We combine the analytical theory of nonaffine deformations of noncrystalline solids with numerical Stokesian dynamical simulations to obtain analytical closed-form expressions for the shear modulus of fractal aggregates in shear flows. The proposed framework also provides analytical predictions for the evolution of the fractal dimension d(f) of the aggregate during the aggregation process. This leads to a lower bound on d(f) below which aggregates are mechanically unstable (they possess floppy modes) and cannot survive without restructuring into more compact, higher-d(f) configurations. In the limit of large aggregates, the predicted lower bound is d(f) = 2.407. This result provides the long-sought explanation as to why all experimental and simulation studies in the past consistently reported d(f) greater than or similar to 2.4 for shear-induced colloidal aggregation. The analytical expressions derived here can be used within population balance calculations of colloidal aggregation in shear flows whereby until now the fractal dimension evolution was treated as a free parameter. These results may open up the possibility of developing new microscopic mechanical manipulation techniques to control nanoparticle and colloidal aggregates at the nanoscale.