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Math  /  Geometry

QuestionGiven a = 4 m, and Mo = 40 Nm, A Mo E Mo C G F TT B - a - — a - — a - Plot out the shear and moment diagrams for the beam.

Studdy Solution

STEP 1

1. The beam is statically determinate.
2. The beam is supported by a fixed support at point A and a roller support at point G.
3. The moments M0=40 Nm M_0 = 40 \, \text{Nm} are applied at points C and E in a counterclockwise direction.
4. The beam is divided into three equal segments of length a=4 m a = 4 \, \text{m} .

STEP 2

1. Determine the reactions at the supports.
2. Calculate the shear force along the beam.
3. Calculate the bending moment along the beam.
4. Plot the shear force diagram.
5. Plot the bending moment diagram.

STEP 3

Determine the reactions at the supports:
- Since the beam is statically determinate and symmetrical, calculate the reaction forces at A and G using equilibrium equations. - Sum of vertical forces: ∑Fy=0 \sum F_y = 0 - Sum of moments about A: ∑MA=0 \sum M_A = 0
Intermediate_Step_1.1: Calculate the reaction at G:
- Consider the moments about A to find the reaction at G. - ∑MA=0 \sum M_A = 0 : −M0+RG×3a=0 -M_0 + R_G \times 3a = 0 - RG=M03a=40 Nm12 m=103 N R_G = \frac{M_0}{3a} = \frac{40 \, \text{Nm}}{12 \, \text{m}} = \frac{10}{3} \, \text{N}
Intermediate_Step_1.2: Calculate the reaction at A:
- Use the sum of vertical forces: RA+RG=0 R_A + R_G = 0 - RA=−RG=−103 N R_A = -R_G = -\frac{10}{3} \, \text{N}

STEP 4

Calculate the shear force along the beam:
- Consider sections of the beam between supports and points of applied moments. - Calculate shear force at each segment boundary.
Intermediate_Step_2.1: Calculate shear force between A and C:
- Shear force VAC=RA=−103 N V_{AC} = R_A = -\frac{10}{3} \, \text{N}
Intermediate_Step_2.2: Calculate shear force between C and E:
- Shear force changes due to moment at C. - VCE=VAC=−103 N V_{CE} = V_{AC} = -\frac{10}{3} \, \text{N}
Intermediate_Step_2.3: Calculate shear force between E and G:
- Shear force changes due to moment at E. - VEG=VCE+RG=0 V_{EG} = V_{CE} + R_G = 0

STEP 5

Calculate the bending moment along the beam:
- Consider sections of the beam and calculate moment at each segment boundary.
Intermediate_Step_3.1: Calculate moment between A and C:
- Moment at A: MA=0 M_A = 0 - Moment at C: MC=M0=40 Nm M_C = M_0 = 40 \, \text{Nm}
Intermediate_Step_3.2: Calculate moment between C and E:
- Moment at E: ME=M0=40 Nm M_E = M_0 = 40 \, \text{Nm}
Intermediate_Step_3.3: Calculate moment between E and G:
- Moment at G: MG=0 M_G = 0

STEP 6

Plot the shear force diagram:
- Plot shear force values calculated at each segment boundary.

STEP 7

Plot the bending moment diagram:
- Plot bending moment values calculated at each segment boundary.
The shear and moment diagrams have been plotted based on the calculations.

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