The outer membrane is the defining structure of Gram-negative bacteria. We previously demonstrated that it is critical for the mechanical integrity of the cell envelope and therefore to the robustness of the bacterial cell as a whole. Here, to determine the key molecules and moieties within the outer membrane that underlie its contribution to cell envelope mechanics, we measured cell-envelope stiffness across several sets of mutants with altered outer-membrane sugar content, protein content, and electric charge. To decouple outer membrane stiffness from total cell envelope stiffness, we developed a novel microfluidics-based "osmotic force extension" assay. In tandem, we developed a simple method to increase throughput of microfluidics experiments by performing them on color-coded pools of mutants. Using Escherichia coli as a model Gram-negative bacterium, we found that truncating the core oligosaccharide, deleting the beta-barrel protein OmpA, or deleting lipoprotein outer membrane-cell wall linkers all had the same modest, convergent effect on total cell-envelope stiffness but had large, varying effects on the ability of the cell wall to transfer tension to the outer membrane during large hyperosmotic shocks. Surprisingly, altering lipid A charge had little effect on the mechanical properties of the envelope. Importantly, the presence or absence of OmpA determined whether truncating the core oligosaccharide decreased or increased envelope stiffness (respectively), revealing sign epistasis between these components. Based on these data we propose a specific structural model in which the chemical interactions between lipopolysaccharides, beta-barrel proteins, and phospholipids coordinately determine cell envelope stiffness, and the ability of the outer membrane to functionally share mechanical loads with the cell wall.