Lines of influence (construction mechanics): meaning and definition

How to build influence lines? Construction mechanics is based on the kinematic method of Lagrange. Its main essence lies in the fact that in a system that is in a state of complete equilibrium, the resultant of all forces on minor displacements is zero.

Specificity of the method

To construct the influence lines of the reaction, bending moment, transverse force for a given beam section, a certain algorithm of actions is used. To begin with, the connection is deleted. In addition, the lines of influence of the internal force are removed, the necessary force is introduced. As a result of such manipulations, the given system will be a mechanism possessing one degree of freedom. In the direction in which the internal force is considered, a slight displacement is introduced. Its direction should be similar to the internal effort, only in this case positive work will be done.

Examples of constructions

On the basis of the principle of displacements, the equation of equilibrium is written down, when solving it, the influence lines are calculated, the necessary effort is determined.

Let us consider an example of such calculations. We build the lines of influence of the transverse force in some section A. In order to cope with the task, it is necessary to plot the displacement diagram of this beam from a single displacement in the direction of the retracted force.

Formula for determining the effort

The lines of influence are constructed using a special formula. It connects the required force, the magnitude of the concentrated force that acts on the beam, with the area of the figure formed by the line of influence and the axis of the diagram under load. And also with the exponent of the bending moment and tangent of the angle of the line of influence of forces and the neutral axis.

If the direction of the distribution load and the concentrated force coincide with the direction of the mobile unit force, they have a positive value.

The bending moment is positive when its direction coincides with the clockwise movement. The tangent will be positive when the angle of rotation is less than a right angle. When performing calculations, the ordinate and area of the influence line are used with their signs. Construction mechanics is based on the statistical method of plotting diagrams.

Definitions

Let's give the basic definitions that are necessary for the performance of qualitative drawings and calculations. The line of influence is the line that connects the internal effort and the movement of a single mobile force.

The ordinates show a change in the analyzed internal force appearing at a certain point on the beam while traveling along the length of a unit force. They show the variation at different points of the internal force under consideration, provided that an external fixed load is used. The statistical variant of the construction is based on the recording of the equilibrium equations.

Two options for building

The construction of influence lines in beams and bending moment is possible in two cases. The force can be placed to the right or left relative to the cross-section used. In the left position, from the cross section of the force, when choosing the calculations, the forces that will act to the right are chosen. With her right action, they are counted on the left.

Multi-span beams

In bridges, for example, when transferring an external load to the load-carrying part of the entire building structure, auxiliary beams are used. The main beam is the one that is the bearing base. Cross beams are considered to be located at the right angle to the main one.

Auxiliary (single-span) beams are called, to which an external load is applied. This type of transmission to the main load beam is considered to be the nodal one. The panel is considered to be the site located between the two nearest nodes. And they are represented in the form of points of the main axis, to which the transverse beams are suitable.

Features

What is the influence line? The definition of this term in the beam is associated with a graph that shows the change in the analyzed factor when the unit force moves along the beam. In its quality, a transverse force, a bending moment, a supporting reaction can act. Any ordinate of the influence lines demonstrates the size of the analyzed factor at the time when the force is located above it. How to build the beam influence lines? A statistical method is based on the compilation of statistical equations. For example, for a simple beam located on two hinged supports, a force moving along the beam is characteristic. If you choose a certain distance, on which it functions, you can build the influence lines of the reaction, draw up the equation of moments, construct a two-point graph.

Next, a curve is plotted for the effect of shear force, for this purpose the ordinates of the influence lines in the section are used.

Cinematographic method

It can be based on the movements of the line of influence. Examples of such graphs can be found in cases where a beam is shown without support, so that the mechanism can move in the positive direction.

For the construction of the influence line of a certain bending moment, it is necessary to cut the hinge into the available cross-section. In this case, the resulting mechanism will rotate by a unit angle in the positive direction.

The construction of the influence line for a transverse force is possible when inserting into the section of the slider and spreading the beam per unit in the positive direction.

You can use the cinematic method to build lines of bending moment and lateral force in the cantilever beam. Taking into account the left side's immobility, such a beam considers motion only for the right-hand side in the positive direction. Thanks to the lines of influence according to the formula, you can calculate any effort.

Calculations for the cinematographic method

In the kinematic calculations, a formula is used that relates the number of support rods, the number of spans, hinges, and the degree of freedom of the task. If the number of degrees of freedom is zero when substituting given values, the problem can be statistically determined. If this indicator has a negative value, the problem is statistically unattainable, with a positive value of degrees of freedom, the geometric construction is performed.

In order to make it more convenient to make calculations, to have a visual representation of the features of the operation of disks in a multi-span beam, a floor plan is constructed.

For this, all the initial hinges that are present in the beam are changed to hinge-fixed supports.

Varieties of beams

Several types of multi-span beams are assumed. Specificity of the first type is that in all spans, with the exception of the first, hinged-mobile supports are used. If instead of hinges use supports, one-span beams will be formed, in which each will rest on the console next to it.

For the second type, alternating spans are characteristic, which have two hinge-movable supports, with spans without supports. In this case, the floor scheme on the console of the central beams is based on the beams-inserts.

In addition, there are also such beams in which the two previous types are combined. To ensure the statistical definability of the girders, a horizontal link is carried between the support on the right-hand neighboring girder. The lower floor in the floor scheme will be represented by the main beam, and the secondary beams are used for the upper floor.

Diagrams of internal force factors

With the help of a step-by-step scheme, it is possible to construct the diagram for an individual beam starting from the upper floor and ending with lower constructions. After the construction of the power internal factors for the upper floor is completed, it is necessary to change all the found values of the reaction of the supports to opposite forces in the direction, then apply them in the floor plan to the lower floor. When constructing on it, the epures use a given load of forces.

After the construction of the diagrams of the power internal factors is completed, a statistical check of the full multi-span beam is carried out. When checking, the condition must be fulfilled, according to which the sum of all the reactions of the supports and the given forces is zero. It is also important to conduct an analysis of compliance with the differential dependence for individual sections of the beam used.

In a graph that expresses the law of the reaction of a support or force internal factor in a specific section of a building, the function of the location of the moving individual cargo is called the influence line. To build them, apply the equation of statistics.

Graphic constructions are used to determine the force internal factors for calculating the responses of supports in certain lines of influence.

Value of calculations

In a broad sense, construction mechanics is regarded as a science that develops calculation methods and principles for testing structures and structures for stability, strength, and rigidity. Due to high-quality and timely calculations for strength, it is possible to guarantee the safety of the erected structures, their total resistance to internal and external forces.

To achieve the desired result, a combination of economy and durability is used.

Calculations for stability allow us to identify critical indicators of external influences that guarantee the preservation of a given form of equilibrium and position in a deformed state.

Calculations for stiffness consist in revealing of various variants of deformations (sediment, deflections, vibrations), because of which the full operation of structures is excluded, there is a threat of strength of structures.

In order not to encounter emergency situations, it is important to carry out such calculations, to analyze the compliance of the obtained indicators with the maximum permissible values.

At present, construction mechanics employs a huge number of various reliable calculation methods that have undergone detailed tests in construction and engineering practice.

Taking into account the constant modernization and development of the construction industry, including its theoretical base, we can talk about the application of new reliable and quality methods for constructing drawings.

In a narrow sense, construction mechanics is associated with theoretical calculations of rods, bars, which form a structure. As a basis for building mechanics are fundamental physics, mathematics, experimental research.

Calculation schemes, which are used in construction mechanics for stone, reinforced concrete, wooden, metal structures, allow to avoid misunderstandings during the erection of buildings and structures. Only with the proper preliminary construction of drawings can we talk about the safety and reliability of the created structures. The construction of lines of influence in the beams is quite a serious and responsible measure, because the life of people depends on the accuracy of actions.