Structural idealization and formulation of problems

Structural idealization

Structural idealization is the process of replacing a real structure with a simplified analytical model that keeps its essential behaviour but can be analysed by the theory of structures. Members are reduced to lines along their centroidal axes, supports to idealized types (fixed, hinged, roller), joints to rigid or pinned connections, loads to point, uniformly distributed or equivalent loads, and the material to a linear elastic one. For example, an RCC building frame is analysed as a plane frame of centre lines with fixed bases, and a roof truss as a pin-jointed truss with loads applied at the joints.

Steps in identification and formulation of the problem

1. Define the function and requirements. State what the structure must do (carry floor or traffic loads, span a gap) and the conditions it must satisfy: strength, stiffness and stability.
2. Idealize the structure. Draw the line diagram with member properties ($E$, $A$, $I$), joint types and support types.
3. Idealize the loads. Collect dead, live, wind, seismic and temperature or settlement effects and arrange them into load cases.
4. Check stability and determinacy. Check that the supports and members make a geometrically stable system, then find $D_s$ and $D_k$.
5. Select the method of analysis. Use equilibrium alone if determinate; otherwise choose the force method (unknowns are redundants, $D_s$) or the displacement method (unknowns are joint displacements, $D_k$), whichever has fewer unknowns.
6. Write the governing equations. Equilibrium, compatibility and force-displacement (constitutive) relations. Solve for the unknown forces or displacements.
7. Find the final results and check. Draw the axial force, shear force and bending moment diagrams, find deflections, and verify equilibrium and compatibility. Compare the results with the strength and stiffness limits.