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I am working the following problem:

The simply supported beam shown in Fig. 12-12a is subjected to the concentrated force P. Determine the maximum deflection of the beam. EI is constant.

The text later says:

By inspection of the elastic curve, Fig. 12-12b, the maximum deflection occurs at D, somewhere within region AB. Here the slope must be zero.

Why does the author say that, by inspection, the maximum deflection occurs at D? How do we know that? He didn't show any work or explain why it should be at D and not somewhere else.

If I consider the maximum slope to occur at a point 2 m from A, then I will take EI(dv2/dx2) = 0, then my answer is, -2((x2)^2) + 12x2 -44/3 = 0, then x = 4.29 m ....

here's the full question . in hree , we can notice that there are 2 sets of slope equation that we can use . Which is equation 5 & 7 . In equation 5 , we will get 1.633 as in the working . ( the author use by 'inspection' the max deflection occur at region AB)

However , when as @Jmac stated , we dont know where is the position where the max deflection is located , how can we use equation 5 to solve ?

why We dont have to consider equation , which involve region DC ??? 22 in 33

solving equation 7 = 0 , i have x = 5.23 , 3 and 0.763 , which is correct ?

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    $\begingroup$ No, it says the maximum deflection occurs at D. Therefore the slope at D is zero. The slope must be zero by the "common-sense" argument that if it was not zero, a point close to D would have a bigger deflection than D. Or you can make a rigorous argument using theorems that are proved in a calculus course. $\endgroup$
    – alephzero
    Dec 3 '16 at 16:51
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    $\begingroup$ $D$ is defined as the point where the maximum deflection occurs. Notice that the location of $D$ is defined as "somewhere within $AB$". I agree that using the term "by inspection" is odd, though. $\endgroup$
    – Wasabi
    Dec 4 '16 at 2:06
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    $\begingroup$ For point loading, I think the max deflection only occurs at the point of loading when it's loaded at the centre (and both ends are supported). Consider that the place you are loading, it's closer to one support than the other. That means that it's closer to a point resiting the load, so the maximum deflection ocurs somewhere else. The case of being loaded in the middle is a special case because you are the same distance from both supports, so the maximum deflection is at the point loading. $\endgroup$
    – JMac
    Dec 5 '16 at 10:48
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    $\begingroup$ @kelvinmacks Not necessarily at "another side"; just further from the support. Consider if you were to place the load just to the left of point C. It couldn't really deflect the beam that much; there's a support right beside it. It would bend the beam a lot though; so the maximum deflection would be somewhere to the left of the load. The load is causing bending along with displacement. This bending is what makes the beam deform more further to the left. $\endgroup$
    – JMac
    Jan 4 '17 at 11:57
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    $\begingroup$ @kelvinmacks I honestly don't really get where you're getting lost there. They use equation 5 to solve for the location of D (by finding where the slope is 0). $\endgroup$
    – JMac
    Jan 4 '17 at 14:14
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Engineers, by design, are lazy. If you are looking at a beam, where you can see that under a given load it deflects most at a certain point, then you can say that it deflects most at that point. It's like when you have symmetry conditions to only have to figure out half of the numbers for a beam, and extrapolating.

Textbook writers say 'by inspection' when they can use their engineering intuition and what they believe to be common sense in order to save labor.

But if you only had the beam, some forces, and we knew that the beam would bend downwards, bottom out, and start to bend back up to the other end, it's intuitive that if you rolled a marble from either end of the beam, it would roll down and settle on the lowest point, where it had a max deflection and a slope of zero.

Now, if there was a beam with multiple forces pointing up and down, with the beam looking like a ride at Disney Land, then that would be a different story. Then, you would have to go through more calculations and really get that engineering intuition into action.

But most often, engineering intuition is knowing how to use the minimum computation for the required results. The author simply said, there is only one global and local minimum, and said, "close enough for government work."

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