The collection of loads is one of the first stages of the design process. Since loads are an external concepts, logically they should not depend on either the design scheme or the chosen method of analysis. They are purely input data for calculations. However, there are situations where the load magnitude is closely related to a specific calculation methodology. This refers to the design of infrastructural objects.

Hello! You are on the engineering resource Dystlab, and I'm Vitalii. Today, we will consider how to determine the forces in the section of a beam bridge using influence lines.

Railway beam bridges

Beam bridges are the most common structures for organizing transportation routes. The span structures on such bridges are usually simple standalone beams. They are manufactured industrially and installed on supports at the construction site. Assembly is carried out using cranes or special engineering equipment.

Due to their small length and relatively low weight, such structures are used both for covering the main (riverbed) spans of the bridge and for approaches.

Live moveable loads by trains

Currently, we are interested in how beam bridge structures work under load and how they determine factors of stress-strain state in similar systems.

In reality, railroad rolling stock (wagons, locomotives) creates concentrated pressure on the rail track. However, thanks to the bridge deck, this pressure is distributed and applied to the main supporting elements of the bridge structure in a more or less distributed manner. And although the calculation of the bridge under the action of concentrated or short-distributed loads is possible, it is not always feasible due to the variety of available transport. Plus, railway parks are constantly replenished with new, more modern units.

Therefore, the engineering methodology outlined in national bridge design standards operates with a single, so-called equivalent load. From a physics point of view, an equivalent load is a regular uniformly distributed pressure. But the advantage of this approach is that the designer does not need to think about which particular transport to choose for calculations. It should always be a conditional equivalent load.

We will be guided by Ukrainian bridge design standards, namely Appendix B DBN V.1.2-15: 2009 "Transport Structures. Bridges and Culverts. Loads and Actions" (PDF). This methodology has hardly changed compared to previous standards — DBN V.2.3-14-2006 or its Soviet predecessor — SNiP 2.05.03-84 "Bridges and Culverts". The only thing that has changed is the adjective "normative" to "characteristic" to provide closer integration with European building standards Eurocode.

About influence lines

In one of our previous discussions, we covered influence lines as one of the methods for determining shear forces and bending moments. In bridge engineering, this approach is mandatory since the temporary load from moving vehicles is attached to the parameters of the influence line.

This may seem a bit unfamiliar, especially if you have not designed infrastructure objects before. Indeed, the calculation of a wooden beam in a cottage or a column of a metal hangar does not limit us in calculation methods. Loads on these elements are assumed to be either concentrated or distributed, based on design standards or relevant material properties. We can then manipulate these loads as we see fit — applying them to the traditional "diagram" model or loading the influence lines. In other words, here the load does not depend on the configuration of the calculation scheme or methods of analysis.

However, with bridges, this is not the case.

According to building codes, live loads are directly dependent on the influence line. Therefore, the two stages — design of the calculation scheme and determination of the load — in bridge design are interrelated. However, do not let that intimidate you. I have been involved in the design of various structures and can attest from my own experience that the influence line method is practically applicable to all types of structural systems. Once you have mastered it, you will be able to significantly expedite your work.

The length of the loading influence line

Here is what the table for determining the intensity of equivalent load \( \nu \) looks like according to Ukrainian bridge design standards:

The first column represents the length of the loading influence line \( \lambda \). This is the length of the segment on which the live load acts. In the case of a reaction or a bending moment, it is essentially the calculated length of the span structure, while in the case of a shear force at any section (but not at the support) — it is the longest length of the segment from the support to that section:

Position of the peak

The parameter \( \alpha \) determines the position of the highest ordinate of the influence line. It is dimensionless and its value ranges from 0 to 0.5. To determine it, the following steps should be taken:

1. Find the distance from the nearest left support to the section (distance \( a \))
2. Find the distance from the nearest right support to the section (distance \( b \))
3. Select the smaller of these two distances
4. Divide this minimum distance by the length of the loading influence line \( \lambda \)

As we can see, for bending moment at any section, the parameter \( \alpha \) is between 0 and 0.5. The parameter \( \alpha = 0.5 \) only relevant for the center of the span, where \( a = b = \dfrac{l}{2} \). For the shear force at any section, including at the support, \( \alpha = 0 \), since the loaded segment of the influence line has the shape of a rectangular triangle.

Live load class

The load intensity is given in the table for two cases: K=1 and K=14, where K is the live load class that is applied to a particular section of the railway. In general, major bridge structures must be designed for load class K=14.

Calculation examples

Example 1

Problem: determine the bending moment in \( \frac{1}{4} \) of the span. The span (design length) of the beam is 8.8 meters. The class of live load is K=14.

Parameters of the influence line for bending moment: \( \lambda = l = 8.8 \) m, \( \alpha = 0.25 \). We determine the intensity of the vertical equivalent load \( \nu \) by interpolation:

1. Interpolate "vertically" using the parameter \( \lambda \)
2. The intensity of load using \( \alpha = 0 \): \( \frac{ \nu_1 - 256.4 }{ 250.2 - 256.4 } = \frac{ 8.8 - 8.0 }{ 9.0 - 8.0 } \)
3. Obtained value \( \nu_1 = 251.44 \) kN/m
4. The intensity of load using \( \alpha = 0.5 \): \( \frac{ \nu_2 - 224.4 }{ 218.9 - 224.4 } = \frac{ 8.8 - 8.0 }{ 9.0 - 8.0 } \)
5. Obtained value \( \nu_2 = 220.00 \) kN/m
6. Interpolate "horizontally" using the parameter \( \alpha \): \( \frac{ \nu - 251.44 }{ 220.00 - 251.44 } = \frac{ 0.25 - 0 }{ 0.5 - 0 } \)
7. Obtained value \( \nu = 235.72 \) kN/m

Alternatively, linear interpolation can be performed using the free Dystlab application (see link below):

The area of the influence line for the bending moment is equal to the area of the corresponding triangle:

\( \omega = \dfrac{1}{2} y l = \dfrac{1}{2} \dfrac{a b}{l} l = \dfrac{0.25 l \cdot 0.75 l}{2} = 7.26 \) m².

Determine the bending moment by loading the entire area of the influence line with the equivalent load:

\( M = \nu \times \omega = 235.72 \times 7.26 = 1711.33 \) kN m.

Example 2

Problem: determine the reaction (maximum shear force) for the same bridge span structure.

Parameters of the influence line: \( \lambda = l = 8.8 \) m, \( \alpha = 0 \). Determine the intensity of the equivalent vertical load \( \nu \) using the same method as in Example 1:

Calculate the area of the influence line for the reaction:

\( \omega = \dfrac{l}{2} = \dfrac{8.8}{2} = 4.4 \) m.

Determine the reaction by loading the influence line with the equivalent load:

\( Q = \nu \times \omega = 251.44 \times 4.4 = 1106.34 \) kN.

Notes on the examples

1. In the above examples, we are working with a characteristic (unfactored, called "normative") load values. To obtain the design factored loads, you need to take into account the relevant coefficients (reliability factor, dynamic coefficient, etc.).
2. In the above examples, we are talking about the total live load. To determine the portion of the load that applies on a single supporting element, the partial factor must be taken into account (if the span structure has two main girders, the factor is 0.5).

The digital solution for bridge design from Dystlab

Alright, that was the theory.

But if you're a practitioner and need ready-made solutions, I have great news for you: Dystlab has developed a dedicated application for determining forces and moments in the railway beam bridge:

Ukraine

0045. [DBN] Application for determining shear forces and bending moments in railway bridge structures (Ukraine)

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0044. Linear cross-interpolation. Calculator

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This calculator is built in TechEditor and works within the same environment. To determine the shear force and bending moment, you only need to enter three parameters:

• K — the class of the live railway load;
• L — the design length of the beam (span);
• x — the coordinate of the section at which you obtain the force or moment (within 0 to L).

Note: as in other Dystlab developments, you can use any unit of measurement, such as mm, cm, in, ft, etc.

This digital solution is available for download on the Dystlab Store, an online hub for engineers. If you're interested in similar or other technological solutions, please let us know. We would be happy to develop the necessary calculators, help automate calculations, and accelerate your company's workflow.

Additionally, I would like to remind you that TechEditor Pro enables you to create your own applications.

Best of luck!

Vitalii Artomov

"I am working to make «Made in Ukraine» a global symbol of quality and style"

CEO, co-founder of Dystlab, developer of TechEditor. Engineer, scientist, Ph.D. with over 20 years of experience in structural analysis and automation of engineering calculations. I advise engineering companies in Ukraine, Europe, and North America.

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