A gravity theory called scalar-tensor-vector gravity (STVG) has been recently developed and succeeded in solar system, astrophysical and cosmological scales without dark matter [J. W. Moffat, J. Cosmol. Astropart. Phys. 03, 004 (2006)]. However, two assumptions have been used: (i) $B(r)=A^{-1}(r)$, where $B(r)$ and $A(r)$ are $g_{00}$ and $g_{rr}$ in the Schwarzschild coordinates (static and spherically symmetric); (ii) scalar field $G=Const.$ in the solar system. These two assumptions actually imply that the standard parametrized post-Newtonian parameter $gamma=1$. In this paper, we relax these two assumptions and study STVG further by using the post-Newtonian (PN) approximation approach. With abandoning the assumptions, we find $gamma eq1$ in general cases of STVG. Then, a version of modified STVG (MSTVG) is proposed through introducing a coupling function of scalar field G: $theta(G)$. We have derived the metric and equations of motion (EOM) in 1PN for general matter without specific equation of state and $N$ point masses firstly. Subsequently, the secular periastron precession $dot{omega}$ of binary pulsars in harmonic coordinates is given. After discussing two PPN parameters ($gamma$ and $beta$) and two Yukawa parameters ($alpha$ and $lambda$), we use $dot{omega}$ of four binary pulsars data (PSR B1913+16, PSR B1534+12, PSR J0737-3039 and PSR B2127+11C) to constrain the Yukawa parameters for MSTVG: $lambda=(3.97pm0.01)times10^{8}$m and $alpha=(2.40pm0.02)times10^{-8}$ if we fix $|2gamma-beta-1|=0$.
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