4.65 determine the reactions at b and c when a 30 mm 60 mm 40 mm 100 mm 60 mm 250 n fig. p4.65

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09-12-2024

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Determining Reactions at B and C: A Step-by-Step Guide (Fig. P4.65)

This article provides a detailed explanation of how to determine the reactions at points B and C for the statically determinate beam shown in Figure P4.65. The figure depicts a beam with a complex geometry and loading, requiring a systematic approach to solve. We'll break down the problem into manageable steps, using principles of statics.

Understanding the Problem (Fig. P4.65):

Figure P4.65 shows a beam subjected to a vertical load of 250 N. The beam's dimensions are provided: 30 mm, 60 mm, 40 mm, 100 mm, and 60 mm. These dimensions define the geometry and influence the moment arms involved in calculating reactions. The goal is to find the vertical reactions at support points B and C. These supports are likely pin and roller supports, implying vertical reactions only (no horizontal reactions).

Step-by-Step Solution:

1. Free Body Diagram (FBD):

The first crucial step is to create a free body diagram (FBD) of the beam. This diagram isolates the beam from its supports and shows all external forces acting on it. This includes:

• The 250 N downward force.
• The vertical reaction force at B (let's call it R_B).
• The vertical reaction force at C (let's call it R_C).

2. Equations of Equilibrium:

To determine R_B and R_C, we'll apply the equations of static equilibrium. For a planar system like this beam, these are:

• ΣF_y = 0: The sum of vertical forces equals zero.
• ΣM_A = 0 (or ΣM_B or ΣM_C): The sum of moments about any point equals zero. Choosing a point where an unknown reaction force acts simplifies the equation.

3. Applying the Equations:

• ΣF_y = 0: R_B + R_C - 250 N = 0

• ΣM_B = 0 (Choosing point B to eliminate R_B): This requires calculating the moment arm of the 250 N force relative to point B. The dimensions provided (30 mm, 60 mm, 40 mm, 100 mm, 60 mm) are crucial for determining this distance. Let's assume the 250 N force acts at a distance 'x' from point B. Then:

  -250N * x + R_C * (Total Length - Distance to C) = 0.

(Note: the "Total Length" and "Distance to C" need to be calculated from the given dimensions in Fig P4.65. These are crucial values that depend on the exact beam configuration shown in the figure, which is not reproducible in text format. A clear image or diagram is needed for precise calculations.)

4. Solving the Simultaneous Equations:

Now we have two equations with two unknowns (R_B and R_C). Solving these simultaneously (either through substitution or elimination) will yield the values of the reactions at points B and C.

5. Interpretation of Results:

The calculated values of R_B and R_C represent the magnitude and direction of the reaction forces at supports B and C. Positive values indicate upward forces (resisting the downward load), while negative values indicate downward forces (which is unusual for a simple beam support).

Conclusion:

Determining the reactions at B and C for the beam in Figure P4.65 involves a systematic application of static equilibrium equations. The precise numerical solution depends on the exact geometry of the beam as depicted in the figure, requiring careful calculation of moment arms using the provided dimensions. Remember that a clear, accurate drawing of Figure P4.65 is essential to obtain the correct numerical results. Without the image, only the methodology can be explained.