Numerical Problems Based on the Cantilever Beam:
Draw the SF and BM diagram for the beam as shown in the figure given below. Also indicate principal values on diagrams.
Sol.: Let reaction at the support A be R_{AV}, R_{AH} and M(counter clock wise), First find support reaction
For that,
ΣV = 0
R_{AV } -2 -3 -3 = 0, R_{AV} = 8 ...(1)
Taking moment about A,
ΣM_{A} = 0
-M + 2 X 1 + 3 X 3 + 3 X 5 = 0
M = 26KNm ...(2)
ΣH = 0
R_{AH }= 0
Calculation for Shear force Diagram
Draw section line, here total 3 section line, which break the
load R_{AV} and 2KN(in between A and B),
2KN and 3KN(Between B and C),
3KN and 3KN (Between C and D). Take left portion of the beam
Take section 1-1
Force on left of section 1-1 is RAV
SF_{1-1} = 8KN (constant value)
Constant value means that the value of shear force at both the nearest point of the section is equal that is
SF_{A} = SF_{B } = 8KN ...(4)
Take section 2-2
Forces on the left of section 2-2 is R_{AV} and 2KN
SF_{2-2} = 8 - 2 = 6KN (constant value)
Constant value means that the value of shear force at both the nearest point of the section is equal that is
SF_{B} = SF_{C} = 6KN ...(5)
Take section 3-3
Forces on the left of section 3-3 is RA, 2KN, 3KN
SF_{3-3} = 8 - 2 - 3 = 3KN (having constant value)
Constant value means that the value of shear force at both the nearest point of the section is equal that is
SF_{C} = SF_{D} = 3KN ...(6)
Calculation for Bending moment Diagram
Assume
Distance of section 1-1 from A is X_{1}
Distance of section 2-2 from A is X_{2}
Distance of section 3-3 from A is X_{3}
Take section 1-1, taking moment about the section 1-1
BM_{1-1} = 8.X_{1}
It is Equation of straight line (Y = mX + C), inclined linear.
Inclined linear means that the value of bending moment at both the nearest point of section is varies with
It is Equation of the straight line (Y = mX + C), inclined linear.
Inclined linear means that the value of bending moment at both the nearest point of section varies with
That is inclined line 20 to 26
Plot the BM_{D } with the help of obtained bending moment values. The SF_{D } and BM_{D } is shown in the figure