Table 3 Equations of each energy for energy conservation law.

Kinetic energy \(E_{k}\)

\(E_{k} = 0.5m_{h} V_{i}^{2} = m_{h} gh_{d}\)

Potential energy \(E_{p}\)

\(E_{p} = \left\{ {\begin{array}{*{20}l} {\left( {m_{be} + m_{h} } \right)g\delta } & {{\text{for }}\delta < \delta_{y} } \\ {\left( {m_{be} + m_{h} } \right)g\delta + \left( {m_{bp} + m_{h} } \right)g\left( {\delta - \delta_{y} } \right)} & {{\text{for }}\delta < \delta_{y} } \\ \end{array} } \right.\)

Deformation energy \(E_{d}\)

\(E_{d} = \int_{0}^{\delta } {P(\delta )} d\delta\)

Spalling energy \(E_{s}\)

\(E_{s} = \left\{ {\begin{array}{*{20}l} 0 & {{\text{for }}\varepsilon_{c} < \varepsilon_{cu} } \\ {0.2f_{td} bc_{c} k_{s} \left( {L_{p} + 2C_{c} } \right)} & {{\text{for }}\varepsilon_{cs} < \varepsilon_{yd} } \\ \end{array} } \right.\)

\(k_{s} = (300/h)^{0.25} \le 1\)

Energy loss \(E_{l}\)

\(E_{l} = E_{k} - \frac{1}{2}(m_{be} + m_{h} )V_{c}^{2} = \frac{{m_{be} }}{{m_{be} + m_{h} }}E_{k}\)