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 X₁

Distance of section 2-2 from A is X₂

Distance of section 3-3 from A is X₃

Take section 1-1, taking moment about the section 1-1

BM_1-1 = 8.X₁

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