Your calculation is incorrect. The general idea is that money you contribute to your pension isn't subject to income tax (at the time you put it in - it is taxed later when you take it out). In your case a contribution of that size will reduce your taxable income below the threshold for higher-rate tax, so you will only get a "40%" refund on the amount above the higher-rate tax threshold.
Note that you don't save the national insurance (NI) when contributing to your SIPP. If your employer offers a pension with a salary sacrifice scheme it might have been possible to contribute to that and also save NI. If your employer operated such a scheme they would have told you. It works by them reducing your gross salary by a certain amount and sending that amount direct to the pension scheme they offer. NI then gets charged on the salary after this "sacrifice" and you also don't have to make any claims for tax relief.
Coming back to the SIPP contriubtion, if you have an income of £60,000 and want your gross (total) pension contribution to be £40,000, then your effective taxable income is £20,000, so that's what you'd end up paying tax on. The income tax you would pay on that is 20% of the amount above the personal allowance (£12,500), i.e £1,500. The income tax on £60,000 is 20% up to the higher-rate threshold (£50,000) and 40% above that, so £7,500+£4,000 = £11,500. So your £40,000 gross contribution would save you £10,000 in tax.
Allowances and tax bands are for the 2020-21 tax year; note that you you have to make pension contributions in the relevant tax year so you can't now do it for 2019-20.
The mechanics of such a contribution are that you need to make a "net" contribution to the SIPP of 80% of your desired gross contribution, and the SIPP provider will then claim the other 20% (which is basic rate tax) from the government. So you would pay £32,000, and they would claim £8,000 from the government. This is called "relief at source".
The reason it works this way is that when you are organising a pension yourself, rather than your employer doing it, you have to make the contribution yourself. Since your employer has already deducted tax via PAYE, the now overpaid tax has to be returned somehow. If all contributors had to reclaim the tax themselves then a lot of people would have to contact HMRC individually. On the other hand if the pension scheme had to reclaim all the tax themselves then they would have to know your exact tax situation. The current mechanics are a compromise: the pension scheme can always reclaim basic rate tax, and higher-rate taxpayers have to do some extra work themselves.
The rest of the tax relief is £2,000, the difference between higher-rate tax and basic-rate tax on the £10,000 of your income which was originally subject to it. You need to reclaim this directly from HMRC after the end of the tax year, by writing to them or filling in a tax return. That would leave your final net contribution at £30,000.
A plan to make a contribution of £40K this year only makes sense if are planning on making very large contributions year after year for a while to build up a big pension, or you expect something to change such as the government removing tax relief completely or your salary changing.
Otherwise I would suggest only making a gross contribution of £10K for several years. That would probably maximise the tax relief you get. For example if your plan was to contribute £40K this year and nothing for the next three years, you will save £10K in tax. But if you contribute £10K for each of those four years, you will save £4K in tax each year for a total of £16K, because your contributions will always come out of money that would otherwise have been taxed at 40%. Of course your actual plans and salary will vary, but that's the general idea.
If you have children (and earn more than the other parent) then making a £10K pension contribution will actually save you more money because of the way child benefit works. This is because while people with a taxable income of £50K get the full amount of child benefit, but people with a taxable income of £60K get no child benefit (or have to repay it via the tax system). In between the two taxable incomes, the amount reduces in proportion.
So for example if you have 2 children and hence would receive child benefit of £1,855, then making a pension contribution of £10K will also mean that you receive the full amount whereas without the pension contribution you will receive nothing. In effect your marginal tax rate is actually 58.55% in the income band between £50K and £60K, all of which you can recover by making the pension contribution. Going back to my contrived 1 years versus 4 years example above, the numbers would become £11,855 versus £23,420.