## 4 Degradation of Compressive Strength with Time  
The resistance of timber structures is mainly dependent on the effects of temperature, relative humidity, loading, and the compressive strength of the material. When these factors are assumed to be independent, a multivariate time-dependent model on the bearing capacity of compression timber member can be formulated, considering the load duration and the degree of timber decay, as [36]  
\[f_{N}(t)=f_{0}\cdot\varphi_{f}(\tau,t)\cdot\varphi_{\Delta}(\delta(t)) \tag{10}\]  
where \(f_{N}(t)\) is the compressive strength at time, \(t\); \(f_{0}\) is the initial compressive strength of timber; \(\varphi_{f}(\tau,t)\) is the time-dependent model of the design strength, including the load duration effect and stress level, \(\tau\), defined as the ratio of actual stress to compressive strength; \(\varphi_{\Delta}(\delta)\) is the time-dependent function of the geometric parameter of component sections with decay [4]; and \(\delta(t)\) is the time-dependent function of the thickness of the decay layer of timber, as shown in Equation (13). A model of the cumulative damage of timber, considering effects of temperature and humidity, can also be derived based on a large number of experiments and [PERSON] cumulative damage model, as [36, 37]  
\[\varphi_{f}(\tau,t)=\frac{1}{B}\ln\{1+[1-t\cdot\exp(-A+\tau B)](\exp B-1)\} \tag{11}\]  
where parameters \(A=13.9526\) and \(B=25.0498\), with the stress level, \(\tau\), usually not exceeding 0.35 in a traditional timber structure [36]. The time-dependent function of the geometric characteristics of a component section under a long-term load can be obtained from [36] as  
\[\varphi_{\Delta}(\delta)=\left(1-\frac{2\delta}{D}\right)^{2} \tag{12}\]  
where \(D\) is the diameter of the compressed timber member. The time-dependent function of the thickness of the decay layer of timber may be taken as [38]  
\[\delta(t)=\frac{\delta_{0}}{2}\left(1+\frac{t}{T_{0}}\right)^{\alpha} \tag{13}\]  
where \(\delta_{0}\) is the existing thickness, \(T_{0}\) is the age of the compression component up to the present, and \(\alpha\) is a parameter of the rate of degeneration of the thickness of the metamorphic layer [4]. The time-dependent model for a log of the new CL, as of today, is discussed below. Table 4 shows that \(f_{0}=35.21\) MPa for the new CL. \(D=149.5\) mm is the mean value of the new CL logs from Table 1. Other parameters and coefficients are assumed to be the same \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Serial** & \(f_{\text{c,mean}}\) & \(K\) & \(\tilde{f}_{\text{c}}\) & \(\tilde{f}_{\text{m-cBP}}\) & \(\delta_{\text{M}}\) \\
**Number** & **(MPa)** & & **(MPa)** & **(MPa)** & **(MPa)** & **(\%)** \\ \hline Aged CL-1 & & & & & \\ \((t=100)\) & 41.98 & 0.65 & 50.27 & 32.68 & 22.15 \\ Aged CL-2 & & & & & \\ \((t=100)\) & 29.56 & 0.65 & 39.91 & 25.94 & 12.25 \\ Aged CL-3 & & & & & \\ \((t=100)\) & 19.37 & 0.65 & 34.00 & 22.10 & 14.09 \\ New CL-4 & & & & & \\ \((t=0)\) & 36.76 & 0.47 & 39.61 & 18.62 & 49.35 \\ New CL-5 & & & & & \\ \((t=0)\) & 33.66 & 0.47 & 39.61 & 18.62 & 44.68 \\ \hline \hline \end{tabular} Note: The difference \(\delta_{\text{M}}\) is calculated by \(\delta_{\text{M}}=[f_{\text{m}}-f_{\text{m-CBP}}]/f_{\text{m}}\%\).  
\end{table} 
Table 7: Estimation of the compressive strength of the log samples, according to [35]. as those in [36], with \(T_{0}\) assumed to be one year without any decay layers, \(\delta_{0}=0.05\) mm, and power index \(\alpha=1.0\). Substituting all of the above parameters into Equations (11)-(13), the time-dependent model for parallel-to-grain compressive strength can be obtained, as shown in Figure 8a. The curves show that the rate of degradation of parallel-to-grain compressive strength becomes larger with the stress levels \(\tau\) with larger residual strength.