# Bending moment of countinuous tapered beam

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1. Jul 20, 2015

### Daumantas

Hello,

I cant find the formula for bending moment calculation of continuous tapered beam. the cross section is I (H) form, where web is 179.5 cm flange is 67 cm, web thickness is 1.5 cm and flange thickness is 2.5 cm. And this beam changes its height to 89.7 cm all other dimensions is the same. Beam is 60 m long and has two spans and three supports at 0, 30 and 60 meters. The load is distributed which intensity is 89.329 kN/m. The beam mass is not taken in to acuont. I have designet and tested this beam with two programs and the bending moment is the same. Examples is atached

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• ###### 3Momentas kintamo aukscio sijos2 su robot.pdf
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2. Jul 20, 2015

### SteamKing

Staff Emeritus
The bending moment of a beam depends only on the end conditions, the support conditions, and the loading.

If you have a continuous beam (one with more than two supports), then the shape of the beam will affect the calculation of the fixed-end moments for each span.

With tapered beams, I don't know of any simple calculations/formulas you can apply. It looks like you'll have to revert to first principles to analyze this beam.

If your previous calculations of the reactions and moments produce static equilibrium, I would say that the beam bending moments you have calculated are also correct, but I couldn't say for certain unless you provide these results (reactions and moments at each support).

3. Jul 20, 2015

### Daumantas

I have one idea to simplify this beam to one span where one support is fixed in movement and rotation and other support is free to move in x direction and rotate. But i dont understand how to involve this tapper affect? Because when i am changing the bigger's cross section size the the bending moment changes and when i change the tapper distance in longitudinal direction it also changes. I think it should some how involve second moment of inertia. But i dont even imagine how this equation should look like...

I am attaching the picture with reaction forces and supports types. There is three supports, in the middle suppor is free to rotate and movement in x and z directions is fixed, side supports is free to rotate and move to x directions but fixed to move in z direction. Reactions: in the middle support it's 3504.02 kN and moment at this support is 12364.05 kN. On both side supports reactions is 927.74 kN and bending moment is zero.

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• ###### reactions and bending moment.png
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4. Jul 21, 2015

### SteamKing

Staff Emeritus
Your original beam's reactions and moments are OK.

The problem with a tapered beam is that there are really no simple formulas with which to do a calculation of the reactions and bending moments for a beam which is supported in a statically indeterminate manner, like this beam on three supports.

Tables of prismatic (non-tapered) beams on 3 or more supports can be prepared for certain loading conditions, but even these are calculated numerically, and the reactions and bending moments are given as factors of the UDL loading, for example.

For an example of this, look at p. 7 of the tables below. This page contains formulas for three-span and four-span continuous beams under various loadings:

http://faculty.arch.tamu.edu/media/cms_page_media/4198/NS8-2beamdiagrams.pdf

Because continuous beams are solved by ensuring the compatibility of displacements between the segments of the beam, and by ensuring that the rotation of the beam is the same on either side of the intermediate supports, the factor EI for the beam is involved in making these conditions occur, due to its influence in determining the slope and deflection of a beam.

If you change EI by either eliminating or extending the taper, the reactions and bending moments will necessarily change, because you have made the beam relatively stiffer or more flexible, depending on how the taper is changed. If you want to find out how the reactions and moments change, then you will have to re-analyze the beam from scratch, and compare the results for each design.