How to build lines of influence? Structural mechanics is based on the kinematic method of Lagrange. Its main essence is that in a system that is in a state of complete equilibrium, the result of all forces at minor displacements is zero.
Method specificity
To construct lines of influence of the reaction, bending moment, shear force for a given section of the beam, a certain algorithm of actions is used. First, delete the connection. In addition, the lines of influence of the internal force are removed, the necessary force is introduced. As a result of such manipulations, a given system will be a mechanism with one degree of freedom. In the direction where the internal force is considered, a slight movement is introduced. Its direction should be similar to internal effort, only in this case will positive work be done.
Building Examples
On the basis of the principle of displacements, write the equation of equilibrium, when solving it, calculate the lines of influence, determine the necessary effort.
Consider an example of such calculations. We draw the lines of influence of the transverse force in a certain section A. To cope with the task, it is necessary to plot the displacements of this beam from a single displacement in the direction of the removed force.
Formula for determining effort
The construction of lines of influence is carried out using a special formula. It connects the desired effort, the amount of concentrated force that acts on the beam, with the area of the figure formed by the line of influence and the axis of the plot under load. And also with an indicator of the bending moment and the tangent of the angle of the line of influence of efforts and the neutral axis.
If the direction of the distribution load and the concentrated force coincide with the direction of the moving unit force, they have a positive value.
The bending moment will be a positive value when its direction coincides with the clockwise movement. The tangent will be positive if the angle of rotation is less than the right angle. When carrying out the calculations, the magnitude of the ordinates and the area of the influence line are used with their signs. Structural mechanics is based on the statistical method of plotting.
Definitions
We give the basic definitions that are necessary to perform high-quality drawings and calculations. The line of influence is the line that connects the internal force and the movement of a single moving force.
The ordinates show a change in the analyzed internal force, which appears at a certain point on the beam when moving along the length of a single force. They show the change at different points of the considered internal force provided that an external stationary load is used. The statistical version of the construction is based on writing 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 located on the right or left relative to the used section. With a left arrangement from the cross section of the force, during the calculations, forces are selected that will act to the right. With her right action, they count according to the left forces.
Multi-span beams
In bridges, for example, when transferring external loads to the bearing part of the entire building structure, auxiliary beams are used. The main beam is called the one that is the supporting base. Cross beams are considered to be located at a right angle to the main one.
Auxiliary (single-span) are called beams, to which an external load is applied. This version of the transfer to the main load beam is considered nodal. A panel is a site located between the two nearest nodes. And they are presented in the form of points of the main axis, to which the transverse beams fit.
Features
What is a line of influence? The definition of this term in a beam is connected with a graph that shows the change in the analyzed factor when a unit force moves along the beam. In its quality, a transverse force, a bending moment, a support reaction can act. Any ordinate of the influence lines shows the size of the analyzed factor at that moment in time when the force is located above it. How to build a line of influence of the beam? The statistical method is based on the compilation of statistical equations. For example, a simple beam located on two hinged supports is characterized by a force moving along the beam. If you select a certain distance at which it operates, you can build lines of influence of the reaction, make an equation of the moments, build a two-point graph.
Next, a curve of the impact of the transverse force is constructed, for this, the ordinates of the influence lines in the section are used.
Cinematic way
An influence line can be constructed based on the movements. Examples of such graphs can be found in cases where they depict a beam without support, so that the mechanism can move in a positive direction.
To build a line of influence of a certain bending moment, it is necessary to embed a hinge into an existing section. In this case, the resulting mechanism will rotate by a unit angle in the positive direction.
The construction of the line of influence with a transverse force is possible by inserting into the cross section of the slider and pushing the beam apart by one in the positive direction.
Using the cinematic method, it is possible to construct lines of bending moment and shear force in a cantilever beam. Given the immobility of the left side in a similar beam, only movement for the right side in the positive direction is considered. Thanks to the influence lines according to the formula, any effort can be calculated.
Cinematic Calculations
In the calculations by the kinematic method, 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 equal to zero when substituting the given values, the problem can be statistically determined. If this indicator has a negative value, the task is not statistically feasible; if the degrees of freedom are positive, a geometric construction is performed.
In order to make it more convenient to carry out calculations, to have a visual representation of the features of the disks in a multi-span beam, build a floor scheme.
To do this, replace all the original hinges available in the beam with articulated-fixed supports.
Varieties of beams
Several types of multi-span beams are contemplated. The specificity of the first type is that in all spans, with the exception of the first, articulated-movable supports are used. If you use supports instead of hinges, single-span beams will be formed in which each will rest on the console next to it.
The second type is characterized by the alternation of spans, which have two pivotally movable supports, with spans without supports. In this case, the floor scheme on the center beam console is based on the insert beams.
In addition, there are also beams in which the two previous types are combined. To ensure statistical definability of the insert beams, the horizontal connection between the support is transferred to the neighboring right beam. The lower floor in the floor scheme will be represented by the main beam, and secondary beams are used for the upper floor.
Diagrams of internal power factors
Using a phased scheme, you can build plots for an individual beam starting from the top floor and ending with the lower structures. After the construction of internal force factors for the upper floor is completed, it is necessary to change all the found reaction values of the supports to opposite in the direction of the force, then attach them in the floor scheme to the lower floor. When building diagrams on it, they use a given load of forces.
After the construction of the diagrams of the force 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 reactions of the supports and the given forces is equal to 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 a change in the reaction of a support or a force internal factor in a specific (given) section of a building, the function of the location of the moving individual load is called the line of influence. To build them apply the statistics equation.
Graphic constructions are used to determine the internal force factors for calculating the reactions of supports along certain influence lines.
Value calculations
In a broad sense, structural mechanics is considered as a science that is engaged in the development of calculation methods and principles for testing structures and structures for stability, strength, as well as rigidity. Thanks to high-quality and timely strength calculations, it is possible to guarantee the safety of the erected structures, their full resistance to internal and external forces.
To achieve the desired result, a combination of profitability and durability is used.
Stability calculations 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.
Stiffness calculations consist in identifying a variety of deformation options (sediments, deflections, vibrations), due to which the full-fledged operation of structures is excluded, and the structural strength is threatened.
In order to avoid emergency situations, it is important to carry out such calculations, to analyze the compliance of the obtained indicators with the maximum permissible values.
Currently, structural mechanics uses a huge variety of reliable calculation methods, which have passed detailed tests of construction and engineering practice.
Given the constant modernization and development of the construction industry, including its theoretical base, we can talk about the use of new reliable and high-quality methods for constructing drawings.
In a narrow sense, structural mechanics is associated with the theoretical calculations of the rods, beams that form the structure. Fundamental physics, mathematics, and experimental studies are used as the basis for structural mechanics.
The design schemes that are used in structural mechanics for stone, reinforced concrete, wooden, metal structures, allow to avoid misunderstandings during the construction of buildings and structures. Only with the correct preliminary construction of drawings can we talk about the safety and reliability of the facilities being created. The construction of lines of influence in beams is a rather serious and responsible measure, because people's lives depend on the accuracy of actions.