  We update gamma-ray burst (GRB) luminosity relations among certain spectral
and light-curve features with 139 GRBs. The distance modulus of 82 GRBs at
$z>1.4$ can be calibrated with the sample at $z\leq1.4$ by using the cubic
spline interpolation method from the Union2.1 Type Ia supernovae (SNe Ia) set.
We investigate the joint constraints on the Cardassian expansion model and dark
energy with 580 Union2.1 SNe Ia sample ($z<1.4$) and 82 calibrated GRBs data
($1.4<z\leq8.2$). In $\Lambda$CDM, we find that adding 82 high-\emph{z} GRBs to
580 SNe Ia significantly improves the constrain on
$\Omega_{m}-\Omega_{\Lambda}$ plane. In the Cardassian expansion model, the
best fit is $\Omega_{m}= 0.24_{-0.15}^{+0.15}$ and $n=0.16_{-0.52}^{+0.30}$
$(1\sigma)$, which is consistent with the $\Lambda$CDM cosmology $(n=0)$ in the
$1\sigma$ confidence region. We also discuss two dark energy models in which
the equation of state $w(z)$ is parametrized as $w(z)=w_{0}$ and
$w(z)=w_{0}+w_{1}z/(1+z)$, respectively. Based on our analysis, we see that our
Universe at higher redshift up to $z=8.2$ is consistent with the concordance
model within $1\sigma$ confidence level.
