If a transformer is rated for a maximum winding current of 10 A, would it be permissible to operate it with a 10 A of input current when the secondary voltage to the load is 40 V.

 

You have not provided enough information. Transformers do not create power that isn't there. They are often used to step AC voltage up or down. Whatever happens to the voltage, the effect on current is opposite.
For example
If I have a step down transfomer that converts 120VAC into 60 VAC, and the load on the secondary side draws 2 Amperes, then the input current on the primary side will be 1 Ampere.
120VAC * 1 Ampere = 120 Watts
60VAC * 2 amperes = 120 Watts.
That's for an ideal transformer. You will actually get a bit less power out than 120 Watts for a non-ideal transformer.

If you try to draw more current than the transformer can supply, a number of things might happen including temperature rise, changes in winding resistance, and core saturation. You don't want to be doing those things. Trust me on this.

 

Sorry

Vin=115
it is a single phase variable auto-transformer i.e. 120-140, so it would step down form 0-120 and step up from 120-140

 

So 10 Amperes at 115 VAC = 1150 Watts. In all cases the output power will be less regardless of output voltage.

 

so you're saying it is not permissible?

 

If a transformer is rated for a maximum winding current of 10 A, would it be permissible to operate it with a 10 A of input current when the secondary voltage to the load is 40 V.

..... secondary voltage to the load is 40 V.. at what load current? Theoretically, the load current can be 1150 / 40 = 28 Amps. BUT your transformer winding is WILL NOT handle that current - it's rated at 10 Amps. So, you can have a load current of 10 Amps @ 40 Volts = 400 Watts. This works out to be 400 / 115 = 3.5 Amps approx at the input.

so you're saying it is not permissible?

...so it's not possible.