Calculation of beams on the deflection. Maximum deflection of the beam: formula

The beam is an element in engineering, which is a rod that loads forces acting in a direction perpendicular to the rod. The activity of engineers often involves the need to calculate the deflection of the beam under load. This action is performed in order to limit the maximum deflection of the beam.

Types

Today, beams made of different materials can be used in construction. It can be metal or wood. Each case involves different beams. In this case, the calculation of beams on the deflection may have some differences that arise on the basis of the difference in structure and materials used.

Wooden beams

Today's individual construction implies a wide application of beams made of wood. Almost every structure contains wooden floors. Beams made of wood can be used as bearing elements, they are used in the manufacture of floors, and also as supports for floors between floors.

It's no secret that the wooden, as well as the steel beam, has the property of flexing under the influence of load forces. The deflection arrow depends on which material is used, the geometric characteristics of the structure in which the beam is used, and the nature of the loads.

Admissible deflection of the beam is formed from two factors:

• Correspondence of deflection and admissible values.
• Possibility to operate the building taking into account the deflection.

The strength and rigidity calculations carried out during the construction make it possible to estimate as efficiently as possible what loads the building can withstand during operation. Also, these calculations allow you to find out exactly what will be the deformation of the structural elements in each particular case. Perhaps no one will argue with the fact that detailed and most accurate calculations are part of the responsibilities of civil engineers, but using several formulas and the skill of mathematical calculations, you can calculate all the necessary quantities yourself.

In order to make the correct calculation of the deflection of the beam, one must also take into account the fact that in the construction of the concept of rigidity and strength are inseparable. Based on the strength calculation data, it is possible to proceed with further calculations regarding rigidity. It is worth noting that the calculation of the deflection of the beam is one of the indispensable elements in calculating the rigidity.

Pay attention to the fact that for carrying out such calculations, it is best to use large-scale calculations yourself, using simple enough schemes. In doing so, it is also recommended to make a small margin in the larger side. Especially if the calculation concerns the bearing elements.

Calculation of beams on the deflection. Work algorithm

In fact, the algorithm by which such a calculation is done is simple enough. As an example, consider a somewhat simplified scheme for calculating, while omitting some specific terms and formulas. In order to calculate the beams for deflection, it is necessary to perform a number of actions in a certain order. The calculation algorithm is as follows:

• A calculation scheme is compiled.
• The geometrical characteristics of the beam are determined.
• The maximum load for this element is calculated.
• In case of necessity, the strength of the beam is checked on the bending moment.
• The maximum deflection is calculated.

As you can see, all the actions are quite simple and quite feasible.

Drawing up the design scheme of the beam

In order to make a calculation scheme, it does not require much knowledge. For this it is sufficient to know the size and shape of the cross-section of the element, the span between the supports and the method of support. The span is the distance between two supports. For example, you use beams as support beams overlapping for bearing walls of the house, between which 4 m, then the span will be equal to 4 m.

Calculating the deflection of a wooden beam, they are considered to be freely supported elements of the structure. In the case of an overlap beam , a circuit with a load that is evenly distributed is adopted for calculation. It is denoted by the symbol q. If, however, the load is of a concentrated nature, then a circuit with a concentrated load, designated F, is taken. The magnitude of this load is equal to the weight that will put pressure on the structure.

Moment of inertia

The geometric characteristic, which was called the moment of inertia, is important in calculations for the deflection of a beam. The formula allows us to calculate this value, we will give it a little lower.

When calculating the moment of inertia, one must pay attention to the fact that the size of this characteristic depends on the orientation of the element in space. In this case, an inverse relationship is observed between the moment of inertia and the magnitude of deflection. The smaller the moment of inertia, the greater the deflection value and vice versa. This dependence can be easily traced in practice. Each person knows that the board laid on the edge, bends much less than a similar board, which is in a normal position.

The calculation of the moment of inertia for a beam with a rectangular section is made by the formula:

J = b * h ^ 3/12, where:

B is the width of the section;

H is the cross-sectional area of the beam.

Calculation of the maximum load level

Determination of the maximum load on the structural element is made taking into account a number of factors and indicators. Usually, when calculating the load level, take into account the weight of 1 running meter of the beam, the weight of 1 square meter of overlap, the load on the temporary overlap and the load from the partitions per 1 square meter of overlap. The distance between the beams, measured in meters, is also taken into account. For an example of calculating the maximum load on a wooden beam, we take the averaged values, according to which the overlap weight is 60 kg / m², the temporary load on the overlap is 250 kg / m², the partitions will weigh 75 kg / m². The weight of the beam itself is very simple to calculate, knowing its volume and density. Suppose that a wooden beam with a cross-section of 0.15x0.2 m is used. In this case, its weight will be 18 kg / m. Also, for example, we take the distance between the beams of the overlap equal to 600 mm. In this case, the coefficient we need is 0.6.

As a result of calculating the maximum load, we obtain the following result: q = (60 + 250 + 75) * 0.6 + 18 = 249 kg / m.

When the value is obtained, you can proceed to calculate the maximum deflection.

Calculation of the maximum deflection value

When the beam is calculated, the formula displays all the necessary elements. It should be noted that the formula used for calculations can have a slightly different form if the calculation is carried out for different types of loads that will affect the beam.

First we bring to your attention the formula used to calculate the maximum deflection of a wooden beam with distributed load.

F = -5 * q * l ^ 4/384 * E * J.

Note that in this formula, E is a constant value, which is called the modulus of elasticity of the material. For wood this value is equal to 100 000 kgf / m².

Continuing the calculations with our data used for the example, we obtain that for a beam made of wood with a cross-section of 0.15 x 0.2 m and a length of 4 m, the maximum deflection under the action of a distributed load is 0.83 cm.

We draw attention to the fact that when the deflection is calculated taking into account the scheme with concentrated load, the formula takes the following form:

F = -F * l ^ 3/48 * E * J, where:

F is the force of pressure on the bar.

Also, we draw attention to the fact that the value of the modulus of elasticity used in the calculations may differ for different types of wood. The influence is exerted not only by the species of the tree, but also by the shape of the timber. Therefore, a single beam of wood, glued beam or rounded log will have different elastic moduli, and hence different values of maximum deflection.

You can pursue different goals by calculating the beams to the deflection. If you want to know the limits of deformation of the structural elements, then after the calculation of the deflection arrows you can stop. If your goal is to establish the level of compliance of the found indicators with construction norms, then they should be compared with data that are placed in special normative documents.

I-beam

Note that I-beam beams are used somewhat less often because of their shape. However, it should also be remembered that such an element of construction can withstand much greater loads than a corner or channel, an alternative to which can be an I-beam.

Calculation of the deflection of an I-beam is worthwhile if you are going to use it as a powerful structural element.

Also, we draw your attention to the fact that it is not possible to calculate the deflection for all types of I-beam beams. In which cases is it possible to calculate the deflection of an I-beam? There are 6 such cases, which correspond to six types of I-beams. These types are:

• A single-span beam with a uniformly distributed load.
• Console with rigid sealing at one end and evenly distributed load.
• A beam from one span with a console on one side, to which a uniformly distributed load is applied.
• Single-span beam with hinged support type with concentrated force.
• One-span hingedly supported beam with two concentrated forces.
• Console with a hard seal and concentrated force.

Metal beams

Calculation of the maximum deflection is the same, be it a steel beam or an element of another material. The main thing is to remember those quantities that are specific and constant, such as the modulus of elasticity of the material. When working with metal beams, it is important to remember that they can be made of steel or from I-beams. The deflection of a metal beam made of steel is calculated taking into account that the constant E in this case is 2 · 105Mpa. All other elements, like the moment of inertia, are calculated by the algorithms described above.

Calculation of the maximum deflection for a beam with two supports

As an example, consider the scheme in which the beam is on two supports, and concentrated force is applied to it at an arbitrary point. Until the moment when the force was applied, the beam was a straight line, but under the influence of force it changed its appearance and became a curve due to deformation.

Suppose that the XY plane is the plane of symmetry of the beam on two supports. All loads act on the beam in this plane. In this case, the fact is that the curve obtained as a result of the action of force will also be in this plane. This curve is called the elastic line of the beam or the deflection line of the beam. Algebraically solve the elastic line of the beam and calculate the deflection of the beam, the formula of which will be constant for beams with two supports, can be as follows.

Deflection at a distance z from the left support of the beam with 0 ≤ z ≤ a

F (z) = (P * a ² * b ² ) / (6E * J * l) * (2 * z / a + z / bz ³ / a ² * b)

Deflection of the beam on two supports at a distance z from the left support for a ≤ z ≤l

F (z) = (-P * a ² * b ² ) / (6E * J * l) * (2 * (lz) / b + (lz) / a- (lz) ³ / a + b ² ), where P is the applied force, E is the modulus of elasticity of the material, J is the axial moment of inertia.

In the case of a beam with two supports, the moment of inertia is calculated as follows:

J = b ₁ h ₁ 3/12, where b ₁ and h ₁ are the values of the width and height of the section of the beam used, respectively.

Conclusion

In conclusion, we can conclude that self-calculation of the maximum deflection of beams of different types is fairly simple. As it was shown in this article, the main thing is to know some characteristics that depend on the material and its geometric characteristics, and also carry out calculations on several formulas in which each parameter has its own explanation and is not taken from nowhere.