# A Case Study on the Rehabilitation of a Fire-Damaged Structure
## 4 Numerical Analysis  
This section discusses the results of the finite element analysis on a typical sample girder and two typical columns in the Battery Room #6 performed to investigate their remaining structural capacity. As target structural components, Girder #2 and Columns #3 and #4 were selected. (Refer to Figure 1 for identification of these members.) Girder #2 was chosen because it transfers the load from adjoining slabs directly to the columns. As illustrated in Figure 14, different damaged models were considered for the two target columns as they have different shapes of exposure to fire, _i.e._, a shape of channel for Column #3 and a shape of angle for Column #4. The following assumptions are made in the course of modeling. They are based on the results of the on-site inspection and tests on the sampled core cylinders (see Table 1) and rebar coupons (see Table 2): \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Specimen No.** & **1-1** & **1-2** & **1-3** & **2-1** & **2-2** & **3-1** & **3-2** & **3-3** \\ \hline Diameter (mm) & 94.0 & 93.7 & 93.7 & 93.8 & 93.9 & 93.9 & 93.8 & 93.9 \\ Length (mm) & 80.2 & 99.9 & 93.9 & 91.3 & 127.7 & 144.9 & 184.7 & 71.5 \\ Length/diameter & 0.853 & 1.066 & 1.002 & 0.973 & 1.360 & 1.543 & 1.969 & 0.761 \\ Correction factor & 0.936 & 1.025 & 1.001 & 0.989 & 1.118 & 1.164 & 1.245 & 0.889 \\ Max. load (kN) & 250.7 & 198.6 & 210.3 & 194.8 & 225.2 & 174.3 & 256.8 & 277.2 \\ Measured strength & 36.1 & 28.8 & 30.5 & 28.2 & 32.5 & 25.2 & 37.2 & 40.0 \\ Estimated strength per BS1881 (MPa) & 33.8 & 29.5 & 30.5 & 27.9 & 36.4 & 29.3 & 46.3 & 35.6 \\ \hline \end{tabular}
\end{table} 
Table 1: Test results of sampled core cylinders. \begin{table}
\begin{tabular}{c c c c} \hline
**Specimen No.** & **Yield Strength (MPa)** & **Tensile Strength (MPa)** & **Elongation (\%)** \\ \hline \(\phi\)25 & 523 \(\pm\) 19 & 656 \(\pm\) 22 & 23 \(\pm\) 1.01 \\ \(\phi\)12 & 381 \(\pm\) 16 & 507 \(\pm\) 22 & 32 \(\pm\) 1.40 \\ \(\phi\)14-1 & 255 \(\pm\) 10 & 403 \(\pm\) 7 & 17.47 \(\pm\) 0.56 \\ \(\phi\)14-2 & 516 \(\pm\) 21 & 590 \(\pm\) 10 & 8.37 \(\pm\) 0.42 \\ \(\phi\)14-3 & 267 \(\pm\) 11 & 415 \(\pm\) 7 & 19.39 \(\pm\) 0.59 \\ \(\phi\)14-1 & 237 \(\pm\) 10 & 395 \(\pm\) 7 & 12.15 \(\pm\) 0.48 \\ \hline \end{tabular}
\end{table} 
Table 2: Test results of sampled rebar coupons. 1. Rebar is undamaged;
2. The depth of damaged concrete is about 50 mm as shown in Figure 14;
3. The residual strength of the damaged concrete is approximately 60% compared with the original capacity. A material and geometric nonlinear finite element analysis was conducted on the target structural components using Abaqus/Standard v6.12 (Dassault Systemes Simulia Corp., Providence, RI, USA, 2012), which is a well-known commercial finite element analysis package. As a concrete material model, the so-called \"concrete damaged plasticity model\" was used. In this model, the behavior of concrete under compression is defined by [PERSON]'s model, which is discussed in ([PERSON], 1998), while its behavior under tension is defined by a bi-linear tension softening curve with a peak strength corresponding to 10% of its compressive strength. The compressive strength, elastic modulus and Poisson's ratio of concrete are 30 MPa, 52.4 GPa and 0.15, respectively. For rebar, a bi-linear stress-strain relationship was applied without considering strain hardening to conservatively evaluate the remaining capacity of the fire-damaged structural components. Their tensile yield strength and elastic modulus are 500 MPa and 206,000 MPa, respectively. Figure 15 shows the numerical models of the target structural components including rebar arrangement. The target concrete girders and columns were modeled using 8-node solid elements with reduced integration (C3D8R). The rebar was modeled using truss elements (T3D2) with the embedded option in ABAQUS.