# Maximum deflection of a beam with both ends fixed and distributed load

I'm designing a part, and I'm trying to figure out what distance a slot should be from the bottom of the part. For simplicity in analysis, I'm approximating the region between the bottom of the slot and the bottom of the part as a beam that is fixed on both sides. This beam will have a uniform load over the entire span.

I've found this equation to find the maximum deflection of the beam $y_{max}$, given the load $w$, beam length $l$, elasticity modulus $E$, thickness $b$, and distance between the slot and the bottom of the part (or beam height) $h$.

$$y_{max}=-\frac{w l^4}{32bh^3}$$

I'm trying to figure out what $h$ ought to be, so I rewrote it in terms of $h$.

$$h=-\frac{1}{2}\left(\frac{wl^4}{4Eby_{max}}\right)^{1/3}$$

I already know what $l$, $E$, and $b$ are, and my plan is to write $h$ as a function of $w$, but I need to eliminate $y_{max}$. I've tried finding it using the formula for bending stress ($\sigma=My/I$), but that resulted me in an identity for $y_{max}$. I tried finding $y_{max}$ by solving the ODE $\ddot{y}=M/(EI)$, but that got me the first formula that's above.

It would be awesome to have another formula for $y_{max}$, but a better/different method of determining what $h$ should be would be just as good.

• I don't understand what you are asking. You seem to be asking for some way to define y_max. This will be a limit that is set by what your part can tolerate. Figure out what the maximum deflection is that you can allow and use that. Or add some more background to your question. – hazzey Mar 3 '16 at 22:06
• The problem above lies where to get the value of ymax. Maximum deflection are normally given and decided by which code you follow for certain design criteria. Example, Philippine codes specify deflection limits up to 1/360 of the span length, such that: ymax = L / 360 – Jem Eripol Aug 19 '17 at 1:58