# How can we know "by inspection" the location of maximum deflection of a simply supported beam with a point load?

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

• 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. Dec 3 '16 at 16:51
• $D$ 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.
– Wasabi
Dec 4 '16 at 2:06
– JMac
Dec 5 '16 at 10:48
• @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.
– JMac
Jan 4 '17 at 11:57
• @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).
– JMac
Jan 4 '17 at 14:14