Vibration and Shock Handbook 14

Vibration and Shock Handbook 14 Every so often, a reference book appears that stands apart from all others, destined to become the definitive work in its field. The Vibration and Shock Handbook is just such a reference. From its ambitious scope to its impressive list of contributors, this handbook delivers all of the techniques, tools, instrumentation, and data needed to model, analyze, monitor, modify, and control vibration, shock, noise, and acoustics. Providing convenient, thorough, up-to-date, and authoritative coverage, the editor summarizes important and complex concepts and results into “snapshot” windows to make quick access to this critical information even easier. The Handbook’s nine sections encompass: fundamentals and analytical techniques; computer techniques, tools, and signal analysis; shock and vibration methodologies; instrumentation and testing; vibration suppression, damping, and control; monitoring and diagnosis; seismic vibration and related regulatory issues; system design, application, and control implementation; and acoustics and noise suppression. The book also features an extensive glossary and convenient cross-referencing, plus references at the end of each chapter. Brimming with illustrations, equations, examples, and case studies, the Vibration and Shock Handbook is the most extensive, practical, and comprehensive reference in the field. It is a must-have for anyone, beginner or expert, who is serious about investigating and controlling vibration and acoustics.

14 Reinforced Concrete

14.3 Beams under Harmonic Excitations 14-18

Mechanical Properties † Design for Machine Vibration

14.4 Design for Explosions/Shocks 14-21

Column † Shear Walls

Summary

This chapter concerns vibration and shock in reinforced concrete structures, addressing how to design reinforcedconcrete structures that will be subjected to vibration or shock, for example, due to a missile attack The chapter isintended to apply vibration theory to concrete structures Section 14.1 describes the fundamental properties ofvibration theory Section 14.2 proposes analytical models for reinforced concrete structures that are subjected toflexure, shear, and dynamic loading Section 14.3 discusses reinforced concrete beams under harmonic excitations,including mechanical properties and designs for machine vibration Section 14.4 indicates the step-by-stepprocedure for designing reinforced concrete columns or walls that will be subjected to seismic loads or shocks,respectively

14.1 Introduction

14.1.1 Basic Concepts

Reinforced concrete structures are capable of free and forced vibration when disturbed from theirequilibrium configuration(see Chapter 1andChapter 2) Free vibration takes place when a structurevibrates under the action inherent to the structure itself without being impressed by external forces Thestructure under free vibration vibrates at its natural frequency, which is one of the dynamic properties ofthe structure Forced vibration takes place under the excitation of an external force and at the frequency

of the exciting force, which is independent of the natural frequency of the structure When the frequency

of the exciting force coincides with the natural frequency of a structure, resonance occurs anddangerously large amplitudes may result Therefore, the calculation of natural frequency and theexamination of resonance are of practical importance Also, in reality, the influence of viscous dampening

on a vibrating structure should be taken into account In this section, the fundamental dynamicproperties are introduced

14-1

where A and B are constants depending on the initiation of the motion and w is a quantity denoting

a physical characteristic of the system The substitution of Equation 14.2 or Equation 14.3 intoEquation 14.1 gives

Determine the natural frequency of the system shown in Figure 14.2, consisting of a weight of

W ¼ 230 kN attached to a horizontal reinforced concrete cantilever beam through the coil spring k2:

mk

mÿ

NN

FIGURE 14.1 Free-body diagrams: (a) showing single-DoF; (b) showing only external forces; (c) showing external and inertial forces.

The cantilever beam has a cross section of 305 mm

ðhÞ £ 152 mm ðbÞ; modulus of elasticity

E ¼ 30 kN/mm2, and a length L ¼ 3 m The coil

spring has stiffness k2¼ 200 N/mm

Solution

The deflection, D, at the free end of a cantilever

beam that is acted upon by a static force, P, at the

free end is given by

D ¼ PL33EIThe corresponding spring constant, k1, is then

k1¼ P

3EI

L3

where I ¼ ð1=12Þbh3 (for rectangular section)

when the contribution of reinforcing bars is

neglected The cantilever and the coil spring of

this system are connected as springs in series

Consequently, the equivalent spring constant is

ke¼ 171 N=mmThe natural frequency for this system is then given by Equation 14.6 as

W = 230 kN(a)

#2@89 mm f'c = 39 N/mm

2A's = 142 mm2

As = 852 mm2

k2= 200 N/mm

FIGURE 14.2 System for Example 14.1.

the y-direction gives the differential equation of

motion

m€y þ c_y þ ky ¼ 0 ð14:8Þ

To solve Equation 14.8, we try y ¼ C ept:

Substituting this function into Equation 14.8

results in the equation

mCp2eptþ cCp eptþ kC ept¼ 0 ð14:9Þ

After cancellation of the common factors,

Equation 14.9 reduces to an equation called the

characteristic equation for the system, given by

2m

2

2 km

s

ð14:11ÞFor a system oscillating with critical damping, the expression under the radical in Equation 14.11 isequal to zero, that is

ccr2m

2

where ccrdesignates the critical damping value (Clough and Penzien, 1993; Paz, 1997)

In general, the damping of the system is expressed as

wherej is the damping ratio of the system

Note: Often the symbolz is used instead of j to denote the damping ratio

The dampening value depends on structural materials According to Dowrick (1987), the values forconcrete structures are listed in Table 14.1 It can be seen from this table that for reinforced andprestressed concrete the values are 5 and 2%, respectively

Example 14.2

A reinforced concrete beam consists of a weight of 5.0 kN and a stiffness of k ¼ 4:0 kN/cm Find

1 the undamped natural frequency

2 the damping coefficient

mÿ

yk

free-TABLE 14.1 Typical Damping Ratios for Concrete Structures

Type of Construction Damping j ; % of Critical Concrete frame, with all walls of flexible construction 5

Concrete frame, with stiff cladding and all internal walls flexible 7

Concrete frame, with concrete or masonry shear walls 10

Concrete and/or masonry shear wall buildings 10

Prestressed concrete 2

Notes: (1) The term frame indicates beam and column bending structures as distinct from shear structures (2) The term concrete includes both reinforced and prestressed concrete in buildings For isolated prestressed concrete members such as in bridge decks, damping values less than 5% may be appropriate, e.g., a value of 1 to 2% may apply if the structure remains substantially uncracked.

1 w ¼pffiffiffiffiffik=m¼pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið4:0 kN=cm £ 980 cm=sec2Þ=5:0 kN¼ 28:0 rad=sec:

2 c ¼jccr¼ 0:05 £ 2 £pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið5:0 £ 4:0Þ=980¼ 14 N sec=cm:

14.1.4 Damped Harmonic Excitation

Now consider the case of one-DoF system in Figure 14.4, vibration due to an external load of a sinefunction under the influence of viscous damping The differential equation of motion is obtained byequating to zero the sum of the forces in the free-body diagram of Figure 14.4(b) Hence,

The total response is then obtained by summing the complementary solution and the particular solution

yðtÞ ¼ e2jwtðA cos wDt þ B sin wDtÞ þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiystsinð4t 2 uÞ

ð1 2 r2Þ2þ ð2rjÞ2

where r is the frequency ratio that is equal to

forced vibration frequency,4; divided by natural

frequency, w:

Note: A further discussion of this topic is found

inChapter 2

By examining the transient component of the

response, it may be seen that the presence of

the exponential factor, e2 j wt, will cause this

component to vanish, leaving only the

steady-state motion, Y; which is given by the second term

of Equation 14.16

The ratio of the steady-state amplitude of Y to

the static deflection yst; defined above, is known as

the dynamic magnification factor, D:

D ¼ yY

st ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

ð1 2 r2Þ2þ ð2rjÞ2

D varies with the frequency ratio, r; and the

damping ratio, j: Equation 14.17 is plotted in

Figure 14.5 (Paz, 1997; Clough and Penzien,

1993) It can be seen from this figure that

0:8 # r # 1:2 is in the resonance zone

There-fore, for design, r 1:2 or r , 0:8 is required to

Fosin ωt

FIGURE 14.4 (a) Damped oscillator harmonically excited; (b) free-body diagram.

14.2 Analytical Models

14.2.1 Model-Based Simulation

When a structure is analyzed for its vibrational characteristics, it first must be presented by a simplemodel that reflects its mechanical properties adequately In many analyses, mass is assumed to beconcentrated at the nodes of the models By using this assumption, a single-story structure can besimplified as a single-DoF system subjected to a time-varying force, FðtÞ: In general, dynamic models ofreinforced concrete structures depend on the structural systems Figure 14.6 indicates the structuralsystems and the corresponding dynamic models (Mo, 1994)

14.2.2 Flexural Behavior

Using the trilinear theory (Mo, 1992), the primary curve (the load–deflection curve) of a reinforced/prestressed concrete beam can be determined The trilinear theory is described as follows To find theload–deflection curve of a beam, first the moment–curvature relationship of each section needs to bedetermined The trilinear moment–curvature relationship is shown inFigure 14.7 The first branch ofthe trilinear curve represents the behavior of the reinforced concrete section until flexural cracking

ðMc;ccÞ: The second branch describes the behavior from the cracking until the yielding of thelongitudinal steel ðMy;cyÞ: The third branch gives the postyield behavior until flexural failure ðMu;cuÞ:For a given cross section, the shape of the moment–curvature curve can be determined by using thefollowing equations Basically, the parabola–rectangle stress–strain curve of concrete specified in theCEB code (1978) and the elastic–plastic stress–strain curve of steel are used in the computation

Framed shearwall Single-degree-of-freedom

m

(a)

(b)

(c)FIGURE 14.6 Structures and corresponding dynamic models.

Moment–curvature curve:

1 Cracking state

Mc¼ bh62fr¼ bh62 7:5qffiffiffif0c

ð14:18Þwhere

c¼ concrete compressive strength in psi

fr¼ concrete rupture strength in psi

1c

My¼ yielding moment

yy ¼ yielding curvature

1sc¼ compression steel strain

1c¼ concrete strain when tension steel yields

1y¼ steel yielding strain

d ¼ effective depth

d0¼ distance between surface of concrete compression block and center of compression steel

xy¼ distance between surface of concrete compression lock and neutral axis when tensionsteel yields

A0

s¼ area of compression steel

Es¼ steel Young’s modulus

Ec¼ concrete Young’s modulus

3 Ultimate state (Figure 14.9)

cu¼ curvature corresponding to ultimate moment

xu¼ distance between surface of concrete compression block and neutral axis at ultimate state

fs¼ steel stress at ultimate state

As¼ area of tension steel

Load–deflection curve:

Once the trilinear moment–curvature is found,

we can convert it into the load–deflection curve

Curvature diagram:

1 Cracking state (Figure 14.11)

uc¼ 12

L2

L

2uA¼ 23

uc¼ rotation at point A at cracking state

dc¼ midspan deflection at cracking state

2 Yielding state (Figure 14.12)

uy ¼ ðarea of triangleÞ þ ðarea of

L3¼ L1þ ccL2L2

2 þ ðcy2ccÞL

2

2L23

ccL2þ 1

uy¼ rotation at point A at yielding state

dy¼ midspan deflection at yielding state

3 Ultimate state (Figure 14.13)

uu¼ ðarea of triangleÞ þ ðarea of first trapezoidÞ þ ðarea of second trapezoidÞ

¼ 1

2L1ccþ 1

2ðccþcyÞL2þ 1

du¼ ðfirst moment of triangleÞ þ ðfirst moment of first trapezoidÞ

þ ðfirst moment of second trapezoidÞ

uu¼ rotation at a point A at ultimate state

du¼ midspan deflection at ultimate state

In this section, the maximum concrete strain at ultimate state ð1cuÞ is assumed to be 0.0035according to the CEB code (1978) If the ACI code (2002) is employed, the value is 0.003 However,

in seismic structures there will be more stirrups In these situations, reinforced concrete beams areconfined Therefore, the maximum concrete strain at ultimate state for confined concrete can be used

b ¼ beam width

lc¼ distance from the critical section to the point of contraflexure

rv¼ ratio of volume of confining steel (including the compression steel) to volume of concrete confined

fyv¼ yielding stress of confining steel

14.2.3 Shear Behavior

Structural walls in a frame building should be so proportioned that they possess the necessary stiffnessneeded to reduce the relative inter-story distortions caused by explosion-induced motions Such walls aretermed structural (or shear) walls because their behavior is governed by shear if the ratio of height tolength is less than unity Their additional function is to reduce the possibility of damage to nonstructuralelements that most buildings contain

Buildings stiffened by structural walls are considerably more effective than rigid frame buildingswith regard to damage control, overall safety, and integrity of the structure This performance is due

to the fact that structural walls are considerably stiffer than regular frame elements and thus canrespond to or absorb the greater lateral forces induced by the earthquake motions, while controllinginter-story drift

The past three decades saw a rapid development of knowledge regarding shear in reinforcedconcrete Various rational models that are based on the smeared-crack concept can satisfy Navior’sthree principles of mechanics of materials (i.e., they satisfy stress equilibrium, straincompatibility, and constitutive laws of materials) These rational or mechanics-based models onthe “smeared-crack level” (in contrast to the “discrete-crack level” or “local level”) include: thecompression field theory (CFT) (Vecchio and Collins, 1981); the rotating-angle softened trussmodel (RA-STM) (Belarbi and Hsu, 1994, 1995; Pang and Hsu 1995); the fixed-angle softened trussmodel (FA-STM) (Pang and Hsu, 1996; Hsu and Zhang, 1997; Zhang and Hsu, 1998); thesoftened membrane model (SMM) (Hsu and Zhu, 1999; Zhu, 2000); and the cyclic SMM(Mansour, 2001)

Vecchio and Collins (1981) proposed the earliest rational theory, CFT, to predict the nonlinearbehavior of cracked reinforced concrete membrane elements However, the CFT is unable to take intoaccount the tension stiffening of the concrete in the prediction of deformations because the tensile stress

of concrete was assumed to be zero

The RA-STM, a rational theory developed at the University of Houston (UH) in 1994–1995, has twoadvantages over the CFT (1) The tensile stress of concrete is taken into account so that the deformationscan be correctly predicted (2) The average stress–strain curve of steel bars embedded in concrete isderived on the “smeared crack level” so that it can be correctly used in the equilibrium and compatibilityequations, which are based on continuous materials.

By 1996, the UH group reported that the FA-STM was capable of predicting the “concretecontribution” ðVcÞ by assuming the cracks to be oriented at the fixed angle

Other significant advancements include the improvements on the softened truss models angle and fixed-angle) As they were, these models could predict the ascending response curves ofshear panels, but not the post-peak descending curves By incorporating two new Hsu/Zhu ratiosinto the FA-STM, a new SMM was established (Hsu and Zhu, 1999; Zhu, 2000), which cansatisfactorily predict entire response curves, including both the ascending and the descendingbranches

(rotating-More recently, Mansour et al (2001a) tested 15 reinforced concrete panels under reversed cyclicstresses Tests results showed that the orientation of the steel grids in a panel has an importanteffect on the shear stiffness, the shape of the hysteretic loops, the shear ductility, and the energydissipation capacity of the panel The cyclic SMM proposed by Mansour et al (2001a) is able topredict rationally the pinching effect in the hysteretic loops, the shear ductility, and the energydissipation capacity of the panels In this chapter, only RA-STM will be introduced, as describedbelow

14.2.3.1 Principle of Transformation

The stresses in a membrane element are best analyzed by the principle of stress transformation(Hsu, 1993) Figure 14.14(a) shows a concrete element in the stationary l–t coordinate system,defined by the directions of the longitudinal and transverse steel To find the three stress components invarious directions, a rotating d–r coordinate system is introduced in Figure 14.14(b) The d–r axeshave been rotated counterclockwise by an angle ofa with respect to the stationary l–t axes The threestress components in this rotating coordinate system are sd; sr; and tdr (or trd) The relationshipbetween the rotating stress components,sd;sr; andtdr; and the stationary stress components,sl;st; and

tlt; is the stress transformation This relationship is a function of the anglea:

The relationship between the rotating d– r axes and the stationary l–t axes is shown by thetransformation geometry in Figure 14.14(c) A positive unit length on the l axis will have projections ofcosa and 2sin a on the d and r axes, respectively A positive unit length on the t axis should give

t l

cosa

11

cosa

−sina sina

(a) Stationary t–l axes

and stresses usingbasic sign convention

(a = 0)

(b) Rotating r–d axes

(rotate colckwise by an

counter-angle a)

(c) Transformationgeometry

projections of sina and cos a: Hence, the rotation matrix ½R is

2sina cos a

ð14:38ÞThe relationship between the stresses in the l–t coordinate ½slt and the stresses in the d –r coordinate

14.2.3.2 Equilibrium Equations

When one studies a concrete element reinforced orthogonally with longitudinal and transverse steel bars,

as shown in Figure 14.15(a), the three stress componentssl;st; andtlt are the applied stresses on thereinforced concrete element viewed as a whole The stresses on the concrete strut itself are denoted asslc;

stc; and tltc; as shown in Figure 14.15(b) The longitudinal and transverse steel provide the smearedstresses ofrlflandrtft; as shown in Figure 14.15(c)

It is significant to recognize the difference between the two sets of stresses:sl;st; and tlt for thereinforced concrete element andslc;stc; andtltfor concrete struts Both sets of stresses ðsl;st;tltandslc;

stc;tltcÞ satisfy the transformation equations In summing the concrete stresses and the steel stresses inthe l and t directions, a fundamental assumption is made according to Hsu (1993) It is assumed that the

steel reinforcement can take only axial stresses Any possible dowel action is neglected Hence, thesuperposition principle for concrete and steel becomes valid and gives the general equilibrium equationsfor reinforced concrete:

Peter’s tests could not delineate the variables that govern the softening parameter because of technicaldifficulties in the biaxial testing of large panels The quantification of the softening phenomenon,therefore, did not occur for almost two decades, when a unique “shear rig” test facility was built in 1981

by Vecchio and Collins (1981) Based on their tests of 17 panels, each 89 cm2and 7 cm thick, theyproposed a softening parameter that was a function of the ratio of the tensile principal strains to thecompression principal strain, 1r=1d:

The discovery and the quantification of this softening phenomenon have allowed a majorbreakthrough in understanding the shear problem in reinforced concrete During the past 20 years, anumber of diverse analytical models have been proposed according to the test results (Peter, 1964;Robinson and Demorieux, 1968; Vecchio and Collins, 1981; Schlaich et al 1982; Schlaich and Schafer,1983; Vecchio and Collins, 1986; Eibl and Neuroth, 1988; Miyahara et al 1988; Kollegger and Mehlhorn,1990; Mikame et al 1991; Ueda et al 1991; Hsu, 1993; Vecchio and Collins, 1993; Vecchio et al 1994;Belarbi and Hsu, 1995) The effect of these softening models on low-rise framed shear walls is studied by

Mo and Rothert (1997) In this section, the softening model proposed by Belarbi and Hsu (1994, 1995) isbriefly introduced

The original softening model derived from test

data proposed the use of a softening parameterz;

where z is a function of the ratio of principal

tensile strain to principal compressive strain

ð1r=1dÞ: The proposed model by Belarbi and Hsu

(1995) involves modification of the Hognestad

parabola (Figure 14.16), which is used as the base

curve describing the uniaxial compressive

1cr¼ strain at cracking of concrete ¼ 0.00008

fcr¼ 3:75qffiffiffif0c

Stress–strain relationship of steel The stress–strain curve of a steel bar in concrete relates the averagestress to the average strain of a long bar crossing several cracks, whereas the stress–strain curve of a barebar relates the stress to the strain at a local point (Okamura and Maekawa, 1991) In other words, a steelbar in concrete is stiffened by the tensile stress of the concrete If the tensile strength of concrete isneglected, as it is in the most of truss models, the following equations are used:

where

Es¼ modulus of elasticity of steel bars

fly¼ yield stress of longitudinal steel bars

1ly¼ yield strain of longitudinal steel bars

It was recommended by Belarbi and Hsu (1995) that both the tensile strength of concrete, presented inthe previous section, and the average stress–strain curve of steel stiffened by concrete, be taken intoaccount In this model, the following equations are used for describing the stress–strain relationship

of steel:

If 1lEs# f0ly; fl ¼ Es1l ð14:56Þ

HognestadParabola

If 1lEs# f0

ly; fl ¼ 1 2 2 2a=458

1000rl ½ð0:91 2 2BÞflyþ ð0:02 þ 0:25BÞEs1l ð14:57Þwhere

CD

tf

FIGURE 14.17 A framed shear wall.