Alan,
Exactly which D1 code are you working to?
Mankenberg

Alan

I'm going to assume your talking about D17

The 0.005 concavity limitation doesn't apply (in the sense that there shall be no concave fillets) because there is specific language in the code text and the commentary that determines exactly how fillet welds are to be measured.

But it does apply,,,,, and we shall see how.

But let's step away from the code for a moment and look at your industry.  Jet engines (which I assume these welds are going on) have a significant percentage of welded joints in gas flow paths and are required to have concave fillet profiles by the OEM   This logically tells us that the 0.005 concavity limitation does not apply in the sense that there can be no concave fillets.  (fillets exceeding a concavity 0.005 below a flat profile)

So we must determine how the Table 6.1 and it's 0.005 limitation on Underfill/Concavity actually does apply

Now back to the book,......  D17 Figure 6.1 (A)  Desirable fillet weld profiles.......... They draw a picture for you of a concave fillet and they also use that drawing to show you how to measure it (which is different than how fillets are measured in A3.0) See commentary C3.3.3 second paragraph

Fillet welds are measured by fillet size... They may be flat, convex or concave in profile as long as they are within the limitations of the code. (see Max convexity requirements in Fig. 6.1 between (A and C)

The 0.005 concavity limit may be applied to  fillets that are undersized  ....... This is how the Table 6.1 limitation on Underfill/Concavity applies tp your situation, when we determine how undersized a concave fillet may be..... 

The moral of the story.... in my opinion

A fillet may be concave..... (D17 drew us a concave fillet picture and called it "desireable")

A fillet shall be properly sized (which includes the limitation of 0.005 of Table 6.1)

The fillet shall be measured as described in Figure 6.1 (with commentary C.3.3 to inform our measuring technique)

So it does apply....... But it must be applied in proper context.

Obviously you are making welds on thin material.  That aside......

A fillet weld does not get it's strength from the leg size.  Instead it's strength is throught the cross section i.e. from the root to the face.  This is why codes are particular about excess concavity and why they make gauges to check for it.

Example;

Say you are welding on 1/2 inch plate and the weld size is called out as 3/8.  Assuming you weld both sides of the joint (assume a T-joint) this fillet weld should outperform the strength of the base metal.  the 3/8" fillet weld was called out not because of leg size but because of the throat.  When an engineer calls out a leg size he is doing so because it is much easier for the welder to estimate leg size under the hood rather than throat depth.  If you were to take that 3/8 fillet and divide it into two equal right triangles you would find that the throat measures about .25 or 1/4 inch.  When you add up the throat on one side of the joint and add it to the throat on the other side you get .5 or 1/2 inch, the thickness of the plate.

You may have heard that fillet weld size should be equal to the plate thickness, this is incorrect.  The total throat depth should be equal to the plate thickness.  This is why a weld size may be doubled when welded only from one side.

If you were to weld a 3/8 fillet and it had excessive concavity the .25 throat depth would not be met no matter the leg size therefore the weld would not meet code and would not meet the design criteria.

This is why your customer is insisting that you meet this criteria.  You may be able to do a little testing due to the thin wall and show that indeed the weld will hold up to whatever the design intent is.  If you can prove this and then present it to the engineer, he can overide the requirment provided documentation can be produced that shows the weld does indeed meet all design criteria.