Steel Buildings in Europe

Part 4: Detailed Design 4 – 5 V V V V k k h 1 1 1 H H H 2 2 2 ( / h ) H 1    First order effects k  1 = H 1 Second order effects k  2 = H 1 + V (  2 / h ) = H 2 Figure 2.2 First and second order effects in a pinned braced frame The equilibrium expression for the second order condition may be rearranged as: H 2 = H 1        V kh / 1 1 Thus, it can be seen that, if the stiffness k is large, there is very little amplification of the applied horizontal force and consideration of first order effects only would be adequate. On the other hand, if the external horizontal force, H 1 , is kept constant while the value of total vertical force V tends toward a critical value V cr (= kh ), then displacements and forces in the restraint tend toward infinity. The ratio V cr / V , which may be expressed as a parameter  cr is thus an indication of the second order amplification of displacements and forces in the bracing system due to second order effects. The amplifier is given by:          cr 1 1/ 1  EN 1993-1-1 [1] presents both general rules and specific rules for buildings. In order to cover all cases, § 5.2.1 of that code considers the applied loading system, F Ed , comprising both horizontal forces H Ed and vertical forces V Ed . The magnitudes of these forces are compared to the elastic critical buckling load for the frame, F cr . The measure of frame stability,  cr is defined as Ed cr F F . Although F cr may be determined by software or from stability functions, the Eurocode provides a simple approach to calculate  cr directly in § 5.2.1(4)B:                H,ED Ed Ed cr   h V H where:  cr is the factor by which the design loading would have to be increased to cause elastic instability in a global mode H Ed is the design value of the horizontal reaction at the bottom of the storey to the horizontal loads and the equivalent horizontal forces