Пресс-релиз популярных книг

Авторы: 111 А Б В Г Д Е Ж З И Й К Л М Н О П Р С Т У Ф Х Ц Ч Ш Щ Э Ю Я

Книги: 164 А Б В Г Д Е Ж З И Й К Л М Н О П Р С Т У Ф Х Ц Ч Ш Щ Э Ю Я

На сайте 111 авторов, 92 книг, 72 статей, 5913 глав.

14.2 Analytical Models

Back

14.2.1 Model-Based Simulation

When a structure is analyzed for its vibrational characteristics, it first must be presented by a simple

model that reflects its mechanical properties adequately. In many analyses, mass is assumed to be

concentrated at the nodes of the models. By using this assumption, a single-story structure can be

simplified as a single-DoF system subjected to a time-varying force, FðtÞ: In general, dynamic models of

reinforced concrete structures depend on the structural systems. Figure 14.6 indicates the structural

systems 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 the

load – deflection curve of a beam, first the moment – curvature relationship of each section needs to be

determined. The trilinear moment – curvature relationship is shown in Figure 14.7. The first branch of

the 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 the

longitudinal steel ðMy ; cy Þ: 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 the

following equations. Basically, the parabola – rectangle stress – strain curve of concrete specified in the

CEB code (1978) and the elastic – plastic stress – strain curve of steel are used in the computation.

Structure Dynamic model

Framed shearwall Single-degree-of-freedom

Single-degree-of-freedom

One-story frame

Simply supported beam

m m

(a)

(b)

(c)

FIGURE 14.6 Structures and corresponding dynamic models.

14-6 Vibration and Shock Handbook

Moment – curvature curve:

1. Cracking state

Mc ¼

bh2

fr ¼

bh2

􀀍

7:5

ffiffiffi

f 0c

q 􀀎

ð14:18Þ

where

cc ¼

EI ð14:19Þ

and

Mc ¼ cracking moment

f 0c ¼ concrete compressive strength in psi

fr ¼ concrete rupture strength in psi

cc ¼ cracking curvature

b ¼ beam width

h ¼ beam depth

E ¼ concrete Young’s modulus

I ¼ moment of inertia

2. Yielding state (Figure 14.8)

1c ¼ 1y

d 2 xy ð14:20Þ

1sc ¼ 1y

xy 2 d0

d 2 xy ð14:21Þ

My ¼ k1 f 0cbxy ðd 2 0:375xy Þ

xy 2 d0

1cðEs 2 EcÞA0s

ðd 2 d0Þ ð14:22Þ

and

cy ¼

xy ð14:23Þ

where

k1 ¼

1 2

􀀏 􀀐

ð14:24Þ

My ¼ yielding moment

yy ¼ yielding curvature

1sc ¼ compression steel strain

1c ¼ concrete strain when tension steel yields

1y ¼ steel yielding strain

d ¼ effective depth

d0 ¼ distance between surface of concrete compression block and center of compression steel

xy ¼ distance between surface of concrete compression lock and neutral axis when tension

steel yields

A0s

¼ area of compression steel

Es ¼ steel Young’s modulus

Ec ¼ concrete Young’s modulus

Xy esc

ey = 0.00207 (for fy = 60 ksi)

FIGURE 14.8 Strain diagram at yielding state.

(yc, Mc)

(yy, My)

(yu, Mu)

FIGURE 14.7 Trilinear moment – curvature curve.

Reinforced Concrete Structures 14-7

3. Ultimate state (Figure 14.9)

xu $ d 0

then

Mu ¼ 0:81f 0cbxuðd 2 0:41xuÞ þ

xu 2 d 0

xu £ 0:0035ðEs 2 EcÞA0s

ðd 2 d 0Þ ð14:25aÞ

xu , d0

then

Mu ¼ fsAsðd 2 0:41xuÞ þ

d 0 2 xu

xu £ 0:0035EsA0s ðd 0 2 0:41xuÞ ð14:25bÞ

cu ¼

0:0035

xu ð14:26Þ

where

Mu ¼ ultimate moment

cu ¼ curvature corresponding to ultimate moment

xu ¼ distance between surface of concrete compression block and neutral axis at ultimate state

fs ¼ steel stress at ultimate state

As ¼ area of tension steel

Load – deflection curve:

Once the trilinear moment – curvature is found,

we can convert it into the load – deflection curve

(Figure 14.10).

M ¼

; [ P ¼

L ð14:27Þ

where

M ¼ moment

P ¼ load

L ¼ beam length

es fs As

0.41Xu

0.81f ′c b Xu

f ′s A′s

d′ 0.0035

FIGURE 14.9 Stress and strain diagrams at ultimate state.

M M

P/2 P/2

FIGURE 14.10 Simple beam with a concentrated load.

14-8 Vibration and Shock Handbook

Curvature diagram:

1. Cracking state (Figure 14.11)

uc ¼

EI ¼

McL

4EI ð14:28Þ

dc ¼ uA

uA ¼

McL2

12EI

ð14:29Þ

where

uc ¼ rotation at point A at cracking state

dc ¼ midspan deflection at cracking state

2. Yielding state (Figure 14.12)

uy ¼ ðarea of triangleÞ þ ðarea of

trapezoidÞ

L1cc þ

2 ðcc þ cy ÞL2 ð14:30Þ

dy ¼ ðfirst moment of triangleÞ þ ðfirst moment of trapezoidÞ

L1cc þ L3

2 ðcc þ cy ÞL2 ¼

ccL21

2 ðcc þ cy ÞL2L3 ð14:31Þ

where

L3 ¼ L1 þ

ccL2

2 þ ðcy 2 ccÞ

2L2

ccL2 þ

L2ðcy 2 ccÞ

ð14:32Þ

uy ¼ rotation at point A at yielding state

dy ¼ midspan deflection at yielding state

3. Ultimate state (Figure 14.13)

uu ¼ ðarea of triangleÞ þ ðarea of first trapezoidÞ þ ðarea of second trapezoidÞ

L1cc þ

2 ðcc þ cy ÞL2 þ

2 ðcy þ cuÞL4 ð14:33Þ

du ¼ ðfirst moment of triangleÞ þ ðfirst moment of first trapezoidÞ

þ ðfirst moment of second trapezoidÞ

L1cc þ L3

2 ðcc þ cy ÞL2 þ L5

2 ðcy þ cuÞL4

ccL21

2 ðcc þ cy ÞL2L3 ð14:34Þ

2 ðcy þ cuÞL4L5

where

L3 ¼ L1 þ

ccL2

2 þ ðcy 2 ccÞ

ccL2 þ

L2ðcy 2 ccÞ

ð14:35Þ

A C B

FIGURE 14.11 Curvature diagram at cracking state.

L1 L2 A C B

FIGURE 14.12 Curvature diagram at yielding state.

A C

L1 L2 L4

FIGURE 14.13 Curvature diagram at ultimate state.

Reinforced Concrete Structures 14-9

L5 ¼ L1 þ L2 þ

cy L4

2 þ ðcu 2 cy Þ

cy L4 þ

L4ðcu 2 cy Þ

ð14:36Þ

uu ¼ rotation at a point A at ultimate state

du ¼ midspan deflection at ultimate state

In this section, the maximum concrete strain at ultimate state ð1cuÞ is assumed to be 0.0035

according 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 are

confined. Therefore, the maximum concrete strain at ultimate state for confined concrete can be used

as follows (Dowrick, 1987):

1cu ¼ 0:003 þ 0:02

􀀏 􀀐

rv fyv

138

􀀏 􀀐2

ð14:37Þ

where

b ¼ beam width

lc ¼ distance from the critical section to the point of contraflexure

rv ¼ ratio of volume of confining steel (including the compression steel) to volume of concrete confined

fyv ¼ 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 stiffness

needed to reduce the relative inter-story distortions caused by explosion-induced motions. Such walls are

termed structural (or shear) walls because their behavior is governed by shear if the ratio of height to

length is less than unity. Their additional function is to reduce the possibility of damage to nonstructural

elements that most buildings contain.

Buildings stiffened by structural walls are considerably more effective than rigid frame buildings

with 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 can

respond to or absorb the greater lateral forces induced by the earthquake motions, while controlling

inter-story drift.

The past three decades saw a rapid development of knowledge regarding shear in reinforced

concrete. Various rational models that are based on the smeared-crack concept can satisfy Navior’s

three principles of mechanics of materials (i.e., they satisfy stress equilibrium, strain

compatibility, and constitutive laws of materials). These rational or mechanics-based models on

the “smeared-crack level” (in contrast to the “discrete-crack level” or “local level”) include: the

compression field theory (CFT) (Vecchio and Collins, 1981); the rotating-angle softened truss

model (RA-STM) (Belarbi and Hsu, 1994, 1995; Pang and Hsu 1995); the fixed-angle softened truss

model (FA-STM) (Pang and Hsu, 1996; Hsu and Zhang, 1997; Zhang and Hsu, 1998); the

softened 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 nonlinear

behavior of cracked reinforced concrete membrane elements. However, the CFT is unable to take into

account the tension stiffening of the concrete in the prediction of deformations because the tensile stress

of concrete was assumed to be zero.

14-10 Vibration and Shock Handbook

The RA-STM, a rational theory developed at the University of Houston (UH) in 1994 – 1995, has two

advantages over the CFT. (1) The tensile stress of concrete is taken into account so that the deformations

can be correctly predicted. (2) The average stress – strain curve of steel bars embedded in concrete is

derived on the “smeared crack level” so that it can be correctly used in the equilibrium and compatibility

equations, which are based on continuous materials.

By 1996, the UH group reported that the FA-STM was capable of predicting the “concrete

contribution” ðVcÞ by assuming the cracks to be oriented at the fixed angle.

Other significant advancements include the improvements on the softened truss models (rotatingangle

and fixed-angle). As they were, these models could predict the ascending response curves of

shear panels, but not the post-peak descending curves. By incorporating two new Hsu/Zhu ratios

into the FA-STM, a new SMM was established (Hsu and Zhu, 1999; Zhu, 2000), which can

satisfactorily predict entire response curves, including both the ascending and the descending

branches.

More recently, Mansour et al. (2001a) tested 15 reinforced concrete panels under reversed cyclic

stresses. Tests results showed that the orientation of the steel grids in a panel has an important

effect on the shear stiffness, the shape of the hysteretic loops, the shear ductility, and the energy

dissipation capacity of the panel. The cyclic SMM proposed by Mansour et al. (2001a) is able to

predict rationally the pinching effect in the hysteretic loops, the shear ductility, and the energy

dissipation capacity of the panels. In this chapter, only RA-STM will be introduced, as described

below.

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 in

various directions, a rotating d – r coordinate system is introduced in Figure 14.14(b). The d – r axes

have been rotated counterclockwise by an angle of a with respect to the stationary l – t axes. The three

stress components in this rotating coordinate system are sd ; sr ; and tdr (or trd ). The relationship

between the rotating stress components, sd ; sr ; and tdr ; and the stationary stress components, sl; st ; and

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

The relationship between the rotating d – r axes and the stationary l – t axes is shown by the

transformation geometry in Figure 14.14(c). A positive unit length on the l axis will have projections of

cos a and 2sin a on the d and r axes, respectively. A positive unit length on the t axis should give

t d

r r

(+)

st (+)

(+)

1 cosa

cosa

−sina

sina

(a) Stationary t–l axes

and stresses using

basic sign convention

(a = 0)

(b) Rotating r–d axes

(rotate countercolckwise

by an

angle a)

geometry

tlt

(+)

tdr

(+)

tdr

(+)

trd

(+)

trd

(+)

tlt

(+)

ttl

(+)

ttl

(+)

FIGURE 14.14 Transformation of stresses.

Reinforced Concrete Structures 14-11

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

½R􀀉 ¼

cos a sin a

2sin a cos a

" #

ð14:38Þ

The relationship between the stresses in the l – t coordinate ½slt 􀀉 and the stresses in the d – r coordinate

½sdr 􀀉 is

½slt􀀉 ¼ ½R􀀉T½sdr 􀀉½R􀀉 ð14:39Þ

where

½slt􀀉 ¼

sl tlt

ttl st

" #

ð14:40Þ

½sdr􀀉 ¼

sd tdr

trd sr

" #

ð14:41Þ

If the d – r axes are defined as the principal axes, tdr must vanish. Introducing the reinforced concrete

sign convention, performing the matrix multiplications and noticing that tlt ¼ ttl and tdr ¼ trd gives the

following equations when only the concrete struts are considered.

slc ¼ sd cos2a þ sr sin2a ð14:42Þ

stc ¼ sd sin2a þ sr cos2a ð14:43Þ

tltc ¼ ð2sd þ sr Þ sin a cos a ð14:44Þ

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 components sl; st ; and tlt are the applied stresses on the

reinforced concrete element viewed as a whole. The stresses on the concrete strut itself are denoted as slc ;

stc ; and tltc ; as shown in Figure 14.15(b). The longitudinal and transverse steel provide the smeared

stresses of rlfl and rt ft ; 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 the

reinforced concrete element and slc ; stc ; and tlt for concrete struts. Both sets of stresses ðsl; st ; tlt and slc ;

stc ; tltc Þ satisfy the transformation equations. In summing the concrete stresses and the steel stresses in

the l and t directions, a fundamental assumption is made according to Hsu (1993). It is assumed that the

= +

Reinforced

concrete

(a) Concrete

struts

(b) Steel

reinforcement

(c)

(−) tuc

pt f1

(+)

(−)

tltc

(−)

stc

(+)

slc

(+)

FIGURE 14.15 Stress condition in reinforced concrete.

14-12 Vibration and Shock Handbook

steel reinforcement can take only axial stresses. Any possible dowel action is neglected. Hence, the

superposition principle for concrete and steel becomes valid and gives the general equilibrium equations

for reinforced concrete:

sl ¼ sd cos2a þ sr sin2a þ r l fl ð14:45Þ

st ¼ sd sin2a þ sr cos2a þ r t ft ð14:46Þ

tlt ¼ ð2sd þ sr Þ sin a cos a ð14:47Þ

14.2.3.3 Compatibility Equations

The same principle of transformation for stresses can be applied to strains. Therefore, the following

compatibility equations can be derived:

1l ¼ 1d cos2a þ 1r sin2a ð14:48Þ

1t ¼ 1d sin2a þ 1r cos2a ð14:49Þ

glt

2 ¼ ð1d þ 1r Þ sin a cos a ð14:50Þ

14.2.3.4 Constitutive Laws

Softened compression stress – strain relationship of concrete. The truss model has been applied to treat the

shear and torsion of reinforced concrete since the turn of the 20th century. However, the prediction based

on the truss model consistently overestimated the shear and torsional strengths of tested specimens. This

nagging mystery has plagued researchers for over half a century. The source of this difficulty was first

understood by Peter (1964). He realized that a reinforced concrete panel element subjected to tension is

actually subjected to biaxial compression – tension stresses. Viewing the action as a two-dimensional

problem, he discovered that the compressive strength in one direction was reduced by cracking due to

tension in the perpendicular direction. After applying the softening effect of concrete struts to the nine

test panels, Peter concluded that a reduction of 15% of the effective compressive strength should be taken

into account in biaxial compression – tension stresses. Apparently, the mistake in applying the truss

model theory before 1964 was the use of the compressive stress – strain relationship of concrete that

was obtained from the uniaxial test of standard cylinders without considering the two-dimensional

softening effect.

Peter’s tests could not delineate the variables that govern the softening parameter because of technical

difficulties 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 cm2 and 7 cm thick, they

proposed a softening parameter that was a function of the ratio of the tensile principal strains to the

compression principal strain, 1r =1d :

The discovery and the quantification of this softening phenomenon have allowed a major

breakthrough in understanding the shear problem in reinforced concrete. During the past 20 years, a

number 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) is

briefly introduced.

Reinforced Concrete Structures 14-13

The original softening model derived from test

data proposed the use of a softening parameter z;

where z is a function of the ratio of principal

tensile strain to principal compressive strain

ð1r =1d Þ: 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

response of concrete.

sd ¼

zf 0c 2

z10

􀀏 􀀐

z10

􀀏 􀀐2 􀀒 􀀓

; 1d =z10 # 1

zf 0c 1 2

1d =z10 2 1

2=z 2 1

􀀏 􀀐2 􀀒 􀀓

; 1d =z10 . 1

8>>><

>>>:

ð14:51aÞ

z ¼

0:9 ffiffiffiffiffiffiffiffiffiffiffiffi

1 þ 6001r p ð14:51bÞ

Tensile stress – strain relationship of concrete. From the tests involving panels subjected to shear, it was

clear that the tensile stress of concrete, sr ; is not zero as assumed in the simple truss model. Based on the

tests of 35 full-size panels (Hsu, 1993), a set of formulas were recommended as follows:

If 1r # 1cr; sr ¼ Ec1y ð14:52Þ

If 1r . 1cr; sr ¼ fcr

1cr

􀀏 􀀐0:4

ð14:53Þ

where

Ec ¼ 47;000

ffiffiffi

f 0c

; and both f 0c and

ffiffiffi

f 0c

are in pounds per square inch

1cr ¼ strain at cracking of concrete ¼ 0.00008

fcr ¼ 3:75

ffiffiffi

f 0c

Stress – strain relationship of steel. The stress – strain curve of a steel bar in concrete relates the average

stress to the average strain of a long bar crossing several cracks, whereas the stress – strain curve of a bare

bar relates the stress to the strain at a local point (Okamura and Maekawa, 1991). In other words, a steel

bar in concrete is stiffened by the tensile stress of the concrete. If the tensile strength of concrete is

neglected, as it is in the most of truss models, the following equations are used:

If 1t # 1ty ; fl ¼ Es1l ð14:54Þ

If 1l . 1ly ; fl ¼ fly ð14:55Þ

where

Es ¼ modulus of elasticity of steel bars

fly ¼ yield stress of longitudinal steel bars

1ly ¼ yield strain of longitudinal steel bars

It was recommended by Belarbi and Hsu (1995) that both the tensile strength of concrete, presented in

the previous section, and the average stress – strain curve of steel stiffened by concrete, be taken into

account. In this model, the following equations are used for describing the stress – strain relationship

of steel:

If 1lEs # f 0ly ; fl ¼ Es1l ð14:56Þ

Hognestad

Parabola

ζeo

ζ fc

′

ζ=fn (ed /er)

′

eo 2eo −ed

−sd

FIGURE 14.16 Compression softening models with

Hognestad curve.

14-14 Vibration and Shock Handbook

If 1lEs # f 0ly ; fl ¼ 1 2

2 2 a=458

1000rl

􀀒 􀀓

½ð0:91 2 2BÞfly þ ð0:02 þ 0:25BÞEs1l􀀉 ð14:57Þ

where

B ¼ ð1=rlÞðfcr=fly Þ1:5 ð14:58Þ

f 0ly ¼ ½1 2 ð2 2 a=458Þ=1000rl􀀉ð0:93 2 2BÞfly ð14:59Þ

14.2.3.5 Solution Procedures

Figure 14.17 shows a framed shear wall. This kind of shear wall will be analyzed in this section. As

discussed by Hsu and Mo (1985), in the design of low-rise structural walls, the boundary elements are

reinforced to resist the applied bending moment, while the webs are designed to resist the applied shear

force. The size and shape of the boundary elements do not have a significant influence on the shear

behavior, as long as they are sufficient to carry the required bending moment. The effect of the boundary

elements on structural walls has been studied by Mo and Kuo (1998). Owing to the restriction of the

boundary elements, the strain of transverse steel in low-rise framed shear walls can be neglected, as

verified by the PCA tests; i.e., 1t ¼ 0: Therefore, adding Equation 14.48 and Equation 14.49 gives

1r ¼ 1l 2 1d ð14:60Þ

Inserting 1r sin2a ¼ 1r 2 1r cos2a into Equation 14.20 gives

cos2a ¼

1r 2 1l

1r 2 1d ð14:61Þ

Substituting Equation (14.60) and Equation (14.61) into Equation 14.45 results in

fl ¼

sl 2 sd ð21d Þ

ð1l 2 21d Þ

2 sr ð1l 2 1d Þ

ð1l 2 21d Þ

􀀒 􀀓

ð14:62Þ

hw I A I

HOR. DIR.

VER. DIR.

(a) General view

(b) Wall element

τb

τb cot α

σdb cos α

cot α

cos α

τ(1)

FIGURE 14.17 A framed shear wall.

Reinforced Concrete Structures 14-15

Neglecting the tensile strength of concrete, i.e., sr ¼ 0; gives

fl ¼

sl 2 sd ð21d Þ

ð1l 2 21d Þ

􀀒 􀀓

ð14:63Þ

For low-rise framed shear walls, the average shear stress t on the horizontal cross section is defined as

t ¼

bd ð14:64Þ

where d is the effective depth, which is defined as the distance between the centroids of the longitudinal

bars in the two flanges, b is the width of the web, and V is the horizontal shear force. The deflection at the

top of the shear wall, d; is determined by

d ¼ gh ð14:65Þ

where h ¼ height of the shear wall.

Based on the softened truss model theory presented above, the algorithm is shown in Figure 14.18

(Mo and Jost, 1993; Mo and Shiau, 1993) and is explained below.

1. Select a given 1d :

2. Assume a value of 1l:

3. Calculate 1r from Equation 14.60.

4. Calculate z using Equation 14.51b.

5. Calculate sd from Equation 14.51a.

6. Calculate sr from Equation 14.52 or

Equation 14.53.

7. Calculate fl from Equation 14.62 or

Equation 14.63.

8. Check fl using Equation 14.54 and

Equation 14.55 or from Equation 14.56

to Equation 14.59.

9a. If the calculated value for fl determined in

Step 8 is not sufficiently close to the value

shown in step 7, repeat steps 2 to 7.

9b. If the calculated value for fl determined

in Step 8 is sufficiently close to

the value shown in Step 7, proceed to

calculate t (or V) and g (or d) from

Equation 14.47 (or Equation 14.64) and

Equation 14.49 (or Equation 14.65),

respectively. This will provide one set of

solutions.

Select other values of 1d and repeat Steps 1 to 9

for each 1d : This will provide a number of sets of

quantities. From these sets of quantities, one can

plot the shear stress vs. distortion curve (or shear

force vs. deflection curve), the longitudinal steel

strain vs. deflection curve, and the longitudinal

steel strain vs. concrete strain curve. In general, the

maximum 1d value can be chosen as 0.003 with an

increment of 0.00005.

Yes

Select ed

Estimate el

Calculate er

Calculate sd

Calculate sr

Calculate fl

Calculate t, g, n, d

Calculate z

check if

the error for fl

is acceptable

check if

ed > 0.003

END

FIGURE 14.18 Algorithm for framed shear wall

analysis.

14-16 Vibration and Shock Handbook

14.2.4 Time-History Analysis

To accurately determine the dynamic behavior of

concrete structures, the time-history analysis

(Clough and Penzien, 1993; Paz, 1997) is

preferred. This section will show an example for

single-DoF systems, such as simple beams,

torsional box tubes, spandrel beams, continuous

beams, one-story frames and one-story framed

shear walls. All of these structures will be

discussed later.

In time-history analysis, a framed shear wall

can be modeled as a nonlinear single-DoF system

(Figure 14.19). The dynamic incremental equilibrium

is shown in Figure 14.19(c).

The equation of the equilibrium is

mDy€i þ ciDy_i þ kiDyi ¼ DFi ð14:66Þ

where m is the mass at the top.

ci and ki are calculated for values of velocity and

displacement corresponding to time t and

assumed to remain constant during the increment

of time Dt: Incremental acceleration, incremental

velocity, and incremental displacement are Dy€i;

Dy_i; and Dyi; respectively.

To perform the step-by-step integration of

Equation 14.66, the linear acceleration method is

employed. In this method, it is assumed that the

acceleration may be expressed by a linear

function of time during the time interval Dt:

Let ti and tiþ1 ¼ ti þ Dt be, respectively, the

designation for the time at the beginning and at

the end of the time interval Dt: When the

acceleration is assumed to be a linear function of

time for the interval of time ti to tiþ1 ¼ ti þ Dt;

as shown in Figure 14.20, the acceleration may be

expressed as

y€ðtÞ ¼ y€i þ

Dy€i

Dt ðt 2 tiÞ ð14:67Þ

Integrating Equation 14.67 twice with respect to time between the limits ti and t ¼ ti þ Dt and

using the incremental displacement Dy as the basic variable gives

Dy€i ¼

Dt2 Dyi 2

yi 2 3y€i ð14:68Þ

and

Dy_i ¼

Dyi 2 3y_i 2

y€i ð14:69Þ

(a)

(b)

(c)

m F(t)

F(t)

k y

ΔFi

mΔyi kiΔyi

ciΔyi

··

FIGURE 14.19 (a) Framed shear wall with a mass m at

the top; (b) model for a nonlinear single-DoF system, and

force, the incremental damping force, the incremental

spring force and the incremental external force.

ÿi

Δÿi

Δt

ti ti+1

ÿi+1

FIGURE 14.20 Linear variation of acceleration during

time interval.

Reinforced Concrete Structures 14-17

The substitution of Equation 14.68 and Equation 14.69 into Equation 14.66 leads to the following

form of the equation of motion:

Dt2 Dyi 2

y_i 2 3y€i

􀀏 􀀐

þ ci

Dyi 2 3y_i 2

y€i

􀀏 􀀐

þ kiDyi ¼ DFi ð14:70Þ

Transferring all the terms containing the unknown incremental displacement, Dyi; to the left-hand

side gives

k􀀊iDyi ¼ DF􀀊i ð14:71Þ

where

k􀀊i ¼ ki þ

Dt2 þ

3ci

Dt ð14:72Þ

and

DF􀀊i ¼ DFi þ m

y_i þ 3y€i

􀀏 􀀐

þ ci 3y_i þ

y€i

􀀏 􀀐

ð14:73Þ

It should be noted that Equation 14.71 is equivalent to the static incremental-equilibrium equation

and may be solved for the incremental displacement by simply dividing the equivalent incremental

load, DF􀀊i; by the equivalent spring constant k􀀊i:

The displacement yiþ1 and the velocity y_iþ1 at time tiþ1 ¼ ti þ Dt are

yiþ1 ¼ yi þ Dyi ð14:74Þ

and

y_iþ1 ¼ y_i þ Dy_i ð14:75Þ

The acceleration y€iþ1 at the end of the time step is obtained directly from the differential equation of

motion to avoid the errors that generally might tend to accumulate from step to step. It follows

y€iþ1 ¼

m ½Fðtiþ1Þ 2 ciþ1y_iþ1 2 kiþyi þ1􀀉 ð14:76Þ

where the coefficients ciþ1 and kiþ1 are now evaluated at time tiþ1.

After the displacement, velocity, and acceleration have been determined at time tiþ1 ¼ ti þ Dt;

the procedure just outlined is repeated to calculate these quantities at the following time step, tiþ2 ¼

tiþ1 þ Dt:

In general, sufficiently accurate results can be obtained if the time interval is taken to be no longer than

one tenth of the natural period of the structure (Clough and Penzien, 1993; Paz, 1997).

Пресс-релиз популярных книг

14.2 Analytical Models

Популярные книги

Популярные статьи

Навигация