Structural failure will occur when the energy impact to the system exceeds the capacity of energy absorption of the structure. The energy absorption within the elastic range has already been studied by Drs. Muto and Umemura. The author has tried to expand this theory beyond the yield point. By the principle of energy conservation, the following equation generally holds. Energy input=Absorbed energy+Kinetic energy+Feed back energy. A: Elastic behavior 1. Let N, M and Q denote axial force, bending moment and shearing force, respectively, in a structural member, Elastic energy=[numerical formula] where ∫applies to whole length of each member, Σ denotes the summation of all members, and k means the characteristic modulus. 2. When the external load is in equilibrium with resisting force of the structure (restoring force plus inertia force, neglecting both energy dissipation and effect of Poisson's ratio), U=φ_e=EA⊿ where U=energy input φ_e=elastic energy A=sectional area ⊿=deflection 3. When the more energy is dissipated, the less stress is induced in the structure due to the equal energy input, hence the structure becomes safer, and vice-versa. B: Plastic behavior 1. General consideration The absorption energy of the structure (φ_p) is the sum of each absorption energy of the members, and in plastic range N_p, M_p and Q_p are constant, then φ_p=[numerical formula] Summarily the absorption energy in static condition is as follows: Absorption energy=Elastic a.e.+Plastic a.e.+creep a.e. 2. Simple example (a) Simple beam with uniform load: U=[numerical formula] where l=span θ_0=end rotation at elastic limit θ=end rotation at plastic limit (b) Cantilever with uniform load: U=[numerical formula] (c) Fixed beam with uniform load: U=[numerical formula] (d) Reinforced Concrete beam: Above equations can be used, and [numerical formula] where Z=plastic section modulus Z_e=elastic section modulus [numerical formula] [numerical formula](by Dr. Umemura) θ=0.02〜0.05 radian (by Dr. Ban) By the calculation in simple examples, the auther has shown that the elastic absorption is very much less than the plastic absorption energy, and in these examples the ratio were about 1.0% in the care of the reinforced concrete beam and about 0.24% in the case of the I beam.
