It is difficult to estimate the redundant strength remaining in a building on fire, since the fire conditions such as their ranges and locations may be different in each case. A large-scale fire occurred on the New York World Trade Center 7 (WTC-7) after the 9.11 terrorist attacks in 2001. The building continued to burn for 7 hours, and ended in a total collapse. The official report on the collapse investigation of WTC-7, released by the National Institute of Standards and Technology (NIST) in 2005, had suggested that the total collapse was triggered by a severe damage of an important column of the building called the key element. On the other hand, there is also an example of a high-rise building on fire that avoided a total collapse, regardless of its main structure continuously heated by fire for a long time; the Windsor building of Madrid in 2005. From the viewpoint of preventing the fire-induced collapse of buildings, it is necessary to clarify the relationship between various structural parameters of buildings, fire locations, their ranges, and the scale of collapse.<br> The purpose of this study is to predict collapse risks of buildings on fire using a key element index (KI). The KI is defined as the ratio of the ultimate yield strengths of the structure with one column eliminated and the initial, undamaged structure. It indicates the contribution of a structural column to the vertical capacity of the structure. We investigated the relationship between the sum of KI values in various cases of fire range and the sum of the height of remains after the collapse. We applied an adaptively shifted integration (ASI) - Gauss code utilizing linear Timoshenko beam elements to investigate the collapse behaviors of a ten-story steel framed building model with various fire patterns. Fracture contact, contact release and re-contact algorithms were implemented in the code. The reduction curves of elastic modulus and yield strength of steel related to elevated temperature, shown by NIST, were adopted to consider the heat effect of fire. Thermal expansion of materials was also considered.<br> From the numerical results of the fire-induced collapse analyses, it is found that the scale of collapse highly depends on fire ranges and locations. The risk of total collapse tends to increase when a large range of fire occurs in the lower layer. In addition, the risk increases much higher in those cases when fire occurs at the peripheral area of the building, compared to those cases at the inner area. The sum of heights of remains seems to depend on the sum of KI values, and there is a specific threshold of the sum of KI values that relates to the initiation of a large scale collapse. Therefore, the sum of KI values may be used to predict and take measures to avoid the risk of collapse under various fire conditions.