danago

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**A simply supported footbridge is to span a total of 7 meters. The load on the beam is a distributed force of 5kN/m plus its weight. Taking into account the beams weight, what dimensions should be selected for the beam?**

Ive been given a table of beam dimensions along with their values for elastic modulus, second moment, measurements of mass per length etc.

I started by coming up with a the bending moment as a function of x, the distance from the end of the beam:

[tex]

M(x) = \left\{ {\begin{array}{*{20}c}

{(17.5 + 0.5W)x - 2.5x^2 } & {x \in [0,3.5)} \\

{3.5W + (17.5 - 0.5W)x - 2.5x^2 } & {x \in [3.5,7]} \\

\end{array}} \right.

[/tex]

Where W is the wight of the beam.

I then found that the maximum bending moment occurs at x=3.5, so:

[tex]

M_{\max } = 30.625 + 1.75W

\]

Given this, i want to choose a beam with dimensions such that the maximum stress in the beam does not exceed the yield strength, where the maximum stress is given by:

[tex]

\sigma _{\max } = \frac{{M_{\max } y_{\max } }}{I} = \frac{{M_{\max } }}{z}

[/tex]

Where z is the elastic section modulus. In this case, i will assume the maximum yield stress to be 60% of the actual yield stress, as in my mechanics class we are generally told to use a safety factor of 0.6.

[tex]

\sigma _{\max } = 150MPa = 150 \times 10^6 Pa

[/tex]

With these restrictions in place, i should find an elastic section modulus such that:

[tex]

z \ge \frac{{M_{\max } }}{{\sigma _{\max } }} = \frac{{30625 + 1750W}}{{150 \times 10^6 }}

[/tex]

Now my table gives values for z in the units mm^3, so i converted it to mm^3 by multiplying by 1000^3.

[tex]

z \ge \frac{{612500 + 35000W}}{3}

[/tex]

Now the table gives different values for the beams mass per unit length (kg/m), so i can calculate the weight of different beams and then put it into the equation above, but everytime i do so, i get answers that seem way too high.

Does anyone have any input? Im not even sure if ive approached the question in the correct way. I think whats causing me problems is the fact that the weight isnt neglected like most other problems i need to solve.

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