Hi. This is my pure economy tank Udyr build. This is pure theorycraft, based on what gives the most effective health for the price.

Refined the calculation for MR stacking. It's actually a 71% defence so madred's bloodrazor only deals 1.16% damage. It would take 87 hits to kill you with a bloodrazor.

Changed item options for late game.

Udyr's base Magic Resist at level 18 is 34.25 points lower than his armour. The runes seek to rectify this problem. Health quints are simply great for his early game, while the magic penetration makes jungling easier and faster.

This is a basic jungling setup. Nothing special to comment about. Be sure not to forget to change your masteries because having no awareness nor buff extension can really hurt your jungle

Always choose smite.

The second spell is yours to choose. I enjoy ghost since it gives you unparalleled chasing ability and movement speed. Otherwise, pick exhaust or ignite if the other team has hard dps/mundo. Although if you don't want to choose ghost think about levelling bear stance early.

Simple route.
Either go Golems [smite]>Wraiths>Wolves (Repeat) Golem[smite]>Lizard
or
Golem[smite]>wolves>wraiths>Lizard[smite]>Golems

Start with phoenix and then choose tortoise. This is the fastest jungling skill sequence, only use tortoise if you have enough mana to go straight back to phoenix. If not, then just keep on phoenix.

Remember to gank

Who not to gank:
Kassadin
Leblanc
Anivia
Tristana (with flash)
Akali (with flash)
Ezreal (with flash)
Vladimir

Don't waste your time trying to kill the solo mid by ganking if they're one of these. Even if you make them lose their flash, the surprise is gone and they'll probably be sensible enough not to push again. Champion like anivia/vladimir are generally ungankable.

Try to gank kassadin before level 6, otherwise don't bother.

Above is a simple chart I made, displaying the effective health that each item gives and highlighting some of the best ones.

As you can see Aegis of the Legion is clearly an amazing item, along with the Heart of Gold and the Warden's Mail. The other items are situational.

The first three are for jungling. After that the game will develop into little skirmishes. Even without any offensive items you dominate any 1 v 1 fight if you have three levels in tiger stance. This skill literally shreds people. After that, people outdamage you so start focusing on tanking (tortoise stance) or if you haven't chosen ghost then bear.

You generally don't need to level bear as the stun doesn't improve and the movement speed rise is negligible.

Finally. The important part!
So you want to spawn, grab a
1)Cloth armour and five health pots, standard jungling build
2)Basic shoes
3)Madred's razor - this helps your jungling and is great for tiger stance
4)Wriggle's lantern or Madred's Bloodrazor and start building your shoes

-At this point you can decide. I prefer choosing the lantern because it gives you lifesteal for countering harass before a skirmish, it's cheap and you get a free ward
-Madred's Bloodrazor gives you far more damage however it hugely slows down your 'tankiness' and the lack of tank in you will display in small early skirmishes, you'll be as squishy as anyone else, even with tortoise stance
~For shoes either build ninja tabi if you've had a bad start/no ganks, or merc treads if you have had ganks. Both give the same effective health, one is just cheaper and lets face it. You'll be GLAD if they CC you. That's just a waste of an ability.

5)Aegis of the Legion. No debate. An amazing item. Incredibly cost effective, even ignoring the aura
6)Here is where you decide what you want to do. You have three choices; Banshee's veil, Randuin's Omen or Warmog's armour
-Banshee's Veil: Amazingly cost effective item and you generally should be getting it since most damage output in the game is magic. The passive is free so you don't need to worry if it's situational or not.
-Randuin's Omen: The components (HoG and Warden's) are absolutely amazing however the overall item with it's passive is overpriced and it's only truly used in large teamfights. On the other hand since you are going to be getting close to others it can be useful. I would recommend buying the heart of gold and warden's and then only finishing it last.
-Warmog's: "This guy's an idiot! Warmog's wtf? lol noob". No. Warmog's gives a huge amount of effective health and Udyr is a good farmer. It's extremely easy to stack it up. Health is generally underrated by most players but health gives far more EHP in a situation such as a champion has 1700 health and 30 Armour/MR. You may think that armour/mr is better but calculations show otherwise. Plus it combats both forms of damage equally.

7)If you want, get another one of the items. If you got randuin's, I strongly recommend that you get warmog's. Your armour/health balance won't be great otherwise.

8)If this was any other champion I would laugh at them getting a guardian angel. It's horribly overpriced and overrated. However having done calculations which I've double checked thoroughly, it's proven to be a good item at this point in the game once you have all your other ones. It's passive is terrible however the balance of MR/Armour is perfect

What you should consider getting:
Spirit Visage: If you're doing badly, it's simply a cheap, good item
Sunfire Cape: A good mix of health/armour. The passive and price is still great.
Atma's impaler: If you hate Guardian Angel and you're stomping the other team

Counters:
Thornmail: Most DPS. Not tryndamere.
Guardian Angel: Master Yi. Most people don't realise that his ultimate is fairly short and it requires kills to reset. Guardian Angels on the whole team totally wrecks him.
Frozen heart: Any high speed attackers. Do not get this vs high damage, slow attackers such as Corki/Xin/Tryndamere
Quicksilver sash: Malzahar/Warwick/Mordekaiser
Warmogs Armour: Any burst champion.

What never to get:
Force of Nature: Horribly, horribly overpriced. Only cost effective at 2700 health and even then there are better items such as banshee's

Why Madred's Bloodrazor does not counter a Warmog's armour:
It will still take 25 hits to kill you. Furthermore it is magic damage. With the MR stats from your items you're taking that down to 1.16% health per hit. It's suddenly become a terrible item! If you want to have fun. Play Mundo with 5 Warmog's and a Force of Nature with full MR/level runes. You will be unkillable.

What to do lategame:
This build should be more than enough for lategame however if you see that you're running around with 4000 gold and one of every potion then here is what you can do.
1)Replace Wriggles with a more offensive item. I recommend the bloodthirster because the extra damage will help you even more with making a carry run away because you've stunned them and taken 2/3s of their life. However, ONLY do this if you have enough money to sell your boots for example, place 3 wards and then re-buy boots. You NEED wards.
2)If your teammates have been smart enough to buy defensive items such as banshee's on Ashe, go ahead and sell aegis. If not, keep it because it's extremely valuable. I recommend replacing it with Force of Nature (because now you can actually make it cost effective, finally) if they have a heavily magic team, Thornmail if they are mostly physical (and you didn't buy this earlier) or Sunfire for a balanced team. Sunfire is great because it will balance out your AR/MR stats and give you more health.
What you could do, if they are ALL burst champions (why haven't you won the game yet) is buy another warmog's, since burst damage is countered by health. Although this situation will almost never arise, if they are all burst then you should have won by 30:00, even if you were losing early game.
3)Sell your ninja tabi (if you got them) for merc treads, however this should be done more mid game, and if the opposing team is not totally physical

Sorry if you don't like this visually. Please don't comment and rate based on that. This was meant to be pure theorycrafting, I hope you learn from it.

~Mukilab