Explosion Protection of Buildings

Explosion Protection of Buildings Author: Miroslav Mynarz

Explosion Protection of Buildings Introduction to the Problems of Determination of Building Structure's Response

3 Classification of actions According to response of the structure: Static (no significant accelerations originate in the structure) Dynamic (significant accelerations originate in the structure)

4 Classification of actions According to its nature: Force (including moments) Deformation or inematic (forced by displacement or rotation)

5 Types of actions Deterministic: Periodical (repeating regularly) Non-periodical (repeating irregularly)

6 Types of actions Periodical: Harmonic (simple) y = A sinωt e. g. rotary motion of unbalanced movement of machine advance motion of piston in cylinder simple tone of musical instrument movement of a bell etc.

7 Types of actions Periodical : Compact of i-harmonic components F(t) = F st + F i sin(ω i t + φ i ) (Fourier transform) e. g. movement of a floor loaded by more machines movement of bridge dec loaded by more vehicles compound tones, noises etc.

8 Types of actions Non-periodical: Impuls (impuls of various shape) I = t F (τ) Δτ e. g. effects of explosions or blast waves push of drop hammers, ramming of piles acoustic stroes etc.

9 Types of actions Non-periodical: Non-periodical force F t Combination: Periodical impulses F t e. g. effects of flat wheels to bridge forging of repeated pushes of drop hammer

0 Types of actions Non-deterministic (random, stochastic): Stationary (steady, still) Non-stationary (unsteady, unstable)

Dynamic ~ time variable load Dynamic load load whose size, direction or place is changing in time. Response of building structure to dynamic load final deformations of the structure (deflections, displacements or rotations) and their derivation in time (velocity, acceleration) as well as internal forces (normal and shear forces or bending moments) and stresses are variable in time.

Dynamic ~ time variable load Determination of dynamic response analysis of the structure at dynamic load. Methods depend on character of the load: deterministic load, non-deterministic load.

3 Dynamic ~ time variable load Deterministic procedures: time course is nown; material and shape of the structure is nown; boundary conditions (for example founding of the structure) are defined.

4 Dynamic ~ time variable load Non-deterministic procedures: time course of load is not nown fully; character of load is variable; character of material properties of the structure is variable.

5 Analysis of the response of the building structure Determination of deformation quantities displacements, deflections; rotations; their time derivations. Following force quantities are derivated from deformation quantities internal forces; stress.

6 Building structure Number of degrees of freedom (number of components of possible displacements or rotations of structure s mass in particular points which are not precluded by sliding supports): real structures infinite number of degrees of freedom; solid body in 3-D space (point mass in space) 6 degrees of freedom; solid slab in plane (point mass in plane) 3 degrees of freedom. For each degree of freedom, equation of equilibrium (equation of motion) can be written.

7 Building structure Characteristics of the structure (material, structural arrangement and boundary conditions): stiffness; mass of inertia (weight); damping properties. Tuning of the structure spectrum of natural frequencies and corresponding free vibration mode shapes (function of stiffness and mass of the structures).

8 Building structure Mode shape (antinode versus node): natural mode shape; free vibration; forced vibration.

9 Quantities displacement (deflection) s(t)=s 0 f(t) rotation φ(t)=φ 0 f(t) velocity of vibration v(t)=ds/dt φ(t)=dφ/dt acceleration of vibration a(t)=d s/dt φ(t)=d φ /dt momentum of (point) mass H(t)=m v(t) momentum of body inertia force H(t)= V v(t) dm F(t)= m a(t)

0 Quantities in direction: of coordinate axes; related to the structure (e. g. in the middle of a span); related to trajectory (e. g. radial, tangential, centrifugal, centripetal). Equation of motion (summary condition of equilibrium of forces in direction of appropriate degree of freedom): displacement; rotational (spin, angular).

Quantities Newton's second law: F(t) = d/dt (m ds/dt) = m d s/dt = m a(t) F(t) force vector s(t) trajectory (displacement, rotation) ~ position vector p(t) = s(t) internal forces in the structure (elastic) m a(t) P(t) inertia forces in the structure (volume) external forces (load)

Quantities p(t) = s(t) m a(t) P(t) internal forces in the structure (elastic) inertia forces in the structure (volume) external forces (load) d Alembert's principle (motion of a point mass ~ a body) : P(t) - m d s/dt = 0

3 Equations of motion Structure under dynamic load: The course of load F(t) is variable in time whereas time change of load is usually in order of seconds or milliseconds. More precise definition of time change of load results from comparison of time change of load F(t) and natural periods of the structure T(i). When time changes of load are long-term, thus in order of hours, days, years, we usually do not tal about dynamic load but about variability of static load.

4 Equations of motion Equations of motion (control equations of the system): Equations of motion (control equations of the system) are derivated from equations of equilibrium in appropriate possible direction of motion (deflection or rotation) which is denoted as degrees of freedom: F I (t) + F D (t) + F E (t) = F(t)

5 Equations of motion Time change of displacement y(t) is considered as a solution of equation of equilibrium, inertia forces F I (t) = m y (t) thus product of mass m and acceleration of vibration y (t) (from Newton's second law); damping forces F D (t) = C y (t) are usually formulated as product of dumping constant C and velocity of vibration y (t); elastic forces F E (t) = y (t) product of stiffness of the structure and displacement of vibration y(t).

6 Equations of motion M y (t) + C y (t) + K y (t) = F where M, C and K are mass, damping and stiffness matrixes of the system; y, y, y are vectors of acceleration, velocity and displacement of vibration; F is a vector of the right sides (load of single parts points of the structure).

7 Equations of motion Members of a mass matrix after discretization of a system consist of masses of moving point masses and mass moments of inertia of rotating point masses. Particular elements m ij of a mass matrix can be determined as forces in a point i corresponding to a unit acceleration y in a point j.

8 Equations of motion Mass matrix: When using software, consistent mass matrix is usually used; it means a mass matrix with nonzero elements even at non-diagonal positions. More simply, generally for manual calculation or computation of small systems or systems with limited number of degrees of freedom, only lumped mass matrix is used it is diagonal matrix with zero elements at non-diagonal positions. It means that inertia forces in i-th point of the system generated by the load applied to this point do not induce inertia response of other points.

9 Equations of motion Let s divide bent beam to several particular elements where mass of single elements is focused into nodes of discrete system. If diagonal matrix is used for calculation then at each position of diagonal, mass of particular node is placed there and it performs only advanced motion. If we tae into account that masses in nodes can also rotate then mass matrix has nonzero elements even at non-diagonal positions it is considered as consistent mass matrix.

30 Equations of motion Damping matrix: For compilation of damping matrix C it is usually assumed that damping is directly proportional to velocity of motion (see formulation of damping forces of equation of motion) this damping is denoted as viscous damping. Then single elements of damping matrix correspond to force in node i at action of unit vector of velocity y in point j: F D = c y

3 Equations of motion In software, implementation of damping to equation of motion with help of coefficients a and b is used with advantage. In this case, damping matrix C is more simply considered to be linear combination of mass matrix and stiffness matrix C = a M + b K Simplification consists especially in determination of damping size with only two parameters a and b that are valid for all frequency components of vibration.

3 Equations of motion De facto, damping of vibrations is usually higher on higher frequencies it means that calculated dynamic response of the structure can be actually different.

33 Equations of motion Stiffness matrix: Particular elements of this matrix correspond to static stiffnesses of single nodal points of 3D discretizated system at unit displacements (and also rotations) of these relevant nodal points. As well as in the cases of mass and damping matrixes, single elements of stiffness matrix ij correspond to a force (moment) in a node i at action of unit displacement (rotation) y in point j: F E = y

34 Equations of motion Calculation decomposes in two cases: - equation without right side (unloaded structure): calculation of natural vibration (tuning of the structure); - equation with right side (loaded structure): calculation of forced vibration. There are many methods for solution of free and forced vibration. The user of particular software should now used method of solution. It is not advisable to use the software as a blac box through which the user gains the results and believes them absolutely. He or she should then now the possibilities of the method in so far to estimate its proximities bringing into the results of response calculation.

35 Equations of motion Development to the free vibration mode shapes: In this case, it highly depends on a number of mode shapes used for calculation. For small amount of mode shapes (up to 5 or 0 shapes), the calculation could involve only 50 to 60 % intensity of a real response. According to Eurocode 8, minimum number of considered global mode shapes is: 3 n or T 0. s n is a number of floors above the foundation or solid subsoil.

36 Equations of motion Damping of vibrations: A size of damping can affect even calculated quantities of the response especially in the case when the response of the structure is close to resonant frequency. It is usually assumed that damping is proportional to the velocity of vibration. This damping is called viscous. Most software generally enables the implementation of viscous damping to the calculation. Viscous damping is also suitable approximation to other assumptions about damping (e. g. hysteretic damping that is proportional to the deflection of vibrations).

37 Equations of motion Damping of vibrations: For determination of a level of viscous damping, it is suitable to result from parameters measured at particular structure. Damping is frequency-dependent. Generally, structural damping is lowest on lowest natural frequencies and damping is increasing with increasing frequency. Relation between the damping ratio D and the logarithmic decrement of damping δ follows: D δ f ω i i δ π

38 Equations of motion Damping of vibrations: - chosen values of the logarithmic decrement of damping δ and the damping ratio D (according to ČSN 73 003) Material Constructive system, element D (%) Steel Masonry Timber Concrete engineering structures (building, halls) towers, smoestacs, high-rise structures single elements columns, beams springs (wound) vertical bric masonry (walls, partition walls) masonry vaults in steel beams stone masonry on lime mortar stone masonry on cement-lime mortar glued beams nailed beams roof trusses tower structures compression elements without cracs elements with tension (without cracs) elements with cracs slab and rib floors crane beams and columns high prefabricated houses 0.08-0,0 0.0 0.08 0.5 0.05 0. 0.34 0.7 0. 0.06 0.5 0.6 0.0 0.5 0.0 0.5 0.5 0.8 0.6 0..7.59.7.9.39 0.80.9 5.4.7.75 0.95.39.55 3.9.39.59.39 3.98 4.46.55.75

deflection y VŠB Technical University of Ostrava 39 Equations of motion Experimental methods for determination of damping: deflection in resonance frequency of excitation f

40 VŠB Technical University of Ostrava Experimental methods for determination of damping: Equations of motion the logarithmic decrement of damping m n n n n y y ln m y y ln δ the damping ratio from rezonant gain pea st y y D method of the width of the resonance curve f f f f D

4 Equations of motion Implementation of a size of damping to the calculation: In software, Rayleigh formulation of damping is generally implemented (through method of integration of equations of motion). It results from solution of matrix equation of motion whose damping matrix C is considered as a linear combination of mass matrix M and stiffness matrix K: C = α M + β K

4 Equations of motion Implementation of a size of damping to the calculation: For a system with one degree of freedom: α D ω β It could be derived even for a system with more degrees of freedom it is then necessary to now relevant number of natural frequencies. D ω

43 Equations of motion Implementation of a size of damping to the calculation: At calculation of systems with more dominant frequency components of excitation or at excitation with general time course of a load, it is necessary to decide which size of damping will be used for which frequency component. At other frequency components whose damping is not nown more precisely a size of response is burdened with bigger or smaller error.

44 Standards requirements for determination of dynamic response of the structure Standards usually do not specify process of calculation and chosen method. For example, standards (ČSN 73 0036, Eurocode 8 or UBC 997 etc.) enable simplified (seismic this procedure can be applied even to the blast effects) calculation of simpler structures at use of cantilever model (the object is replaced by a bar model cantilever with masses focused in single floors).

45 Standards requirements for determination of dynamic response of the structure Simplification of geometry (numerical model) versus calculation method Cantilever model is suitable for high slender structures but not for structures with wide plan (e. g. industrial halls), underground structures, indented buildings or structures with dominant effect in vertical direction (bridges, huge cantilever length of floors in buildings). In such cases it is correct to use 3D or at least D model and to perform dynamic computation on these models. Simplification of input data: material characteristic and load.

46 Damping of system with one degree of freedom Natural undamped vibration: F in F re m d v dt d v m dt v t t t v m mass of a system t 0 - inertia force - restoring force of the spring v(t) vertical displacement of the spring stiffness (the spring constant)

47 Damping of system with one degree of freedom v Natural undamped vibration: t A sinω t B cosω t v sinω t A 0 0 0 0 0 0 φ0 0 v0 cos φ0 B v sinφ 0 0 0 v A B 0 0 0 φ 0 B A 0 0 B φ arcg 0 A 0 0

48 Damping of system with one degree of freedom Initial conditions: t dv v t ω A cosω t B sinω t 0 0 0 0 0 dt v0 ω0 cos ω0t φ0 v0 ω0 sin ω0t φ t π d v v t ω A sinω t B cosω t 0 0 0 0 0 dt ω t φ v ω sinω t φ π v0 ω0 sin 0 0 0 0 0 0 0

49 Damping of system with one degree of freedom π T0 π f ω 0 0 m ω 0 m f 0 0 T 0 ω π π m Natural frequency f 0 = f ()

50 Damping of system with one degree of freedom v Natural damped vibration: d v m dt d v dt t t ω dv b dt b dv dt t t v ω t 0 t 0 For solution in shape of v(t)=e αt it has characteristic equation: with roots α α ω α ω b 0 0 0 ω ω ω, b b 0 ωbt ωbt t e A sinω t B cosω t v e sinω t d d d d v 0 d φ 0

5 Damping of system with one degree of freedom Following three cases can occur: - critical damping, when ω b = ω 0, b r = ; characteristic equation has one real, double root: α, =-ω b, - overdamping, when ω b > ω 0, b r > ; characteristic equation has two different real roots: α ω ω ω, b b 0 - underdamping, when ω b < ω 0, b r < ; characteristic equation has two imaginary complex conjugate roots: α ω i ω ω ω iω, b 0 b b d ω d ω ω 0 b is natural angular (circular) frequency of damping vibration.

5 Damping of system with one degree of freedom υ ln v v ω f b ωbt d d πω ω d b υ πω ω d b b r π b r υ πb r the damping ratio b r = D

53 Damping of system with one degree of freedom Harmonic excitation undamped vibration: m d v dt t v v t v t sinωt F sinωt v F F ω0 v st mω m ω ω ω ω 0 0 v st δ

54 Damping of system with one degree of freedom Harmonic excitation undamped vibration: force force resonance deflection deflection

55 Damping of system with one degree of freedom Harmonic excitation undamped vibration: v m d v dt d v dt t t ω c b dv dt t t dv dt v ω 0 v t t Ft t F m t A sinωt B cosωt v sinωt φ v F A t B F sinωt tgφ B A

56 VŠB Technical University of Ostrava Damping of system with one degree of freedom Harmonic excitation undamped vibration: 0 0 4 η b η η F ω ω ω ω ω ω m F A r b 0 4 b η η b η F ω ω ω ω ωω m F B r r b b 4 4 η π υ η η υ π υ η δ

57 Damping of system with one degree of freedom Harmonic excitation undamped vibration: π Under-resonant area for η= resonance δ υ over-resonant area. v v st F mω ω b m ω ω ω 0 ω ω 0 4ω ω b 0 v st δ F 4ω ωb

58 Damping of system with two degrees of freedom Simple beam undamped vibration: m m d v dt t v d v dt t t v t 0 v t v t 0 ω, m m 4 m m m m a v v 0 0 m ω m ω a v v 0 0 m ω m ω

59 Damping of system with two degrees of freedom Simple beam undamped vibration: I m d ζ dt t d u dt t a ζζ uu a ζ u m m t ut 0 ζu t ζ t 0 uζ I m a b

60 Damping of system with two degrees of freedom Simple beam undamped vibration: EI b b ζζ 4 6 3 l l l 6EI b EI 3 l l uu l ζu ω, uu m ζζ I 4 uu m ζζ I uζ mi a a i i I m

6 VŠB Technical University of Ostrava Damping of system with two degrees of freedom 0 t v t v dt t v d m Simple beam undamped vibration: 0 t v t v dt t v d m 4 m m m m m m ω,

6 Damping of system with three degrees of freedom

63 Than you for your attention.