Plants sustain itself with slender stems upright to reach sometimes over 100 meter height. To support its own weight, plant cell wall makes nanocomposite material composed of densely packed crystalline cellulose as tensile resistant element, whose estimated Young’s modulus (150 GPa) is the highest among all biomolecules. A historical common-sense assertion considers that the particular hydrogen bonds along the fiber direction are responsible to its high rigidity. To verify this long standing myth, we estimated elastic tensors using density functional theory (DFT) and different hypothetic analogues. Replacing hydroxyl groups with fluorine results in similar packing without hydrogen bonds, and dispersion interaction can be suppressed by leaving the correction term out of the calculation. Both hydrogen bonds and dispersion interactions contribute substantially to stiffness in the transverse direction, reducing stiffness by 30 to 70%, whereas their contribution to longitudinal stiffness is moderate (21%). Stiffness along the chains is dominated by the covalent bonds in the axial direction and limitations of reorientation due to the compact packing of macromolecular chains. Our study reveals the molecular nature of the high modulus of cellulose and provides a basis for understanding cellulose structure-property relationships.