UN NUEVO INFORME DE FÍSICA ELÉCTRICA

Application of Differential Equations to Deflection in Beams.

Application Related to Civil Engineering.

Juan David Pachón Corredor1
Codigo: 40161032

1Grupo 2. Ingenieria Civil, Universidad De La Salle.

[pic 1]

1. Problem statement

• The beam of the figure 1, is an IPE-160 and is subjected to the indicated concentrated load of 30kN, E=calculate by the method of the differential equation of the elastic line:[pic 2]

A. The equation of the elastic line of the beam.
B.  Turns of sections A and B.
C.  Maximum Arrow.

1.  Approach

• For subsection A, The test specimen specifies that the differential equation of the elastic line must be used, to carry out the problem approach, we have as basis the following differential equation, also called the elastic curve equation:

[pic 3]

Where:

• v(x): Represents the arrow, ordinate (y-axis) or vertical displacement, with respect to the position without loads.
• x: Is the abscissa (X axis) on the beam.
• Is the bending moment on the abscissa.[pic 4]
•  Is the second moment of area or moment of inertia of the cross section.[pic 5]
• E: Is the modulus of elasticity of the material.

• For subsection B, it is required a calculation of moments, or turns of a force, for that we use the equation of the curve or elastic line of the previous section, with the help of the derivative and the values ​​found in the previous section. Thus, the following basic solution equations are obtained based on Figure 1, and the results of subsection A:

• [pic 6][pic 7]

The above equations are better understood by the posterior diaphragm and will allow us to calculate the turns in sections a and b of the figure.

• For subsection C, the maximum arrow in a beam is nothing other than the maximum defect that it can withstand, using the turns or moments and the equation of its elastic curve in the anterior incisors, we can calculate it using all the values ​​found. This in terms of coordinates.

1. Analysis

The graph of the situation presented would be the following:[pic 8]

[pic 9]
Figure 1. Graph of the situation.

• For subsection A, following basic principles of static the first thing we must do for incniso A is to calculate the reactions in the system depending on the type of support, so that the system is in equilibrium, because that is the necessary condition to proceed to apply the equation of the elastic curve with the corridas, the force and the modulus of elasticity of the beam.
• For section B, with the equilibrium system rections and the elastic curve equations, we will proceed to calculate the turns or moments by a system of equations tenidneo in account the forces, and the distances present in the system.
• Section C is the one that most interests us because based on all the procedures applied in the previous sections with the help of differential equations, we will obtain the point where the maximum or maximum defexion in the beam would occur due to the load and reactions that it supports.

1. Solution [pic 10]

1. Calculations of reactions, equilibrium equations:

[pic 11]

Equation of the elastic curve  : [pic 12]

[pic 13][pic 14][pic 15][pic 16][pic 17]

• The boxes in red correspond to the equations of elastic curve found.

1. With the above operations, the turns in a and b are obtained, thus:[pic 18]

The above calculation corresponds to the turns requested by the problem.

1. This is the most important, because here lies the true application of the differential equations in beams, using the above data, we get:

[pic 19]The previous calculation corresponds to the calculation of the maximum arrow of the beam studied.[pic 20][pic 21]

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