DC Sizing

Physical or virtual? Below is the minimum I recommend

1/2 Core for each DC
4 GB RAM
2X Disks, NTDS on secondary otherwise writeback caching is disabled on the system drive

https://docs.microsoft.com/en-us/windows-server/get-started/system-requirements

What is CA design? Where is CRL/OCSP?

its a virtual domain controller. enterprise root ca on one of the domain controller. crl published in AD.

I need a documented formula for DC sizing

download Active Directory IPD from Microsoft

http://go.microsoft.com/fwlink/?LinkId=157704

It is for 2008 R2 but still applies to 2016  / 2012

It will guide you through out DC sizing, placement and configuration etc

IPD is not available for 2012 server onwards

we have a case where a site of 1400 users. we have 2 domain controllers with 2 cores and 12 gb ram.

the domain controllers started shutting down unexpectedly.

we increates it to 4 cores and 24 gb ram and since then the dc's are fine.

I am asking a formula because I designed a dc with 2 cores and 12 gb and turned out to be an issue.

we have a design review process where we have to validate the deigh and sizing using vendor documents. thays why I askes

the domain controllers started shutting down unexpectedly.

Clearly, you have an issue with your environment. I have worked on no less than 300 domains from a handful to hundred on thousands of endpoints and I never had the requirement to have a 24GB RAM DC

Ideally DC sizing is depends upon OS basic system requirement, number of authenticating user base

Upto 10K users base you should have 2 DCs with 4 cores each and sufficient memory, memory calculation is like installed memory should be able to load entire AD database, this is best suitable for small or mid sized setups but not useful for lakhs of user base where AD database grown beyond 32 GB etc

Normally upto 10K user base, you should have 2 DCs with 4 cores CPU each and 8GB minimum memory, check IPD for complete details

Did you examine the resource utilization before upgrading?  Was the CPU hammered?  Was the RAM used up?  What kind of research did you do to solve the problem or you just guessed that more resources were required (because no Windows system I've ever seen ever just "shuts down" because of low resources... becomes unstable and slow, sure, but not shuts down.  

Were these physical or virtual?

Albeit years ago, I managed a domain of 1000 users on two DCs with FAR less resources and there were never any issues with the DCs performance or shutting down.

well we opened a case with VMware. they suggested this

we then opened a case with ms and they kept analiying the logs for ever. they suggested we install new dc.

its then we increased the memory