CONCEPT OF SHEAR FORCE , BENDING MOMENT AND RELATIONS BETWEEN THE SHEAR FORCE , BENDING MOMENT AND THE RATE OF LOADING.

Before going in to the discussion about shear force and bending moments I need you all the readers to understand the main concept behind the calculation of shear force and bending moment and what are the main basic types of structures in civil engineering terms which can help to simplify our understanding of the structures.

Basically the structure is divided in to two categories namely statically determinate structures and statically indeterminate structures.

1)  STATICALLY DETERMINATE STRUCTURES.

A structure which can be analyzed with the help of three equilibrium conditions (i.e.) is called as a statically determinate structure.

These Conditions are: Σ H = 0, Σ V = 0 and Σ M = 0. It should be noted that the results of analysis are independent of the material from which the structure has been fabricated.

Example – simply supported beam, cantilever beam, overhanging beam, 3 hinged arch etc.

2) STATICALLY INDETERMINATE STRUCTURES.

A structure which cannot be analyzed with the help of threw equilibrium conditions or a structure which required more conditions to solve or analyze the structures are called as statically indeterminate structures.

Examples – propped cantilever, two hinged arch, fixed beams etc...

So talking about in determinate structures, they are more constraints by their supports or members (redundancies) than required to obtain statically determinate structures. Therefore equilibrium conditions as well as additional conditions are required to analyze indeterminate structures.

Though there are many complications in analyzing of statically indeterminate structures, but at the same time they are more useful and advantageous then statically determinate structures due to following reasons.

1. The Maximum stresses in statically indeterminate structures are generally lower as compare to statically determinate structures.

2. Indeterminate structures have more members and support reaction than determinate structures so if the part of member or support indeterminate structures fails, the entire structure will not necessarily collapse, and in fact the loads will be re distributed to adjacent members or supports.

3. Statically indeterminate structures have higher stiffness (i.e. deformation will be less or small)

DETERMINACY AND STABILITY

A stable structure remains stable for any imaginable system of loads. Therefore, the types of loads, their number and their points of application are not considered

When deciding the stability or determinacy of the structure. A given structure considered externally determinate if the total number of reaction components is equal to the equations of equilibrium available. In other words:

I hope you guys have a clear understanding of the structures, so now let’s get into the shear force and bending moment diagram.

CONCEPT OF SHEAR FORCE.

Shear force at any cross-section of the beam is the algebraic sum of all vertical forces acting on the beam on the right and left side of the section.

That means shear force is sum of all the forces on the right and left side of any cross section of a structure or a beam.

In simple word we can say that if you take any cross section of a beam then the forces action on the right side and left side should have equal values.

 

SING CONVENTIONS FOR THE SHEAR FORCE.

POSITIVE SHEAR

An upward force to the left of the section and downward force to the right side of the section is taken as positive shear

And

NEGATIVE SHEAR

A Downward force to the left of the section and an upward for force the right side of the section is taken as negative shear.

CONCEPT OF BENDING MOMENT.

Bending moment at any cross section of the beam is the algebraic sun of the moments of all the forces acting on the right side or left side of the section.

In simple words we can say that if you take any cross section of a beam then the sum of moments on the right and left side of the section should have equal values.

 

SIGN CONVERSATION FOR BENDING MOMENT.

POSITIVE BENDING MOMENT

Clockwise moment to the left of the section and an anticlockwise moment to the right of the section is taken as positive bending moment.

NEGATIVE BENDING MOMENT

Anticlockwise moment to the left of the section and clockwise moment to the right of the section is taken as negative bending moment.

From the above figure we can see that the forces acting on the upward direction on both side of section produce positive moment

And we can clearly see that due to such moments the deflection of a structure create sagging pattern hence called as sagging moment.

From the above figure we can see that the forces acting in the downward direction on both side of section produce negative moment (referring to sign conventions).

And therefore we can clearly see that due to such moments the deflection of a structure creates hogging pattern hence called as hogging moment.

RELATIONSHIPS BETWEEN SHEAR FORCE, BENDING MOMENT AND LOADING

To understand the relationship between shear forces, bending moment and loading, with the help of simple example of beam which is illustrated in the image above.

With reference to image, the beam is loaded with uniformly distributed load (w), which is spread throughout the length (L) of the beam.

So let’s take a section at point C shown in the second figure, and calculate the vertical shear force which is acting at point C.

\[ {Shear ~force~ at~ point ~C} = {~summation ~of~ all ~the~ forces~ acting }\] \[ {~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~} {~on ~the ~left~ side ~the~section}\] \[ {V}_{C} = {(Σ ~{F}_{V}) _{L}}\] \[ {V}_{C} = {({R}_{1}) - {w ~ x}}\] \[ { Where~ R_{1} ~and ~R_{2} } = {w~L\over {2}}\] \[ {V}_{C} = {({{w~L\over {2}})} - {w ~ x}}\]

To calculate moment at point C , we have to apply the conditions of equilibrium which is ….

Moment at point C = summation of all the moment acting on the right or left side of the section

\[ { Moment ~ at~ point ~C} = {~summation ~of~ all ~the~ moments~ acting }\] \[ {~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~} {~on ~the ~left~ side ~the~ section}\] \[ {M}_{C} = {(Σ ~{M}_{C}) _{L}}\] \[ {M}_{C} = {({R}_{1} ~X~{x}}) - {w ~ x}~ X ~({x\over 2})\] \[ {M}_{C} = {({{w~L\over {2}})} ~X~{x}} - {w ~ x}~ X ~({x\over 2})\] \[ {M}_{C} = {({{w~L~x\over {2}})} - ({wx^{2}\over 2})}\]

Now, if we differentiate the MOMENT with respect to x,

We get,

\[ {d\over d_x } ~ {M} = {d\over d_x } ({{w~L~x}\over 2}) - {d\over d_x } ({w ~ x^2\over 2})\] \[ {dM\over d_x } ~ = {d_x\over d_x } ({{w~L}\over 2}) - {d~ x^2\over d_x } ({w \over 2})\] \[ {dM\over d_x } ~ = {1 } ({{w~L}\over 2}) - { 2x } ({w \over 2})\] \[ {dM\over d_x } ~ = ({{w~L}\over 2}) - ({wx })\]

But we have found out the value of shear force as ,

\[ {V}_{C} = {({{w~L\over {2}})} - {w ~ x}}\]

Therefore we can say that,

The value of shear force is equal to the value of derivation of bending moment

Or in other words

“The rate of change of the bending moment with respect to x is equal to the shearing force, or the slope of the moment diagram at the given point is the shear at that point.”

\[ {V}_{C} ={dM\over d_x }\]

Now, if we differentiate the SHEAR FORCE with respect to x,

We get,

\[ {d\over d_x } {V} = {d\over d_x } ({w~L\over 2}) - {d\over d_x } {w ~ x}\] \[ {dV\over d_x } = {d\over d_x }{1} ({w~L\over 2}) - {d\over d_x } {w ~ x}\] \[ {dV\over d_x } = { 0} - {d\over d_x } {w }\] \[ {dV\over d_x } = - {d\over d_x } {w }\] \[ {dV\over d_x } = {Load }\]

Note = here the negative sign indicates the load is acting downward in direction as per the sign conventions which we use to calculate the shear force

Therefore we can say that the differentiation of shear force is equal to the value of load acting on it

Or in other words,

The rate of change of the shearing force with respect to x is equal to the load or the slope of the shear diagram at a given point equals the load at that point.

 THANK YOU...!!

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