Second order effects

Imperfections

Imperfections Frame imperfection Member imperfection: always to be allowed for Member imperfection: only for slender members (rare) in sway frames, otherwise it is covered in the relevant buckling curve

Frame imperfection The frame imperfection is as follows:

Frame imperfection Frame imperfection can be replaced by an equivalent closed system of horizontal forces applied at the floor levels (including the foundation level).

Methods of structural analysis

Choice between elastic analysis and plastic analysis Elastic analysis can always be used. Plastic analysis allowed only when one meets the restrictions on steel properties, cross-section classification, restraints (at or near plastic hinges) and, if needed, on joint ductility.

Frame classification

Braced and unbraced frames A frame without bracing is unbraced. A frame with bracing is classified as braced only when the braced structure is sufficiently stiff laterally.

Braced and unbraced frames Frame without bracing is unbraced Frame with bracing is unbraced when: braced when: where are the lateral flexibilities of the structure with and without the bracing respectively

Sway and non-sway Non-sway frame when: Sway frame when: where: is the design value of the total vertical load applied to the structure and: is the elastic critical vertical load producing sway instability of the structure

Elastic critical load The following approach may be used: Approximate procedure: where: d is the sway at the top of storey i h is the height of storey i H and V are the total horizontal and vertical reactions respectively at the bottom of storey i

Frame classification Decision on the use of bracing or not influences sway classification Preliminary member sizing and estimates of column vertical loads: use to provide an indication of the sway classification using:

Frame classification and type of frame analysis 1st order 2nd order

Second order analysis for sway frames “True” second order analysis includes both the sway effect (P-D effect) and the P-d effect (member deflection effect) by modifying the member stiffness terms. Load is applied in increments. Iteration and convergence procedures are required. For most sway frames, only the P-D effect (sway effect) needs to be allowed for. The concept of the Equivalent Lateral Load Method may also be used. Eurocode 3: indirect methods often possible.

Sway frame : 2nd order effects Alternatives to a “true” 2nd order analysis: 1st order analysis + “Amplified Sway Moment” method when: 1st order analysis + “Sway Mode Buckling Length” method (20% sway moment increase)

Check for joint stiffness when rigid joints are assumed Joint must meet “rigid joint” classification criteria

Check of initial semi-rigid joint “good guess” stiffnesses Semi-rigid joint stiffnesses can vary within a fairly wide range without significantly altering the frame resistance

Fixity factor approach: check for joint stiffnesses