In this paper, we investigate the problem of linear joint probabilistic constraints. We assume that the rows of the constraint matrix are dependent and the dependence is driven by a convenient Archimedean copula. Further we assume the distribution of the constraint rows to be elliptically distributed, covering normal, $t$, or Laplace distributions. Under these and some additional conditions, we prove the convexity of the investigated set of feasible solutions. We also develop two approximation schemes for this class of stochastic programming problems based on second-order cone programming, which provides lower and upper bounds. Finally, a computational study on randomly generated data is given to illustrate the tightness of these bounds.