What is Design

By design it is meant that selection of sizes of members to resist the forces and moments found by analysis of the structure. It is designed because the resulting structure should be safe and economical. Analysis of statically determinate structures (simply supported beams, cantilever truss etc) is simple. But the analysis of statically indeterminate structures is difficult (portal frames and multistory building). It requires the cross sectional properties which is initially unknown. By thumb rule assume sections and analysis is done, if it found not satisfactory reanalysis is done.

Limitations of Design Method

Finding forces and checking the member sizes for strength, stability, deflections, has limitations

1)     Strength and stability interaction of individual member and structure is not considered. In practice effective length concept is used.

2)     For non-rectangular and irregular frames the effective length factor could not find easily by elastic analysis.

3)     When limit state method is used the serious limitation occurs. There is no verification of the compatibility condition. The strength equations in the design codes do not consider the system compatibility.

In the design of the structural system inelastic analysis for strength and stability must be done. Because the structure will behave non linearly.

Methods of Analysis

It is very difficult to model the actual behavior of a structure. Various methods of analysis

1)     Elastic analysis

2)     Linear buckling analysis

3)     Inelastic analysis

4)     Inelastic buckling analysis

5)     Plastic analysis

Elastic analysis

Slope deflection method and moment distribution method are examples of elastic analysis

It is assumed that material is linear elastic. Analysis may be done based on the equilibrium in the under deformed geometry of the structure. Assumptions

1)     Materials behave linearly

2)     The members behave linearly  member instability is ignored

3)     The frame instability is ignored frame behaves linearly

Linear Buckling Analysis

The elastic buckling load is calculated. The loads at which displacements occur is called buckling load. The effective lengths of columns are found. Effective length/least radius of gyration is found, which should be in the allowable limits. Effective length is calculated from end conditions and critical load can be calculated from Eulers equation. 

Inelastic buckling Analysis

By modifying structural stiffness matrix one can obtain critical loads that represent the actual limit state of the structure in a better manner. Column stiffness is reduced due to axial compression and residual stresses. Computer programs are also available to calculate inelastic critical loads of frames.

Sources of non linearity

The behavior of frames is non linear due to the following effects. Material effect that influence the steel frame design

1)     Gradual yielding due to bending, while bending induced by external forces the cross section will remain elastic only up to small strains, when it increases the beam yield gradually and finally become plastic. The gradual yielding leads to plastic hinges. If plastic hinges is formed the structure may collapses for small loads.

2)     Yielding due to residual stresses, due to residual stresses also the member become perfectly plastic and stiffness will be reduced. Residual stresses may result from manufacturing/fabrication processes.

Conclusion 

If a structure is accidently over loaded, which may be due to earth quake plastic hinges may form and stiffness of the member or frame is reduced, the structure will fail with a lesser force than designed.