  Because the constant extension ensemble of single chain molecule is not
always equivalent with constant force ensemble, a model of double-stranded
conformations, as in RNA molecules and $\beta$-sheets in proteins, with fixed
extension constraint is built in this paper. Based on polymer-graph theory and
the self-avoiding walks, sequence dependence and excluded-volume interactions
are explicitly taken into account. Using the model, we investigate
force-extension curves, contact distributions and force-temperature curves at
given extensions. We find that, for the same homogeneous chains, the
force-extension curves are almost consistent with the extension-force curves in
the conjugated force ensembles. Especially, the consistence depends on chain
lengths. But the curves of the two ensembles are completely different from each
other if sequences are considered. In addition, contact distributions of
homogeneous sequence show that the double-stranded regions in hairpin
conformations tend to locate at two sides of the chain. We contribute the
unexpected phenomena to the nonuniformity of excluded-volume interactions of
the region and two tails with different lengths. This tendency will disappear
if the interactions are canceled. Finally, in constant extension ensemble, the
force-flipping transitions conjugated with re-entering phenomena in constant
force ensemble are observed in hairpin conformations, while they do not present
in secondary structure conformations.
