\documentclass[border=10pt]{standalone}
\usepackage{pgfplots, amsmath}
\usepackage{filecontents}

\definecolor{color1}{rgb}{0.9,0.9,0.9}
\definecolor{color2}{rgb}{0.7,0.7,0.7}
\definecolor{color3}{rgb}{0.5,0.5,0.5}
\definecolor{color4}{rgb}{0.3,0.3,0.3}
\definecolor{color5}{rgb}{0.1,0.1,0.1}

%\pgfplotsset{every axis plot/.append style={line width=1pt}}
\pgfplotsset{every axis plot/.append style={line width=1pt},compat=1.8}

\begin{document}
	\begin{tikzpicture}
	\begin{axis}[
	xlabel=$t$,
	ylabel= $\sigma_h$,
	width = 7cm,
	height = 5cm,
	ymin = -0.45,
	%ymin = 1,
	%domain = 3.5:450,
	%transpose legend,
	%legend columns=-1,
	%legend style={at={(0.5,1.30)},anchor=north},
	minor tick style={draw=none},
	legend entries ={$C=0$, $C=0.25$, $C=0.5$,$C=0.75$,$C=1$},
	legend style={font=\tiny,at={(1,0)},anchor=south east,cells={anchor=west}},
	]
	\addplot +[color1,mark=none] table[x index=0,y index=1] {data_P_0.txt};
	\addplot +[color2,mark=none] table[x index=0,y index=1] {data_P_25.txt};
	\addplot +[color3,mark=none] table[x index=0,y index=1] {data_P_50.txt};
	\addplot +[color4,mark=none] table[x index=0,y index=1] {data_P_75.txt};
	\addplot +[color5,mark=none] table[x index=0,y index=1] {data_P_100.txt};
	%\legend{P=0, P=0.25, P=0.5,P=0.75,P=1}
	\end{axis}
	\end{tikzpicture}
\end{document}