Influence Lines, Virtual Work, and Structural Analysis Methods

Influence Lines and Structural Analysis

Influence lines (IL) and their use in structural analysis. Methods of IL construction.
Internal forces and bending moments caused by moving loads can be solved with the help of influence lines. Influence line S graphs the variation of a quantity S at a specific point x on a beam or truss caused by a unit load F=1 placed at any point u along the structure.
Methods for constructing the influence line:

• Analytical method (creating equations using equilibrium conditions)
• Kinematic method (Müller-Breslau Principle; using principal of virtual work)
• Combination of above-mentioned methods
• Numerically (based on definition, i.e. tabulation of the influence values for multiple points along the structure)

Virtual Work and Maxwell-Betti Theorem

Principle of virtual work, Maxwell-Betti reciprocal work theorem.
Principle of virtual work (virtual load): If a virtual load is applied on a structure, the external virtual work is equal to the whole inner virtual work.
Maxwell-Betti reciprocal work theorem: for a linear elastic structure subjected to two sets of forces the virtual work done by the set P through the displacements produced by the set Q is equal to the virtual work done by the set Q through the displacements produced by the set P. Principle of the unit dummy force method for calculation of structural displacements.

Combining Theorems

Combine Maxwell-Betti reciprocal theorem with the principle of virtual work.

• Consider a unit virtual load (unit force or unit moment).
• Place unit load to the position and direction corresponding to analyzed displacement.

Real beams - bending is a predominant loading component (depending on dimensions and direction of load). How to calculate ?

• Analytical integration
• Numerical integration (e.g. Simpson's rule)
• Vereshchagin's rule
• Tables

Vereshchagin's Rule

Vereshchagin's rule – what it is used for, conditions of validity.
It is used to calculate the unit dummy force method. Limitations:

• One of the functions must be constant or linear.
• The value of Mt or Mt must be taken from the constant or linear function.
• If both are constant and linear then

Degrees of Freedom and Static Indeterminacy

Degrees of freedom of a structure, external/internal restraints, static indeterminacy.
Degrees of freedom m - number of independent motions (displacements + rotations) that are allowed to the body:

• Rigid body in the space: m = 6
• Rigid body in the plane: m = 3

Stability must be ensured by restraints.
Restraints of the body r:

• External (supports)
• Internal (hinges, rods)

Depending on the degree of static uncertainty, the structures can be:

• Ns = 0: Supporting of the body is statically and kinematic determinate, immobility is ensured.
• Ns > 0: Supporting of the body is statically indeterminate and kinematic overdeterminate, immobility is ensured.
• ns

Force Method and Three-Moment Equation

Principle of the Force method and the Three-moment equation method.
The force method or the method of consistent deformation is based on the equilibrium of forces and compatibility of structures. The method entails first selecting the unknown redundants for the structure and then removing the redundant reactions or members to obtain the primary structure.
The three moment equation expresses the relation between bending moments at three successive supports of a continuous beam, subject to a loading on a two adjacent span with or without settlement of the supports.

Symmetry in Structural Analysis

Use of symmetry in structural analysis, symmetric/antimetric load, boundary conditions
Geometry of the structure and boundary conditions are symmetric: Only one half of the structure can be solved, which helps in reducing the number of unknowns to solve for. Symmetric and antisymmetric loading - both load cases can be solved but the symmetry boundary conditions differ for each of them.

Symmetric load: through the support

through the middle of the field

Antisymmetric load: through the support

through the middle of the field