Girder w/ Parabolic Haunch. What equation describes the haunch depth? 1

[UtahAggiePE]

This has been driving me crazy. So I'm dealing with load rating continuous highway girders over bents that have parabolic haunches at the interior locations. The problem is, what equation do you used to describe the depth of the section?

 

[BAretired]

Pretty straightforward. Assume beam of length L with parabolic haunch at each end; haunch has length a and height b.

Dx = Depth of beam at x from Left End

When x < a, Dx = d + [(x-a)/a][sup]2[/sup]*b
When x > (L-a), Dx = d + [(x-L+a)/a][sup]2[/sup]*b
When a < x < (L-a), Dx = d

For any x, Dx = d + (x<b)[(x-a)/a][sup]2[/sup]*b + (x>(L-a))[(x-L+a)/a][sup]2[/sup]*b

The Boolean expressions shown in red take the value 0 if false and 1 if true.

BA

 

[BAretired]

Oops, I meant to write:

When a > x > (L-a), Dx = d

BA

 

[BAretired]

Damn! When x is between the haunches, the depth of the beam is d. I was correct the first time.

BA

 

[UtahAggiePE]

Thanks for your help. You would think it would be straightforward and simple but the question we were running into is was this a curve based on y=x^2 between x=0 and 1 (as yours is) or is it a curve based on using the upper half of the y=sqrt(x). Both are the same curve but really it comes down to what part of the curve you are going to use! Thanks again.

 

[UtahAggiePE]

Correction...just in case I'm not being clear. The question is how you will orient the curve and which part will act as the base. I agree it seems clear that it is an inverted y=x^2 curve. Thanks very much.

 

[BAretired]

I'm not sure I understand the difficulty. The graph of the equation
y-k = a(x-h)[sup]2[/sup]
is a parabola with vertex at the point (h,k). The parabola opens upward if a>0 and downward if a<0. You want yours opening downward.

The vertex is the point where the curve meets the straight line representing the beam depth between haunches.

The equation
x-h = a(y-h)[sup]2[/sup]
is a parabola which opens to the right or to the left, depending on the sign of a, but that is not what you want here.


BA

 

[BAretired]

The equation
x-h = a(y-h)[sup]2[/sup]
is a parabola which opens to the right or to the left, depending on the sign of a, but that is not what you want here.

That equation should have read:
x-h = a(y-k)[sup]2[/sup] with vertex at point (h,k).

It could be used but the vertex would be at the support and the curve would meet the underside of beam at an angle rather than tangentially.

If the haunched beams are existing, I believe y=a.x[sup]2[/sup] is the more likely configuration for the haunches but this could be confirmed by visual examination.

BA