Why Raavan could not pick Shiva's Dhanush

In ramayanam, Rama and Lakshmana reached Mithila along with their guru Vishwamithra for sita's swayamvaram.

When they reached there, they had some time to see around the place. Rama and Lakshmana visited a garden at janaka's place where Sita was already worshipping Parvathi Devi.

It is said that then and there, Rama was in love by the beauty of Sita. It was like entire world had bowed down to her beauty.

On the other hand, when Sita first laid her eyes on Rama, she wished that only he should be her husband.

By now on King janaka's invite, Ravana had already arrived for the swayamvaram. The rumour spread around that as Ravana was the great devotee of Lord Shiva, only he could pick the bow and pull the bow strings.

Sita knew of his evil mannerisms, and was worried if those rumours would come true. So, the night before the swayamvaram, Sita went up to her mother Queen Sunaina, and expressed her fear. Her mother advised her to visit Gauri temple and perform the GauriShankar puja. She spent the whole night and day worshipping Goddess Parvathi Devi to bless her.

LordShiva and his wife Gauri are worshipped as ideal couple. So, whoever, with pure heart, seeks blessings from this couple is blessed with harmonious relationship and heart-wished life partner. Thus, Gauri puja is being devoted as a manifestation to seek the ideal couple's blessing.

The next morning, as per King Janaka's condition all the kings and princes tried their best to pick the holy sacred ‘Shiva Dhanassu’. Ravana had managed to pick up the bow, but only to drop it later.

After this, Vishwamithra ordered Rama to go ahead and break the bow. It is said that Sita quitely prayed to God for Rama's success. Rama impeccably broke the bow in to two halves, which leaded to “SitaRama kalyanam”.

It is believed that only the blessings of Goddess Parvathi turned the fate of Sita, and Lord Rama was announced as the honorable winner of the swayamvaram.

So, this is the reason behind the Ravana's failure in Sita swayamvaram. Though the Ravana was the great devotee of Lord Shiva he couldn't pick the bow, only because of Sita's strong desire to marry Rama by the blessings of Gauri Devi.

Also

First of all, who said Ravana is all powerful and he is almighty,there are many people or vanar’s who’re more powerful then Ravana to name few: Hanuman, Angad, and his father Bali, and finally Sri Rama in the same era.

Second, even Sri Rama is the great devotee of Lord Shiva, as he himself told to his wife Sita (after marriage), that he is not her Swamy (husband), “Hum sab ka swamy hai Paremeshwar” (I am not your Parameshwar, please don’t pray to me, if you really want to pray, then please pray to that Parameshwar, who is my Swamy). This shows humbleness of Sri Rama.

Third, Hanuman, the all powerful and almighty (who can move mountans and crush lanka), who is more powerful than Rama or anyone in that era, also without him, Sri Rama would not able to win the battle with Ravana, but he himself (Hanuman) said, I am simple vanar, without your name (Rama), I cannot do anything. It’s your name that is giving me strength, that shows humbleness.

Now, coming to Great Ravana (who is great shiva bhakt), has filled himself with illusion of arrogance and ego, and without taking the permissions of Lord Shiva, he tried to lift the BOW, and failed. But if you observe, Sri Rama, had payed his patronage to that great Shiva and then lifted the BOW, and he was successful.

There is another incident with Ravana, which has happened in past, same with his arrogance and ego, Ravana had tried to lift Kailasha parvat itself where Lord Shiva is sitting with his Consort, he was failure then also. Then only he sang the holy “Shiva Tandava Stotra” in asking pardon from shiva for his arrogance and requesting Shiva to remove his foot from the Kailasha mountain to remove his clutched hand struck below the mountain. later as we all know Lord Shiva is all merciful and Bholanatha, given him blessing and also given him Chandra haas sword, as lord shiva got melted with his Shiva Tandava Stotra. (So this is the story)