11.3 Construction and interpretation of shear and moment diagrams
4 min read•july 30, 2024
and diagrams show internal forces in beams. They help us see where a beam might break or bend too much. These diagrams are key tools for engineers designing safe structures.
Building these diagrams involves calculating reactions, plotting equations, and finding critical points. Understanding how loads affect shear and moment helps us interpret the diagrams and predict beam behavior under different conditions.
Shear Force and Bending Moment Diagrams
Graphical Representations and Internal Forces
Shear force and bending moment diagrams are graphical representations of the internal forces and moments acting on a beam along its length
Shear force is the internal force that acts perpendicular to the beam's axis
Bending moment is the internal moment that causes the beam to bend
Concentrated loads (point loads), distributed loads, and moments can be applied to a beam
The magnitudes and locations of these loads and moments affect the shear force and bending moment diagrams
Constructing Diagrams and Sign Conventions
The sign convention for shear force and bending moment diagrams follows the right-hand rule
force acts upward
Positive bending moment causes compression on the top of the beam
Constructing shear force and bending moment diagrams involves:
Calculating the reactions at the supports
Determining the equations for shear force and bending moment along the beam
Plotting these equations
Discontinuities in the shear force diagram occur at concentrated loads and reactions
Discontinuities in the bending moment diagram occur at concentrated moments
The slope of the shear force diagram represents the acting on the beam
The slope of the bending moment diagram represents the shear force
Critical Points on Diagrams
Locations and Significance of Critical Points
Critical points on shear force and bending moment diagrams include:
Locations of zero shear force
Maximum or minimum shear force
Zero bending moment (inflection points)
Maximum or minimum bending moment
Points of zero shear force indicate the locations of maximum or minimum bending moment
The slope of the bending moment diagram changes sign at these points
Points of maximum or minimum shear force occur at:
Concentrated loads
Reactions
Discontinuities in the distributed load
Points of zero bending moment (inflection points) indicate the locations where the beam changes from positive to negative curvature or vice versa
Identifying Critical Points
Points of maximum or minimum bending moment occur where the shear force diagram crosses the zero axis
The slope of the bending moment diagram is zero at these points
To identify critical points:
Analyze the shear force diagram for zero crossings, maxima, and minima
Examine the bending moment diagram for zero crossings, maxima, minima, and inflection points
Consider the locations of concentrated loads, reactions, and discontinuities in the distributed load
Interpretation of Diagrams
Insights from Shape and Magnitude
The shape and magnitude of shear force and bending moment diagrams provide insights into the internal forces and moments acting on the beam
They help identify critical regions that may require special attention in design
The magnitude of the shear force at any point represents the net internal shear force acting at that cross-section of the beam
The magnitude of the bending moment at any point represents the net internal moment causing bending at that cross-section of the beam
Regions with high magnitudes of shear force or bending moment indicate areas of the beam that experience significant internal stresses
Deflected Shape and Curvature
The curvature of the bending moment diagram indicates the beam's deflected shape
Positive curvature corresponds to sagging (beam bending downward)
Negative curvature corresponds to hogging (beam bending upward)
Sudden changes in the slope of the shear force or bending moment diagrams indicate the presence of:
Concentrated loads
Reactions
Moments acting on the beam
Analyzing the curvature and slope changes helps understand the beam's behavior under loading
Load, Shear, and Moment Relationships
Interconnections and Derivatives
The load, shear force, and bending moment diagrams are interconnected
Understanding their relationships is crucial for analyzing beam behavior
The distributed load acting on the beam is equal to the slope of the shear force diagram at any point
The shear force at any point is equal to the slope of the bending moment diagram at that point
The second derivative of the bending moment diagram with respect to the beam's length is equal to the distributed load acting on the beam
Discontinuities and Integration
Concentrated loads and reactions appear as discontinuities or jumps in the shear force diagram
The magnitude of the jump is equal to the load or reaction
Concentrated moments appear as discontinuities or jumps in the bending moment diagram
The magnitude of the jump is equal to the applied moment
By integrating the shear force diagram, the bending moment diagram can be obtained
By integrating the bending moment diagram, the beam's deflection can be determined
These relationships allow for the calculation of internal forces, moments, and deflections at any point along the beam
Key Terms to Review (18)
Area under the curve: The area under the curve refers to the space between a curve on a graph and the horizontal axis, which can be used to determine quantities such as total force, work done, or accumulated load. In the context of shear and moment diagrams, this area provides valuable information about the internal forces and moments acting on a structural element, allowing for an understanding of how the structure will behave under various loads.
Bending Moment: A bending moment is a measure of the internal moment that induces bending in a beam or structural element when external loads are applied. It reflects how much a beam wants to bend in response to these loads, which is crucial in understanding how structures respond to forces and maintaining their integrity.
Cut section: A cut section is a conceptual tool used in mechanics to analyze internal forces within a structure by 'cutting' through it to expose the internal components. This method is essential for understanding how external loads affect the internal shear and moment distributions, leading to the construction of shear and moment diagrams that are critical for structural analysis.
Distributed Load: A distributed load is a force applied uniformly over a length of a structural element, such as a beam, rather than at a single point. This type of loading is crucial in understanding how structures respond to various forces, as it influences shear forces, bending moments, and ultimately the stability and safety of structures.
Equilibrium Equations: Equilibrium equations are mathematical statements that describe the condition of a body in static equilibrium, where the sum of all forces and moments acting on it is zero. These equations are essential for analyzing structures and components to ensure they can withstand applied loads without movement or deformation, connecting various concepts like distributed forces, free-body diagrams, and shear and moment diagrams.
Fixed support: A fixed support is a type of boundary condition in structural engineering that restrains a structure at a specific point, preventing both translational and rotational movement. This means the structure cannot move up, down, or sideways, and it cannot rotate about the support point, effectively anchoring it in place. The presence of a fixed support has significant implications for analyzing forces, moments, and deflections within a structure.
Free-body diagram: A free-body diagram is a graphical representation used to visualize the forces acting on an object, showing all external forces and moments applied to that body while isolating it from its surroundings. This tool is essential for analyzing the equilibrium and motion of structures, facilitating the understanding of how forces interact in various scenarios.
Inflection Point: An inflection point is a point on a curve where the curvature changes sign, meaning it transitions from concave up to concave down or vice versa. In the context of shear and moment diagrams, these points are crucial as they indicate where the bending moment changes from increasing to decreasing or vice versa, signaling important changes in the behavior of beams under load.
Method of sections: The method of sections is a technique used in statics to analyze the internal forces in a truss by making a cut through the truss and applying equilibrium equations to the resulting sections. This method allows for the determination of member forces without having to analyze the entire structure, which simplifies calculations and provides a clear understanding of how loads are transferred within the truss.
Negative moment: A negative moment refers to a twisting effect that tends to cause clockwise rotation about a point or axis, which contrasts with a positive moment that causes counterclockwise rotation. Understanding negative moments is crucial for analyzing the behavior of structures under applied loads, as they help in determining how forces interact and affect stability. Negative moments often appear in shear and moment diagrams, where they indicate regions of bending that can lead to potential failure in materials.
Point Load: A point load is a force applied at a specific location on a structural element, resulting in concentrated stress at that point. This type of load is crucial in analyzing how structures respond to various forces, particularly in understanding how it affects the overall stability and strength of beams, trusses, and frames.
Positive Shear: Positive shear is defined as the internal shear force that acts parallel to the cross-section of a beam, causing it to tend to slide one part of the beam over another. In a shear and moment diagram, positive shear is typically represented above the zero axis, indicating a tendency for the material above the section to move rightward relative to the material below. Understanding positive shear is essential when constructing and interpreting shear and moment diagrams, as it helps visualize how forces are distributed within a structural element.
Reaction Forces: Reaction forces are the forces that occur at supports or connections in a structure due to external loads applied to that structure. These forces balance out the applied loads, ensuring the equilibrium of the structure, and are crucial when analyzing how structures respond to different loading conditions. Understanding reaction forces is essential for constructing shear and moment diagrams, as these diagrams visually represent how internal forces change along a beam or structural element.
Roller support: A roller support is a type of structural support that allows a beam or structure to rotate and move horizontally while resisting vertical loads. This flexibility enables structures to accommodate thermal expansion and other movements, making roller supports essential in various engineering applications.
Shear Force: Shear force is a measure of the internal force acting along a cross-section of a structural element, which is perpendicular to its longitudinal axis. It plays a crucial role in determining how structures respond to applied loads, and understanding it is essential when analyzing different types of loading conditions, distributed forces, and the behavior of beams and frames.
Shear-moment relationship: The shear-moment relationship describes the connection between shear forces and bending moments along a beam. It illustrates how the distribution of shear force influences the bending moment experienced by a structure, revealing critical insights into its behavior under loading. Understanding this relationship is essential for constructing accurate shear and moment diagrams, which visually represent these forces and moments throughout the beam's length.
Slope of the diagram: The slope of the diagram refers to the rate of change of shear or moment values along a beam as depicted in shear and moment diagrams. This slope represents how the internal forces change in response to applied loads, providing critical insights into the behavior of structures under various loading conditions.
Static Equilibrium: Static equilibrium occurs when an object is at rest and all forces acting on it are balanced, resulting in no net force or moment acting on it. This condition ensures that the object remains in a stable state without any movement or rotation, which is crucial for understanding various engineering principles such as force distribution, load analysis, and structural integrity.