is partly positive and partly negative.If we plot the deflected shape of the beam just below the bending moment
This diagram shows that L.H.S of the beam „sags' while the R.H.S of the beam „hogs'
The point C on the beam where the curvature changes from sagging to hogging is a point of contraflexure.
It corresponds to a point where the bending moment changes the sign, hence in order to find the point of contraflexures obviously the B.M would change its sign when it cuts the X-axis therefore to get the points of contraflexure equate the bending moment equation equal to zero.The fibre stress is zero at such sections
Note: there can be more than one point of contraflexure