! Missing } inserted.<inserted text>} \end{gather}

Question

\begin{gather}
\sigma_1$ = $\dfrac{M_1.y}{I_x}$ + $\dfrac{F_z}{A}\tag{1}\\
\sigma_2 = \dfrac{M_2.y}{I_x} + \dfrac{F_z}{A}\tag{2}\\
M_1 = F_y.L_1\tag{3}\\
M_1 = F_y.(L_1 + L_2)\tag{4}\\
\sigma_3 = \dfrac{M_3.y}{I_y} + \dfrac{F_z}{A}\tag{5}\\
\sigma_4 = \dfrac{M_4.y}{I_y} + \dfrac{F_z}{A}\tag{6}\\
M_3 = F_y.L_1\tag{7}\\
M_4 = F_y.(L_1 + L_2)\tag{8}
\end{gather}

Answer 1

• The immediate source of the crash is the presence of the four $ symbols. Either get rid of them or replace them with \$ -- in case you need to display $ symbols, which doesn't seem likely at all.

• In addition, you should get rid of the eight \tag statements -- they're redundant. And, you should either omit the . ("period", aka "full stop") symbols entirely or, at the very least, replace them with \cdot. (I favor the first option.)

• You're in display math mode by default inside a gather environment. Thus, replace all \dfrac statements with \frac.

• Optional: Change all 8 instances of = to &= and replace gather with `align.

• Optional (but suggested by @egreg, and hence to be taken seriously): Make the vertical distances between the rows equal by inserting (typographic) struts in the lines with a \frac term. In the solution below, look for the definition and use of \mystrut.

\documentclass{article}
\usepackage{amsmath} % for 'align' environment
%% Define a typographic strut:
\newcommand\mystrut{\vphantom{\frac{My}{I_y}}}
\begin{document}

\begin{align}
\sigma_1 &= \frac{M_1y}{I_x} + \frac{F_z}{A}\\
\sigma_2 &= \frac{M_2y}{I_x} + \frac{F_z}{A}\\
M_1      &= F_yL_1          \mystrut\\
M_1      &= F_y(L_1 + L_2)  \mystrut\\
\sigma_3 &= \frac{M_3y}{I_y} + \frac{F_z}{A}\\
\sigma_4 &= \frac{M_4y}{I_y} + \frac{F_z}{A}\\
M_3      &= F_yL_1          \mystrut\\
M_4      &= F_y(L_1 + L_2)  \smash[b]{\mystrut}
\end{align}

\end{document}