Which principle states that the influence line for a function (reaction, shear, moment) is to the same scale as the deflected shape of the beam when the beam is acted on by the function?

Question

Prepp Team · Accepted Answer

Understanding Influence Lines in Structural Analysis

Influence lines are essential tools in structural analysis. They graphically show how a specific function (like reaction at a support, shear force at a point, or bending moment at a point) varies as a unit load moves across the structure. Influence lines help engineers determine the position of live loads that will cause the maximum value of a particular function, which is crucial for design.

The Muller-Breslau Principle Explained

The question asks about a principle that connects the influence line of a function to the deflected shape of a beam. This principle is known as the Muller-Breslau Principle. It provides a simple and intuitive way to draw influence lines.

The principle states the following:

The influence line for a function (such as reaction, shear, or bending moment) at a specific point in a structure is to the same scale as the deflected shape of the structure when it is subjected to a generalized displacement corresponding to that function, applied at the location of the function.

Let's break down what this means:

To draw the influence line for a reaction at a support, remove the restraint corresponding to the reaction (e.g., remove vertical restraint for a vertical reaction) and apply a unit displacement in the direction of the reaction at that support. The resulting deflected shape of the beam is the influence line for that reaction.
To draw the influence line for shear force at a point, cut the beam at that point and allow a unit relative displacement (shear deformation) to occur while maintaining equilibrium. The resulting deflected shape is the influence line for the shear force at that point. Specifically, the two cut ends are moved relative to each other by a unit amount (one going up, the other down) such that the angle between the tangents remains unchanged.
To draw the influence line for bending moment at a point, introduce a hinge at that point and apply a unit relative rotation (bending deformation). The resulting deflected shape is the influence line for the bending moment at that point. The rotation is applied by rotating the two cut ends relative to each other by a unit angle.

The Muller-Breslau principle is based on the principle of virtual work or Betti's Law/Maxwell's Reciprocal Theorem. It's a powerful qualitative method for sketching influence lines accurately without complex calculations, especially useful for indeterminate structures where analytical methods can be more involved.

Applying the Muller-Breslau Principle

Using the Muller-Breslau principle to draw influence lines involves these general steps:

Identify the function (reaction, shear, or moment) and its location for which the influence line is needed.
Remove the support or cut the structure corresponding to that function.
Apply a unit displacement (or rotation) in the direction of the function at that location.
Sketch the resulting deflected shape of the structure. This deflected shape, drawn to a certain scale, is the influence line for the function. The ordinates of the deflected shape represent the value of the function when a unit load is placed at that point on the original structure.

For statically determinate structures, the influence lines drawn using this principle will be straight lines. For indeterminate structures, they will be curved.

Why Other Principles Are Not Applicable Here

Von Mises: The Von Mises criterion is a yield criterion used in materials science and structural mechanics to predict when a material will yield under complex stress conditions. It is not related to drawing influence lines or structural deflections in the context of this question.
Rankine: Rankine's theory is associated with earth pressure calculations (Rankine's theory of earth pressure) and column buckling (Rankine-Gordon formula). It is not related to the principle described for influence lines.
Maxwell: Maxwell's Reciprocal Theorem states that the deflection at point A due to a unit load at point B is equal to the deflection at point B due to a unit load at point A. While the Muller-Breslau principle is based on Maxwell's theorem or virtual work, the principle that *directly* relates the influence line shape to the deflected shape due to a specific generalized displacement is the Muller-Breslau principle itself. Maxwell's theorem establishes reciprocity, which underpins the validity of Muller-Breslau.

Therefore, the principle that directly states the relationship between an influence line and a deflected shape caused by a corresponding displacement is the Muller-Breslau principle.

Revision Table: Key Structural Principles

Principle	Description	Relevance to Influence Lines
Muller-Breslau Principle	Influence line shape for a function is the deflected shape due to a corresponding unit generalized displacement.	Directly used to draw influence lines qualitatively and determine their shape.
Maxwell's Reciprocal Theorem	Deflection at A due to load at B equals deflection at B due to load at A.	Provides the theoretical basis for the Muller-Breslau principle.
Principle of Virtual Work	External virtual work equals internal virtual work.	Another theoretical basis for the Muller-Breslau principle.
Von Mises Criterion	Yield criterion for ductile materials under multi-axial stress.	Not related to influence lines or deflection shapes for this purpose.
Rankine Theory	Earth pressure or column buckling.	Not related to influence lines or deflection shapes for this purpose.

Additional Information on Influence Lines and Muller-Breslau

Influence lines are crucial for:

Determining maximum effects (reaction, shear, moment) due to moving loads like vehicles on bridges.
Placing standard live loads (like AASHTO truck loads or uniformly distributed loads) to produce the worst-case scenario for design.
Understanding how loads at different points on a structure contribute to the value of a specific function at a given point.

The Muller-Breslau principle makes drawing influence lines significantly easier, especially for complex structures or when a qualitative understanding is sufficient. It transforms the problem of finding ordinates by moving a unit load into a problem of sketching a deflected shape due to an imposed displacement.

For example, to draw the influence line for the vertical reaction at support A of a simple beam, you would remove the vertical support at A (making it a roller or allowing vertical movement) and apply a unit upward displacement at A. The resulting triangular deflected shape of the beam would be the influence line for the reaction at A.