In these blog, we will study the shear force and bending moment of different beams with different type of load. But before starting, we should be clear about the beams,loads and their type. For these please go through my previous blog BEAMS ,LOADS AND ITS TYPES once.
Shear force: The algebric sum of all the vertical loads acting on either left or right side of section or member.
Bending moment: The algebric sum of all momentum due to vertical load acting on either left/right side of member or section.
Shear force diagram: The diagram which indicate the point at which beam is breaking into 2 pieces.
Bending moment diagram: The diagram which indicate the point at which beam is bending.
Sign convention for shear force &bending moment:
1.Shear force:
- If we started from left of section then take all upside forces positive (+ve) and all downward forces negative (-ve).
- If we started from right side of section then take all upside forces negative(-ve) and all downward forces (+ve).
2.Bending moment:Take all the upside forces positive (+ve) and all downward forces negative (-ve).
Important points for shear force and bending moment:
- consider the left or right side of portion of the member.
- The shear force between any two vertical loads will constant. Thus the shear force diagram is horizontal between the loads.
- The shear force diagram will increase or decrease suddenly by vertical line where there is point load.
- The bending moment at the two supports of simply supported beam and the free end of cantilever beam is zero.
Equilibrium condition for planer structure(XY-plane):
- All the horizontal forces will be zero.
- All the vertical forces will be zero.
- All the momentum will be zero.
We will use these 3 conditions for calculating shear forces and bending moments for all types.
Method(overview):
- Firstly measure all forces acting on the beam, distance between supports,distance of the point where force is acting in case of point load from one end supports.
- Then apply the equilibrium conditions on it which will give relations between the reaction forces(act upward) Ra and Rb.
- Then solve it to get the value of Ra and Rb.
- Use these value of Ra and Rb to calculate the value of shear force and bending at every point where force is acting(in case of point load) and at ends too.
- We also calculate the point where shear force is zero in case of unifrom distributed load .The bending moment is maximum at these point.
- Then draw the shear force and bending moment diagram according to the values.
For all type of beams, shear force diagram and bending moment of different type of loads follow different law i.e.cubic law, linear law or parabolic law as listed below:
Type of load. SF. BM
- Point load. Rectangular. Linear
- U.D.L. linear. Parabolic
- U.V.L. Parabolic. Cubic
Now let us take a example of cantilever beam with uniformly distributed load 2KN/m from at left end upto 2m and point load of 10KN at free end of beam.Total distance is 5m
a.First of all apply equilibrium conditions
1. Sum of all horizontal forces are zero i.e. Ha=0
2.Sum of all vertical forces are zero i.e.
Ra-2×2-10=0 => Ra=14 KN
Rc =0 as there is no support at end support C.
3.Sum of momentum is zero i.e.
Ra×5-2×2×(2/2+3)+Ma=0
=>Ma= -54KNm
b. Now calculate shear force and bending moment at point A,B,C which give value
Ra=14KN, Rb=10KN,Rc=0
BMa=-54KNm, BMb= -30KNm,BMc=0KNm
Now draw the shear force and bending moment diagram as shown below.
These is how you easily calculate the value of shear force and bending moment.
For any queries,leave the message in comment section.
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