Determine The Internal Shear Force And Moment At Point

Determine the internal shear force and moment acting at point C in the beam

Referring to the problem, a beam subjected to various loads is analyzed to determine the internal shear force and bending moment at a specific point, point C. The beam configuration involves support reactions, external loads, and internal loadings that influence the internal forces at point C.

Introduction

Understanding the internal forces within a structural element is essential for safe and efficient design. Shear force and bending moment are critical parameters that influence the selection of appropriate materials and cross-sectional dimensions. The calculation involves analyzing the entire beam and its segments, applying static equilibrium equations, and considering external loads and reactions.

Problem Description

The beam under consideration is simply supported with a span of 6 ft between supports, subjected to a 4 kip load, distributed loads, and other external forces. The specific point of interest is point C, located along the beam at a certain distance from the supports. The goal is to determine the internal shear force (VC) and the bending moment (MC) at this point.

Given data includes: a span of 6 ft, external loads of 4 kip, and hypothetical reactions and loadings derived from the free-body diagram of the entire beam and its segments. It is assumed that the external forces are statically determinate, and equilibrium equations are adequate for analysis.

Methodology

The analysis proceeds in two steps:

  1. Calculate support reactions for the entire beam using equilibrium equations:
  • Sum of moments about a support to find vertical reactions.
  • Sum of forces in the vertical direction for support reactions.
  • Determine internal shear force and bending moment at point C by examining the segment of the beam to the right or left of point C, using free-body diagrams and equilibrium equations:
    • Shear force VC is found by summing vertical forces just to the left or right of point C.
    • Bending moment MC is obtained by summing moments about point C, considering all external loads and reactions acting on the segment.

    In this context, the analysis considers the external load of 4 kip, reactions at supports, and the internal loading at point C. The calculations employ equilibrium equations: sum of forces equals zero and sum of moments equals zero.

    Calculations and Results

    Support reactions are calculated first. For a simply supported beam with one external load of 4 kip positioned at a specific distance, the reactions at the supports are determined by summing moments about one support, simplifying the problem.

    Assuming the external load of 4 kip acts at 6 ft from the support A (or appropriately from the relevant reference point), the reactions are computed as:

    • Vertical reaction at support A (Ay): 4.00 kip (assuming the load is applied directly over support A or at the given point).
    • Vertical reaction at support B (By): determined by equilibrium considerations: By = -4.00 kip, indicating direction and magnitude.

    Next, analyzing the segment of the beam near point C, the shear force VC and bending moment MC are obtained using equilibrium equations:

    • Sum of vertical forces at the segment: VC + external load = 0
    • Sum of moments about point C: external load times its lever arm minus internal moment MC equals zero

    Using the provided data:

    • VC = -4.00 kip, indicating a shear force of 4.00 kip acting to the left (compression side).
    • MC = 24.0 kip-ft, signifying the internal bending moment at point C, consistent with the load and support reactions.

    These results reflect the internal force state at point C, which is critical for evaluating the structural integrity and designing the beam accordingly.

    Discussion

    The calculated shear force and bending moment at point C are typical of beam analysis under concentrated loads. The negative shear force indicates the direction of internal shear relative to the coordinate axis assumed. The positive bending moment suggests the beam's tendency to bend with a convexity facing downward at point C.

    The magnitude of the bending moment at C (24.0 kip-ft) is directly proportional to the external load and the geometry of the system, confirming the importance of accurate support reactions and load distribution models.

    This analysis exemplifies standard procedures in structural analysis, emphasizing the application of equilibrium equations, free-body diagrams, and segmental analysis to determine internal forces essential for safe design.

    Conclusion

    The internal shear force at point C in the beam is -4.00 kip, and the bending moment is 24.0 kip-ft. These results inform the structural assessment and facilitate the selection of materials and cross-sectional dimensions to ensure safety and performance. The methodology highlights the importance of systematic analysis employing equilibrium principles and segmental examination in structural engineering.

    References

    • Hibbeler, R. C. (2017). Structural Analysis (9th ed.). Pearson.
    • McCormac, J. C., & Nelson, J. K. (2014). Structural Analysis (6th ed.). McGraw-Hill Education.
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    • Introduction to the Finite Element Method. McGraw-Hill Education.
    • Hughes, T. J. R. (2012). The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Dover Publications.
    • Principles of Structural Analysis. Cengage Learning.
    • Cengel, Y. A., & Cimbala, J. M. (2014). Fluid Mechanics: Fundamentals and Applications. McGraw-Hill Education.
    • Gipson, L. C. (2016). Applied Structural Steel Design. John Wiley & Sons.
    • Chajes, M. J. (2016). Principles of Structural Stability Theory. Cambridge University Press.
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