How to Fix a Piano String That broken

Piano Strings
The average modern piano has over 230 strings under a combined tension of 15 to 20 tons. A concert grand piano may have a combined string tension of up to 30 tons. Pianos made in the eighteenth century were not as powerful and used low-tension wire made from an alloy different from the wire used today.

Piano String : Scale design
The scale design of a piano refers to the calculations the piano manufacturer used to determine the pitch, diameter, length, and the tension of each wire. Good quality pianos usually have a better scale design that involves a lot of engineers and scientists taking many measurements and crunching the numbers. The piano is then built in the laboratory and is tested by listening to it. If the piano does not sound good, the design team does more calculations, makes another test piano and listens to it. This continues either until the piano manufacturer is completely pleased with the results or until the research budget runs out!

Piano Bass Strings
If the bass strings were made of plain steel wire, the lowest notes would have a string length of around 25 feet long! Since the piano has to be able to fit into normal-sized living rooms, the designers had to achieve a lower pitch using shorter wires. Getting a lower pitch with shorter wire requires using a larger diameter wire. Unfortunately, if a wire is used that has too large of a diameter, the wire will break under the required tension. The solution is to use a smaller steel core wire and then to use another wire wrapped around the core wire to add mass. The extra mass makes the wire behave as though it is a larger diameter wire without causing the problems of string breakage.
Originally, the wire that was used to wrap around the core wire was made of iron. Later, some pianos were made with aluminum-wrapped bass strings, and now copper wrapping is used. Plain steel music wire is used throughout the tenor and treble sections of the piano.
A common question is "Why does the piano have only one or two wires for each note in the bass section and three wires in the rest of the piano?" The short answer is that the number of wires used for each note helps determine the volume of that note. A large bass string can produce much more volume than a smaller plain wire. The volume is balanced by using more wires for each note in the treble section and fewer wires for each note in the bass section.

Broken Piano Strings
Occasionally a wire will be broken and need to be replaced. If the string is not replaced promptly, it can cause uneven wear on the hammer that will lead to additional repairs being necessary. A string can break for a number of reasons. In most cases of string breakage, there usually is the presence of rust that weakens the string. The string might break because of a kink or a bend in the wire, or there may be a defect in the wire. In tropical areas or areas with high humidity, pianos are made using tinned wire to prevent excess rust. Strings frequently become brittle with age, and the splintered ends of a broken wire can testify to that. Concert instruments and other pianos that receive a lot of heavy use are notorious for popping strings.
When a piano is being tuned and a string breaks, it is usually due to a weakness in the string as listed above. Sometimes the wire can also be broken by a piano tuner who does not use a proper technique in using the tuning lever. When the broken wire is examined and the wire shows evidence of "necking down" (the wire being overstretched to the point that the wire is damaged) it often is the fault of the tuner. If a piano is flat in pitch because it has been neglected, the pitch raise can cause rusty strings to break. However, I have successfully raised pitch on a number of old pianos that had very rusty strings and were more than a whole note flat in pitch, and often the entire tuning will proceed without any of the strings breaking. Other times I may be tuning a much newer piano that is not flat, and a string will just decide to break.

Piano String Repairs
When a pinao string breaks, there are several repair options available. A broken string can be repaired by tying a tuner's knot to splice the old wire to a short piece of new wire. The advantage of such a repair is that an older wire that is spliced will often match the tone of the surrounding strings better than if it had been replaced. In addition, a string that has been repaired this way will stabilize much more quickly than a new wire. Tying a broken string eliminates the need to return several times to retune the string back to the correct pitch. The drawback is that a wire with a knot in it doesn't match the other wires in appearance.

Piano String Replacement
My preference is to replace a broken wire with a new wire. Piano wire comes in long coils that are several hundred feet long. I use a micrometer to choose the appropriate gauge of wire, cut off the appropriate length, and do the replacement. When a bass string breaks, you have the option of having the string replaced with a universal bass string or a custom-made bass string.
A universal bass string set has a number of different sizes of core wires with different sizes of copper wrapping. The replacement is cut to the correct length, and then the wrap wire is "unraveled" until the beginning and end of the wrapping matches the other bass strings on the piano. The problem with using universal bass strings is that often the tone of the new piano string is decidedly different from the surrounding strings. It is rare to find a universal bass string that will perfectly match the core diameter and wrap diameter of the broken wire. Also, I have personally had a problem with universal bass strings breaking shortly after they are installed.
A string maker will make a custom bass string precisely to the correct size. The string maker already has a list of stringing scales for many current production pianos, but an older or obscure brand of piano may not be in their files. To find the correct size of wire requires either sending the broken string to the string maker, making a paper pattern, or taking several measurements of the broken string. Custom-made bass strings involve more work, take more time, and are more expensive, but it is my preference to use them, because they give the advantage of matching the tone of the existing strings much more closely.

Piano Restringing
An older piano often will develop a lot of rust on the strings, or it may have a problem with breaking strings. The bass strings often become tubby and "dead" because dirt and debris will be caught in the coils and make the strings not as flexible. The bass strings can sometimes be rejuvenated by working to get the junk out of the coils, but it is preferable to replace the entire set.
When restringing a piano, the tuning pins and strings are usually replaced as a unit at the same time. When the strings are removed, they are carefully measured and the scale design is calculated to determine what size of strings should be used for replacement. Good quality newer pianos often will have an acceptable scale design, but old or obscure piano designs can often be improved upon, especially in the bass section. The tone of the piano can often be improved by "tweaking" the scale design.
Although piano wire is made of steel, it does have some elasticity and does stretch. A new piano, or one that has been restrung, will continue to stretch and go flat for quite some time. If you have ever replaced a set of strings on your guitar, you know what I am talking about! After a piano has been restrung and returned to the customer's house, it needs to be tuned a minimum of four times that first year and a minimum of twice a year thereafter. It is my opinion that a piano will stabilize much more quickly the more frequently it is tuned.