A function that represents the load effect (force or displacement) at a point in the structure as a unit action moves along a path or over surface.
Influence Function:
- Superposition of all the load effects yields
- Action A = P1* η (x1)+P2* η (x2)+--- +Pn*η (xn) = ΣPi*η (xi) = ΣPi*ηi (5.1)
- Linear behavior is a necessary condition for application of Equation 5.1, that is, the influence coefficients must be based on a linear relationship between the applied unit action and the load effect.
- For statically determinate structures, this relationship typically hold true except for cases of large deformation where consideration of deformed geometry must be considered in the equilibrium formulation.
- The unit action load effect relationship in the statically indeterminate structures is a function of the relative stiffness of the elements. If stiffness changes are due to load application from either material non linearity and/or geometric non linearity (large deflections), then the principle of superposition cannot be applied. In such cases, the use of influence functions is not appropriate and the loads must be applied sequentially as expected in the real structure.
- The analysis of a structure subjected to numerous load placements can be labour intensive and algebraically complex. The unit action must be considered at numerous locations requiring several analyses.
- The Muller-Breslau Principle allows the analyst to study one load case to generate the entire influence function.
- Because the function has the same characteristics whether generated by traversing a unit action or by the Muller-Breslau Principle, many of the complicating features are similar.
- The development of the Muller-Breslau Principle requires the application of Betti’s Theorem. This important energy theorem is prerequisite to the understanding of the Muller-Breslau Principle.
- Consider two force systems P and Q associated with displacements p and q applied to a structure that behaves linear elastically.
- Application of the Q-q system to the structure and equating the work performed by gradually applied forces to the internal strain energy yields
- ½ * Σ Qi * qi = UQq
- where UQq is the strain energy stored in the beam when the loads Q are applied quasi-statically through displacement q.
- Now apply the forces of the second system P with the Q forces remaining in place. Note that the forces Q are now at the full value and move through displacement p due to force P. The work performed by all the forces is
- ½ * Σ Qi * qi + ½ * Σ Qi * pi + ½ * Σ Pi * pi =Ufinal
- where Ufinal is the associated internal strain energy due to all forces applied in the order prescribed.
- Use the same force systems to apply the forces in the reverse order, that is, P first and then Q. The work performed by all forces is
- ½ * Σ Pi * pi + ½ * Σ Pi * qi + ½ * Σ Qi * qi =Ufinal
- If the structure behaves linear elastically, then the final displaced shape and internal strain energy are independent of the order of load application. Therefore, equating Ufinal in Equations 5.5 and 5.6 yields
- Σ Qi * pi = Σ Pi * qi
- In a narrative format, the Betti’s Theorem states
- The product of the forces of the first system times the corresponding displacements due to the second force systems is equal to the forces of the second force system times the corresponding displacements of the first system.
- Although the derivation is performed with reference to a beam, the method is generally applicable to any linear elastic structural system.
- An influence function for an action may be established by removing the constraint associated with the action and imposing a unit displacement. The displacement at every point in the structure is the influence function. In other words, the structure’s displaced shape is the influence function.
- The sense of the displacements that define the influence function must be considered. For concentrated or distributed forces, the translation collinear with the direction of the action is used as the influence ordinate or function. If the applied action is a couple, then the rotation is the associated influence function.
- One of the most useful applications of the Muller-Breslau Principle is in the development of qualitative influence functions. Because most displaced shapes due to applied loads may be intuitively generated in an approximate manner, the influence functions may be determined in a similar fashion.
- Although exact ordinates and/or functions require more involved methods, a function can be estimated by simply releasing the appropriate restraint, inducing the unit displacement, and sketching the displaced shape.
- This technique is extremely useful in determining an approximate influence function that in turn aids the engineer in the placement of loads for the critical effect.
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