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Applying Knovel Math to Real World Problems

Sasha Gurke October 5, 2009

Knovel Math provides fully documented Mathcad worksheets for engineering calculations from trusted reference works, reducing the time it takes to find and solve equations, and document calculations.

In this solution story, we’ll walk through a step-by-step use case showing you how Knovel Math can help design a stage platform.

For more information about Knovel Math, or to view a short video, click here.

Designing a metal platform for a stage

A theatre requires a light metal platform in order to support stage lights and some scenery equipment.

The platform must accommodate up to 2 stage hands. The weight of the lights and scenery equipment may reach 2600 lb and could be assumed to be distributed evenly along the platform. To achieve this, the structure should be 20 ft long and 2.6 ft wide and suspended by 3 cables on each side attached 10 ft apart. A stage engineer must now determine the dimensions of the structural components and the load on the cables.

Modeling The Platform

The engineer envisions a platform made of 2 parallel long beams tied by short cross-beams with a light deck. The deck should be attached in a way that prevents it from being subjected to any bending load born by the beams. The platform will be made from the aluminum alloy 6063-T6 which has a density of 167.6 lb/ft3 and a design tensile yield strength of 25 ksi.
The weight of the lights, scenery equipment, stage hands and the structure is assumed to be distributed evenly between 2 parallel beams with a coefficient of 1.15 introduced to account for the possible unevenness in load distribution. As a result, each of the 2 parallel beams could be calculated as one continuous beam in bending. Moreover, by neglecting elongation of the supporting cables, the beam could be calculated as simply supported.

The maximum bending moment could be near the middle of the span or at the middle of the beam at the point where the cable is attached, if the weight of 2 people is applied to one of the beams near to the middle of the span.

The preliminary calculations have shown that this structure would have an approximate weight of 400 lb, and the cross section of a long beam would be 2.0 in2.

The solution can be reduced to that of a 3-span continuous beam with an evenly distributed load of:equation1

and a concentrated load of W2 in one of the spans. Assuming the weight of a stage hand to be 260 lb, the concentrated load can be calculated as:equation2Consequently, the load diagram of the beam could be drawn as:

equation3

Where w1 – distributed load
______W2 – concentrated load
______l1=l2=10 ft – spans
______a1 – position of the concentrated load

This beam is a multi-span continuous beam that is statically undetermined. The most commonly used solution for these beams utilizes the Three Moment Equation method. The load diagram to be used for the solution is:

image4

Running a basic search on Knovel, the engineer searches using the keywords “three moment equation” (click on the image to run the search)

image5

and retrieves Roark’s Formulas for Stress and Strain (7th Edition), a comprehensive handbook containing a Mathcad-enabled worksheet for Three-Moment Equation in Example 8.3.1 (Nondimesional).

image6

Although Example 8.3.1 differs from the specific problem the engineer wants to solve, he can use it as a basis for his solution. To account for a different loading, the engineer has to modify the equations 1 and 2 in the Example 8.3.1 and use cases 1e, 2e and 3e from Table 8.1 (Also Mathcad-enabled on Knovel.)

The engineer can use these cases to calculate the slopes (to be used in the equations 1 and 2) at the right and left ends of the beam for all span loads. The series of calculations, schematically outlined below, are done in Mathcad by copying the equations from Knovel Math files for cases 1e and 2e (for span a).image7

and for the case 2e (for span b):

image8
Click on the images to zoom in to the details.

The engineer’s goal is to calculate the bending moment at the middle support (M2) as a function of the position of the concentrated load W2:

image9
Once the bending moment is calculated, the engineer can calculate the reactions at the supports by superposing the cases 1e, 2e, and 3e:

image10

Then the engineer can derive the equations for the shear force and the bending moment of the whole beam as a function of their distribution along the beam (x) and the position of the concentrated load (a):

image11

From the calculated maximum bending moment the engineer selects a suitable shape, using Knovel Math file for the case 3 from the Table A.1 of Roark’s to calculate section properties:

image12

The bending moment of parallel beams has a maximum value (1.667 kip ft) when both stage hands are standing on the same side of the platform about 4.05 ft from its end.

To remain stable at this moment, the platform was built with long beams made from an aluminum alloy 6063-T6 channel with standard dimensions RT 1¾×4½×1/8. The maximum load on the middle cable is 1.579 kip when both stage hands are standing on the same side in the middle of the platform.

To see a 7-page report with the complete Mathcad solution click here.

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