How to calculate the Shear Force and Bending Moment Diagram?

DEFINITION OF SHEAR FORCE AND BENDING MOMENT DIAGRAM

These are the most significant parts of structural analysis for design. You can quickly identify the size, type and material of member with the help of shear force and bending moment diagram. Let’s know the whole concept in detail:

SHEAR FORCE DEFINITION: It is an independent parameter. According to the shear force definition, it is the algebraic sum of forces acting either on the left-hand side or right-hand side of the section. A shear force acts in a parallel direction to the large length of the structure.  

Left (+ve) Right Left (-ve) Right

Figure: Sign convention of shear beam

Above figure defines the sign convention of shear force in the beam. If the forces are in the upward direction in the left side and downward direction on the right side of the section of the member, then shear force at that particular point will be positive. If forces are the downward direction in the left side and upward direction in the right side of the section of the member then shear force at that point will be negative.  

WHAT IS BENDING MOMENT? It depends on the length of the member. We can define the bending moment as the algebraic sum of moment of all forces acting on the left side or right side of the section. 

           Left (+ve) Right Left (-ve) Right

Figure: Sign convention of Bending moment

   Sagging Hogging

Above figure defines the sign convention of the bending moment in the beam. If the moment of forces is spinning clockwise in the left side and anticlockwise in the right side of the section of the member, then bending moment at that point will be positive. If the moment of forces is spinning anticlockwise in the left side and clockwise in the right side of the section of the member, then bending moment at that point will be negative.  

APPLICATION OF SHEAR FORCE AND BENDING MOMENT:

One of the most significant applications of shear force and bending moment diagram is that we can calculate how much area of steel is required for the section of the structural member while designing any of the structural members such as beam, column or any other member. It is a necessary calculation for the design of structural member with the help of bending moment, and with the help of shear force, we can also check shear in the member.

Relationship between shear force and bending moment

It is too essential to understand the different relations between shear, loading, and bending moment diagram to solve various types of problems by using the method.  

• At first this is the relationship between a distributed load on the loading diagram and the shear diagram. It is derived that the slope of the shear diagram is equal to the magnitude of the distributed load because a distributed load varies the shear load according to its magnitude. According to Schewedler theorem, the relationship between the magnitude of shear force and distributed load is given by            

• Few straight outcomes of these are that a shear diagram will have a very little change in magnitude when point load is applied to a member and a linearly differing shear magnitude as an outcome of a constant distributed load. Parallelly, it can be shown that the slope of the moment diagram at a given point will be equal to the magnitude of the shear diagram at such distance. The relationship between distributed shear force and bending moment is shown by:

An undeviated result of this is that at every point when the shear diagram intersects zero the moment diagram will have a local maximum or minimum. Also, if the shear diagram is zero over a length of the member, the moment diagram will have an unvarying value over such length. According to calculus, it comes in the knowledge that a point load will conduct to a continuously differing moment diagram, and an unvarying distributed load will lead to a quadratic moment diagram.

For Example – Cantilever Beam with Uniformly Varying Load (UVL)

Shear and Bending moment diagram

Effective depth = Total depth – clear cover – (diameter of bar/2)
Where,
d = Effective depth
D = Total depth

Effective length: Effective length of the cantilever beam
(Effective length) L = clear span of the beam + effective depth of beam /2

Step -1: Shear force calculation
SFB = 0
SFA = 1/2 x L x W

Step – 2: Bending moment calculation
MB = 0
MA = ½ x L x W x 1/3 x L

For Example – Simply Supported Beam with Uniformly Distributed Load (UDL)

Figure: Beam cross- section

Effective depth = Total depth – clear cover – (diameter of bar/2)
Where,
d = effective depth
D = Total depth

Effective length: Effective length of the simply supported beam will be least of the following

  1. Centre to centre distance between the supports:
    L = clear span of the beam + width of support /2 + width of support /2
    2.(Effective length) L = clear span of the beam + effective depth of the beam

Step -1: Calculate Support reactions
∑Fy = 0 Here (+ve) and (- ve)
So, RA + RB = WL (a)
and
∑MA 0
-RB x L + WL x L/2 = 0
RB = WL2/2L
RB = WL/2
Put the value of RB in equation no. (a)
RA + WL/2 = WL
RA = WL – WL/2
RA = WL/2

STEP- 2: Shear force calculation
(SFA)Left = 0
(SFA)Right = WL/2
(SFc)Left =WL/2 – WL/2
(SFc)Left = 0
(SFc)Right = WL/2 -WL/2
(SFc)Right = 0
(SFB)Left =WL/2 – WL
(SFB)Left = -WL/2
(SFB)Right = 0

STEP -3: Bending Moment Calculation
MA = 0
MC = (WL/2 x L/2) –(WL/2 x L/2 x 2)
MC = WL2 /4 – WL2 /8
MC = (2WL2 – WL2) /8

Figure: Bending moment and shear for a different type of beam:

We are considering the UDL load is brick wall load on the simply supported beam.
Where,
W = weight of the wall
L = effective length of the beam
You can put the value in place of W and L you can easily find the maximum bending moment for the beam and slab. This maximum bending moment helps you to find out the effective depth and the area of steel in beam and slab.
For the formula of effective depth checking and Ast (area of steel) calculation you can go in IS 456: 2000 Annex G Clause 38.1-page no. 96.

Hope this will help you in preparation and study of structure analysis If you have any doubts in solving problems of structure analysis stability and determinacy then feel free to write to me! Eye on structures will try to bring you concepts and real-life structure issues. So get ready to build up your strong structure knowledge with us.

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