How to draw lines of influence? Structural 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 insignificant displacements is equal to zero.
Method specificity
To build the lines of influence of the reaction, bending moment, shear force for a given beam section, a certain algorithm of actions is used. First, remove the link. In addition, the lines of influence of the internal force are removed, and the necessary force is introduced. As a result of such manipulations, the given system will be a mechanism with one degree of freedom. In the direction where the internal force is considered, a small displacement is introduced. Its direction should be similar to the internal effort, only in this case positive work will be done.
Examples of constructions
Based on the principle of displacements, the equilibrium equation is written down, when solving it, the lines of influence are calculated, and the required effort is determined.
Let's consider an example of such calculations. We build lines of influence of the transverse force in some section A. Toto 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
Construction of lines of influence is carried out using a special formula. It connects the desired 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 indicator of the bending moment and the 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 moving unit force, they have a positive value.
Bending moment will be positive when its direction is clockwise. The tangent will be positive when the angle of rotation is less than a right angle. When carrying out calculations, the value of the ordinates and the area of the line of influence are used with their signs. Structural mechanics is based on the statistical method of constructing diagrams.
Definitions
Let's 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 displacement of a unit moving force.
The ordinates demonstrate the change in the analyzed internal force that appears at a certain point on the beam when moving along the length of a unit force. They show the change at different points in the internalforce under the condition of using an external fixed load. The statistical version of the construction is based on writing the equilibrium equations.
Two construction options
Construction of influence lines in beams and bending moment is possible in two cases. The force can be located to the right or left of the section used. When the forces are located to the left of the section, during the calculations, the forces that will act to the right are selected. With her right action, they count according to the left forces.
Multi-span beams
In bridges, for example, auxiliary beams are used to transfer the external load to the load-bearing part of the entire building structure. The main beam is called the one that is the supporting base. Transverse beams are considered to be located at a right angle to the main one.
Auxiliary (single-span) beams are called, to which an external load is applied. This option of transferring the load to the main beam is considered nodal. The panel is considered to be the area located between the two nearest nodes. And they are represented as points of the main axis, to which the cross beams fit.
Features
What is the Line of Influence? The definition of this term in a beam is associated with a graph that shows the change in the analyzed factor when a unit force moves along the beam. It can be a transverse force, a bending moment, a support reaction. Any ordinate of the lines of influence demonstrates the sizeanalyzed factor at the time when the force is located above it. How to draw lines of influence of a beam? The statistical method is based on the compilation of statistics 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 at which it functions, you can build lines of influence of the reaction, draw up an equation of moments, build a two-point graph.
Next, a shear force action curve is constructed, for this, the ordinates of the influence lines in the section are used.
Cinematic way
Maybe a line of influence can be built based on the movements. Examples of such graphs can be found in cases where a beam is shown 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 cut a hinge into the existing section. In this case, the resulting mechanism will rotate by a unit angle in the positive direction.
Building a line of influence with a transverse force is possible when inserting into the section of the slider and expanding the beam by one in the positive direction.
You can use the cinematic method to plot bending moment and shear force lines in a cantilever beam. Taking into account the immobility of the left side in such a beam, only the movement for the right side in the positive direction is considered. Thanks to the lines of influence, the formula can calculate any effort.
Calculationsin cinematic mode
When calculating by the kinematic method, a formula is used that relates the number of support rods, the number of spans, hinges, degrees of freedom of the task. If, when substituting the given values, the number of degrees of freedom is equal to zero, the problem can be determined statistically. If this indicator has a negative value, the task is statistically impossible, with a positive degree of freedom, 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 operation of disks in a multi-span beam, a floor diagram is built.
To do this, all the original hinges in the beam are replaced with hinged supports.
Varieties of beams
Several types of multi-span beams are assumed. The specificity of the first type is that in all spans, with the exception of the first, hinged-movable supports are used. If supports are used instead of hinges, single-span beams will be formed, in which each will rest on the adjacent console.
The second type is characterized by the alternation of spans, which have two articulated movable supports, with spans without supports. In this case, the floor plan on the console of the central beams is based on the insert beams.
Besides, there are beams that combine the two previous types. To ensure the statistical definability of the insert beams, the horizontal connection between the support is transferred to the right adjacent beam. Lower floor in floor-by-floorthe scheme will be represented by the main beam, and the secondary beams are used for the upper floor.
Plots of internal force factors
With the help of a step-by-step scheme, you can plot for a single beam starting from the top floor and ending with the bottom constructions. After the construction of internal force factors for the upper floor is completed, it is necessary to change all the found values of the support reaction to forces opposite in direction, then apply them in the floor diagram to the lower floor. When plotting diagrams on it, a given load of forces is used.
After completion of the plotting of force internal factors, a statistical check of the complete multi-span beam is carried out. When checking, the condition must be met, according to which the sum of all reactions of supports and given forces is equal to zero. It is also important to analyze the compliance with the differential dependence for individual sections of the used beam.
In a graph that expresses the law of change in the reaction of a support or a force internal factor in a specific (given) section of a building, the functions of the location of a moving individual load are called the line of influence. To build them apply the equation of statistics.
Graphic constructions are used to determine the force internal factors for calculating the reactions of supports along certain lines of influence.
Calculation value
In a broad sense, building mechanics is considered as a science that develops methods of calculation and principles of verificationstructures and structures for stability, strength, and 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 economy and durability is used.
Stability calculations make it possible to identify critical indicators of external influences that guarantee the preservation of a given form of balance and position in a deformed state.
Rigidity calculations are to identify a variety of deformation options (settlement, deflection, vibration), due to which the full operation of structures is excluded, there is a threat to the strength of structures.
In order to avoid emergencies, it is important to carry out such calculations, analyze the compliance of the obtained indicators with the maximum permissible values.
Currently, structural mechanics uses a huge variety of reliable calculation methods that have been tested in detail by building and engineering practices.
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, building mechanics is associated with theoretical calculations of rods, beams that form a structure. Fundamental physics, mathematics, and experimental studies serve as the basis for structural mechanics.
Design schemes that are used in structural mechanics for stone, reinforced concrete, wood, metal structures, allow you 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 structures being created. Building lines of influence in beams is a rather serious and responsible undertaking, because people's lives depend on the accuracy of actions.
