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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:
<br /> M(x) = \left\{ {\begin{array}{*{20}c}<br /> {(17.5 + 0.5W)x - 2.5x^2 } & {x \in [0,3.5)} \\<br /> {3.5W + (17.5 - 0.5W)x - 2.5x^2 } & {x \in [3.5,7]} \\<br /> \end{array}} \right.<br />
Where W is the wight of the beam.
I then found that the maximum bending moment occurs at x=3.5, so:
<br /> M_{\max } = 30.625 + 1.75W<br /> \]<br /> <br /> 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:<br /> <br /> <br /> \sigma _{\max } = \frac{{M_{\max } y_{\max } }}{I} = \frac{{M_{\max } }}{z}<br /><br /> <br /> 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. <br /> <br /> <br /> \sigma _{\max } = 150MPa = 150 \times 10^6 Pa<br /><br /> <br /> With these restrictions in place, i should find an elastic section modulus such that:<br /> <br /> <br /> z \ge \frac{{M_{\max } }}{{\sigma _{\max } }} = \frac{{30625 + 1750W}}{{150 \times 10^6 }}<br /><br /> <br /> Now my table gives values for z in the units mm^3, so i converted it to mm^3 by multiplying by 1000^3.<br /> <br /> <br /> z \ge \frac{{612500 + 35000W}}{3}<br /><br /> <br /> 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. <br /> <br /> Does anyone have any input? I am not even sure if I've approached the question in the correct way. I think what's causing me problems is the fact that the weight isn't neglected like most other problems i need to solve.
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:
<br /> M(x) = \left\{ {\begin{array}{*{20}c}<br /> {(17.5 + 0.5W)x - 2.5x^2 } & {x \in [0,3.5)} \\<br /> {3.5W + (17.5 - 0.5W)x - 2.5x^2 } & {x \in [3.5,7]} \\<br /> \end{array}} \right.<br />
Where W is the wight of the beam.
I then found that the maximum bending moment occurs at x=3.5, so:
<br /> M_{\max } = 30.625 + 1.75W<br /> \]<br /> <br /> 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:<br /> <br /> <br /> \sigma _{\max } = \frac{{M_{\max } y_{\max } }}{I} = \frac{{M_{\max } }}{z}<br /><br /> <br /> 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. <br /> <br /> <br /> \sigma _{\max } = 150MPa = 150 \times 10^6 Pa<br /><br /> <br /> With these restrictions in place, i should find an elastic section modulus such that:<br /> <br /> <br /> z \ge \frac{{M_{\max } }}{{\sigma _{\max } }} = \frac{{30625 + 1750W}}{{150 \times 10^6 }}<br /><br /> <br /> Now my table gives values for z in the units mm^3, so i converted it to mm^3 by multiplying by 1000^3.<br /> <br /> <br /> z \ge \frac{{612500 + 35000W}}{3}<br /><br /> <br /> 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. <br /> <br /> Does anyone have any input? I am not even sure if I've approached the question in the correct way. I think what's causing me problems is the fact that the weight isn't neglected like most other problems i need to solve.
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