2019

Structural Analysis

Example Series

EX09: Application of Moment Influence Line in

Continuous Beams

This document is a written version of video EX09, which can be found online at

the web addresses listed below.

Educative Technologies, LLC http://www.Lab101.Space

https://www.youtube.com/c/drstructure

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Structural Analysis – EX09

Consider a highway bridge consisting of four spans, as shown in Figure 1. We wish to determine the maximum negative moment at the interior support B due to vehicular loads.

Figure 1: A highway bridge with four continuous spans

We are going to solve the problem in three steps. First, we draw the influence line for the bending moment at support B. Second, we use the drawn influence line to determine the load pattern that causes the negative moment at B to reach its maximum value. Third, we load the beam using the pattern obtained in the previous step and analyze the structure in order to determine the maximum negative moment at the support.

Step 1: Draw the Influence Line for the Bending Moment at Support B

To draw the moment influence line for support B, we place a fictitious hinge at point B, and then we draw the deformed shape of the beam due to a positive moment placed at the hinge. If you are not familiar with this qualitative approach for drawing influence lines, please review Lectures SA16 through SA18.

Figure 2: The moment influence line for an interior support of a continuous beam

Figure 2 shows the resulting deformed shape of the beam. Segments AB bends downward since there is a counterclockwise moment applied at its right end. The same is true or segment BC - it bends downward due to the clockwise moment placed at its left end. Consequently, the beam bends upward in segment CD, and downward in segment DE.

Note that since the beam is statically indeterminate, the diagram is drawn as a curve. If the beam was statically determinate, the influence line would have consisted of line segments only.

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In segments AB, BC, and DE the diagram is drawn below the x-axis (it has negative values) because these segments are bending downwards. Therefore, any loads placed on these segments result in a negative moment to develop at support B.

Step 2: Determine the Critical Load Pattern

Generally, the analysis and design of structural systems are governed by a set of specifications prepared by professionals who are tasked with providing guidelines for ensuring the safety of built structures. In the United States, the American Association of State Highway and Transportation Officials (AASHTO) has published a set of specifications that are widely in use for the analysis and design of highway bridges.

According to the AASHTO specifications, in order to determine the maximum negative moment at an interior support in a bridge, two types of loads need to be placed on the structure: a uniformly distributed load with a magnitude of 9.5 kN/m, and a pair of concentrated loads each having a magnitude of 80 kN.

The distributed load simulates the load due to a series of trucks moving across the bridge in tandem. The concentrated loads simulate the weights of two individual trucks strategically placed on the bridge to induce the maximum negative moment at the support.

Since the influence line has negative values in segments AB, BC, and DE, in order to produce the maximum negative moment at B, we need to place the AASHTO-specified uniformly distributed load on these segments, as depicted in Figure 3.

Figure 3: The distributed load pattern resulting in the development of the maximum negative moment in an interior support of a bridge

Also, a concentrated load of 80 kN needs to be placed at either side of support B, at the point where the influence diagram has its peak value (see Figure 4).

Figure 4: The AASHTO specified concentrated loads for the development of the maximum negative moment in an interior support of a bridge

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Since the influence line is nonlinear, the locations of the concentrated loads, shown in Figure 4, are not known; they need to be determined. These distances are labeled x1 and x2 in the figure.

We are going to determine the two distances using a search technique. To determine x1, we can place a unit load at various points on segment AB and analyze the beam for each load position. Then, by comparing the moment values at B due to these loads, we can accurately approximate the position at which the influence line attains its peak value in the segment. The same process can be used to determine x2.

To facilitate the calculations involved in this analysis, we are going to use a simple application program that we have developed for analyzing continuous beams using the slope-deflection method. The application can be downloaded and used for educational purposes from: http://Lab101.Space/inotes.asp

To determine x1, let’s start by placing a unit load at the mid-point of segment AB and determine the resulting bending moment at B.

According to the slope-deflection method, as implemented in the application program, the bending moment at B due to a unit load placed 9 meters to the left of point A is: 1.81 kN-m. Three screenshots of the application showing the beam definition, the load definition, and the calculated moment are given in Appendix A (see Figures A1 through A3).

Now, let’s move the load 1 meter to the right of its current position and determine the resulting bending moment at B. When the unit load is placed 10 meters to the right of A, the bending moment at B becomes: -1.85 kN-m.

Again, increment the load position by 1 meter and calculate the resulting bending moment. When the load is located 11 meters to the right of A, the bending moment at B becomes: -1.85 kN-m.

The above values are shown in Table 1. A close examination of the data reveals that the bending moment at B reaches its maximum value when the unit load is somewhere between 10 and 11 meters to the right of point A. Note that at the mid-point of this interval, at x = 10.5, the bending moment at B is -1.86 kN-m.

1x

BM

9.00 -1.81 10.00 -1.85 10.50 -1.86 11.00 -1.85

Table 1: Bending moment values at an interior support due to a unit load placed at various

locations on the beam segment to the left of the support

Let’s assume that -1.86 is a close enough approximation for the peak moment value. Therefore, we can conclude that when a unit load is placed 10.5 meters to the right of A, the bending moment at B due to the load reaches its maximum negative value.

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We can repeat this process for segment BC in order to determine x2. If we place a unit load at the mid-point of the segment, the bending moment at B becomes -1.33 kN-m. Table 2 shows the values for the bending moment at various load positions in segment BC.

2x

BM

9.00 -1.33 8.00 -1.39 7.00 -1.42 6.00 -1.40

Table 2: Bending moment values at an interior support due to a unit load placed at various

locations on the beam segment to the right of the support

According to the pattern shown in the table above, we can conclude that the moment at B reaches its maximum negative value when the unit load is located between 6 and 8 meters to the right of B.

Although we can continue the search in order to pinpoint the load location with a higher accuracy, for our illustrative purposes, let’s take 7 as the answer. That is, if a unit load is placed 7 meters to the right of point B, the bending moment at B reaches its maximum negative value.

Figure 5 shows the two concentrated load positions at which the negative moment at B reaches its maximum value.

Figure 5: The concentrated load positions for calculating the maximum negative moment at an

interior support for a highway bridge

As depicted in Figure 6, if we combine the two load patterns, we get the critical load pattern for determining the absolute maximum negative moment at B.

Figure 6: The critical load pattern for calculating the maximum negative moment at the left

interior support of a 4-span bridge

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Step 3: Analyze the Beam

Knowing the critical load pattern, we can use the application program to analyze the beam under the combined loads in order to determine the bending moment value at B.

Figures A4 and A5 show the load definitions, and Figure A6 shows the resulting moment value at B. It is -633.18 kN-m.

In summary, per AASHTO specifications, the maximum negative bending moment at interior support B is: 633.18 kN-m.

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Appendix A

Figure A1: The definition of a continuous beam in an interactive beam analysis tool

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Figure A2: The definition of a unit concentrated load in an interactive beam analysis tool

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Figure A3: An internal bending moment due to a unit load shown in interactive beam analysis tool

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Figure A4: The definition of two concentrated loads for causing the maximum negative bending moment at an interior support shown in interactive beam analysis tool

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Figure A5: The definition of the uniformly distributed loads for causing the maximum negative bending moment at an interior support shown in interactive beam analysis tool

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Figure A6: Maximum negative internal moment at an interior support due to the AASHTO-specified loads shown in interactive beam analysis tool