Theory of Structures II (ENCE 252): Slope Deflection Method

The Slope Deflection Method Explained

The Slope Deflection Method is a classical displacement-based approach for analyzing statically indeterminate beams and frames. Unlike force methods, which treat redundant forces as unknowns, this method treats the rotations (slopes) and displacements of the joints as the primary unknowns. It’s a precursor to modern matrix methods of structural analysis and provides a powerful way to handle structures with a high degree of kinematic indeterminacy.

The core of the method lies in relating the moments at the ends of each member to the rotations and displacements of its ends and any external loads applied to the member. This is achieved through the “slope-deflection equations.” Once these equations are established for every member, joint equilibrium equations are formulated and solved to find the unknown joint rotations and displacements. Finally, the member end moments and support reactions are calculated.

The general form of the slope-deflection equation for a member AB is expressed as: $$ M_{AB} = M_{FAB} + \frac{2EI}{L}(2\theta_A + \theta_B – \frac{3\Delta}{L}) $$ Where $M_{AB}$ is the final moment, $M_{FAB}$ is the fixed-end moment, $E$ is the modulus of elasticity, $I$ is the moment of inertia, $L$ is the member length, $\theta_A$ and $\theta_B$ are the slopes at ends A and B, and $\Delta$ is the relative displacement (settlement).