Structural analysis begins with the fundamental distinction between statically determinate and indeterminate structures. A structure is statically determinate when all internal forces and support reactions can be determined using only the equations of static equilibrium, ΣF = 0 and ΣM = 0. When a structure carries more unknowns than the available independent equilibrium equations can resolve, the difference defines its degree of static indeterminacy, which equals the number of redundant forces that must be analyzed through additional methods such as compatibility of deformations or energy principles.
For indeterminate beams and frames, the moment distribution method provides an iterative approach to finding member end moments. Each beam member is assigned a stiffness factor, expressed as k = 4EI/L for a member whose far end is fixed, and k = 3EI/L when the far end is pinned, where E is the elastic modulus, I is the moment of inertia, and L is the member length. As unbalanced moments at a joint are distributed to connected members in proportion to their stiffnesses, half of the moment carried to the near joint is also transferred to the far joint, an effect captured by the carry-over factor of 0.5 for members with fixed far ends. The iteration continues until the unbalanced moments at every joint become negligibly small, indicating equilibrium has been achieved.
The principle of superposition states that for linear elastic structures, the total response (deflection, bending moment, or shear) produced by multiple loads equals the algebraic sum of the responses produced by each load acting independently. Building on this, the virtual work method calculates deflections in trusses and beams by applying a virtual unit load at the point and direction of interest and integrating the product of real and virtual internal work along the member. Once the elastic range is exceeded and a section reaches the full plastic moment Mp, a plastic hinge forms, allowing rotation without further resistance. This capacity for rotation enables moment redistribution in beams and frames, a concept central to plastic analysis and capacity-based design.
Slenderness governs how a column fails under compression. The slenderness ratio is expressed as KL/r, where K is the effective length factor based on end conditions, L is the unbraced length, and r is the radius of gyration. Long, slender columns buckle elastically at the Euler critical load, given by P_cr = π²EI / (KL)², before the material reaches its yield strength. Stocky columns, by contrast, fail by crushing at or near the material yield stress, so design codes distinguish between the elastic buckling range and the inelastic or short-column range to determine the governing failure mode.