# Is the Qmax of an T-Beam equal if calculated both above and below the neutral plane?

I was doing a question involving a T-Beam question wrong since I was selecting the wrong area to use for $Q_{max}$ when trying to maximise the shear stress.

$$\large{\tau}= \frac{VQ}{It}$$

A friend argued that the area both above and below the Neutral Axis should be equal but despite trying multiple times, I have been unable to get the same result.

Is this statement correct? (and if not, why)

It would seem by argument of equal values both above and below the beam that perhaps it would be correct, even if the beam is unsymmetrical.

• Is your question about finding the neutral axis or finding Qmax?
– hazzey
Mar 12, 2015 at 18:33
• @hazzey If I can use the either side of the neutral axis to find Q and get an equivalent answer. Mar 12, 2015 at 18:35
• I'm still confused as to what you are asking. Maybe you need another diagram. Your equation is the calculated shear stress at a particular location. Which areas are you interested in?
– hazzey
Mar 13, 2015 at 0:17

The answer to the headline question is yes, it is equal. It physically has to be equal, since you can't have two different values of a particular directional stress at one point.

The quoted equation is for shear stress, where V = total shear force at the location in question; Q = first moment of area 'beyond' the point considered; t = thickness in the material perpendicular to the shear; I = Moment of Inertia of the entire cross sectional area.

The problem seems to be the assumption that the cross-sectional area above and below the neutral axis is equal (ie, the embedded question "the area both above and below the Neutral Axis should be equal ... is this statement correct" to which the answer is no)

Neutral axis is the line about which the first moment of area each side is equal, not the line about which the areas each side are equal.

Applying that equation above and below the neutral axis, V an I are properties of the whole section so identical wherever you do the calculation on this cross-section, t is a function of the level at which the calculation is done so identical in both applications, and Q is equal by definition (that's the definition of the neutral axis), so the answer has to be identical since all the numbers substituting into the equation are identical.

Example: for the diagram shown, if T=10, W=60, H=70, then the tee shown has a cross-section of 1200, and both flange and stem each have area 600. However, the neutral axis is NOT at the bottom surface of the flange (ie, is not where there is equal area above and below). In this example, the neutral axis is at 22.5 from the top surface, or 12.5 below the level at which the areas are equal either side. So there's 725 area above the neutral axis, and 475 below the neutral axis.

Calculating for above the neutral axis, there are two rectangular regions,

Q = 10 x 60 x 17.5 + 12.5 x 10 x 6.25 = 11281.25


Calculating for below the neutral axis,

Q = 47.5 x 10 x 23.75 = 11281.25


I = 552500 and t = 10 at the neutral axis, so whatever value of V is used, $\tau$ is the same.