Assessing destructive capacity of impulse loads

The paper gives an overview of the methodology to find dynamic coefficients of impulse loads typical of various emergencies (explosive blast, massive body falling, hydraulic shock, etc.). Criteria are described under which dynamic coefficients are determined by impulse impact alone irrelevant of its form. Correlations are quoted to find dynamic coefficients for impact loads that accompany such emergencies.


Introduction
It is the accepted practice of construction design that non-static loads are reduced to equivalent static loads.Calculations of dynamic impact on building structures use the following methodology.Dynamic load is substituted with its equivalent static load found by this formula [1][2][3]: where P max с -maximum dynamic load value; К d -dynamic coefficient equal to maximum value of dynamic function Т(t) that describes motion of the structure (its sections shifting) in time.
As is apparent from (2.2.1), to substitute dynamic load with its equivalent load we need to know dynamic factor K d .
For example, dynamic coefficient Кd for a triangular-shaped load is frequently known from diagrams (Fig. 1) available in many reference manuals on strength calculation of structures, where T and T 1 -duration of dynamic load and the time it takes to peak; Z Кfrequency of main oscillation pitch in the structure, found using the methods of construction mechanics.
The diagram means that if T 1 tends to zero (saw-tooth load form), then the dynamic coefficient's numeric value approaches 2. Considering this, designers double any sawtooth load when they do static calculations.However, this approach is wrong for short-term (impulse) loads.This is the issue discussed in this paper.

Fig. 1. Diagrams of dynamic coefficient Кd
Emergency situations create momentary loads rather often.Discharge of explosive material or impact of a massive body falling (e.g.transport containers falling in a cooling pool of a nuclear power plant) or hydraulic shock etc. creates loads that last for milliseconds at very high pressures.
Let us examine the dynamic factor methodology for such loads.This discussion will adhere to the approaches accepted in structural dynamics.
We know that the structure's movement at the elastic deformation stage is described by the equation: ), ( where Z -circular frequency of the structure's own vibrations круговая частота; f(t) - T(t)=S(t)/S МАКС -relative movement of the structure, or dynamic function; Smax -offset of the structure responding to static load -Рmax.
The solution to equation (1) with zero initial conditions appears as: ). ) sin( ) ( cos To find the dynamic coefficient we need to know the frequency of the structure's own oscillation pitch -Z, expressed with formula [4]: where L -structure's span; В -bend rigidity: , where Е -dynamic modulus of elasticity, I -axial moment of inertia, М к -linear mass of structure: М q g к , where qlinear load from explosion, plus load of the structure's own weight, Dfactor quoted from reference literature (e.g., [4]), g -free fall acceleration.Equation ( 1) can be solved by solving numerically this system of differential equations of the first order: This method has the following disadvantage.Calculations assume that the process of each element's oscillation uses a certain frequency (typically that of the pitch), although in reality oscillations occur across the entire spectrum of natural frequencies of the elastic element.Moreover, arbitrary limiting conditions can be set with approximation in this method, but to calculate strength of building structures in emergency, when maximum possible loads are assumed as inputs, even though they are unlikely, this approach to calculate K d is justified to an extent.
Equation ( 1) was integrated for several load types and for different ratios of load duration (Т load ) to the structure's own oscillation period (Т 0 ).Fig. 2 gives values of dynamic coefficients for different ratios of load duration (Т load ) to the structure's own oscillation period (Т 0 ), and for five load types.Evidently, the temporal nature of the load is an influence on the dynamic factor only for load duration to oscillation period ratios greater than 0.5*Т 0 .This fact is actually illustrated in Fig. 1, plotted based on rather lengthy loads (relative to the structure's own oscillation period), but not on impulse loads.
Fig. 3 gives values of dynamic coefficients for different load duration to oscillation period ratios, К=0 for shock load, and К=1 for a load type that only has a buildup phase.Conditionally these can be regarded as extreme forms: shock load and smooth load.As stated above (see Fig. 2), for ratios Т НАГРУЗКИ /Т 0 <0.5, the form of load does not affect the numeric values of dynamic coefficients.They are mainly influenced by pressure impulse: . This is seen from the results in Fig. 4 that quotes dynamic coefficients for triangle-shaped loads (I=0.5*Pmax*Тlaod) and for rectangle-shaped loads (I=Pmax*Тload).Fig. 4.Dynamic coefficients for ratios Т НАГРУЗКИ /Т 0 <0.5 and for triangle-and rectangle-shaped loads Calculation demonstrated that dynamic coefficient for brief loads (Т load /Т 0 <0.5) can be approximately found using this formula: where I -pressure impulse -

Conclusions
The paper discusses specifics of calculating dynamic coefficients for impulse loads typical of many emergency situations.
Criteria are described under which dynamic coefficients depend only by shock load impulse but not by its shape form.
The paper gives ratios needed to calculate dynamic coefficients for shock loads that accompany emergency situations (discharge of explosive materials, massive body falling, hydraulic shock etc.)

Fig. 2 .Fig. 3 .
Fig. 2. Dynamic coefficients for five load types ) produces a function of the structure's movement in time responding to force f(t).The maximum value of displacement function T(t) is dynamic coefficient K d .