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February 5th, 2018, 03:13 AM   #21
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Yes I mean the constants of integration, I have read the thread but am unsure of how to determine the values of a and b, I think the method is by simultaneous equation but I don’t know how to do this
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February 5th, 2018, 08:11 AM   #22
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OK from post#11


$\displaystyle EI\frac{{{d^2}y}}{{d{x^2}}} = M = {\rm{first}}\;{\rm{expression}}$


We get the slope from the first integration


$\displaystyle EI\frac{{dy}}{{dx}} = {\rm{first}}\;{\rm{integral}} + A$

and the deflection from the second integration.


$\displaystyle EIy = {\rm{second}}\;{\rm{integral}} + Ax + B$


So the second integration leads us to an exquation for the vertical deflection, y.
But it introduces two constants A and B.

However we know the value of the deflection at two points, (the supports).
This is because the beam neither lifts off the supports nor do the supports themselves sink.
So the deflection is zero at these points on the beam.

so y(x=1.4) = y(x=5.6) = 0

So putting these values into the deflection equation we find


$\displaystyle EIy = - \frac{{720 < x{ > ^3}}}{6} + 1198\frac{{ < x - 1.4{ > ^3}}}{6} - 450\frac{{ < x - 2.5{ > ^3}}}{6} - 100\frac{{ < x - 3.6{ > ^4}}}{{24}} + 1022\frac{{ < x - 5.6{ > ^3}}}{6} + 100\frac{{ < x - 5.6{ > ^4}}}{{24}} + Ax + B$



$\displaystyle 0 = - \frac{{720 < 5.6{ > ^3}}}{6} + 1198\frac{{ < 5.6 - 1.4{ > ^3}}}{6} - 450\frac{{ < 5.6 - 2.5{ > ^3}}}{6} - 100\frac{{ < 5.6 - 3.6{ > ^4}}}{{24}} + 1022\frac{{ < 5.6 - 5.6{ > ^3}}}{6} + 100\frac{{ < 5.6 - 5.6{ > ^4}}}{{24}} + A*5.6 + B$

I will do the second value for you

Since these are macaulay brackets, we ignore all terms for which the bracket is negative (or zero)

reducing the equation to


$\displaystyle 0 = - \frac{{720\left( {175.6} \right)}}{6} + 1198\frac{{74.1}}{6} - 450\frac{{29.8}}{6} - 100\frac{{16}}{{24}} + 5.6A + B$


$\displaystyle 0 = - 21072 + 14795 - 2235 - 67 + 5.6A + B$


$\displaystyle 8579 = 5.6A + B$

Can you do the first one

You should have


$\displaystyle 0 = - \frac{{720 < 1.4{ > ^3}}}{6} + 1198\frac{{ < 1.4 - 1.4{ > ^3}}}{6} - 450\frac{{ < 1.4 - 2.5{ > ^3}}}{6} - 100\frac{{ < 1.4 - 3.6{ > ^4}}}{{24}} + 1022\frac{{ < 1.4 - 5.6{ > ^3}}}{6} + 100\frac{{ < 1.4 - 5.6{ > ^4}}}{{24}} + 1.4A + B$


reducing to


$\displaystyle 0 = - \frac{{720 < 1.4{ > ^3}}}{6} + 1.4A + B$

Last edited by studiot; February 5th, 2018 at 08:35 AM.
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February 5th, 2018, 10:13 AM   #23
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So a = 1963.86 and b = -2420.6? Does that sound about right?
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February 5th, 2018, 10:20 AM   #24
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Studiot I would like to thank you for your help on this question, I was able to work my way through it and complete it on time and correctly. Once I had got the values for a and b it was a doddle

Seems like there’s quiet a few people having the same sort of problems as I did on this kind of question, I noticed vincot and Aaronmooy47 also asking for some help. I do find that the resources available when looking at solving beams is sometimes a bit scarce and takes a while to dig out exactly what you’re looking for.

But then again I suppose that’s why people use websites like this so that people like you can pass on your knowledge and give them better guidance than a textbook ever could.
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