Math Problem Statement

w = 38 kN/m . Determine the magnitude of the internal shear force at point C in the beam.

Solution

To determine the internal shear force at point CC in the beam, we need to analyze the given diagram, including the loading conditions and the supports.

Given:

  • The beam has a uniformly distributed load (UDL) w=38kN/mw = 38 \, \text{kN/m}.
  • The UDL acts over a length of 2m2 \, \text{m}.
  • The beam is supported at points AA and BB, with point CC located at the midpoint between these supports.
  • The distances between points AA, BB, and CC are each 2m2 \, \text{m}.

Steps to Solve:

  1. Calculate the total load from the UDL: The UDL covers 2m2 \, \text{m}, so the total load WW is: W=w×2m=38kN/m×2m=76kNW = w \times 2 \, \text{m} = 38 \, \text{kN/m} \times 2 \, \text{m} = 76 \, \text{kN}

  2. Determine the reactions at the supports AA and BB: Since the beam is symmetric and the loading is also symmetric, the reactions at AA and BB should be equal.

    The total downward force on the beam is 76kN76 \, \text{kN}. Therefore, each support reaction is: RA=RB=76kN2=38kNR_A = R_B = \frac{76 \, \text{kN}}{2} = 38 \, \text{kN}

  3. Calculate the shear force just to the left of point CC: Moving from point AA towards CC, the shear force at any section is the reaction at AA minus the UDL up to that point.

    For point CC, located 2m2 \, \text{m} from AA: VC=RAw×2m=38kN76kN=38kNV_C = R_A - w \times 2 \, \text{m} = 38 \, \text{kN} - 76 \, \text{kN} = -38 \, \text{kN}

Conclusion:

The magnitude of the internal shear force at point CC is 38kN38 \, \text{kN}. The negative sign indicates that the shear force is acting downwards at point CC.

Would you like more details on this calculation, or do you have any questions?


Here are 5 related questions to consider:

  1. What would be the shear force at a point between AA and CC but not at CC?
  2. How would the shear force diagram look for this beam?
  3. What is the moment at point CC given the same loading conditions?
  4. How would the reaction forces change if the load were not uniformly distributed?
  5. What is the significance of the shear force sign (positive or negative)?

Tip: When solving beam problems, always start by calculating support reactions before determining internal forces like shear or bending moments.