179

4Analysis of Beams on

Elastic Half-Space

4.1 IntroductionAs mentioned earlier in Chapter 1, some soils under applied pressure behave as elastic material following the rules of the theory of elasticity, and soil models such as elas-tic half-space and elastic layer may produce more realistic practical results compared to results obtained by using Winkler foundation. But analysis of beams supported on elastic half-space is a difficult problem and few works are devoted to this subject. Most known publications are scientific and research papers written by academics and researchers that cannot be used by practicing engineers. Nevertheless, there are some methods that allow performing analysis of simple beams supported on elastic half-space as well as on elastic layer. These methods can be used for hand calculations and computer analysis as well. Chapter 4 presents equations and tables for analysis of free supported beams on elastic half-space developed by Borowicka (1938, 1939) and Gorbunov-Posadov (1940, 1949, 1953, 1984). Theoretical bases of the method are not discussed in this book. The reader can find a detailed explanation of the method in the original publications mentioned above and a good review of the method by Selvadurai (1979). This chapter also includes the analysis of complex beams, such as beams with various boundary conditions, various continuous beams, stepped beams developed by the author of this book (Tsudik 2006), and analysis of frames on elastic half-space including direct methods as well as iterative methods. Tables are developed for beams that meet the following two requirements: a/b ≥ 7 and the width of the beam is nar-row enough so the bending in the transverse direction can be ignored; a is half of the length of the beam and b is half of the width of the beam, as shown in Figures 4.1 and 4.2. Tables are developed for three categories of beams: 1. rigid beams, 2. short beams, and 3. long beams.

In order to find out if the beam belongs to rigid beams, the following parameter is obtained:

/t E a b E I2 10

3 21

r o= -` j: D (4.1)

All tables can be found at the end of the chapter.

SAMPLE C

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180 Analysis of Structures on Elastic Foundations

where E0 is the modulus of elasticity of the soil, o is Poisson’s ratio of the soil, E1 is the modulus of elasticity of the beam material, and I is the moment of inertia of the beam. The beam belongs to rigid beams when t ≤ 0.5. The beam belongs to rigid beams also when a = a/b < 20 and 0.5 ≤ t ≥ 1.

In order to find out if the beam belongs to long beams, parameter L of the system

beam-soil is found as follows: ( )

Lb E

E I2 1

0

12

3o

=-

l where b′ = 2b and coefficients m and

b are obtained as Lam = and

Lbb = . The beam belongs to long beams when b < 0.15

and m > 1 or when b ≤ 0.30 and m > 2 or when b ≤ 0.50 and m > 3.5. If parameters of the beam do not meet requirements for rigid and long beams, the beam belongs to the short beams and analysis of such beams is performed using tables for analysis of short beams. Tables for analysis of short and long beams are shown at the end of the chapter. Now, we can start with the analysis of rigid beams. Tables and equations presented in Chapter 4 are developed only for analysis of simple free supported beams.

The author of this book (Tsudik 2006) proposed to use the method of forces for analysis of complex beams that includes simple beams with various boundary con-ditions and continuous beams, including beams with various intermediate and end supports.

4.2 Analysis of Rigid BeamsTables 4.1–4.5 are developed for analysis of rigid beams on elastic half-space for two types of loads: concentrated vertical loads and moments applied to the center of the beam. If a concentrated vertical load P0 is applied to the center of the beam, the fol-lowing equations are used:

ppb aP

0

0

$=

l, , , ,M MQ P a Y Y

E aP

tgQ P 1 00 0 0 0 0

0

20! $ $ $ $o {== =

-= (4.2)

Figure 4.1

Figure 4.2

SAMPLE C

HAPTER

Analysis of Beams on Elastic Half-Space 181

In equations 4.2, p is the soil pressure, Q is the shearing force, M is the moment, and Y is the settlement of the beam. Coefficients p

0, Q

0, M

0, and Y 0 are dimensionless and

taken from Table 4.2. Rotation of the beam for this type of load, tg{ = 0. If the beam is loaded with a concentrated moment m0 equations 4.3 are used:

1 1, ,

,

Qam

M m

Y tgE

xam

tg tgE a

m

ppb am

1 1

00

10

2

20

10

2

0

0

1 20 !

! $

!

{ o { { o

= =

=-

=-

= Q Ml

_

`

a

bbb

bb

(4.3)

In these equations, x = x′/a, where x′ is the distance of the cross section from the cen-ter of the beam and is always positive regardless of where the cross section is located, whether at the left or right half of the beam. Two signs (±) mean that the upper sign belongs to the right half and the lower sign belongs to the left half of the beam. Tables 4.2 and 4.4 do not contain information for beams with 7 ≤ a < 10. In order to obtain the soil pressure, shearing forces, and moments in this case, it is recommended to use data for a = 10. However, settlements and rotations of the beam, when a < 10, cannot be obtained from these tables and have to be found from equations 4.4 shown below:

E

KAP

tgE a

K m

1

1

p

x

0

2

00

0

2

31$

~ o

{ o

=-

=-

_

`

a

bbb

bb

(4.4)

In 4.4, coefficient K0 is dimensionless and is obtained from Table 4.1; A is the area of the foundation. Coefficient K1 is also dimensionless and obtained from Table 4.1; Kx is the moment applied to the beam in the direction of the length. Table 4.1 is built using specified values of these two coefficients. Equations 4.4 are obtained for totally rigid individual rectangular foundations and used here for beam analysis when a < 10. Analysis of the beam is performed as follows. Each half of the beam is divided into 10 sections, as shown in Figure 4.3. Since the loads are symmetrical or asymmetrical, coef-ficients are given only for the right half of the beam starting from point 0.

If the beam has a series of vertical concentrated loads, they have to be replaced with one load and one moment applied to the center of the beam. If the beam has

Figure 4.3

SAMPLE C

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182 Analysis of Structures on Elastic Foundations

a distributed load located at any part of the beam, this load is replaced with a series of concentrated vertical loads and all these loads, in turn, are also replaced with one moment and one vertical load applied to the center of the beam. The total moment applied to the center of the beam m0 is equal to the sum of all moments produced by vertical loads Pi and moments mi applied to the beam. The total vertical load P0 is equal to the sum of all vertical loads Pi including distributed loads q(x). The moment is positive if applied clockwise. When P0 and m0 are found, the soil pressure is obtained as follows:

0 a ap pp

P m0

1 20!=

b bl l (4.5)

Coefficients p0 and p

1 are obtained from Tables 4.2 and 4.4, respectively, taking into

account the actual value of parameters a = a/b and x = x′/a at each point of the beam. Vertical concentrated loads Pi, distributed loads q(x), and moments applied to the beam will produce shear forces Q(x). The shearing force at any point of the beam is obtained from equation 4.6:

Q QQ x Pam

Q0 0 1

0ext

!= + -` j (4.6)

where Q0 and Q

1 are found from Tables 4.2 and 4.4, respectively, and Qext for the

right half of the beam is equal to P Px 0-/ . For the left half of the beam Q Pxext

=/ , where Px/ includes all vertical loads located at distance x between the left end of the beam and the point the shear is obtained. Vertical concentrated loads Pi, distributed loads q(x), and moments mi will also produce moments that are obtained from equa-tion 4.7:

M M P a M m M0 0 1 0 ext!= + (4.7)

In 4.7 coefficients M0 and M

1 are obtained from Tables 4.2 and 4.4, respectively. Mext

for the right half of the beam are found from equation 4.8:

M M m P ax mx x 0 0ext= + + -// (4.8)

where Mx/ is the sum of all moments produced by all vertical concentrated (Pi) and distributed loads q(x) located at left from point x; mx/ is the sum of all concentrated moments applied to the beam and located at left from point x.

For the left half of the beam:

M M mx xext= +// (4.9)

The settlement of the beam is obtained from equation 4.10:

gY Y P t xam

aE1

0 0 10

0

2

! {o

=-

e`o

j (4.10)

The soil pressure, shear forces, moments, settlements, and rotations of the beam can be found using equations presented above and Tables 4.1–4.5. A simple numerical ex-ample that illustrates analysis of rigid beams supported on elastic half-space is shown next.

SAMPLE C

HAPTER

Analysis of Beams on Elastic Half-Space 183

Example 4.1Given: A reinforced concrete beam loaded with a vertical load P = 200t is shown in Figure 4.4. The length of the beam is 2a = 10m, the cross section is equal to b′ ∙ h = 2m ∙ 1.5m(h) = 3m2 (b′ = 2b), and the moment of inertia of the beam section is equal

to h . .I12 12

2 1 5 0 5625m3 3

3$= = =

bl . The beam is loaded with a vertical concentrated

load equal to 200t. The modulus of elasticity of the soil is E = 300kg/cm2 = 3000t/m2; Poisson’s ratio m = 0.4 and the modulus of elasticity of the concrete is E1 = 3,000,000t/m2. Find the soil pressure, shearing forces, and moments. Build the soil pressure, shear, and moment diagrams.

Figure 4.4

Solution:

1. Find parameter t = rEa3b/[2(1 − m2)EI] = 3.14 ∙ 3,000 ∙ 53 ∙ 1/(1.68 ∙ 0.5625 ∙ 3,000,000) = 0.415. Since t < 0.5, the beam is rigid.

2. Find ba

15 5a = = = and coefficients p

0 from Table 4.2. As can be seen, the

closest parameter a = 10. So all coefficients are obtained for a = 10. The soil pressure diagram is shown in Figure 4.5.

3. Find the soil pressure. The soil pressure is obtained, for convenience, only in all other points of the beam.

. ./ /pp P b a 0 439 200 1 5 17 56t/m0 0 0

$ $ $= = = ; p0.1 = 0.440 ∙ 40 = 17.6t/m; p0.2 = 0.442 ∙ 40 = 17.68t/m; p0.3 = 0.446 ∙ 40 = 17.84t/m; p0.4 = 0.445 ∙ 40 = 18.2t/m; p0.5 = 0.462 ∙ 40 = 18.48t/m; p0.6 = 0.475 ∙ 40 = 19t/m; p0.7 = 0.498 ∙ 40 = 19.92t/m; p0.8 = 0.541 ∙ 40 = 21.64t/m; p0.9 = 0.632 ∙ 40 = 25.28t/m; p1.0 = 0.842 ∙ 40 = 33.68t/m

Figure 4.5 The soil pressure diagram

SAMPLE C

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184 Analysis of Structures on Elastic Foundations

The total soil pressure applied to the beam is equal to:

. 198.2p p0 5 tip

p

0.

.

0 1

1 0

+ =/ , that is close to the load applied to the beam P0 = 200t. The

difference is only 0.9%. The shear diagram is built analogously, as shown in Figure 4.6. Shear forces are found below using the second equation from 4.2:

.QQ P 0 5 200 100t0 0 0

$ $=- =- =- ; Q0.1 = −0.456 ∙ 200 = −91.2t; Q0.2 = −0.412 ∙ 200 = −52.4t; Q0.3 = −0.367 ∙ 200 = −73.4t; Q0.4 = −0.322 ∙ 200 = −64.4t; Q0.5 = −0.277 ∙ 200 = −55.4t; Q0.6 = −0.230 ∙ 200 = −46.0t; Q0.7 = −0.182 ∙ 200 = −36.4t; Q0.8 = −0.130 ∙ 200 = −26.4t; Q0.9 = −0.072 ∙ 200 = −14.4t; Q1.0 = 0.

Obtained moments and the moment are shown below. The moment diagram is shown in Figure 4.7.

. .MM P a 0 2703 200 5 270 3tm0 0 0

$ $ $ $= = = ; M0.1 = 0.2225 ∙ 200 ∙ 5 = 222.5tm; M0.2 = 0.1791 ∙ 200 ∙ 5 = 179.1tm; M0.3 = 0.1401 ∙ 200 ∙ 5 = 140.1tm; M0.4 = 0.1056 ∙ 200 ∙ 5 = 105.6tm; M0.5 = 0.0756 ∙ 200 ∙ 5 = 75.6tm; M0.6 = 0.0502 ∙ 200 ∙ 5 = 50.2tm; M0.7 = 0.0295 ∙ 200 ∙ 5 = 29.5tm; M0.8 = 0.0139 ∙ 200 ∙ 5 = 13.9tm; M0.9 = 0.0037 ∙ 200 ∙ 5 = 3.7tm; M1.0 = 0

Taking into account that a = 5 < 10, the settlement of the beam is obtained from the first equation 4.4:

. .,

.E

KAP1

3 0001 0 4 0 77

10 2200 0 964cmp

0

2

00

2

$ $$

~ o=-

=-

=

The soil pressure, shear, and moment diagrams are shown in Figures 4.5, 4.6, and 4.7, respectively.

Figure 4.6 Shear diagram

Figure 4.7 Moment diagram

SAMPLE C

HAPTER

Analysis of Beams on Elastic Half-Space 185

Example 4.2The given beam is loaded with asymmetrical loads as shown in Figure 4.8, assuming P1 = 100t M1 = 10t 3x m

1=l x 2m

2=l . Find the shear forces and rotation of the beam.

Figure 4.8

Solution:

1. The total moment applied to the center of the beam is equal to:

175tm25 100m M P a x 5 30 1= - - = - - =-l` `j j

2. The total vertical concentrated load applied to the center of the beam is equal to:

P0 = P = 100t

3. The shear forces are obtained below. For the right half of the beam we have:

. ( . )( )

.Q Q P Qam

Q 0 5 100 0 7095175

25 185t0 0 0 1

0ext

$ $= + - =- + --

=-

. ( . )( )

.Q Q P Qam

Q 0 456 100 0 7045175

20 96t.0 1 0 0 1

0ext

$ $= + - =- + --

=-

. ( . )( )

.Q Q P Qam

Q 0 412 100 0 6865175

17 19t.0 2 0 0 1

0ext

$ $= + - =- + --

=-

. ( . )( )

.Q Q P Qam

Q 0 367 100 0 6585175

13 67t.0 3 0 0 1

0ext

$ $= + - =- + --

=-

. ( . )( )

.Q Q P Qam

Q 0 322 100 0 6175175

10 605t.0 4 0 0 1

0ext

$ $= + - =- + --

=-

. ( . )( )

.Q Q P Qam

Q 0 277 100 0 5635175

7 995t.0 5 0 0 1

0ext

$ $= + - =- + --

=-

. ( . )( )

.Q Q P Qam

Q 0 230 100 0 4965175

5 64t.0 6 0 0 1

0ext

$ $= + - =- + --

=-

SAMPLE C

HAPTER

186 Analysis of Structures on Elastic Foundations

. ( . )( )

.Q Q P Qam

Q 0 182 100 0 4135175

3 545t.0 7 0 0 1

0ext

$ $= + - =- + --

=-

. ( . )( )

.Q Q P Qam

Q 0 130 100 0 3115175

2 115t.0 8 0 0 1

0ext

$ $= + - =- + --

=-

. ( . )( )

.Q Q P Qam

Q 0 072 100 0 1805175

0 900t.0 9 0 0 1

0ext

$ $= + - =- + --

=-

Q1 = 0

For the left half of the beam:

. ( . )( )

.Q Q P Qam

Q 0 5 100 0 7095175

25 185100 t0 0 0 1

0ext

$ $= + - = --

- =-

. ( . )( )

.Q Q P Qam

Q 0 456 100 0 7045175

100 29 76t.0 1 0 0 1

0ext

$ $= + - = --

- =--

. ( . )( )

.Q Q P Qam

Q 0 412 100 0 6865175

100 35 79t.0 2 0 0 1

0ext

$= + - = + --

- =--

.100 40 27t=-. ( . )( )

Q Q P Qam

Q 0 367 100 0 6585175

.0 3 0 0 1

0ext

$= + - = + --

--

. ( . )( )

.Q Q P Qam

Q 0 322 100 0 6175175

100 46 205t.0 4 0 0 1

0ext

$= + - = + --

- =--

. ( . )( )

.Q Q P Qam

Q 0 277 100 0 5635175

47 401t.0 5 0 0 1

0ext

$= + - = + --

=-

. ( . )( )

.Q Q P Qam

Q 0 230 100 0 4965175

40 36t.0 6 0 0 1

0ext

$= + - = + --

=-

. ( . )( )

.Q Q P Qam

Q 0 182 100 0 4135175

32 655t.0 7 0 0 1

0ext

$= + - = + --

=-

. ( . )( )

.Q Q P Qam

Q 0 130 100 0 3115175

23 885t.0 8 0 0 1

0ext

$= + - = + --

=-

. ( . )( )

.Q Q P Qam

Q 0 072 100 0 1805175

13 5t.0 9 0 0 1

0ext

$= + - = + --

=-

Q−1.0 = 0.

Taking into account that a = 5 < 10, rotation of the foundation is obtained using the second equation from 4.4.

The shear diagram is shown in Figure 4.9.

. . .,

tgE a

K m13 000

1 0 4 1 452175

0 00888x

0

2

31

2

3$ $ ${ o=-

=- -

=-` j

m0 = mx = −175tm. K1 = 1.45 is obtained from Table 4.1 for a = 5.

SAMPLE C

HAPTER

Analysis of Beams on Elastic Half-Space 187

4.3 Short Beam AnalysisAnalysis of short beams is performed using Tables 4.6–4.13 that are developed for analysis of beams with concentrated vertical loads P and moments m. Table 4.14 is developed for analysis of short beams with a uniformly distributed load q along the length of the beam. The beam is divided into 20 sections of equal length. The length of each section is equal to 0.1 a where a is half of the beam’s length, as shown previ-ously in Figure 4.3. Although the tables are developed for beams with a/b = 10, they produce good practical results when 7 ≤ a/b ≥ 15.

Tables 4.6–4.9 and equations 4.11 allow obtaining moments, soil pressures, shear forces, and settlements of the beam due to applied vertical load P. These equations look as follows:

p ~,M MPa pb aP Q QP

E aP1

0

2

~o

= = = =-

l

` j

(4.11)

In order to use Tables 4.6–4.9 and equations 4.11, we have to obtain parameter t from

equation 4.1 and two other parameters add = and x x

a=l , where d is the distance

between the point the load is applied and the center of the beam, and x′ is the distance from the center of the beam and the point in which M, p, Q, and ~ are obtained. It is important to mention that coefficients M and Q have to be multiplied by 10−3; coef-ficients p and ~ have to be multiplied by 10−2.

Tables 4.10–4.13 and equations 4.12 allow calculating moments, shear forces, soil pressures, and settlements of the beam due to applied concentrated moments m:

a~p, , ,M Mm Q Q

am p

bm

E am1

20

2

2

~o

= = = =-

l

` j

(4.12)

Figure 4.9 Shear diagram

SAMPLE C

HAPTER

188 Analysis of Structures on Elastic Foundations

The use of Tables 4.10–4.13 and equations 4.12 requires the following data: parameter

t that is obtained from equation 4.1, parameters add = and x x

a=l , where d is the distance

between the center of the beam and the point the moment m is applied, and x′ is the distance between the center of the beam and the point in which M, Q, p, and ~ have to be found. If the distance d is not divisible by 0.1a, the concentrated load (vertical load or moment) should be moved to the closest point the beam is divided. All tables are developed for the following numerical values of t: 1, 2, 5, and 10. If t < 0.75 analy-sis is performed using tables for rigid beams (Tables 4.1–4.5), when t > 10 analysis is performed using Tables 4.15–4.19 for analysis of long beams.

All numerical values of M and Q in the tables, for convenience, are multiplied by 1000 and all numerical values of p and Y are multiplied by 100. If, for example, in tables for M or Q the values are shown as equal to −045, the actual value of this coef-ficient is equal to −0.045, while in tables for p and Y it is equal to −0.45.

In Tables 4.6–4.9, the shearing force Q is shown in bold at all points of the beam, where the vertical load P is applied. This shear is equal to the shear QL

at left from the point the load is applied. The shear at right from this point is found as follows:

R LQ Q 1= - . When the beam is loaded at any point with a moment m as shown in tables 4.10–4.13, the moment shown bold is the moment at left from the point where the moment is applied and is equal to ML . The moment at right from that point is equal to M M 1L +=R . Two numerical examples illustrate the use of Tables 4.6–4.9 and equations 4.11 and Tables 4.10–4.13 and equations 4.12.

Example 4.3The given beam loaded with a vertical concentrated load is shown in Figure 4.10. The data are: the modulus of elasticity of the soil E0 = 3,000t/m2, the total length of the beam 2a = 12m, b′ = 2b = 1.6m, Poisson’s ratio o = 0.40, the modulus of elasticity of the beam material E1 = 2,600,000t/m2, the height of the beam h = 1m, the load P = 100t applied to the beam, and x′ = 2.4m at right from the center. Build the moment and shear diagrams.

Figure 4.10

To find out the category the beam belongs to, we find parameter t:

. . . . . .

/

, / , ,

E a b E I2 1

3 14 2 300 6 0 8 1 68 2 600 00012

1 6 1 2 14 0 5

t 03 2

1

33

$ $ $ $ $ $ 2

r o= -

= =

` j: D

SAMPLE C

HAPTER

Analysis of Beams on Elastic Half-Space 189

Since t = 2.14 > 0.5, the beam is not rigid. Since . ,

, , .

.L1 6 2 300

2 3 000 00012

1 6 1

6 013

3

3

$

$ $

= = ,

and b = b/L = 0.8/6.013 = 0.133 < 0.15, but m = a/L = 6/6.013 = 0.997 < 1, the beam belongs to short beams.

Solution:

Using Table 4.7 for t = 2 and d = d/a = 2.4/6 = 0.4, we can find the moments, shear forces, soil pressure, and settlements in all 21 points of the beam. Below, the moments are obtained using the first equation from 4.11:

.M MPa M M100 6 10 0 63$ $ $= = =- .M−1 = 0, M−0.9 = 0.6tm, M−0.8 = −1.2tm, M−0.7 = −1.8tm, M−0.6 = −2.4tm, M−0.5 = −1.8tm, M−0.4 = 0, M−0.3 = 3.0tm, M−0.2 = 8.4tm, M−0.1 = 16.2tm, M−0.0 = 27.0tm, M0.1 = 40.80tm, M0.2 = 58.20tm, M0.3 = 79.20tm, M0.4 = 105.0tm, M0.5 = 74.40tm, M0.6 = 49.20tm, M0.7 = 28.80tm, M0.8 = 13.80tm, M0.9 = 3.60tm, M1.0 = 0.

The moment diagram is shown in Figure 4.11.The shear diagram can be built analogously. Shear forces are obtained using the

third equation from 4.11:

.Q QP Q Q100 10 0 13$= = =-

Q−1 = 0, Q−0.9 = −11 ∙ 0.1 = −1.1t, Q−0.8 = −15 ∙ 0.1 = −1.5t, Q−0.7 = −11 ∙ 0.1 = −1.1t, Q−0.6 = 0 ∙ 0.1 = 0, Q−0.5 = 17 ∙ 0.1 = 1.7t, Q−0.4 = 42 ∙ 0.1 = 4.2t, Q−0.3 = 72 ∙ 0.1 = 7.2t, Q−0.2 = 109 ∙ 0.1 = 10.9t, Q−0.1 = 152 ∙ 0.1 = 15.2t, Q0 = 202 ∙ 0.1 = 20.2t, Q0.1 = 258 ∙ 0.1 = 25.8t, Q0.2 = 320 ∙ 0.1 = 32.0t, Q0.3 = 338 ∙ 0.1 = 33.8t, Q0.4 = 461 ∙ 0.1 = 46.1t, Q0.5 = −462 ∙ 0.1 = −46.2t, Q0.6 = −382 ∙ 0.1 = −38.2t, Q0.7 = −299 ∙ 0.1 = −29.9t, Q0.8 = −213 ∙ 0.1 = −21.3t, Q0.9 = −117 ∙ 0.1 = −11.7t, Q1 = 0.

The shear diagram is shown in Figure 4.12. Numerical values of the moments and shear forces are found for all 21 points of the beam, but shown, for convenience, in both diagrams only at all other points.

Figure 4.11 Moment diagram

SAMPLE C

HAPTER

190 Analysis of Structures on Elastic Foundations

Example 4.4The given beam is shown in Figure 4.13. Data: t = 2,

.

. .ad

5 42 7 0 5d = = = , m = 240tm.

Find the soil pressure and settlements of the beam.

Figure 4.13

By introducing the given numerical data into equations 4.12 we find:

p p p

~~ ~

, ,.

.. .

,

.. .

.

,

QM M Q QM m Qam p

b am

E am

2405 4240 44 44

1 4 5 4240 5 88

12 100 5 40 84 240 0 0032921

2 2

02

2

2

$

$$~

o

= = = = = = = =

=-

= =

l

` j

Now, the soil pressure and settlements of the beam can be found.

p−1.0 = −5.88 ∙ 2.1 = −12.3480t/m, p−0.9 = −5.88 ∙ 1.43 = −8.4080t/m, p−0.8 = −5.88 ∙ 1.08 = −6.3504t/m, p−0.7 = −5.88 ∙ 0.87 = −5.1156t/m, p−0.6 = −5.88 ∙ 0.73 = −4.2924t/m, p−0.5 = −5.88 ∙ 0.60 = −3.5280t/m, p−0.4 = −5.88 ∙ 0.49 = −2.8800t/m, p−0.3 = −5.88 ∙ 0.38 = −2.2344t/m, p−0.2 = −5.88 ∙ 0.27 = −1.5876t/m, p−0.1 = −5.88 ∙ 0.17 = −1.0000t/m, p−0.0 = −5.88 ∙ 0.07 = −0.4116t/m, p0.1 = 5.88 ∙ 0.04 = 0.2352t/m, p0.2 = 5.88 ∙ 0.15 = 0.8820t/m, p0.3 = 5.88 ∙ 0.28 = 1.6464t/m, p0.4 = 5.88 ∙ 0.43 = 2.5284t/m, p0.5 = 5.88 ∙ 0.58 = 3.4104t/m, p0.6 = 5.88 ∙ 0.75 = 4.4100t/m, p0.7 = 5.88 ∙ 0.95 = 5.5860t/m, p0.8 = 5.88 ∙ 1.21 = 7.1148t/m, p0.9 = 5.88 ∙ 1.60 = 9.4080t/m, p1.0 = 5.88 ∙ 2.32 = 13.6416t/m.

Settlements of the beam are found analogously:

Figure 4.12 Shear diagram

SAMPLE C

HAPTER

Analysis of Beams on Elastic Half-Space 191

..

. .,

mE a

m0 329

12 100 5 40 84 240 0 0032921 cm

02

2

2$$ ~~ ~

o~ ==

-= =

` j

~−1.0 = −0.32921 ∙ 0.87 = −0.286cm, ~−0.9 = −0.32921 ∙ 0.90 = −0.296cm, ~−0.8 = −0.32921 ∙ 0.94 = −0.395cm, ~−0.7 = −0.32921 ∙ 0.96 = −0.316cm, ~−0.6 = −0.32921 ∙ 0.98 = −0.322cm, ~−0.5 = −0.32921 ∙ 0.99 = −0.326cm, ~−0.4 = −0.32921 ∙ 1.00 = −0.329cm, ~−0.3 = −0.32921 ∙ 0.98 = −0.322cm, ~−0.2 = −0.32921 ∙ 0.94 = −0.395cm, ~−0.1 = −0.32921 ∙ 0.87 = −0.286cm, ~0.00 = −0.32921 ∙ 0.76 = −0.250cm, ~0.1 = −0.32921 ∙ 0.60 = −0.198cm, ~0.2 = −0.32921 ∙ 0.39 = −0.128cm, ~0.3 = −0.32921 ∙ 0.12 = −0.0395cm, ~0.4 = 0.32921 ∙ 0.23 = 0.076cm, ~0.5 = 0.32921 ∙ 0.67 = 0.221cm, ~0.6 = 0.32921 ∙ 1.14 = 0.375cm, ~0.7 = 0.32921 ∙ 1.58 = 0.520cm, ~0.8 = 0.32921 ∙ 2.02 = 0.665cm, ~0.9 = 0.32921 ∙ 2.44 = 0.665cm, ~1.0 = 0.32921 ∙ 2.87 = 0.945cm.

Soil pressure and settlements’ diagrams are shown in Figures 4.14 and 4.15, respec-tively. Numerical values of the soil pressure and settlements, for convenience, are not shown at all points of the beam. Moment and shear diagrams can be built analogously.

Figure 4.14 Soil pressure diagram

Figure 4.15 Diagram of the settlements

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192 Analysis of Structures on Elastic Foundations

4.4 Analysis of Short Beams with Uniformly Distributed LoadsTables and equations for such type of analysis are developed only for beams loaded completely from the left to the right ends, as shown in Figure 4.16.

Figure 4.16

Equations for obtaining M, Q, p, and Y are shown below:

p, ,,M aq p q Y YE

qM a q Q Q 12

0

2o= = =

-= (4.13)

Coefficients M , Q , p , and Y are taken from Table 4.14. Since the load applied to the beam is symmetrical, all coefficients are given only for the right half of the beam. Coefficient Q for the left half of the beam should be taken with the opposite sign.

Parameter x has the same meaning as in the tables shown earlier and is equal to xax

=l ,

where x′ is the distance from the center of the beam and the point coefficients M , Q , p , and Y are found. A numerical example is shown below.

It is important to mention that rotations of short beams at all points, except the left and right ends, are found from the following formula: tg{ = (Yi+1 − Yi−1)/0.2a. Rotations of the beam at the ends are found as follows:

tg{ = (Yi − Yi−1)/0.1a

where Yi is the settlement of the beam at point i.

Example 4.5The given beam is 10m long, parameter t = 10m. The beam is loaded with a uniformly distributed load q = 100t/m, coefficient t = 10. Build the moment and soil pressure diagrams.

Table 4.14 and equations 4.13 are used to build the moment and soil pressure dia-grams shown in Figures 4.17 and 4.18, respectively. Because of the beam symmetry both diagrams are built only for the right half of the beam.

Let us check the equilibrium of vertical loads applied to the beam. The total uni-formly distributed load applied to the beam is equal to 100 ∙ 5 = 500t. The total soil pressure is equal to:

98 ∙ 0.25 + 2 ∙ 97 ∙ 0.5 + 3 ∙ 96 ∙ 0.5 + 93 ∙ 0.5 + 95 ∙ 0.5 + 100 ∙ 0.5 + 110 ∙ 0.5 + 134 ∙ 0.25 = 498t

As shown, the difference between the total applied load and total soil pressure is neg-ligible: (498 ≈ 500).

SAMPLE C

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Analysis of Beams on Elastic Half-Space 193

4.5 Analysis of Long BeamsAs mentioned earlier, the beam belongs to the category of long beams when:

b < 0.15 and m > 1 or b ≤ 0.3 and m > 2 or b ≤ 0.5 and m > 3.5, where Lbb = and

aL

m = 1

Lb E

E I2

0

12

3m

=-

l

` j. Tables are developed only for analysis of beams

with concentrated vertical loads. If the beam is loaded with distributed loads, these loads have to be replaced with a series of concentrated vertical loads. When a moment is applied to the beam, the beam should be analyzed as a short beam. Analysis is per-formed using Tables 4.17–4.21 and the following equations:

p , , ,pLP M MPL Q QP Y Y

E LP1

0

2

$m= = = =-

(4.14)

Coefficients p , M , Q , and Y are obtained from Tables 4.17–4.21 that are developed for the following numerical values of b: 0.025, 0.075, 0.15, 0.3, 0.5. Analysis is per-formed in several steps:

Step 1: Choose the right table for analysis taking into account the following:

When 0.01 ≤ b ≤ 0.04, use Table 4.17 for b = 0.025.When 0.04 < b ≤ 0.10, use Table 4.18 for b = 0.075.

Figure 4.17 Moment diagram

Figure 4.18 Soil pressure diagram

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194 Analysis of Structures on Elastic Foundations

When 0.10 < b ≤ 0.20, use Table 4.19 for b = 0.15.When 0.20 < b ≤ 0.40, use Table 4.20 for b = 0.30.When 0.40 < b ≤ 0.70, use Table 4.21 for b = 0.50.

If the beam has several vertical loads Pi, as shown in Figure 4.19, analysis is performed for each load and final results are obtained by superposition. Analysis of long beams is based on analysis of infinite and semi-infinite beams.

Figure 4.19

Step 2: After the table for analysis is chosen, the so-called conditional distances be-tween each load and the left and right ends of the beam have to be found. Conditional distances, ali and ari, are obtained as follows:

Ld

Ld

lili

riria a= =

where dli and dri are the actual distances between the load and the left and right ends, respectively. Both types of distances are shown in Figure 4.19. When performing anal-ysis of long beams two cases are possible:

1. When b ≤ 0.2 and one of two values (ali, ari) is less than 1, or when b > 0.2 and one of the same values is less than 2. In this case analysis of the beam is per-formed using the smallest of the two a. The beam is analyzed as a semi-infinite beam.

2. When b ≤ 0.2 and one of two values (ali, ari) is larger than 1, or when b > 0.2 and one of the same two values is larger than 2. In this case analysis is per-formed using a = ∞. The beam is analyzed as an infinite beam.

Example 4.6Given: A beam shown in Figure 4.20, L = 2m b = 0.075 al = 0.4 ar = 2.2 E0 = 180kg/cm2 P = 60t o0 = 0.3 l = 6.2m.

Figure 4.20

SAMPLE C

HAPTER

Analysis of Beams on Elastic Half-Space 195

Find the soil pressure and settlements of the beam.

Solution:

p p ppLP

20060000 300= = = . .

,Y YY

E LP1

1801 0 3

20060 000

1 50

02 2

$m

=-

=-

=

Analysis is performed in Table 4.15.The soil pressure and settlements’ diagrams are shown in Figures 4.21 and 4.22,

respectively. The soil pressure is shown in kg/m, the settlements in cm.

Figure 4.21 Soil pressure diagram

Figure 4.22 Settlements diagram

Example 4.7The given beam is shown in Figure 4.23.

L = 3m, b = 0.3 . .33 6 1 2la = = , . .

34 2 1 4ra = = , Q QQ P 80$= =

Find the shear forces and build the shear diagram.Analysis is performed in Table 4.16. Using Table 4.20 for b = 0.3 and al = 1.2, nu-

merical values of Q are found and final shear forces are obtained. The shear diagram is shown in Figure 4.24. The soil pressure, moments, and settlements of the beam can be obtained analogously.

SAMPLE C

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196 Analysis of Structures on Elastic Foundations

4.6 Analysis of Complex BeamsAll tables presented and used in this chapter are developed for analysis of free sup-ported beams. When the ends of the beam are restrained against various deflections, analysis becomes more complex. For example, analysis of the beam shown in Figure 4.25 is performed easily using one of the tables presented in this chapter. Analysis of the beam shown in Figure 4.26, with one restraint against vertical deflection at the left end and two other restraints against vertical deflection and rotation at the right end, requires solving a system of three equations that reflect the actual boundary condi-tions. Analysis, in principle, is not different from the analysis used in Chapter 3 for finite beams on Winkler foundation. The same method is applied to analysis of beams supported on elastic half-space using tables and equations introduced above. Analysis of the beam shown in Figure 4.26 is performed as follows: After removal of all three restraints and replacing them with unknown reactions X1, X2, and X3, the beam will look as shown in Figure 4.27.

Figure 4.23

Figure 4.24 Shear diagram

Figure 4.25 Figure 4.26

SAMPLE C

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Analysis of Beams on Elastic Half-Space 197

Now, the beam is analyzed applying the given load P and all three unknown re-actions successively assuming that all unknown reactions are equal to one unit. All analyses are performed using tables and equations presented earlier. Using the results of these analyses, the following system of equations is written:

X X XX X XX X X

000

P

P

P

11 1 12 2 13 3 1

21 1 22 2 23 3 2

31 1 32 2 33 3 3

d d d d

d d d d

d d d d

+ + + =

+ + + =

+ + + =

_

`

a

bb

bb (4.15)

In this system, dik is deflection of the beam at point i due to one unit load applied at point k, diP is the deflection of the beam at point i due to the given load P. By solv-ing system 4.15, reactions X1, X2, and X3 are found and final analysis of the beam is performed by applying to the beam all given loads and reactions. The same method can be used for analysis of continuous beams, including beams with spring supports. For example, analysis of the beam shown in Figure 4.28 is performed as follows: Af-ter removing all restraints-supports and replacing them with unknown reactions, the given beam will look as shown in Figure 4.29. The total number of unknowns in this case is equal to the number of removed restraints. The system of equations will look as follows:

X X X X XX X X X X

X XC

X X X

X X X X XX X X X X

00

1 0

00

P

P

DP

P

P

3 4 5

3 4 5

33 3 4 5

3 4 5

3 4 55 5 5

11 1 12 13 14 15 1

21 22 2 23 24 25 2

31 32 2 34 35 3

41 42 2 43 44 45 4

51 52 2 53 54

2

1

1

1

1

d d d d d d

d d d d d d

d d d d d d

d d d d d d

d d d d d d

+ + + + + =

+ + + + + =

+ + + + + + =

+ + + + + =

+ + + + + =

e o

_

`

a

bbbb

bbbb

(4.16)

In this system, CD is the rigidity of an elastic support D. All other coefficients in this system have the same meaning as in system 4.15. By solving the system of equations 4.16 all unknown reactions are found. Final analysis of the beam is performed by ap-plying to the beam the given loads and obtained reactions.

As shown, analysis of these types of beams is much more time consuming compared to analysis of beams with free supported ends. Moreover, the method can be used only when the given beam is replaced with one free supported beam with the given loads and unknown reactions, as shown in Figure 4.29. When analysis requires replacement of the given complex beam with several simple beams, the method cannot be used because each simple beam supported on elastic half-space produces settlements not

Figure 4.27

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198 Analysis of Structures on Elastic Foundations

only under its loaded area, but also under other neighboring beams that are not taken into account. This method of analysis is justified for beams on Winkler foundation where all beams work independently without affecting each other, but when the beam is supported on elastic half-space, this method cannot be applied directly. However, we propose a simple method that can resolve this problem. Let us assume we have to analyze a pin-connected beam shown in Figure 4.30. Analysis is performed in the fol-lowing steps:

Step 1: The given beam is divided into two simple beams, as shown in Figure 4.31, and unknown forces of interaction X1 are applied to both beams at points 2. Taking into account that settlements of the beams 1–2 and 2–3 at point 2 are the same, equation 4.17 is written:

X 0( ) ( ) P P1 1 2 1 2 3 1 1 11 2

~ ~ ~ ~+ + + =- -

` j (4.17)

where ~1(1−2) and ~1(2−3) are settlements in direction X1 of the beams 1–2 and 2–3 due to one unit loads applied at point 2; P1 1

~ and P1 2~ are settlements of the beams 1–2

and 2–3 due to the given loads P1 and P2. Reactions X1 obtained from 4.17 are used for separate analyses of both beams.

Step 2: Both beams are analyzed separately as two simple beams.

Step 3: In order to specify analysis of one of the beams, let’s say beam 1–2, by tak-ing into account the soil pressure produced by the beam 2–3, the soil pressure under beam 2–3 is replaced with individual loads-stamps applied to the soil in 21 points. For example, the load applied at point i can be found from equation 4.18:

Pp l20i

i= (4.18)

Figure 4.28

Figure 4.29

SAMPLE C

HAPTER

Analysis of Beams on Elastic Half-Space 199

where pi is the soil pressure per one unit of the beam length 2–3 and l is the length of the beam. By applying loads Pi to the half-space under beam 2–3 and using equa-tions 1.23 and 1.25 from Chapter 1, additional settlements under beam 1–2 are found. These settlements will produce additional moments and shear forces in beam 1–2. As-suming that beam 1–2 is supported at all 21 points on non-yielding supports, as shown in Figure 4.33, and applying the settlements to all supports, analysis of beam 1–2 is performed.

Now, by summing obtained results with the results of the original analysis of beam 1–2, final moments and shear forces in beam 1–2 are found. The same method is used for analysis of beam 2–3.

It is obvious that this method can be applied to analysis of stepped beams and groups of close located foundations.

Figure 4.30

Figure 4.31

Figure 4.32

Additional settlements beam 1-2 Soil pressure diagram beam 2-3

SAMPLE C

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200 Analysis of Structures on Elastic Foundations

4.7 Analysis of Frames on Elastic Half-SpaceAnalysis of continuous foundations supported on the elastic half-space discussed above can take into account the rigidity of the superstructure. A method of analysis of foundations and 2D frames using the frame shown in Figure 4.34 is explained below. Analysis is performed in the following order:

The given system frame-foundation is divided into two parts: frame and foundation, as shown in Figure 4.35. All frame supports are restrained against any deflections. By applying successively to the frame the given loads and one unit deflections at all points of support, the following system of equations is written:

...

.......................................................

...

X r r r rX r r r r

X r r r r

n n P

n n P

n n n nn n nP

1 11 1 12 2 1 1

2 21 1 22 2 2 2

1 1 2 2

~ ~ ~

~ ~ ~

~ ~ ~

= + + + +

= + + + +

= + + + +

_

`

a

bbb

bb

(4.19)

In this system of equations:

Xi is the unknown reaction at point irij is the reaction at point i due to one unit load applied at point j~i is the settlement or rotation at point iriP is the reaction at point i due to the given loads applied to the frame.

Now, by applying the given loads, and unknown reactions acting in the opposite direc-tion to the foundation, all deflections of the foundation can be expressed as follows:

...

.............................................................

...

X X XX X X

X X X

n n P

n n P

n n n nn n nP

1 11 1 12 2 1 1

2 21 1 22 2 2 2

1 1 2 2

~ ~

~ ~

~ ~

D D D

D D D

D D D

= + + + +

= + + + +

= + + + +

_

`

a

bbb

bb

(4.20)

In this system of equations, Dij is deflection of the foundation at point i due to reaction equal to one unit and applied to point j, and ~iP is the deflection of the foundation at point i due to the given loads applied to the foundation.

The number of unknowns can be reduced by introducing reactions Xi from 4.19 into the system of equations 4.20 that leads to a system of equations with only n un-known deflections. In this case, the stiffness method is used. If deflections ~i from the

Figure 4.33 Beam 1–2 supported on 21 supports

SAMPLE C

HAPTER

Analysis of Beams on Elastic Half-Space 201

system of equations 4.20 are introduced into the system of equations 4.19, we again obtain a system of equations with only n unknown reactions. In this case, the method of forces is used.

By solving both systems of equations 4.19 and 4.20 as one system, we find 2n un-knowns: n deflections and n reactions. In this case we use the so-called combined method of analysis. Final analysis of the frame is performed by applying to the frame the given loads and deflections of all supports. Final analysis of the foundation is per-formed by applying to the foundation the given loads and obtained reactions acting in the opposite direction. It should be mentioned that the iterative method is another way to perform analysis. This method was discussed in Chapter 3 in the analysis of frames on Winkler foundation. The only difference between the analyses described in Chapter 3 and analysis in this chapter is the method of obtaining deflections of the foundation.

Figure 4.34 Figure 4.35

α 1 2 3 4 5 6 7 8 9 10

K0 0.88 0.86 0.82 0.79 0.77 0.74 0.73 0.71 0.69 0.67

K1 0.52 0.85 1.10 1.30 1.45 1.60 1.70 1.80 1.90 2.00

Table 4.1 Coefficients K0 and K1SAMPLE C

HAPTER

202 Analysis of Structures on Elastic Foundations

αx

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

P0

10 0.439 0.440 0.442 0.446 0.455 0.462 0.475 0.498 0.541 0.632 0.842

15 0.447 0.448 0.450 0.454 0.459 0.468 0.480 0.500 0.537 0.611 0.773

20 0.452 0.453 0.455 0.458 0.464 0.471 0.483 0.502 0.535 0.600 0.737

30 0.458 0.459 0.460 0.463 0.468 0.475 0.486 0.503 0.532 0.586 0.699

50 0.464 0.465 0.466 0.468 0.473 0.480 0.489 0.503 0.538 0.574 0.665

100 0.469 0.470 0.471 0.473 0.477 0.483 0.481 0.504 0.525 0.561 0.634

Q0

10 −0.500 −0.456 −0.412 −0.367 −0.322 −0.277 −0.230 −0.182 −0.130 −0.072 0

15 −0.500 −0.455 −0.411 −0.365 −0.320 −0.273 −0.225 −0.177 −0.125 −0.068 0

20 −0.500 −0.454 −0.410 −0.363 −0.317 −0.271 −0.223 −0.174 −0.122 −0.066 0

30 −0.500 −0.454 −0.408 −0.362 −0.315 −0.268 −0.220 −0.171 −0.119 −0.063 0

50 −0.500 −0.453 −0.407 −0.360 −0.313 −0.265 −0.217 −0.168 −0.116 −0.061 0

100 −0.500 −0.453 −0.406 −0.359 −0.311 −0.263 −0.214 −0.165 −0.113 −0.058 0

M0

10 0.2703 0.2225 0.1791 0.1401 0.1056 0.0756 0.0502 0.0295 0.0139 0.0037 0

15 0.2672 0.2194 0.1761 0.1373 0.1931 0.0735 0.0486 0.0284 0.0132 0.0035 0

20 0.2654 0.2176 0.1744 0.1357 0.1017 0.0722 0.0475 0.0277 0.0128 0.0034 0

30 0.2634 0.2156 0.1725 0.1340 0.1002 0.0710 0.0465 0.0270 0.0125 0.0033 0

50 0.2615 0.2137 0.1707 0.1323 0.0986 0.0697 0.0455 0.0263 0.0120 0.0031 0

100 0.2596 0.2120 0.1690 0.1307 0.0972 0.0685 0.0446 0.0256 0.0117 0.0030 0

Table 4.2 Dimensionless coefficients for analysis of rigid beams loaded with a verticalconcentrated load P0 applied at the center of the beam

a 10 15 20 30 50 100

Y0

1.081 1,210 1.302 1.431 1.595 1.814

Table 4.3 Dimensionless coefficients Y0 for rigid beams loaded

with a vertical concentrated load P0 applied at the center of the beamSAM

PLE CHAPTER

Analysis of Beams on Elastic Half-Space 203

αx

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

p0

10 0 0.114 0.229 0.347 0.470 0.600 0.746 0.918 1.148 1.506 2.155

15 0 0.119 0.239 0.362 0.490 0.625 0.772 0.943 1.162 1.483 2.024

20 0 0.122 0.245 0.371 0.501 0.638 0.786 0.957 1.169 1.449 1.957

30 0 0.125 0.252 0.381 0.513 0.652 0.802 0.971 1.176 1.415 1.885

50 0 0.129 0.258 0.390 0.525 0.667 0.817 0.984 1.181 1.414 1.820

100 0 0.133 0.265 0.400 0.538 0.680 0.831 0.996 1.86 1.415 1.761

Q0

10 −0.709 −0.704 −0.686 −0.658 −0.617 −0.563 −0.496 −0.413 −0.311 −0.180 0

15 −0.716 −0.710 −0.692 −0.662 −0.619 −0.564 −0.494 −0.408 −0.304 −0.173 0

20 −0.719 −0.713 −0.695 −0.664 −0.621 −0.564 −0.493 −0.406 −0.300 −0.169 0

30 −0.723 −0.717 −0.698 −0.667 −0.622 −0.564 −0.491 −0.403 −0.296 −0.165 0

50 −0.727 −0.721 −0.701 -.0669 −0.623 −0.564 −0.490 −0.400 −0.292 −0.162 0

100 −0.731 −0.724 −0.705 −0.671 −0.624 −0.564 −0.488 −0.397 −0.288 −0.158 0

M0

10 0.5 0.4293 0.3597 0.2924 0.2285 0.1694 0.1163 0.0707 0.0343 0.0095 0

15 0.5 0.4285 0.3584 0.2907 0.2265 0.1672 0.1142 0.0690 0.0332 0.0091 0

20 0.5 0.4283 0.3577 0.2897 0.2253 0.1660 0.1130 0.0679 0.0324 0.0088 0

30 0.5 0.4297 0.3570 0.2887 0.2241 0.1647 0.1119 0.0670 0.0319 0.0084 0

50 0.5 0.4275 0.3563 0.2876 0.2229 0.1635 0.1107 0.0660 −0.0313 0.0083 0

100 0.5 0.4271 0.3556 0.2867 0.2217 0.1622 01095 0.0651 −0.0306 0.0081 0

Table 4.4 Dimensionless coefficients for analysis of rigid beams loaded with a concentrated moment m0 applied at the center of the beam

a 10 15 20 30 50 100

tg1{ 2.088 2.466 3.737 3.122 3.606 4.311

Table 4.5 Coefficients tg1{ for rigid beams loaded with a concen-

trated moment m0SAMPLE C

HAPTER

204

t =

1

Co

ef.

δx

−1

−0.

9−

0.8

−0.

7−

0.6

−0.

5−

0.4

−0.

3−

0.2

−0.

10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

M

0

000

301

002

203

805

908

511

615

219

324

119

315

211

608

505

903

802

201

000

30

Q0

050

096

140

186

234

283

335

388

444

500*

−44

4−

338

−33

5−

283

−23

4−

186

−14

0−

096

−05

00

p05

504

704

504

504

604

805

005

305

505

605

605

605

505

305

004

804

604

504

504

705

5

~08

709

209

710

210

711

111

611

912

212

412

512

412

211

911

611

110

710

209

709

208

7

M

0.1

000

200

701

502

704

306

308

811

715

219

223

718

814

510

807

605

002

901

300

30

Q0

034

068

102

139

180

223

269

319

372

426

482*

−46

0−

403

−34

6−

290

−23

5−

181

−12

7−

069

0

p0.

3603

303

403

603

804

204

504

805

105

405

605

705

705

705

605

50.

540.

5405

606

207

7

~06

907

608

208

809

510

110

611

211

612

112

412

612

712

512

412

211

911

611

310

910

6

M

0.2

000

100

400

901

702

904

306

108

411

214

418

222

617

613

109

306

103

601

700

40

Q0

019

040

066

095

128

165

205

250

300

352

408

467*

−47

3−

412

−35

0−

288

−22

4−

158

−08

50

p01

902

002

302

703

103

503

904

304

705

105

505

705

906

106

106

206

306

506

907

709

6

~05

005

906

607

408

108

909

710

411

111

712

212

713

013

213

413

413

413

213

113

112

9

M

0.3

000

000

100

300

801

402

403

605

307

309

912

916

620

815

711

207

404

402

000

50

Q0

005

015

032

053

079

109

145

185

230

280

335

393

454*

−48

1−

410

−33

0−

270

−19

2−

106

0

p00

200

801

401

902

402

803

303

804

304

805

205

606

005

306

606

907

107

508

109

412

1

~03

304

305

206

006

907

808

709

510

411

212

012

613

213

714

214

414

614

815

015

115

2

M

0.4

000

0−

001

−00

1−

001

001

005

012

023

037

055

079

107

142

184

132

088

052

024

007

0

Q0

−00

7−

007

000

012

031

056

085

121

162

208

260

318

380

448*

−47

940

2−

319

−22

9−

127

0

p−

010

−00

300

301

001

602

202

703

303

804

404

905

506

006

507

007

508

008

609

511

214

5

~01

502

503

604

605

606

607

608

609

610

611

612

413

414

214

915

516

016

516

917

417

8

0.5

0−

001

−00

4−

007

−01

0−

012

−01

3−

011

−00

700

001

202

805

007

711

115

310

206

102

900

80

0−

021

−03

1−

032

−02

7−

015

003

028

058

095

138

188

244

307

376

453*

−46

2−

370

−26

7−

148

0

−02

7−

015

−00

500

200

901

502

102

703

404

004

605

305

906

607

308

008

809

710

913

117

0

001

102

203

304

405

506

607

808

910

111

212

213

414

515

516

517

418

118

919

720

4

0.6

0−

002

−00

6−

012

−01

8−

024

−03

0−

034

−03

5−

034

−02

9−

020

−00

601

404

107

511

807

003

300

90

0−

032

−05

2−

063

−06

5−

059

−04

7−

029

−00

302

906

811

516

923

230

338

447

4*−

425

−30

9−

172

0

−03

9−

026

−01

5−

006

002

009

015

022

029

036

043

051

068

067

075

085

096

108

125

151

198

−01

6−

004

007

020

031

044

056

068

081

094

107

120

134

148

161

174

186

199

210

221

233

0.7

0−

002

−00

8−

017

−02

6−

037

−04

7−

056

−06

3−

069

−07

0−

068

−06

1−

049

−02

9−

002

034

080

038

010

0

0−

044

−07

4−

092

−10

2−

103

−09

7−

085

−06

4−

036

000

044

096

158

230

313

409

519*

−35

2−

198

0

−05

2−

036

−02

4−

014

−00

500

200

901

702

403

204

004

805

706

707

708

910

311

914

017

222

9

−03

0−

018

−00

600

702

003

304

606

007

408

810

211

813

314

916

518

219

921

423

024

626

2

0.8

0−

003

−01

0−

021

−03

5−

049

−06

4−

078

−09

1−

103

−11

1−

116

−11

6−

111

−09

9−

079

−05

0−

010

043

012

0

0−

055

−09

5−

122

−13

9−

147

−14

7−

139

−12

4−

101

−06

9−

028

023

084

157

243

344

462

603*

−22

40

−06

5−

047

−03

3−

022

−01

2−

004

004

011

019

027

036

046

056

067

079

093

109

129

155

194

259

−04

7−

033

−02

0−

006

008

022

036

050

066

081

097

115

132

151

170

190

211

231

252

272

293

0.9

0−

003

−01

3−

026

−04

3−

061

−08

−10

0−

119

−13

6−

152

−16

4−

171

−17

3−

169

−15

6−

133

−10

0−

052

013

0

0−

067

−11

6−

151

−17

5−

190

−19

6−

194

−18

7−

165

−13

7−

099

−05

001

008

417

227

840

455

874

8*0

−07

9−

057

−04

1−

029

−01

9−

010

−00

200

601

402

303

304

305

406

708

109

711

513

817

021

229

3

−06

2−

047

−03

3−

018

−00

401

102

504

105

807

409

211

113

115

217

419

822

224

727

229

832

5

1.0

0−

004

−01

5−

031

−05

1−

073

−09

7−

122

−14

7−

170

−19

2−

213

−22

6−

235

−23

3−

232

−21

7−

189

147

085

0

0−

079

−13

7−

180

−21

2−

233

−24

5−

249

−24

4−

229

−20

5−

170

−12

3−

064

011

102

212

347

512

721

1000

*

−09

2−

067

−05

0−

037

−02

6−

017

−00

800

101

001

902

904

105

306

608

210

012

114

818

423

832

6

−07

6−

061

−04

6−

031

−01

6−

001

016

032

050

068

088

108

130

154

179

206

233

263

293

325

356

Tab

le 4

.6 D

imen

sion

less

coe

ffici

ents

for

ana

lysi

s of

sho

rt b

eam

s lo

aded

with

a c

once

ntra

ted

ver

tical

load

P

SAMPLE C

HAPTER

205

t =

1

Co

ef.

δx

−1

−0.

9−

0.8

−0.

7−

0.6

−0.

5−

0.4

−0.

3−

0.2

−0.

10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

000

301

002

203

805

908

511

615

219

324

119

315

211

608

505

903

802

201

000

30

005

009

614

018

623

428

333

538

844

450

0*−

444

−33

8−

335

−28

3−

234

−18

6−

140

−09

6−

050

0

055

047

045

045

046

048

050

053

055

056

056

056

055

053

050

048

046

045

045

047

055

087

092

097

102

107

111

116

119

122

124

125

124

122

119

116

111

107

102

097

092

087

0.1

000

200

701

502

704

306

308

811

715

219

223

718

814

510

807

605

002

901

300

30

003

406

810

213

918

022

326

931

937

242

648

2*−

460

−40

3−

346

−29

0−

235

−18

1−

127

−06

90

0.36

033

034

036

038

042

045

048

051

054

056

057

057

057

056

055

0.54

0.54

056

062

077

069

076

082

088

095

101

106

112

116

121

124

126

127

125

124

122

119

116

113

109

106

0.2

000

100

400

901

702

904

306

108

411

214

418

222

617

613

109

306

103

601

700

40

001

904

006

609

512

816

520

525

030

035

240

846

7*−

473

−41

2−

350

−28

8−

224

−15

8−

085

0

019

020

023

027

031

035

039

043

047

051

055

057

059

061

061

062

063

065

069

077

096

050

059

066

074

081

089

097

104

111

117

122

127

130

132

134

134

134

132

131

131

129

0.3

000

000

100

300

801

402

403

605

307

309

912

916

620

815

711

207

404

402

000

50

000

501

503

205

307

910

914

518

523

028

033

539

345

4*−

481

−41

0−

330

−27

0−

192

−10

60

002

008

014

019

024

028

033

038

043

048

052

056

060

053

066

069

071

075

081

094

121

033

043

052

060

069

078

087

095

104

112

120

126

132

137

142

144

146

148

150

151

152

0.4

000

0−

001

−00

1−

001

001

005

012

023

037

055

079

107

142

184

132

088

052

024

007

0

0−

007

−00

700

001

203

105

608

512

116

220

826

031

838

044

8*−

479

402

−31

9−

229

−12

70

−01

0−

003

003

010

016

022

027

033

038

044

049

055

060

065

070

075

080

086

095

112

145

015

025

036

046

056

066

076

086

096

106

116

124

134

142

149

155

160

165

169

174

178

M

0.5

0−

001

−00

4−

007

−01

0−

012

−01

3−

011

−00

700

001

202

805

007

711

115

310

206

102

900

80

Q0

−02

1−

031

−03

2−

027

−01

500

302

805

809

513

818

824

430

737

645

3*−

462

−37

0−

267

−14

80

p−

027

−01

5−

005

002

009

015

021

027

034

040

046

053

059

066

073

080

088

097

109

131

170

~0

011

022

033

044

055

066

078

089

101

112

122

134

145

155

165

174

181

189

197

204

M

0.6

0−

002

−00

6−

012

−01

8−

024

−03

0−

034

−03

5−

034

−02

9−

020

−00

601

404

107

511

807

003

300

90

Q0

−03

2−

052

−06

3−

065

−05

9−

047

−02

9−

003

029

068

115

169

232

303

384

474*

−42

5−

309

−17

20

p−

039

−02

6−

015

−00

600

200

901

502

202

903

604

305

106

806

707

508

509

610

812

515

119

8

~−

016

−00

400

702

003

104

405

606

808

109

410

712

013

414

816

117

418

619

921

022

123

3

M

0.7

0−

002

−00

8−

017

−02

6−

037

−04

7−

056

−06

3−

069

−07

0−

068

−06

1−

049

−02

9−

002

034

080

038

010

0

Q0

−04

4−

074

−09

2−

102

−10

3−

097

−08

5−

064

−03

600

004

409

615

823

031

340

951

9*−

352

−19

80

p−

052

−03

6−

024

−01

4−

005

002

009

017

024

032

040

048

057

067

077

089

103

119

140

172

229

~−

030

−01

8−

006

007

020

033

046

060

074

088

102

118

133

149

165

182

199

214

230

246

262

M

0.8

0−

003

−01

0−

021

−03

5−

049

−06

4−

078

−09

1−

103

−11

1−

116

−11

6−

111

−09

9−

079

−05

0−

010

043

012

0

Q0

−05

5−

095

−12

2−

139

−14

7−

147

−13

9−

124

−10

1−

069

−02

802

308

415

724

334

446

260

3*−

224

0

p−

065

−04

7−

033

−02

2−

012

−00

400

401

101

902

703

604

605

606

707

909

310

912

915

519

425

9

~−

047

−03

3−

020

−00

600

802

203

605

006

608

109

711

513

215

117

019

021

123

125

227

229

3

M

0.9

0−

003

−01

3−

026

−04

3−

061

−08

−10

0−

119

−13

6−

152

−16

4−

171

−17

3−

169

−15

6−

133

−10

0−

052

013

0

Q0

−06

7−

116

−15

1−

175

−19

0−

196

−19

4−

187

−16

5−

137

−09

9−

050

010

084

172

278

404

558

748*

0

p−

079

−05

7−

041

−02

9−

019

−01

0−

002

006

014

023

033

043

054

067

081

097

115

138

170

212

293

~−

062

−04

7−

033

−01

8−

004

011

025

041

058

074

092

111

131

152

174

198

222

247

272

298

325

M

1.0

0−

004

−01

5−

031

−05

1−

073

−09

7−

122

−14

7−

170

−19

2−

213

−22

6−

235

−23

3−

232

−21

7−

189

147

085

0

Q0

−07

9−

137

−18

0−

212

−23

3−

245

−24

9−

244

−22

9−

205

−17

0−

123

−06

401

110

221

234

751

272

110

00*

p−

092

−06

7−

050

−03

7−

026

−01

7−

008

001

010

019

029

041

053

066

082

100

121

148

184

238

326

~−

076

−06

1−

046

−03

1−

016

−00

101

603

205

006

808

810

813

015

417

920

623

326

329

332

535

6

SAMPLE C

HAPTER

206

t =

2

Co

ef.

δx

−1

−0.

9−

0.8

−0.

7−

0.6

−0.

5−

0.4

−0.

3−

0.2

−0.

10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

M

0

000

200

701

603

004

707

009

813

217

321

917

313

209

807

004

703

001

600

700

20

Q0

036

072

111

155

202

254

310

371

434

500*

−43

4−

371

−31

0−

254

−20

2−

155

−11

107

203

60

p03

903

503

704

104

504

905

405

806

206

506

606

506

205

805

404

904

504

103

703

503

9

~07

108

008

909

710

611

412

112

713

313

613

713

613

312

712

111

410

609

708

908

007

1

M

0.1

000

100

401

002

003

305

007

209

913

217

121

716

912

809

306

404

102

301

100

30

Q0

021

046

076

110

150

195

246

301

361

424

489*

−44

5−

381

−31

8−

259

−20

3−

152

−10

3−

055

0

p02

002

302

703

103

704

204

805

305

806

206

406

506

506

406

105

705

304

904

805

0−

61

~05

706

607

508

509

410

311

111

912

613

213

613

813

713

412

912

411

711

010

209

508

7

M

0.2

000

000

200

501

001

903

104

706

809

412

616

420

815

911

708

205

303

001

400

40

Q0

007

022

043

070

103

140

183

232

287

348

412

478*

−45

4−

387

−32

2−

258

−19

6−

134

−07

10

p00

501

101

802

403

003

504

004

605

205

806

306

606

706

706

606

506

306

206

206

507

8

~04

005

006

007

108

109

110

111

011

912

713

313

814

114

214

013

713

312

812

311

811

3

M

0.3

000

0−

001

000

002

007

014

025

040

059

084

114

151

194

145

102

067

039

019

005

0

Q0

005

001

014

034

059

089

126

169

219

274

335

400

468*

−46

1−

389

−31

7−

244

−17

1−

093

0

p−

010

001

010

017

022

028

034

040

046

053

058

063

067

069

071

072

072

072

074

083

106

~02

603

604

705

806

807

908

910

011

011

912

813

614

214

614

814

814

714

514

214

013

7

M

0.4

000

1−

002

−00

3−

004

−00

300

000

501

402

704

506

809

713

217

512

408

204

802

300

60

Q0

−01

1−

015

−01

100

001

704

207

210

915

220

225

832

038

846

1*−

462

−38

2−

299

−21

3−

117

0

p−

014

−00

700

000

701

402

102

703

404

004

605

305

906

507

107

507

908

108

409

010

313

4

~01

002

103

204

305

406

607

708

809

911

012

113

114

014

815

515

916

216

316

416

516

5

0.5

0−

001

−00

4−

008

−01

1−

014

−01

5−

014

−01

1−

004

007

023

044

071

106

148

098

058

027

007

0

0−

023

−03

5−

037

−03

2−

021

−00

302

005

108

813

318

424

330

838

146

2*−

451

−35

7−

256

−14

10

−03

1−

017

−00

600

100

801

402

102

703

404

104

805

506

206

907

708

409

009

710

712

516

1

−00

1−

010

021

032

043

055

066

078

090

102

114

126

138

149

159

168

175

181

186

191

196

0.6

0−

002

−00

6−

011

−01

7−

023

−02

8−

032

−03

4−

033

−02

8−

020

−00

601

404

007

411

706

903

300

90

0−

003

−04

9−

059

−06

1−

057

−04

6−

029

−00

502

606

411

116

522

830

038

247

5*−

422

−30

5−

170

0

−03

5−

025

−01

4−

006

001

008

014

020

027

035

042

050

059

067

077

087

098

109

124

149

195

−01

5−

003

008

019

031

042

054

067

079

093

106

120

134

149

163

176

180

200

211

221

231

0.7

0−

002

−00

7−

014

−02

3−

032

−04

1−

049

−05

6−

061

−06

3−

061

−05

5−

043

−02

500

103

608

103

901

00

0−

038

−06

4−

081

−08

9−

091

−08

7−

077

−05

9−

034

−00

203

908

914

821

830

139

851

0*−

358

−20

20

−04

3−

032

−02

1−

012

−00

500

100

701

402

102

903

704

505

406

407

608

910

412

114

217

523

4

−02

4−

013

−00

200

902

103

204

405

707

008

409

811

413

014

716

418

219

921

723

324

926

5

0.8

0−

002

−00

9−

018

−02

9−

041

−05

4−

066

−07

8−

088

−09

7−

102

−10

3−

099

−08

907

1−

044

−00

904

501

20

0−

045

−07

8−

102

−11

7−

126

−12

8−

193

−11

2−

094

−06

8−

032

013

069

138

221

321

440

585*

−23

40

−05

2−

039

−02

8−

019

−01

2−

005

001

007

014

022

030

040

050

062

075

091

109

131

160

203

272

−03

7−

026

−01

5−

004

008

020

032

045

059

074

089

107

125

145

166

188

211

235

258

281

304

0.9

0−

003

−01

0−

021

−03

4−

050

−06

6−

083

−10

0−

116

−13

0−

142

−15

0−

154

−15

2−

142

−12

3−

093

−04

801

40

0−

053

−09

3−

123

−14

5−

158

−16

8−

170

−16

5−

159

−13

3−

103

−06

3−

010

057

140

243

370

522

730*

0

−06

1−

046

−03

4−

025

−01

8−

011

−00

500

100

801

602

503

504

605

907

509

311

414

117

723

131

7

−04

7−

036

−02

5−

014

003

009

021

034

049

064

081

100

120

142

167

193

221

251

281

313

344

1.0

0−

003

−01

2−

024

−04

0−

058

−07

9−

100

−12

2−

164

−16

4−

182

−19

8−

210

−21

5−

217

−20

3−

180

−14

2−

084

0

0−

061

−10

8−

144

−17

2−

193

−20

8−

216

−21

8−

212

−19

8−

174

−13

8−

089

−02

505

916

530

047

169

510

00*

−07

0−

053

−04

1−

032

−02

4−

018

−01

2−

005

002

010

019

029

043

056

074

094

119

151

194

258

361

−05

7−

046

−03

6−

025

−01

4−

002

010

024

038

055

073

093

115

140

168

198

231

167

305

344

385

Tab

le 4

.7 D

imen

sion

less

coe

ffici

ents

for

ana

lysi

s of

sho

rt b

eam

s lo

aded

with

a c

once

ntra

ted

ver

tical

load

P

SAMPLE C

HAPTER

207

t =

2

Co

ef.

δx

−1

−0.

9−

0.8

−0.

7−

0.6

−0.

5−

0.4

−0.

3−

0.2

−0.

10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

000

200

701

603

004

707

009

813

217

321

917

313

209

807

004

703

001

600

700

20

003

607

211

115

520

225

431

037

143

450

0*−

434

−37

1−

310

−25

4−

202

−15

5−

111

072

036

0

039

035

037

041

045

049

054

058

062

065

066

065

062

058

054

049

045

041

037

035

039

071

080

089

097

106

114

121

127

133

136

137

136

133

127

121

114

106

097

089

080

071

0.1

000

100

401

002

003

305

007

209

913

217

121

716

912

809

306

404

102

301

100

30

002

104

607

611

015

019

524

630

136

142

448

9*−

445

−38

1−

318

−25

9−

203

−15

2−

103

−05

50

020

023

027

031

037

042

048

053

058

062

064

065

065

064

061

057

053

049

048

050

−61

057

066

075

085

094

103

111

119

126

132

136

138

137

134

129

124

117

110

102

095

087

0.2

000

000

200

501

001

903

104

706

809

412

616

420

815

911

708

205

303

001

400

40

000

702

204

307

010

314

018

323

228

734

841

247

8*−

454

−38

7−

322

−25

8−

196

−13

4−

071

0

005

011

018

024

030

035

040

046

052

058

063

066

067

067

066

065

063

062

062

065

078

040

050

060

071

081

091

101

110

119

127

133

138

141

142

140

137

133

128

123

118

113

0.3

000

0−

001

000

002

007

014

025

040

059

084

114

151

194

145

102

067

039

019

005

0

000

500

101

403

405

908

912

616

921

927

433

540

046

8*−

461

−38

9−

317

−24

4−

171

−09

30

−01

000

101

001

702

202

803

404

004

605

305

806

306

706

907

107

207

207

207

408

310

6

026

036

047

058

068

079

089

100

110

119

128

136

142

146

148

148

147

145

142

140

137

0.4

000

1−

002

−00

3−

004

−00

300

000

501

402

704

506

809

713

217

512

408

204

802

300

60

0−

011

−01

5−

011

000

017

042

072

109

152

202

258

320

388

461*

−46

2−

382

−29

9−

213

−11

70

−01

4−

007

000

007

014

021

027

034

040

046

053

059

065

071

075

079

081

084

090

103

134

010

021

032

043

054

066

077

088

099

110

121

131

140

148

155

159

162

163

164

165

165

M

0.5

0−

001

−00

4−

008

−01

1−

014

−01

5−

014

−01

1−

004

007

023

044

071

106

148

098

058

027

007

0

Q0

−02

3−

035

−03

7−

032

−02

1−

003

020

051

088

133

184

243

308

381

462*

−45

1−

357

−25

6−

141

0

p−

031

−01

7−

006

001

008

014

021

027

034

041

048

055

062

069

077

084

090

097

107

125

161

~−

001

−01

002

103

204

305

506

607

809

010

211

412

613

814

915

916

817

518

118

619

119

6

M

0.6

0−

002

−00

6−

011

−01

7−

023

−02

8−

032

−03

4−

033

−02

8−

020

−00

601

404

007

411

706

903

300

90

Q0

−00

3−

049

−05

9−

061

−05

7−

046

−02

9−

005

026

064

111

165

228

300

382

475*

−42

2−

305

−17

00

p−

035

−02

5−

014

−00

600

100

801

402

002

703

504

205

005

906

707

708

709

810

912

414

919

5

~−

015

−00

300

801

903

104

205

406

707

909

310

612

013

414

916

317

618

020

021

122

123

1

M

0.7

0−

002

−00

7−

014

−02

3−

032

−04

1−

049

−05

6−

061

−06

3−

061

−05

5−

043

−02

500

103

608

103

901

00

Q0

−03

8−

064

−08

1−

089

−09

1−

087

−07

7−

059

−03

4−

002

039

089

148

218

301

398

510*

−35

8−

202

0

p−

043

−03

2−

021

−01

2−

005

001

007

014

021

029

037

045

054

064

076

089

104

121

142

175

234

~−

024

−01

3−

002

009

021

032

044

057

070

084

098

114

130

147

164

182

199

217

233

249

265

M

0.8

0−

002

−00

9−

018

−02

9−

041

−05

4−

066

−07

8−

088

−09

7−

102

−10

3−

099

−08

907

1−

044

−00

904

501

20

Q0

−04

5−

078

−10

2−

117

−12

6−

128

−19

3−

112

−09

4−

068

−03

201

306

913

822

132

144

058

5*−

234

0

p−

052

−03

9−

028

−01

9−

012

−00

500

100

701

402

203

004

005

006

207

509

110

913

116

020

327

2

~−

037

−02

6−

015

−00

400

802

003

204

505

907

408

910

712

514

516

618

821

123

525

828

130

4

M

0.9

0−

003

−01

0−

021

−03

4−

050

−06

6−

083

−10

0−

116

−13

0−

142

−15

0−

154

−15

2−

142

−12

3−

093

−04

801

40

Q0

−05

3−

093

−12

3−

145

−15

8−

168

−17

0−

165

−15

9−

133

−10

3−

063

−01

005

714

024

337

052

273

0*0

p−

061

−04

6−

034

−02

5−

018

−01

1−

005

001

008

016

025

035

046

059

075

093

114

141

177

231

317

~−

047

−03

6−

025

−01

400

300

902

103

404

906

408

110

012

014

216

719

322

125

128

131

334

4

M

1.0

0−

003

−01

2−

024

−04

0−

058

−07

9−

100

−12

2−

164

−16

4−

182

−19

8−

210

−21

5−

217

−20

3−

180

−14

2−

084

0

Q0

−06

1−

108

−14

4−

172

−19

3−

208

−21

6−

218

−21

2−

198

−17

4−

138

−08

9−

025

059

165

300

471

695

1000

*

p−

070

−05

3−

041

−03

2−

024

−01

8−

012

−00

500

201

001

902

904

305

607

409

411

915

119

425

836

1

~−

057

−04

6−

036

−02

5−

014

−00

201

002

403

805

507

309

311

514

016

819

823

116

730

534

438

5

SAMPLE C

HAPTER

208

t =

5

Co

ef.

δx

−1

−0.

9−

0.8

−0.

7−

0.6

−0.

5−

0.4

−0.

3−

0.2

−0.

10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

M

0

000

000

200

701

402

604

306

609

613

418

013

409

606

604

302

601

400

700

200

00

Q0

011

029

058

096

143

199

263

336

416

500*

−41

6−

336

−26

3−

199

−14

3−

096

−05

8−

029

−01

10

p01

001

402

303

304

305

106

006

907

708

208

408

207

706

906

005

104

303

302

301

401

0

~04

205

807

308

810

211

713

014

215

215

916

215

915

214

213

011

710

208

807

305

804

2

M

0.1

000

000

000

200

601

402

504

206

509

613

317

913

309

606

604

302

601

400

600

20

Q0

−00

100

902

805

709

514

219

926

533

641

750

0*−

418

−33

8−

264

−19

8−

141

−09

6−

060

−03

10

p00

700

401

502

403

304

305

206

107

007

708

108

308

107

707

106

205

104

003

102

903

5

~03

605

006

407

809

210

612

013

214

315

215

916

015

714

913

912

611

209

708

106

605

0

M

0.2

000

100

200

200

100

301

002

203

906

109

112

817

412

809

006

003

702

000

900

20

Q0

−01

2−

011

003

026

056

094

139

195

260

335

415

498*

−41

9−

340

−26

6−

199

−13

9−

087

−04

20

p−

018

−00

500

801

902

703

304

105

006

107

007

808

208

308

107

707

106

405

604

804

304

4

~02

403

705

106

507

909

310

612

013

314

415

416

016

316

015

414

413

312

010

709

308

0

M

0.3

0−

001

−00

3−

006

−00

6−

005

−00

200

501

703

305

608

512

216

712

108

805

202

901

300

30

Q0

−02

1−

024

−01

600

102

305

208

013

719

325

933

241

149

3*−

424

−34

2−

264

−19

3−

129

−06

90

p−

031

−01

103

501

301

902

603

304

205

206

107

007

608

008

308

308

007

506

706

106

208

1

~01

6−

028

041

054

067

080

093

106

120

132

144

153

160

164

162

156

148

138

127

116

105

M

0.4

0−

001

−00

3−

006

−00

9−

010

−01

0−

007

000

010

026

048

076

112

156

109

070

041

019

005

0

Q0

−01

7−

027

−02

9−

022

−00

701

504

608

413

018

524

932

030

948

4*−

428

−34

1−

257

−17

8−

097

0

p−

020

−01

4−

006

002

011

019

027

034

042

050

059

068

076

082

087

088

086

081

079

086

113

~00

1−

013

025

038

050

063

076

090

104

117

131

143

154

163

168

168

165

160

153

146

139

0.5

0−

001

−00

5−

009

−01

3−

017

−01

9−

019

−01

7−

012

−00

201

203

205

909

413

809

005

202

400

60

0−

027

−03

9−

042

−03

9−

030

−01

500

603

607

311

917

323

630

838

947

8*−

428

−33

0−

230

−12

40

−03

6−

019

−00

700

000

601

101

802

503

304

105

005

806

707

608

509

209

709

910

111

114

3

000

010

021

031

042

053

065

077

090

104

118

132

145

158

169

176

180

181

180

178

176

0.6

0−

001

−00

5−

009

−01

5−

020

−02

5−

028

−03

0−

030

−02

6−

019

−00

601

203

807

111

406

703

100

80

0−

025

−04

2−

051

−05

4−

050

−04

2−

029

−00

901

805

409

915

321

829

337

947

7*−

414

−29

6−

162

0

−02

6−

022

−01

3−

005

000

005

010

016

023

032

040

049

059

070

081

092

103

113

124

144

187

010

−00

100

901

902

903

905

106

207

508

910

411

913

615

216

818

319

520

421

121

822

3

0.7

0−

001

−00

5−

010

−01

6−

023

−03

0−

037

−42

−04

7−

048

−04

7−

042

−03

2−

016

007

040

083

040

011

0

0−

026

−04

6−

059

−00

6−

069

−06

8−

063

−05

1−

032

−00

503

007

412

819

627

837

849

5*36

820

70

−02

5−

024

−01

6−

009

−00

4−

001

003

008

015

023

031

039

049

060

074

091

109

127

147

179

244

−01

0−

002

−00

601

302

203

104

005

106

207

509

010

612

314

216

218

220

222

123

825

326

8

0.8

0−

001

−00

5−

011

−01

8−

026

−03

5−

045

−05

4−

062

−06

9−

075

−07

7−

076

−06

8−

055

−03

200

204

901

30

0−

026

−04

9−

066

−07

8−

087

−09

1−

092

−08

9−

080

−06

4−

040

−00

604

110

217

927

739

955

2*−

254

0

−02

6−

024

−02

0−

015

−01

0−

006

−00

300

100

601

202

002

904

005

306

808

710

913

617

121

929

6

−01

9−

012

−00

600

100

901

702

503

504

605

907

409

211

113

315

718

421

224

127

029

932

6

0.9

0−

001

−00

5−

012

−02

0−

030

−04

0−

052

−06

5−

078

−09

0−

102

−11

1−

118

−12

0−

116

−10

3−

080

−04

1−

016

0

0−

028

−05

3−

073

−08

9−

103

−11

4−

123

−12

7−

127

−12

1−

107

−08

3−

046

007

079

176

303

470

693*

0

−03

0−

027

−02

2−

018

−01

5−

012

−01

0−

007

−00

300

301

001

903

004

406

208

411

114

519

125

836

5

−02

1−

016

−01

1−

006

000

007

014

022

033

045

060

077

098

123

151

183

218

257

298

341

384

1.0

0−

001

−00

6−

013

−02

2−

032

−04

5−

060

−07

6−

093

−11

1−

128

−14

5−

160

−17

1−

177

175

−16

1−

132

−08

10

0−

029

−05

6−

079

−10

0−

120

−13

7−

153

−16

6−

175

−17

8−

174

−16

0−

132

−08

8−

021

074

207

388

640

1000

*

−03

0−

028

−02

5−

022

−02

0−

018

−01

7−

014

−01

1−

006

000

008

020

035

055

080

112

154

212

298

433

−02

4−

020

−01

6−

013

−00

9−

004

002

010

019

030

045

063

055

112

144

183

225

273

327

384

444

Tab

le 4

.8 D

imen

sion

less

coe

ffici

ents

for

ana

lysi

s of

sho

rt b

eam

s lo

aded

with

a c

once

ntra

ted

ver

tical

load

P

SAMPLE C

HAPTER

209

t =

5

Co

ef.

δx

−1

−0.

9−

0.8

−0.

7−

0.6

−0.

5−

0.4

−0.

3−

0.2

−0.

10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

000

000

200

701

402

604

306

609

613

418

013

409

606

604

302

601

400

700

200

00

001

102

905

809

614

319

926

333

641

650

0*−

416

−33

6−

263

−19

9−

143

−09

6−

058

−02

9−

011

0

010

014

023

033

043

051

060

069

077

082

084

082

077

069

060

051

043

033

023

014

010

042

058

073

088

102

117

130

142

152

159

162

159

152

142

130

117

102

088

073

058

042

0.1

000

000

000

200

601

402

504

206

509

613

317

913

309

606

604

302

601

400

600

20

0−

001

009

028

057

095

142

199

265

336

417

500*

−41

8−

338

−26

4−

198

−14

1−

096

−06

0−

031

0

007

004

015

024

033

043

052

061

070

077

081

083

081

077

071

062

051

040

031

029

035

036

050

064

078

092

106

120

132

143

152

159

160

157

149

139

126

112

097

081

066

050

0.2

000

100

200

200

100

301

002

203

906

109

112

817

412

809

006

003

702

000

900

20

0−

012

−01

100

302

605

609

413

919

526

033

541

549

8*−

419

−34

0−

266

−19

9−

139

−08

7−

042

0

−01

8−

005

008

019

027

033

041

050

061

070

078

082

083

081

077

071

064

056

048

043

044

024

037

051

065

079

093

106

120

133

144

154

160

163

160

154

144

133

120

107

093

080

0.3

0−

001

−00

3−

006

−00

6−

005

−00

200

501

703

305

608

512

216

712

108

805

202

901

300

30

0−

021

−02

4−

016

001

023

052

080

137

193

259

332

411

493*

−42

4−

342

−26

4−

193

−12

9−

069

0

−03

1−

011

035

013

019

026

033

042

052

061

070

076

080

083

083

080

075

067

061

062

081

016

−02

804

105

406

708

009

310

612

013

214

415

316

016

416

215

614

813

812

711

610

5

0.4

0−

001

−00

3−

006

−00

9−

010

−01

0−

007

000

010

026

048

076

112

156

109

070

041

019

005

0

0−

017

−02

7−

029

−02

2−

007

015

046

084

130

185

249

320

309

484*

−42

8−

341

−25

7−

178

−09

70

−02

0−

014

−00

600

201

101

902

703

404

205

005

906

807

608

208

708

808

608

107

908

611

3

001

−01

302

503

805

006

307

609

010

411

713

114

315

416

316

816

816

516

015

314

613

9

M

0.5

0−

001

−00

5−

009

−01

3−

017

−01

9−

019

−01

7−

012

−00

201

203

205

909

413

809

005

202

400

60

Q0

−02

7−

039

−04

2−

039

−03

0−

015

006

036

073

119

173

236

308

389

478*

−42

8−

330

−23

0−

124

0

p−

036

−01

9−

007

000

006

011

018

025

033

041

050

058

067

076

085

092

097

099

101

111

143

~00

001

002

103

104

205

306

507

709

010

411

813

214

515

816

917

618

018

118

017

817

6

M

0.6

0−

001

−00

5−

009

−01

5−

020

−02

5−

028

−03

0−

030

−02

6−

019

−00

601

203

807

111

406

703

100

80

Q0

−02

5−

042

−05

1−

054

−05

0−

042

−02

9−

009

018

054

099

153

218

293

379

477*

−41

4−

296

−16

20

p−

026

−02

2−

013

−00

500

000

501

001

602

303

204

004

905

907

008

109

210

311

312

414

418

7

~01

0−

001

009

019

029

039

051

062

075

089

104

119

136

152

168

183

195

204

211

218

223

M

0.7

0−

001

−00

5−

010

−01

6−

023

−03

0−

037

−42

−04

7−

048

−04

7−

042

−03

2−

016

007

040

083

040

011

0

Q0

−02

6−

046

−05

9−

006

−06

9−

068

−06

3−

051

−03

2−

005

030

074

128

196

278

378

495*

368

207

0

p−

025

−02

4−

016

−00

9−

004

−00

100

300

801

502

303

103

904

906

007

409

110

912

714

717

924

4

~−

010

−00

2−

006

013

022

031

040

051

062

075

090

106

123

142

162

182

202

221

238

253

268

M

0.8

0−

001

−00

5−

011

−01

8−

026

−03

5−

045

−05

4−

062

−06

9−

075

−07

7−

076

−06

8−

055

−03

200

204

901

30

Q0

−02

6−

049

−06

6−

078

−08

7−

091

−09

2−

089

−08

0−

064

−04

0−

006

041

102

179

277

399

552*

−25

40

p−

026

−02

4−

020

−01

5−

010

−00

6−

003

001

006

012

020

029

040

053

068

087

109

136

171

219

296

~−

019

−01

2−

006

001

009

017

025

035

046

059

074

092

111

133

157

184

212

241

270

299

326

M

0.9

0−

001

−00

5−

012

−02

0−

030

−04

0−

052

−06

5−

078

−09

0−

102

−11

1−

118

−12

0−

116

−10

3−

080

−04

1−

016

0

Q0

−02

8−

053

−07

3−

089

−10

3−

114

−12

3−

127

−12

7−

121

−10

7−

083

−04

600

707

917

630

347

069

3*0

p−

030

−02

7−

022

−01

8−

015

−01

2−

010

−00

7−

003

003

010

019

030

044

062

084

111

145

191

258

365

~−

021

−01

6−

011

−00

600

000

701

402

203

304

506

007

709

812

315

118

321

825

729

834

138

4

M

1.0

0−

001

−00

6−

013

−02

2−

032

−04

5−

060

−07

6−

093

−11

1−

128

−14

5−

160

−17

1−

177

175

−16

1−

132

−08

10

Q0

−02

9−

056

−07

9−

100

−12

0−

137

−15

3−

166

−17

5−

178

−17

4−

160

−13

2−

088

−02

107

420

738

864

010

00*

p−

030

−02

8−

025

−02

2−

020

−01

8−

017

−01

4−

011

−00

600

000

802

003

505

508

011

215

421

229

843

3

~−

024

−02

0−

016

−01

3−

009

−00

400

201

001

903

004

506

305

511

214

418

322

527

332

738

444

4

SAMPLE C

HAPTER

210

t =

10

Co

ef.

δx

−1

−0.

9−

0.8

−0.

7−

0.6

−0.

5−

0.4

−0.

3−

0.2

−0.

10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

M

0

000

0−

001

−00

100

200

902

204

006

610

114

610

106

604

002

200

900

2−

001

−00

800

000

Q0

−00

7−

003

016

048

093

150

221

304

399

500*

−39

9−

304

−22

1−

150

−09

3−

048

−01

600

300

700

p−

008

−00

301

102

603

905

106

407

709

009

010

209

909

007

706

405

103

902

601

1−

003

−00

8

~02

003

905

907

909

811

813

715

416

917

918

417

916

915

413

711

809

807

905

903

902

0

M

0.1

0−

001

−00

3−

004

−00

4−

001

006

019

038

065

101

147

103

068

042

024

013

006

003

001

0

Q0

−01

7−

018

−00

701

505

009

615

623

031

540

850

7*−

395

−30

1−

216

−14

3−

087

−04

8−

027

−01

40

p−

026

−00

900

501

702

804

005

306

708

009

009

709

909

709

008

006

504

702

901

501

102

1

~02

504

205

807

409

110

812

514

215

717

017

918

117

516

214

512

510

408

105

903

601

3

M

0.2

0−

002

−00

5−

008

−00

9−

008

−00

400

301

603

506

209

814

510

106

604

002

201

100

400

10

Q0

−02

5−

033

−02

4−

005

021

055

090

156

230

316

412

512*

−39

1−

299

−21

8−

147

−08

9−

046

−01

80

p−

032

−01

600

201

502

302

903

805

006

508

009

209

910

009

508

707

606

405

103

602

101

7

~01

703

204

706

207

709

310

912

614

315

817

218

118

417

916

715

113

211

209

107

004

9

M

0.3

0−

002

−00

5−

009

−01

2−

013

−01

2−

008

−00

101

303

206

009

714

309

906

403

802

100

900

20

Q0

−00

3−

041

−03

4−

020

−00

202

205

610

316

423

732

241

350

9*−

394

−30

0−

215

−14

4−

090

−04

80

p−

047

−01

900

101

101

602

102

904

005

406

707

908

809

409

709

609

007

906

304

604

106

3

~01

602

804

005

206

507

909

410

912

514

115

717

017

918

217

716

515

013

111

209

207

2

M

0.4

0−

001

−00

3−

007

−01

0−

014

−01

5−

014

−01

1−

002

011

030

057

093

138

093

058

033

015

004

0

Q0

−01

6−

031

−03

8−

036

−02

5−

006

022

058

104

162

231

311

403

502*

−39

7−

300

−21

5−

143

−07

80

p−

016

−01

6−

011

003

006

015

023

032

041

052

063

075

086

096

101

100

092

078

066

067

095

~−

003

009

021

033

046

059

073

088

104

121

138

154

168

178

183

179

170

157

142

126

109

0.5

0−

001

−00

5−

009

−01

3−

017

−02

0−

022

−02

1−

017

−01

000

302

204

808

212

608

004

502

000

50

0−

029

−04

0−

042

−04

0−

034

−02

4−

006

019

054

000

155

222

300

391

491*

−40

4−

299

−20

0−

106

0

−04

1−

018

−00

600

000

400

801

402

103

004

005

006

107

308

509

610

410

610

209

509

612

3

006

014

023

031

041

051

062

074

088

103

119

136

153

168

180

187

187

181

173

163

153

0.6

0−

001

−00

3−

007

−01

2−

016

−02

0−

024

−02

6−

026

−02

4−

018

−00

701

003

406

711

006

403

000

80

0−

018

−03

4−

042

−04

4−

043

−03

8−

030

−01

600

704

008

313

720

328

337

648

2*−

402

−28

0−

151

0

−01

3−

019

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2−

005

000

003

006

011

018

028

038

048

060

073

086

100

112

119

124

136

176

006

008

010

018

026

035

046

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085

101

119

138

157

176

192

204

210

212

212

212

0.7

000

0−

003

−00

6−

010

−01

5−

020

−02

6−

030

−03

4−

036

−03

6−

032

−02

4−

1001

204

208

404

601

10

0−

015

−03

0−

040

−04

5−

049

−05

2−

051

−04

5−

031

−00

902

005

910

917

325

036

148

6*−

372

−20

80

−00

7−

017

−01

2−

007

−00

4−

004

−00

200

301

001

802

603

404

405

607

309

411

513

415

117

925

0

003

008

013

018

024

030

037

046

056

068

082

099

118

139

161

184

206

226

241

253

265

0.8

000

000

200

601

001

502

102

703

404

104

705

205

605

605

204

102

200

805

201

40

0−

010

−02

5−

038

−04

8−

055

−06

1−

066

−06

8−

067

−06

0−

045

−02

001

807

114

424

036

452

4*−

270

0

−00

4−

014

−01

4−

011

−00

8−

007

−00

5−

004

−00

100

401

102

003

104

506

208

310

914

118

123

431

3

−00

8−

004

000

005

009

014

020

028

037

048

062

079

099

123

150

180

214

248

282

313

344

0.9

000

0−

002

−00

5−

009

−01

5−

021

−02

9−

038

−04

7−

058

−06

8−

078

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6−

092

−09

2−

085

−06

7−

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018

0

0−

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059

−07

0−

082

−09

3−

101

−10

6−

104

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070

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029

117

242

416

657*

0

−00

6−

013

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012

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011

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011

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000

7−

002

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030

049

073

105

147

203

285

412

−00

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002

000

001

001

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031

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059

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104

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171

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261

313

267

421

1.0

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0−

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−01

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−03

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041

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068

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4−

100

−11

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137

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142

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2−

078

0

0−

008

−02

0−

033

−04

8−

064

−08

1−

099

−11

8−

136

−15

2−

163

−16

7−

159

−13

5−

086

−00

512

030

858

510

00*

−00

4−

011

013

−01

4−

015

−01

6−

018

−01

9−

018

−01

701

400

800

101

503

506

810

115

322

733

650

7

−00

1−

001

−00

2−

002

−00

2−

002

−00

100

200

601

402

403

905

908

612

016

221

427

534

542

250

4

Tab

le 4

.9 D

imen

sion

less

coe

ffici

ents

for

ana

lysi

s of

sho

rt b

eam

s lo

aded

with

a c

once

ntra

ted

ver

tical

load

P

SAMPLE C

HAPTER

211

t =

10

Co

ef.

δx

−1

−0.

9−

0.8

−0.

7−

0.6

−0.

5−

0.4

−0.

3−

0.2

−0.

10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

000

0−

001

−00

100

200

902

204

006

610

114

610

106

604

002

200

900

2−

001

−00

800

000

0−

007

−00

301

604

809

315

022

130

439

950

0*−

399

−30

4−

221

−15

0−

093

−04

8−

016

003

007

00

−00

8−

003

011

026

039

051

064

077

090

090

102

099

090

077

064

051

039

026

011

−00

3−

008

020

039

059

079

098

118

137

154

169

179

184

179

169

154

137

118

098

079

059

039

020

0.1

0−

001

−00

3−

004

−00

4−

001

006

019

038

065

101

147

103

068

042

024

013

006

003

001

0

0−

017

−01

8−

007

015

050

096

156

230

315

408

507*

−39

5−

301

−21

6−

143

−08

7−

048

−02

7−

014

0

−02

6−

009

005

017

028

040

053

067

080

090

097

099

097

090

080

065

047

029

015

011

021

025

042

058

074

091

108

125

142

157

170

179

181

175

162

145

125

104

081

059

036

013

0.2

0−

002

−00

5−

008

−00

9−

008

−00

400

301

603

506

209

814

510

106

604

002

201

100

400

10

0−

025

−03

3−

024

−00

502

105

509

015

623

031

641

251

2*−

391

−29

9−

218

−14

7−

089

−04

6−

018

0

−03

2−

016

002

015

023

029

038

050

065

080

092

099

100

095

087

076

064

051

036

021

017

017

032

047

062

077

093

109

126

143

158

172

181

184

179

167

151

132

112

091

070

049

0.3

0−

002

−00

5−

009

−01

2−

013

−01

2−

008

−00

101

303

206

009

714

309

906

403

802

100

900

20

0−

003

−04

1−

034

−02

0−

002

022

056

103

164

237

322

413

509*

−39

4−

300

−21

5−

144

−09

0−

048

0

−04

7−

019

001

011

016

021

029

040

054

067

079

088

094

097

096

090

079

063

046

041

063

016

028

040

052

065

079

094

109

125

141

157

170

179

182

177

165

150

131

112

092

072

0.4

0−

001

−00

3−

007

−01

0−

014

−01

5−

014

−01

1−

002

011

030

057

093

138

093

058

033

015

004

0

0−

016

−03

1−

038

−03

6−

025

−00

602

205

810

416

223

131

140

350

2*−

397

−30

0−

215

−14

3−

078

0

−01

6−

016

−01

100

300

601

502

303

204

105

206

307

508

609

610

110

009

207

806

606

709

5

−00

300

902

103

304

605

907

308

810

412

113

815

416

817

818

317

917

015

714

212

610

9

M

0.5

0−

001

−00

5−

009

−01

3−

017

−02

0−

022

−02

1−

017

−01

000

302

204

808

212

608

004

502

000

50

Q0

−02

9−

040

−04

2−

040

−03

4−

024

−00

601

905

400

015

522

230

039

149

1*−

404

−29

9−

200

−10

60

p−

041

−01

8−

006

000

004

008

014

021

030

040

050

061

073

085

096

104

106

102

095

096

123

~00

601

402

303

104

105

106

207

408

810

311

913

615

316

818

018

718

718

117

316

315

3

M

0.6

0−

001

−00

3−

007

−01

2−

016

−02

0−

024

−02

6−

026

−02

4−

018

−00

701

003

406

711

006

403

000

80

Q0

−01

8−

034

−04

2−

044

−04

3−

038

−03

0−

016

007

040

083

137

203

283

376

482*

−40

2−

280

−15

10

p−

013

−01

9−

012

−00

500

000

300

601

101

802

803

804

806

007

308

610

011

211

912

413

617

6

~00

600

801

001

802

603

504

605

707

008

510

111

913

815

717

619

220

421

021

221

221

2

M

0.7

000

0−

003

−00

6−

010

−01

5−

020

−02

6−

030

−03

4−

036

−03

6−

032

−02

4−

1001

204

208

404

601

10

Q0

−01

5−

030

−04

0−

045

−04

9−

052

−05

1−

045

−03

1−

009

020

059

109

173

250

361

486*

−37

2−

208

0

p−

007

−01

7−

012

−00

7−

004

−00

4−

002

003

010

018

026

034

044

056

073

094

115

134

151

179

250

~00

300

801

301

802

403

003

704

605

606

808

209

911

813

916

118

420

622

624

125

326

5

M

0.8

000

000

200

601

001

502

102

703

404

104

705

205

605

605

204

102

200

805

201

40

Q0

−01

0−

025

−03

8−

048

−05

5−

061

−06

6−

068

−06

7−

060

−04

5−

020

018

071

144

240

364

524*

−27

00

p−

004

−01

4−

014

−01

1−

008

−00

7−

005

−00

4−

001

004

011

020

031

045

062

083

109

141

181

234

313

~−

008

−00

400

000

500

901

402

002

803

704

806

207

909

912

315

018

021

424

828

231

334

4

M

0.9

000

0−

002

−00

5−

009

−01

5−

021

−02

9−

038

−04

7−

058

−06

8−

078

−08

6−

092

−09

2−

085

−06

7−

035

018

0

Q0

−01

0−

024

−03

6−

047

−05

9−

070

−08

2−

093

−10

1−

106

−10

4−

093

−07

0−

032

−02

911

724

241

665

7*0

p−

006

−01

3−

013

−01

2−

012

−01

1−

011

−01

1−

010

007

−00

200

601

603

004

907

310

514

720

328

541

2

~−

003

−00

200

000

100

106

601

001

502

203

104

305

907

910

413

517

121

426

131

326

742

1

M

1.0

000

0−

002

−00

4−

008

−01

4−

021

−03

0−

041

−05

4−

068

−08

4−

100

−11

7−

137

−43

−14

8−

142

−12

2−

078

0

Q0

−00

8−

020

−03

3−

048

−06

4−

081

−09

9−

118

−13

6−

152

−16

3−

167

−15

9−

135

−08

6−

005

120

308

585

1000

*

p−

004

−01

101

3−

014

−01

5−

016

−01

8−

019

−01

8−

017

014

008

001

015

035

068

101

153

227

336

507

~−

001

−00

1−

002

−00

2−

002

−00

2−

001

002

006

014

024

039

059

086

120

162

214

275

345

422

504

SAMPLE C

HAPTER

212

t =

1

Co

ef.

δx

−1

−0.

9−

0.8

−0.

7−

0.6

−0.

5−

0.4

−0.

3−

0.2

−0.

10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

M

0

0−

009

−03

2−

066

−11

0−

162

−22

0−

285

−35

3−

426

500*

426

353

285

220

162

110

066

032

009

0

Q0

−16

2−

292

−39

6−

581

−55

2−

614

−66

7−

709

−73

6−

745

−73

6−

709

−66

7−

614

−55

2−

481

−39

6−

292

−16

20

p−

180

−14

5−

115

−09

2−

077

−06

6−

058

−04

8−

035

−01

80

018

035

048

058

066

077

093

115

145

180

~−

190

−17

5−

159

−14

3−

127

−10

9−

091

−07

1−

050

−02

60

026

050

071

091

109

127

143

159

175

190

M

0.1

0−

008

−02

9−

061

−10

2−

151

−20

7−

268

−33

5−

406

−47

9−

553

373

302

235

173

119

072

035

010

0

Q0

−14

6−

270

−37

1−

454

−52

4−

586

−64

1−

685

−72

3−

742

−74

3−

725

−69

3−

644

−58

3−

510

−42

2−

316

−18

00

p−

155

−13

6−

112

−09

1−

075

−06

6−

059

−05

2−

041

−02

7−

010

008

026

041

055

067

080

096

119

154

210

~−

199

−18

4−

168

−15

3−

136

−12

0−

102

−08

3−

062

−03

9−

013

015

045

071

095

119

141

162

183

204

224

M

0.2

0−

007

−02

6−

056

−09

6−

142

−19

6−

255

−31

9−

388

−46

0−

534

−60

831

924

918

412

607

703

701

00

Q0

−13

5−

250

−34

9−

431

−50

2−

564

−61

9−

667

−70

6−

732

−74

4−

739

−71

6−

675

−61

5−

540

−44

8−

338

−19

60

p−

145

−12

5−

107

−09

0−

076

−06

6−

058

−05

1−

043

−03

3−

020

−00

401

403

205

106

808

310

012

316

323

6

~−

189

−17

6−

162

−14

8−

134

−12

0−

104

−08

7−

069

−04

8−

025

001

030

060

088

115

140

164

188

212

236

M

0.3

0−

006

−02

5−

053

−09

0−

135

−18

6−

243

−30

6−

373

−44

3−

516

−59

0−

665

263

196

135

082

040

011

0

Q0

−12

9−

236

−32

8−

409

−48

1−

544

−60

0−

648

−68

8−

719

−73

9−

745

−77

3−

701

−54

8−

573

−47

7−

360

−21

10

p−

143

−11

6−

099

−08

6−

076

−06

7−

060

−05

2−

044

−03

6−

025

−01

300

202

104

306

508

510

513

017

325

9

~−

170

−15

9−

148

−13

7−

125

−11

3−

101

−08

7−

071

−05

3−

033

−01

001

604

607

610

613

416

418

821

424

1

M

0.4

0−

006

−02

4−

052

−08

7−

130

−18

0−

236

−29

7−

362

−43

1−

503

−57

7−

652

725

205

141

086

041

011

1

Q0

−12

7−

229

−31

5−

394

−46

6−

530

−58

6−

633

−67

3−

706

−73

1−

745

−74

372

8−

673

−60

0−

502

−37

8−

219

0

p−

152

−11

1−

092

−08

2−

076

−06

9−

060

−05

1−

044

−03

7−

025

−02

0−

006

012

011

060

086

111

139

182

265

~−

152

−14

3−

134

−12

5−

116

−10

7−

096

−08

5−

071

−05

7−

039

−01

900

403

106

209

512

715

718

621

624

6

0.5

0−

009

−03

2−

063

−09

914

2−

198

−24

4−

302

−36

4−

429

−49

7−

568

−54

0−

711

−78

115

409

604

801

40

0−

175

−27

0−

338

−39

9−

457

−51

0−

557

−59

9−

636

−66

8−

696

−71

5−

721

−71

0−

677

−61

8−

534

−42

0−

258

0

−24

8−

122

−07

6−

063

−05

9−

056

−05

0−

044

−03

9−

035

−03

0−

024

−01

400

102

204

607

109

813

319

134

0

−14

2−

134

−12

7−

120

−11

2−

104

−09

5−

085

−07

3−

059

−04

3−

025

−00

302

205

108

512

015

519

022

325

7

0.6

0−

007

−02

6−

054

−08

8−

130

−17

8−

231

−28

9−

352

−41

8−

488

−56

0−

533

−70

7−

778

−84

409

604

701

30

0−

138

−23

4−

313

−38

3−

417

−50

6−

559

−60

6−

647

−68

2−

710

−72

9−

735

−72

4−

693

−63

6−

548

−42

3−

249

0

−17

3−

112

−08

6−

073

−06

7−

062

−05

6−

050

−01

4−

038

−03

2−

024

−01

300

102

004

307

110

514

720

630

0

−14

6−

139

−13

1−

123

−11

6−

107

−09

8−

087

−07

6−

062

−04

6−

028

−00

601

904

808

112

016

120

124

128

1

0.7

0−

005

−01

9−

041

−07

2−

110

−15

5−

201

−26

4−

326

−39

2−

462

−53

5−

610

−68

5−

758

−82

8−

892

054

015

0

0−

094

−18

4−

268

−34

6−

418

−48

3−

543

−59

6−

644

−68

4−

717

−73

9−

749

−74

5−

722

−67

5−

598

−47

4−

286

0

−09

3−

093

−08

7−

081

−07

5−

068

−06

2−

057

−05

1−

044

−03

6−

027

−01

7−

003

013

034

061

099

151

220

353

−15

1−

143

−13

5−

127

−11

9−

110

−10

1−

190

−07

8−

064

−04

8−

030

−00

801

704

608

011

816

220

825

430

0

0.8

0−

006

−02

3−

048

−08

1−

121

−16

7−

219

−27

6−

339

−40

5−

475

−54

7−

621

−69

4−

766

−83

4−

896

−94

901

40

0−

116

−21

2−

294

−36

6−

430

−49

1−

547

−59

9−

645

−68

3−

712

−73

1−

737

−73

9−

705

−65

7−

578

−45

5−

271

0

−12

8−

105

−08

8−

076

−06

8−

062

−05

8−

054

−04

9−

042

−03

4−

024

−01

300

0−

015

−03

506

209

915

022

232

6

−15

7−

148

−14

0−

132

−12

3−

113

−10

4−

092

−08

0−

066

−04

9−

030

−00

801

804

708

112

016

421

426

732

0

0.9

0−

006

−02

2−

048

−08

1−

120

−16

6−

218

−27

6−

338

−40

4−

474

−54

6−

620

−69

4−

766

−83

4−

896

−94

8−

985

0

0−

115

−21

0−

293

−36

5−

431

−49

1−

547

−59

8−

644

−68

2−

712

−73

1−

738

−73

1−

706

−65

7−

578

−45

6−

274

0

−12

6−

104

−08

8−

077

−06

9−

063

−05

8−

054

−04

8−

042

−03

4−

025

−01

400

001

603

606

209

814

822

133

5

−15

3−

146

−13

7−

129

−12

1−

112

−10

2−

092

−08

0−

066

049

−03

1−

009

016

045

079

117

161

210

265

324

1.0

0−

006

−02

2−

048

−08

1−

120

−16

7−

219

−27

6−

338

−40

4−

474

−54

7−

620

−69

4−

766

−83

4−

896

−94

8−

993

−10

3

0−

116

−21

1−

292

−36

5−

432

−49

2−

548

−59

9−

644

−68

2−

711

−73

1−

739

−73

2−

706

−65

6−

576

−45

5−

274

0

−12

9−

104

−08

7−

077

−06

9−

063

−05

8−

053

−04

8−

041

−03

4−

025

−01

4−

001

016

036

063

098

147

220

336

150

−14

2−

134

−12

7−

118

−11

0−

101

−09

0−

079

−06

5−

049

−03

1−

010

005

044

077

115

158

207

263

324

Tab

le 4

.10

Dim

ensi

onle

ss c

oeffi

cien

ts f

or a

naly

sis

of s

hort

bea

ms

load

ed w

ith a

con

cent

rate

d m

omen

t m

SAMPLE C

HAPTER

213

t =

1

Co

ef.

δx

−1

−0.

9−

0.8

−0.

7−

0.6

−0.

5−

0.4

−0.

3−

0.2

−0.

10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0−

009

−03

2−

066

−11

0−

162

−22

0−

285

−35

3−

426

500*

426

353

285

220

162

110

066

032

009

0

0−

162

−29

2−

396

−58

1−

552

−61

4−

667

−70

9−

736

−74

5−

736

−70

9−

667

−61

4−

552

−48

1−

396

−29

2−

162

0

−18

0−

145

−11

5−

092

−07

7−

066

−05

8−

048

−03

5−

018

001

803

504

805

806

607

709

311

514

518

0

−19

0−

175

−15

9−

143

−12

7−

109

−09

1−

071

−05

0−

026

002

605

007

109

110

912

714

315

917

519

0

0.1

0−

008

−02

9−

061

−10

2−

151

−20

7−

268

−33

5−

406

−47

9−

553

373

302

235

173

119

072

035

010

0

0−

146

−27

0−

371

−45

4−

524

−58

6−

641

−68

5−

723

−74

2−

743

−72

5−

693

−64

4−

583

−51

0−

422

−31

6−

180

0

−15

5−

136

−11

2−

091

−07

5−

066

−05

9−

052

−04

1−

027

−01

000

802

604

105

506

708

009

611

915

421

0

−19

9−

184

−16

8−

153

−13

6−

120

−10

2−

083

−06

2−

039

−01

301

504

507

109

511

914

116

218

320

422

4

0.2

0−

007

−02

6−

056

−09

6−

142

−19

6−

255

−31

9−

388

−46

0−

534

−60

831

924

918

412

607

703

701

00

0−

135

−25

0−

349

−43

1−

502

−56

4−

619

−66

7−

706

−73

2−

744

−73

9−

716

−67

5−

615

−54

0−

448

−33

8−

196

0

−14

5−

125

−10

7−

090

−07

6−

066

−05

8−

051

−04

3−

033

−02

0−

004

014

032

051

068

083

100

123

163

236

−18

9−

176

−16

2−

148

−13

4−

120

−10

4−

087

−06

9−

048

−02

500

103

006

008

811

514

016

418

821

223

6

0.3

0−

006

−02

5−

053

−09

0−

135

−18

6−

243

−30

6−

373

−44

3−

516

−59

0−

665

263

196

135

082

040

011

0

0−

129

−23

6−

328

−40

9−

481

−54

4−

600

−64

8−

688

−71

9−

739

−74

5−

773

−70

1−

548

−57

3−

477

−36

0−

211

0

−14

3−

116

−09

9−

086

−07

6−

067

−06

0−

052

−04

4−

036

−02

5−

013

002

021

043

065

085

105

130

173

259

−17

0−

159

−14

8−

137

−12

5−

113

−10

1−

087

−07

1−

053

−03

3−

010

016

046

076

106

134

164

188

214

241

0.4

0−

006

−02

4−

052

−08

7−

130

−18

0−

236

−29

7−

362

−43

1−

503

−57

7−

652

725

205

141

086

041

011

1

0−

127

−22

9−

315

−39

4−

466

−53

0−

586

−63

3−

673

−70

6−

731

−74

5−

743

728

−67

3−

600

−50

2−

378

−21

90

−15

2−

111

−09

2−

082

−07

6−

069

−06

0−

051

−04

4−

037

−02

5−

020

−00

601

201

106

008

611

113

918

226

5

−15

2−

143

−13

4−

125

−11

6−

107

−09

6−

085

−07

1−

057

−03

9−

019

004

031

062

095

127

157

186

216

246

M

0.5

0−

009

−03

2−

063

−09

914

2−

198

−24

4−

302

−36

4−

429

−49

7−

568

−54

0−

711

−78

115

409

604

801

40

Q0

−17

5−

270

−33

8−

399

−45

7−

510

−55

7−

599

−63

6−

668

−69

6−

715

−72

1−

710

−67

7−

618

−53

4−

420

−25

80

p−

248

−12

2−

076

−06

3−

059

−05

6−

050

−04

4−

039

−03

5−

030

−02

4−

014

001

022

046

071

098

133

191

340

~−

142

−13

4−

127

−12

0−

112

−10

4−

095

−08

5−

073

−05

9−

043

−02

5−

003

022

051

085

120

155

190

223

257

M

0.6

0−

007

−02

6−

054

−08

8−

130

−17

8−

231

−28

9−

352

−41

8−

488

−56

0−

533

−70

7−

778

−84

409

604

701

30

Q0

−13

8−

234

−31

3−

383

−41

7−

506

−55

9−

606

−64

7−

682

−71

0−

729

−73

5−

724

−69

3−

636

−54

8−

423

−24

90

p−

173

−11

2−

086

−07

3−

067

−06

2−

056

−05

0−

014

−03

8−

032

−02

4−

013

001

020

043

071

105

147

206

300

~−

146

−13

9−

131

−12

3−

116

−10

7−

098

−08

7−

076

−06

2−

046

−02

8−

006

019

048

081

120

161

201

241

281

M

0.7

0−

005

−01

9−

041

−07

2−

110

−15

5−

201

−26

4−

326

−39

2−

462

−53

5−

610

−68

5−

758

−82

8−

892

054

015

0

Q0

−09

4−

184

−26

8−

346

−41

8−

483

−54

3−

596

−64

4−

684

−71

7−

739

−74

9−

745

−72

2−

675

−59

8−

474

−28

60

p−

093

−09

3−

087

−08

1−

075

−06

8−

062

−05

7−

051

−04

4−

036

−02

7−

017

−00

301

303

406

109

915

122

035

3

~−

151

−14

3−

135

−12

7−

119

−11

0−

101

−19

0−

078

−06

4−

048

−03

0−

008

017

046

080

118

162

208

254

300

M

0.8

0−

006

−02

3−

048

−08

1−

121

−16

7−

219

−27

6−

339

−40

5−

475

−54

7−

621

−69

4−

766

−83

4−

896

−94

901

40

Q0

−11

6−

212

−29

4−

366

−43

0−

491

−54

7−

599

−64

5−

683

−71

2−

731

−73

7−

739

−70

5−

657

−57

8−

455

−27

10

p−

128

−10

5−

088

−07

6−

068

−06

2−

058

−05

4−

049

−04

2−

034

−02

4−

013

000

−01

5−

035

062

099

150

222

326

~−

157

−14

8−

140

−13

2−

123

−11

3−

104

−09

2−

080

−06

6−

049

−03

0−

008

018

047

081

120

164

214

267

320

M

0.9

0−

006

−02

2−

048

−08

1−

120

−16

6−

218

−27

6−

338

−40

4−

474

−54

6−

620

−69

4−

766

−83

4−

896

−94

8−

985

0

Q0

−11

5−

210

−29

3−

365

−43

1−

491

−54

7−

598

−64

4−

682

−71

2−

731

−73

8−

731

−70

6−

657

−57

8−

456

−27

40

p−

126

−10

4−

088

−07

7−

069

−06

3−

058

−05

4−

048

−04

2−

034

−02

5−

014

000

016

036

062

098

148

221

335

~−

153

−14

6−

137

−12

9−

121

−11

2−

102

−09

2−

080

−06

604

9−

031

−00

901

604

507

911

716

121

026

532

4

M

1.0

0−

006

−02

2−

048

−08

1−

120

−16

7−

219

−27

6−

338

−40

4−

474

−54

7−

620

−69

4−

766

−83

4−

896

−94

8−

993

−10

3

Q0

−11

6−

211

−29

2−

365

−43

2−

492

−54

8−

599

−64

4−

682

−71

1−

731

−73

9−

732

−70

6−

656

−57

6−

455

−27

40

p−

129

−10

4−

087

−07

7−

069

−06

3−

058

−05

3−

048

−04

1−

034

−02

5−

014

−00

101

603

606

309

814

722

033

6

~15

0−

142

−13

4−

127

−11

8−

110

−10

1−

090

−07

9−

065

−04

9−

031

−01

000

504

407

711

515

820

726

332

4

SAMPLE C

HAPTER

214

t =

2

Co

ef.

δx

−1

−0.

9−

0.8

−0.

7−

0.6

−0.

5−

0.4

−0.

3−

0.2

−0.

10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

M

0

0−

008

−03

0−

063

−10

6−

156

−21

4−

278

−34

8−

423

−50

042

334

827

821

415

610

606

303

000

80

Q0

−15

1−

279

−38

2−

466

−54

0−

609

−67

3−

731

−76

4−

777

−76

4−

731

−67

3−

609

−54

0−

466

−38

2−

279

−15

10

p−

157

−14

1−

115

−09

2−

078

−07

1−

067

−06

0−

046

−02

50

025

046

000

067

071

078

092

115

141

157

~−

171

−16

0−

149

−13

7−

126

−11

3−

097

−07

9−

057

−03

10

031

057

079

097

113

126

137

149

160

171

M

0.1

0−

009

−03

2−

068

−11

2−

164

−22

3−

286

−35

4−

424

−49

5−

567

362

294

229

169

116

070

034

009

0

Q0

−16

8−

298

−40

2−

486

−55

5−

611

−65

6−

689

−71

0−

719

−71

4−

697

−66

7−

624

−56

9−

499

−41

4−

309

−17

60

p−

192

−14

7−

115

−09

3−

076

−06

3−

050

−03

9−

027

−01

5−

002

011

024

036

049

062

077

094

117

151

205

~−

192

−18

1−

169

−15

9−

146

−13

2−

116

−09

9−

078

−05

3−

023

013

049

080

108

132

154

175

195

214

233

M

0.2

0−

009

−03

2−

067

−11

1−

162

−22

0−

283

−35

0−

420

−49

1−

563

−63

329

823

217

211

807

103

400

90

Q0

−16

6−

294

−39

7−

481

−55

0−

606

−65

1−

684

−70

6−

718

−71

5−

700

−67

3−

631

−57

6−

506

−42

0−

314

−17

90

p−

190

−14

4−

114

−09

2−

076

−06

2−

050

−03

9−

028

−01

6−

005

008

021

034

048

062

078

095

118

153

210

~−

175

−16

7−

159

−15

1−

142

−13

2−

120

−10

6−

090

−06

9−

045

−01

402

306

109

412

515

317

920

422

825

3

M

0.3

0−

008

−03

1−

066

−10

9−

161

−21

8−

280

−34

7−

416

−48

7−

559

−62

9−

698

236

174

100

072

035

010

0

Q0

−16

4−

291

−39

2−

476

−54

6−

602

−64

7−

680

−70

3−

714

−71

4−

702

−67

6−

637

−58

3−

513

−42

6−

319

−18

30

p−

190

−14

2−

113

−09

2−

076

−06

3−

050

−03

9−

028

−01

7−

006

006

019

032

046

062

078

096

112

155

216

~−

140

−13

6−

133

−12

9−

125

−12

0−

113

−10

4−

093

−07

8−

059

−03

4−

004

033

073

108

141

172

201

230

260

M

0.4

0−

008

−03

1−

065

−10

9−

159

−21

7−

279

−34

5−

414

−48

4−

557

−62

7−

696

−76

217

612

107

303

501

00

Q0

−16

4−

289

−38

9−

473

−54

2−

598

−64

3−

677

−69

9−

711

−71

2−

702

−67

9−

641

−58

9−

519

−43

2−

323

−18

40

p−

191

−14

1−

111

−09

1−

076

−06

3−

051

−03

9−

028

−01

7−

007

009

017

030

045

061

078

097

122

158

217

~−

104

−10

5−

106

−10

6−

106

−10

6−

104

−09

9−

093

−08

3−

069

−05

0−

025

006

045

087

125

162

197

232

267

0.5

0−

009

−03

3−

068

−11

1−

162

−21

9−

280

−34

6−

414

−48

4−

554

−62

5−

693

−75

9−

821

124

075

037

010

0

0−

173

−29

7−

394

−47

4−

540

−59

4−

637

−67

0−

692

−70

4−

705

−69

6−

674

−63

9−

590

−52

3−

438

−33

1−

193

0

−21

0−

143

−10

8−

087

−07

3−

060

−04

9−

038

−02

7−

017

−00

700

401

502

804

305

807

509

512

116

023

2

−08

7−

090

−09

4−

096

−09

8−

099

−10

0−

098

−09

4−

087

−07

6−

060

−03

9−

012

023

067

114

158

202

244

287

0.6

0−

009

−03

2−

067

−10

9−

159

−21

6−

217

−34

3−

411

−48

1−

552

−62

3−

692

−75

8−

820

−87

607

503

601

00

0−

165

−29

0−

388

−47

0−

538

−59

3−

637

−67

1−

694

−70

6−

708

−69

9−

677

−64

3−

593

−52

7−

442

−33

3−

191

0

−19

5−

141

−11

0−

089

−07

4−

062

−05

0−

039

−02

6−

018

−00

700

401

502

704

205

707

509

612

416

222

5

−09

7−

100

−10

2−

104

−10

5−

106

−10

5−

103

−09

9−

091

−08

0−

064

−04

3−

017

018

061

113

170

225

278

332

0.7

0−

008

−03

0−

063

−10

5−

155

−21

1−

272

−33

7−

405

−47

5−

546

−61

7−

686

−75

3−

815

−87

2−

922

038

010

0

0−

156

−27

8−

378

−46

1−

530

−58

8−

633

−66

8−

693

−70

7−

710

−70

1−

681

−64

8−

600

−53

6−

453

−34

4−

199

0

178

−13

7−

110

−09

1−

076

−06

3−

051

−04

0−

030

−01

9−

008

003

014

027

040

055

073

095

125

168

236

−10

8−

109

−11

1−

111

−11

2−

112

−11

1−

108

−10

2−

095

−08

3−

067

−04

6−

019

015

057

109

171

237

303

369

0.8

0−

008

−03

1−

065

−10

7−

157

−21

3−

275

−34

0−

408

−47

8−

549

−62

0−

689

−75

5−

817

−87

4−

923

−96

301

00

0−

161

−28

5−

384

−46

5−

534

−59

0−

635

−66

9−

693

−70

7−

709

−70

0−

678

−64

4−

596

−53

2−

448

−34

0−

196

0

−18

6−

140

−11

0−

090

−07

4−

062

−05

0−

040

−02

9−

019

−00

900

301

502

704

105

607

309

512

416

623

0

−12

0−

120

−12

0−

120

−12

0−

118

−11

6−

112

−10

6−

097

−08

4−

067

−04

5−

017

018

061

113

176

246

328

416

0.9

0−

008

−03

1−

054

−10

7−

157

−21

3−

275

−34

0−

408

−47

8−

549

−62

0−

689

−75

5−

817

−87

4−

923

−96

2−

990

0

0−

160

−28

4−

384

−46

6−

534

−59

0−

635

−66

9−

693

−70

6−

709

−70

0−

679

−64

5−

596

−53

2−

448

−34

0−

196

0

−18

5−

139

−11

0−

090

−07

4−

062

−05

0−

040

−02

9−

019

−00

800

301

502

704

105

607

309

512

416

623

2

−11

4−

115

−11

6−

116

−11

6−

115

−11

3−

110

−10

4−

096

−08

5−

069

−04

8−

020

014

056

108

169

241

326

416

1.0

0−

008

−03

1−

064

−10

7−

157

−21

3−

275

−34

0−

408

−47

8−

549

−62

0−

689

−75

5−

817

−87

4−

923

−96

2−

990

−10

3

0−

168

−28

4−

384

−46

6−

534

−59

0−

635

−66

9−

693

−70

6−

709

−70

0−

679

−64

5−

596

−53

2−

448

−33

9−

196

0

−18

6−

139

−11

0−

090

−07

5−

062

−05

0−

040

−02

9−

019

−00

800

301

502

704

105

607

409

512

416

623

2

−10

6−

108

−10

9−

111

−11

1−

111

−11

1−

108

−10

3−

095

−08

5−

069

−04

9−

022

011

053

103

164

236

320

416

Tab

le 4

.11

Dim

ensi

onle

ss c

oeffi

cien

ts f

or a

naly

sis

of s

hort

bea

ms

load

ed w

ith a

mom

ent m

SAMPLE C

HAPTER

215

t =

2

Co

ef.

δx

−1

−0.

9−

0.8

−0.

7−

0.6

−0.

5−

0.4

−0.

3−

0.2

−0.

10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0−

008

−03

0−

063

−10

6−

156

−21

4−

278

−34

8−

423

−50

042

334

827

821

415

610

606

303

000

80

0−

151

−27

9−

382

−46

6−

540

−60

9−

673

−73

1−

764

−77

7−

764

−73

1−

673

−60

9−

540

−46

6−

382

−27

9−

151

0

−15

7−

141

−11

5−

092

−07

8−

071

−06

7−

060

−04

6−

025

002

504

600

006

707

107

809

211

514

115

7

−17

1−

160

−14

9−

137

−12

6−

113

−09

7−

079

−05

7−

031

003

105

707

909

711

312

613

714

916

017

1

0.1

0−

009

−03

2−

068

−11

2−

164

−22

3−

286

−35

4−

424

−49

5−

567

362

294

229

169

116

070

034

009

0

0−

168

−29

8−

402

−48

6−

555

−61

1−

656

−68

9−

710

−71

9−

714

−69

7−

667

−62

4−

569

−49

9−

414

−30

9−

176

0

−19

2−

147

−11

5−

093

−07

6−

063

−05

0−

039

−02

7−

015

−00

201

102

403

604

906

207

709

411

715

120

5

−19

2−

181

−16

9−

159

−14

6−

132

−11

6−

099

−07

8−

053

−02

301

304

908

010

813

215

417

519

521

423

3

0.2

0−

009

−03

2−

067

−11

1−

162

−22

0−

283

−35

0−

420

−49

1−

563

−63

329

823

217

211

807

103

400

90

0−

166

−29

4−

397

−48

1−

550

−60

6−

651

−68

4−

706

−71

8−

715

−70

0−

673

−63

1−

576

−50

6−

420

−31

4−

179

0

−19

0−

144

−11

4−

092

−07

6−

062

−05

0−

039

−02

8−

016

−00

500

802

103

404

806

207

809

511

815

321

0

−17

5−

167

−15

9−

151

−14

2−

132

−12

0−

106

−09

0−

069

−04

5−

014

023

061

094

125

153

179

204

228

253

0.3

0−

008

−03

1−

066

−10

9−

161

−21

8−

280

−34

7−

416

−48

7−

559

−62

9−

698

236

174

100

072

035

010

0

0−

164

−29

1−

392

−47

6−

546

−60

2−

647

−68

0−

703

−71

4−

714

−70

2−

676

−63

7−

583

−51

3−

426

−31

9−

183

0

−19

0−

142

−11

3−

092

−07

6−

063

−05

0−

039

−02

8−

017

−00

600

601

903

204

606

207

809

611

215

521

6

−14

0−

136

−13

3−

129

−12

5−

120

−11

3−

104

−09

3−

078

−05

9−

034

−00

403

307

310

814

117

220

123

026

0

0.4

0−

008

−03

1−

065

−10

9−

159

−21

7−

279

−34

5−

414

−48

4−

557

−62

7−

696

−76

217

612

107

303

501

00

0−

164

−28

9−

389

−47

3−

542

−59

8−

643

−67

7−

699

−71

1−

712

−70

2−

679

−64

1−

589

−51

9−

432

−32

3−

184

0

−19

1−

141

−11

1−

091

−07

6−

063

−05

1−

039

−02

8−

017

−00

700

901

703

004

506

107

809

712

215

821

7

−10

4−

105

−10

6−

106

−10

6−

106

−10

4−

099

−09

3−

083

−06

9−

050

−02

500

604

508

712

516

219

723

226

7

M

0.5

0−

009

−03

3−

068

−11

1−

162

−21

9−

280

−34

6−

414

−48

4−

554

−62

5−

693

−75

9−

821

124

075

037

010

0

Q0

−17

3−

297

−39

4−

474

−54

0−

594

−63

7−

670

−69

2−

704

−70

5−

696

−67

4−

639

−59

0−

523

−43

8−

331

−19

30

p−

210

−14

3−

108

−08

7−

073

−06

0−

049

−03

8−

027

−01

7−

007

004

015

028

043

058

075

095

121

160

232

~−

087

−09

0−

094

−09

6−

098

−09

9−

100

−09

8−

094

−08

7−

076

−06

0−

039

−01

202

306

711

415

820

224

428

7

M

0.6

0−

009

−03

2−

067

−10

9−

159

−21

6−

217

−34

3−

411

−48

1−

552

−62

3−

692

−75

8−

820

−87

607

503

601

00

Q0

−16

5−

290

−38

8−

470

−53

8−

593

−63

7−

671

−69

4−

706

−70

8−

699

−67

7−

643

−59

3−

527

−44

2−

333

−19

10

p−

195

−14

1−

110

−08

9−

074

−06

2−

050

−03

9−

026

−01

8−

007

004

015

027

042

057

075

096

124

162

225

~−

097

−10

0−

102

−10

4−

105

−10

6−

105

−10

3−

099

−09

1−

080

−06

4−

043

−01

701

806

111

317

022

527

833

2

M

0.7

0−

008

−03

0−

063

−10

5−

155

−21

1−

272

−33

7−

405

−47

5−

546

−61

7−

686

−75

3−

815

−87

2−

922

038

010

0

Q0

−15

6−

278

−37

8−

461

−53

0−

588

−63

3−

668

−69

3−

707

−71

0−

701

−68

1−

648

−60

0−

536

−45

3−

344

−19

90

p17

8−

137

−11

0−

091

−07

6−

063

−05

1−

040

−03

0−

019

−00

800

301

402

704

005

507

309

512

516

823

6

~−

108

−10

9−

111

−11

1−

112

−11

2−

111

−10

8−

102

−09

5−

083

−06

7−

046

−01

901

505

710

917

123

730

336

9

M

0.8

0−

008

−03

1−

065

−10

7−

157

−21

3−

275

−34

0−

408

−47

8−

549

−62

0−

689

−75

5−

817

−87

4−

923

−96

301

00

Q0

−16

1−

285

−38

4−

465

−53

4−

590

−63

5−

669

−69

3−

707

−70

9−

700

−67

8−

644

−59

6−

532

−44

8−

340

−19

60

p−

186

−14

0−

110

−09

0−

074

−06

2−

050

−04

0−

029

−01

9−

009

003

015

027

041

056

073

095

124

166

230

~−

120

−12

0−

120

−12

0−

120

−11

8−

116

−11

2−

106

−09

7−

084

−06

7−

045

−01

701

806

111

317

624

632

841

6

M

0.9

0−

008

−03

1−

054

−10

7−

157

−21

3−

275

−34

0−

408

−47

8−

549

−62

0−

689

−75

5−

817

−87

4−

923

−96

2−

990

0

Q0

−16

0−

284

−38

4−

466

−53

4−

590

−63

5−

669

−69

3−

706

−70

9−

700

−67

9−

645

−59

6−

532

−44

8−

340

−19

60

p−

185

−13

9−

110

−09

0−

074

−06

2−

050

−04

0−

029

−01

9−

008

003

015

027

041

056

073

095

124

166

232

~−

114

−11

5−

116

−11

6−

116

−11

5−

113

−11

0−

104

−09

6−

085

−06

9−

048

−02

001

405

610

816

924

132

641

6

M

1.0

0−

008

−03

1−

064

−10

7−

157

−21

3−

275

−34

0−

408

−47

8−

549

−62

0−

689

−75

5−

817

−87

4−

923

−96

2−

990

−10

3

Q0

−16

8−

284

−38

4−

466

−53

4−

590

−63

5−

669

−69

3−

706

−70

9−

700

−67

9−

645

−59

6−

532

−44

8−

339

−19

60

p−

186

−13

9−

110

−09

0−

075

−06

2−

050

−04

0−

029

−01

9−

008

003

015

027

041

056

074

095

124

166

232

~−

106

−10

8−

109

−11

1−

111

−11

1−

111

−10

8−

103

−09

5−

085

−06

9−

049

−02

201

105

310

316

423

632

041

6

SAMPLE C

HAPTER

216

t =

5

Co

ef.

δx

−1

−0.

9−

0.8

−0.

7−

0.6

−0.

5−

0.4

−0.

3−

0.2

−0.

10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

M

0

0−

008

−02

6−

055

−09

4−

141

−19

6−

260

−33

3−

414

−50

0−

414

333

260

196

141

094

055

026

007

0

Q0

−11

8−

243

−34

4−

427

−50

6−

594

−68

9−

777

−84

0−

864

−84

0−

777

−68

9−

594

−50

6−

427

−34

4−

243

−11

80

p−

092

−13

0−

114

−08

9−

078

−08

3−

093

−09

4−

077

−04

500

004

507

709

409

308

307

808

911

413

009

2

~−

113

−11

7−

120

−12

2−

122

−12

0−

113

−09

9−

078

0045

000

045

078

099

113

120

122

122

120

117

113

M

0.1

0−

004

−01

6−

039

−07

0−

109

−15

5−

210

−27

6−

351

−43

3−

519

395

314

241

176

120

073

036

011

0

Q0

−06

4−

175

−27

4−

351

−42

4−

507

−60

4−

705

−79

2−

848

−86

3−

837

−77

7−

096

−60

6−

514

−42

0−

314

−18

10

p−

002

−10

3−

109

−08

6−

077

−07

6−

091

−10

1−

097

−07

4−

037

006

045

073

087

092

092

097

116

154

210

~−

187

−18

4−

181

−17

8−

173

−16

6−

155

−14

0−

118

−08

7−

045

011

068

112

146

172

193

210

224

237

250

M

0.2

0−

001

−00

9−

025

−05

0−

083

−12

3−

172

−22

9−

297

−37

3−

456

−54

337

028

621

014

408

904

401

30

Q0

−03

3−

118

−20

8−

289

−36

4−

443

−53

2−

627

−72

1−

802

−85

6−

876

−85

7−

800

714

610

500

383

236

0

p01

5−

069

−09

2−

087

−07

7−

076

−08

3−

093

097

−08

9−

070

−03

900

003

907

309

710

811

212

617

930

5

~−

161

−16

1−

163

−16

4−

164

−16

2−

158

−15

0−

136

−11

5−

084

−04

2−

015

073

120

157

188

214

237

259

280

M

0.3

0−

001

−00

7−

020

−04

0−

067

−10

1−

144

−19

6−

257

−32

6−

405

−49

0−

580

330

244

167

102

050

014

0

Q0

−03

9−

095

−16

1−

232

−30

8−

389

−47

3−

561

−65

3−

741

−82

2−

882

−90

8−

880

−82

2−

715

−58

4−

442

−27

30

p−

026

−04

9−

062

−06

9−

074

−07

8−

082

−08

6−

090

−09

1−

086

−07

2−

045

−00

504

308

912

213

714

920

237

2

~−

084

−09

4−

104

−11

4−

123

−13

0−

136

−13

8−

136

−12

7−

111

−08

3−

043

012

071

118

158

193

224

254

283

M

0.4

0−

001

−00

7−

017

−03

3−

055

−08

6−

125

−17

1−

226

−28

0−

363

−44

5−

535

−63

027

618

911

405

501

40

Q0

−04

8−

078

−12

3−

189

−26

7−

348

−42

7−

505

−58

9−

681

−77

8−

868

−93

3−

952

−91

4−

817

−67

6−

504

−29

80

p−

077

−03

0−

034

−05

6−

074

−08

0−

083

−08

5−

086

−08

8−

096

−09

6−

080

−04

500

806

812

215

818

523

638

8

~−

005

−02

4−

062

−06

0−

078

−09

5−

110

−12

2−

130

−13

3−

128

−11

4−

089

−04

900

706

812

016

620

824

828

8

0.5

0−

001

−00

5−

013

−02

5−

044

−07

1−

106

−14

8−

198

−25

7−

325

−40

5−

493

−58

9−

688

218

133

064

017

0

0−

038

−05

7−

094

−15

6−

232

−31

0−

385

−45

9−

540

−63

3−

737

−84

2−

931

−98

1−

976

−90

6−

775

−59

0−

347

0

−06

8−

019

−02

5−

050

−07

1−

073

−07

7−

079

−07

9−

086

−09

9−

107

−10

0−

073

−02

503

710

215

921

128

243

5

028

−00

6−

016

−03

7−

058

−07

9−

098

−11

4−

128

−13

6−

138

−13

2−

116

−08

6−

041

022

092

155

213

270

325

0.6

0−

001

−00

6−

013

−02

3−

040

−06

3−

−93

−13

2−

178

−23

4−

300

−37

6−

461

−55

1−

654

−75

315

407

602

10

0−

034

−05

4−

084

−13

3−

196

−26

7−

343

−42

3−

510

−60

6−

708

−81

0−

902

−97

1−

999

−97

1−

870

−68

3−

389

0

−05

6−

021

−02

2−

039

−05

7−

068

−07

4−

078

−08

3−

091

−09

9−

103

−09

9−

083

−05

1−

003

062

142

233

337

469

−00

4−

023

−04

1−

059

−07

7−

094

−11

0−

124

−13

5−

142

−14

3−

136

−12

0−

092

−04

901

109

218

126

634

742

8

0.7

0−

001

−00

4−

009

−01

9−

034

−05

4−

081

−11

7−

161

−21

5−

278

−35

2−

435

−52

7−

625

−72

6−

824

089

026

0

0−

016

−04

1−

075

−11

9−

173

−23

8−

312

−39

6−

488

−58

6−

687

−78

6−

878

−95

4−

998

−96

5−

939

−77

4−

475

0

−01

0−

020

−02

9−

039

−04

9−

059

−07

0−

079

−08

8−

095

−10

0−

100

−09

7−

085

−06

4−

029

027

110

226

379

580

−03

7−

052

−06

6−

081

−09

6−

110

−12

2−

133

−14

2−

146

−14

6−

139

−12

2−

095

−05

400

408

218

329

440

351

0

0.8

0−

001

−00

3−

009

−01

8−

031

−05

0−

076

−11

0−

153

−20

7−

270

−34

2−

424

−51

4−

611

−71

2−

813

−90

302

80

0−

008

−03

9−

074

−11

2−

159

−22

0−

297

−38

6−

483

−58

2−

679

−77

2−

860

−93

8−

997

−01

8−

972

−82

0−

514

0

017

−02

5−

034

−03

5−

041

−05

3−

069

−08

4−

094

−09

9−

098

−09

5−

091

−08

4−

071

−04

400

709

222

040

064

0

−07

2−

083

−09

4−

105

−11

6−

126

−13

6−

143

−14

8−

149

−14

6−

136

−11

7−

088

−04

501

409

319

432

246

260

1

0.9

000

0−

002

−00

7−

016

−02

9−

049

−07

5−

108

−15

2−

204

−26

7−

340

−42

2−

512

−61

0−

711

−81

0−

902

−97

10

0−

002

−03

1−

068

−11

2−

163

−22

4−

298

−38

4−

478

−57

7−

677

−77

4−

864

−94

3−

998

−99

6−

970

−82

6−

528

0

021

−02

0−

035

−04

0−

046

−05

6−

068

−08

0−

090

−09

7−

099

−09

9−

094

−08

5−

069

−04

000

908

821

039

668

1

−05

6−

068

−08

1−

094

−10

6−

118

−12

9−

138

−14

5−

148

−14

7−

139

−12

3−

095

−05

500

207

917

830

245

662

4

1.0

0−

000

−00

2−

007

−01

6−

029

−04

9−

076

−11

0−

153

−20

5−

258

−34

0−

422

−51

3−

611

−71

2−

812

−90

2−

971

−10

3

0−

007

−03

1−

066

−11

1−

165

−22

9−

303

−38

6−

436

−57

3−

674

−77

4−

868

−94

7−

103

−11

14−

965

−82

0−

528

0

005

−01

7−

030

−04

0−

049

−05

9−

069

−07

8−

087

−09

4−

099

−10

1−

098

−08

8−

069

−03

601

409

020

739

169

1

036

−05

0−

065

−98

0−

094

−10

8−

121

−13

2−

141

−14

6−

147

−14

1−

126

−10

1−

062

−00

606

816

628

944

062

3

Tab

le 4

.12

Dim

ensi

onle

ss c

oeffi

cien

ts f

or a

naly

sis

of s

hort

bea

ms

load

ed w

ith a

con

cent

rate

d m

omen

t m

SAMPLE C

HAPTER

217

t =

5

Co

ef.

δx

−1

−0.

9−

0.8

−0.

7−

0.6

−0.

5−

0.4

−0.

3−

0.2

−0.

10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0−

008

−02

6−

055

−09

4−

141

−19

6−

260

−33

3−

414

−50

0−

414

333

260

196

141

094

055

026

007

0

0−

118

−24

3−

344

−42

7−

506

−59

4−

689

−77

7−

840

−86

4−

840

−77

7−

689

−59

4−

506

−42

7−

344

−24

3−

118

0

−09

2−

130

−11

4−

089

−07

8−

083

−09

3−

094

−07

7−

045

000

045

077

094

093

083

078

089

114

130

092

−11

3−

117

−12

0−

122

−12

2−

120

−11

3−

099

−07

800

4500

004

507

809

911

312

012

212

212

011

711

3

0.1

0−

004

−01

6−

039

−07

0−

109

−15

5−

210

−27

6−

351

−43

3−

519

395

314

241

176

120

073

036

011

0

0−

064

−17

5−

274

−35

1−

424

−50

7−

604

−70

5−

792

−84

8−

863

−83

7−

777

−09

6−

606

−51

4−

420

−31

4−

181

0

−00

2−

103

−10

9−

086

−07

7−

076

−09

1−

101

−09

7−

074

−03

700

604

507

308

709

209

209

711

615

421

0

−18

7−

184

−18

1−

178

−17

3−

166

−15

5−

140

−11

8−

087

−04

501

106

811

214

617

219

321

022

423

725

0

0.2

0−

001

−00

9−

025

−05

0−

083

−12

3−

172

−22

9−

297

−37

3−

456

−54

337

028

621

014

408

904

401

30

0−

033

−11

8−

208

−28

9−

364

−44

3−

532

−62

7−

721

−80

2−

856

−87

6−

857

−80

071

461

050

038

323

60

015

−06

9−

092

−08

7−

077

−07

6−

083

−09

309

7−

089

−07

0−

039

000

039

073

097

108

112

126

179

305

−16

1−

161

−16

3−

164

−16

4−

162

−15

8−

150

−13

6−

115

−08

4−

042

−01

507

312

015

718

821

423

725

928

0

0.3

0−

001

−00

7−

020

−04

0−

067

−10

1−

144

−19

6−

257

−32

6−

405

−49

0−

580

330

244

167

102

050

014

0

0−

039

−09

5−

161

−23

2−

308

−38

9−

473

−56

1−

653

−74

1−

822

−88

2−

908

−88

0−

822

−71

5−

584

−44

2−

273

0

−02

6−

049

−06

2−

069

−07

4−

078

−08

2−

086

−09

0−

091

−08

6−

072

−04

5−

005

043

089

122

137

149

202

372

−08

4−

094

−10

4−

114

−12

3−

130

−13

6−

138

−13

6−

127

−11

1−

083

−04

301

207

111

815

819

322

425

428

3

0.4

0−

001

−00

7−

017

−03

3−

055

−08

6−

125

−17

1−

226

−28

0−

363

−44

5−

535

−63

027

618

911

405

501

40

0−

048

−07

8−

123

−18

9−

267

−34

8−

427

−50

5−

589

−68

1−

778

−86

8−

933

−95

2−

914

−81

7−

676

−50

4−

298

0

−07

7−

030

−03

4−

056

−07

4−

080

−08

3−

085

−08

6−

088

−09

6−

096

−08

0−

045

008

068

122

158

185

236

388

−00

5−

024

−06

2−

060

−07

8−

095

−11

0−

122

−13

0−

133

−12

8−

114

−08

9−

049

007

068

120

166

208

248

288

M

0.5

0−

001

−00

5−

013

−02

5−

044

−07

1−

106

−14

8−

198

−25

7−

325

−40

5−

493

−58

9−

688

218

133

064

017

0

Q0

−03

8−

057

−09

4−

156

−23

2−

310

−38

5−

459

−54

0−

633

−73

7−

842

−93

1−

981

−97

6−

906

−77

5−

590

−34

70

p−

068

−01

9−

025

−05

0−

071

−07

3−

077

−07

9−

079

−08

6−

099

−10

7−

100

−07

3−

025

037

102

159

211

282

435

~02

8−

006

−01

6−

037

−05

8−

079

−09

8−

114

−12

8−

136

−13

8−

132

−11

6−

086

−04

102

209

215

521

327

032

5

M

0.6

0−

001

−00

6−

013

−02

3−

040

−06

3−

−93

−13

2−

178

−23

4−

300

−37

6−

461

−55

1−

654

−75

315

407

602

10

Q0

−03

4−

054

−08

4−

133

−19

6−

267

−34

3−

423

−51

0−

606

−70

8−

810

−90

2−

971

−99

9−

971

−87

0−

683

−38

90

p−

056

−02

1−

022

−03

9−

057

−06

8−

074

−07

8−

083

−09

1−

099

−10

3−

099

−08

3−

051

−00

306

214

223

333

746

9

~−

004

−02

3−

041

−05

9−

077

−09

4−

110

−12

4−

135

−14

2−

143

−13

6−

120

−09

2−

049

011

092

181

266

347

428

M

0.7

0−

001

−00

4−

009

−01

9−

034

−05

4−

081

−11

7−

161

−21

5−

278

−35

2−

435

−52

7−

625

−72

6−

824

089

026

0

Q0

−01

6−

041

−07

5−

119

−17

3−

238

−31

2−

396

−48

8−

586

−68

7−

786

−87

8−

954

−99

8−

965

−93

9−

774

−47

50

p−

010

−02

0−

029

−03

9−

049

−05

9−

070

−07

9−

088

−09

5−

100

−10

0−

097

−08

5−

064

−02

902

711

022

637

958

0

~−

037

−05

2−

066

−08

1−

096

−11

0−

122

−13

3−

142

−14

6−

146

−13

9−

122

−09

5−

054

004

082

183

294

403

510

M

0.8

0−

001

−00

3−

009

−01

8−

031

−05

0−

076

−11

0−

153

−20

7−

270

−34

2−

424

−51

4−

611

−71

2−

813

−90

302

80

Q0

−00

8−

039

−07

4−

112

−15

9−

220

−29

7−

386

−48

3−

582

−67

9−

772

−86

0−

938

−99

7−

018

−97

2−

820

−51

40

p01

7−

025

−03

4−

035

−04

1−

053

−06

9−

084

−09

4−

099

−09

8−

095

−09

1−

084

−07

1−

044

007

092

220

400

640

~−

072

−08

3−

094

−10

5−

116

−12

6−

136

−14

3−

148

−14

9−

146

−13

6−

117

−08

8−

045

014

093

194

322

462

601

M

0.9

000

0−

002

−00

7−

016

−02

9−

049

−07

5−

108

−15

2−

204

−26

7−

340

−42

2−

512

−61

0−

711

−81

0−

902

−97

10

Q0

−00

2−

031

−06

8−

112

−16

3−

224

−29

8−

384

−47

8−

577

−67

7−

774

−86

4−

943

−99

8−

996

−97

0−

826

−52

80

p02

1−

020

−03

5−

040

−04

6−

056

−06

8−

080

−09

0−

097

−09

9−

099

−09

4−

085

−06

9−

040

009

088

210

396

681

~−

056

−06

8−

081

−09

4−

106

−11

8−

129

−13

8−

145

−14

8−

147

−13

9−

123

−09

5−

055

002

079

178

302

456

624

M

1.0

0−

000

−00

2−

007

−01

6−

029

−04

9−

076

−11

0−

153

−20

5−

258

−34

0−

422

−51

3−

611

−71

2−

812

−90

2−

971

−10

3

Q0

−00

7−

031

−06

6−

111

−16

5−

229

−30

3−

386

−43

6−

573

−67

4−

774

−86

8−

947

−10

3−

1114

−96

5−

820

−52

80

p00

5−

017

−03

0−

040

−04

9−

059

−06

9−

078

−08

7−

094

−09

9−

101

−09

8−

088

−06

9−

036

014

090

207

391

691

~03

6−

050

−06

5−

980

−09

4−

108

−12

1−

132

−14

1−

146

−14

7−

141

−12

6−

101

−06

2−

006

068

166

289

440

623

SAMPLE C

HAPTER

218

t =

10

Co

ef.

δx

−1

−0.

9−

0.8

−0.

7−

0.6

−0.

5−

0.4

−0.

3−

0.2

−0.

10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

M

0

0−

006

−01

9−

044

−07

8−

119

−17

0−

234

−31

2−

402

−50

0−

402

−31

2−

234

−17

0−

119

−07

8−

044

−01

9−

006

0

Q0

−07

0−

193

−29

3−

370

−45

6−

570

−70

9−

849

−95

3−

991

−95

3−

849

−70

9−

570

−45

6−

370

−29

3−

193

−07

00

p01

1−

119

−11

6−

085

−07

6−

098

−12

9−

145

−12

8−

075

007

512

814

512

909

807

608

511

611

901

1

~−

029

−05

2−

074

−09

5−

113

−12

6−

132

−12

6−

106

−06

60

066

106

126

132

126

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0.1

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005

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015

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116

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234

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539

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923

116

611

206

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501

20

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010

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4−

208

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0−

350

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9−

584

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6−

873

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1−

983

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7−

841

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8−

594

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3−

386

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9−

167

0

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M

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000

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8−

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M

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553

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30

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601

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154

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263

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231

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801

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013

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204

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353

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564

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6−

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38−

1139

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57−

1078

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5−

689

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70

073

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016

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137

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198

251

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528

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117

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072

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142

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181

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197

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122

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805

412

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6

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579

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719

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011

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013

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074

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2−

217

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399

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76−

1071

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491

0

056

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046

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131

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108

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305

016

328

942

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7

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052

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037

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097

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5−

152

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5−

192

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0−

195

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3−

128

−05

405

718

129

339

950

2

0.7

000

100

300

500

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005

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040

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114

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9−

239

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2−

421

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2−

651

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211

703

40

001

902

401

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8−

103

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4−

262

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7−

488

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1−

765

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2−

1050

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62−

1219

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82−

1002

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50

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011

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276

5

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119

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160

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191

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129

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104

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434

549

864

8

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000

100

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506

600

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001

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226

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2−

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35−

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50

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4

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131

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188

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152

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821

0.9

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200

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000

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30

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016

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956

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193

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172

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304

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7−

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−10

3

003

303

702

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003

−04

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096

−16

4−

248

−34

9−

466

−59

7−

740

−88

7−

1028

−11

48−

1224

−12

19−

1081

−72

30

−05

5−

015

−00

5−

020

−03

3−

046

−06

1−

076

−09

2−

109

−12

4−

138

−14

6−

146

−13

4−

102

−04

206

023

051

097

7

004

−01

5−

034

−05

3−

073

−09

4−

115

−13

6−

156

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5−

189

−19

6−

194

−17

7−

141

−08

001

414

833

056

786

4

Tab

le 4

.13

Dim

ensi

onle

ss c

oeffi

cien

ts f

or a

naly

sis

of s

hort

bea

ms

load

ed w

ith a

mom

ent m

SAMPLE C

HAPTER

219

t =

10

Co

ef.

δx

−1

−0.

9−

0.8

−0.

7−

0.6

−0.

5−

0.4

−0.

3−

0.2

−0.

10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0−

006

−01

9−

044

−07

8−

119

−17

0−

234

−31

2−

402

−50

0−

402

−31

2−

234

−17

0−

119

−07

8−

044

−01

9−

006

0

0−

070

−19

3−

293

−37

0−

456

−57

0−

709

−84

9−

953

−99

1−

953

−84

9−

709

−57

0−

456

−37

0−

293

−19

3−

070

0

011

−11

9−

116

−08

5−

076

−09

8−

129

−14

5−

128

−07

50

075

128

145

129

098

076

085

116

119

011

−02

9−

052

−07

4−

095

−11

3−

126

−13

2−

126

−10

6−

066

006

610

612

613

212

611

309

507

405

202

9

0.1

000

0−

005

−02

1−

015

−07

7−

116

−16

8−

234

−31

5−

407

−50

539

930

923

116

611

206

903

501

20

001

0−

104

−20

8−

250

−35

0−

449

−58

4−

736

−87

3−

961

−98

3−

037

−84

1−

718

−59

4−

483

−38

6−

289

−16

70

156

−08

6−

120

−08

5−

065

−08

1−

118

−14

8−

150

−11

705

701

307

511

412

711

910

309

310

514

319

1

−20

8−

210

−21

3−

215

−21

6−

214

−20

7−

192

−16

7−

126

066

020

107

168

210

236

254

262

266

268

270

0.2

000

300

4−

003

−02

0−

044

−07

7−

119

−17

4−

242

−32

4−

416

−51

438

829

621

4−

145

090

016

014

0

004

5−

030

−12

6−

208

−28

3−

370

−48

1−

612

−75

1−

873

−95

9−

991

−96

2−

877

−75

4−

618

−49

2−

381

−24

80

154

−03

7−

096

−09

0−

076

−07

8−

098

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3−

138

−13

4−

107

−06

1−

002

058

107

134

134

116

111

170

350

−15

1−

195

−19

5−

195

−20

0−

203

−20

4−

199

−18

7−

164

−12

5−

066

−02

010

716

921

224

225

227

728

829

8

0.3

000

300

300

0−

010

−02

8−

053

−05

8−

133

−19

0−

260

−34

2−

436

−53

736

026

217

710

705

401

50

001

4−

014

−06

7−

134

−21

2−

300

−39

8−

508

−63

1−

760

−88

4−

983

−10

33−

1015

−92

4−

778

−61

4−

463

−30

20

017

−01

1−

043

−06

1−

072

−08

3−

093

−10

4−

117

−12

7−

129

−11

6−

078

−01

805

612

316

116

014

519

745

9

−04

4−

062

−08

1−

101

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0−

138

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5−

168

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5−

174

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1−

131

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000

008

114

018

221

323

725

727

6

0.4

000

200

000

0−

005

−01

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037

−06

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104

−15

1−

210

282

−36

9−

470

−57

931

121

012

505

901

50

0−

019

−00

4−

020

−07

9−

162

−25

0−

334

−42

1−

523

−64

9−

797

−94

5−

1061

−11

09−

1068

−94

1−

757

−55

3−

333

0

076

015

005

−03

9−

075

−08

1−

085

−08

9−

092

−11

3−

136

−15

3−

138

−08

7−

005

087

162

199

207

248

464

102

068

034

001

−03

3−

067

−09

9−

129

−15

4−

173

−18

5−

178

−15

6−

110

−03

504

610

715

419

422

926

3

M

0.5

000

200

200

400

3−

005

−02

2−

046

−07

7−

117

−16

7−

231

−31

1−

407

−51

7−

632

255

155

074

019

0

Q0

010

018

010

−01

3−

123

−20

4−

270

−35

3−

443

−56

4−

716

−88

4−

1038

−11

39−

1157

−10

78−

915

−68

9−

407

0

p07

302

801

6−

013

−07

1−

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−07

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079

−07

9−

104

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7−

164

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7−

133

−06

303

012

419

825

132

352

8

~15

411

707

904

200

1−

035

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109

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2−

170

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1−

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−19

7−

172

−12

2−

038

054

129

195

255

316

M

0.6

000

000

0−

001

−00

1−

003

−01

4−

032

−05

8−

092

−13

8−

197

−27

1−

360

−46

4−

579

−69

719

009

302

50

Q0

−01

1−

011

−01

3−

019

−07

4−

142

−21

7−

300

−39

9−

519

−66

1−

816

−97

0−

1100

−11

78−

1176

−10

71−

845

−49

10

p05

601

801

7−

015

−04

6−

064

−07

1−

078

−09

0−

109

−13

1−

151

−15

8−

146

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8−

043

050

163

289

420

567

~08

205

202

2−

007

−03

7−

067

−09

7−

125

−15

2−

175

−19

2−

200

−19

5−

173

−12

8−

054

057

181

293

399

502

M

0.7

000

100

300

500

500

3−

005

−01

8−

040

−07

1−

114

−16

9−

239

−32

2−

421

−53

2−

651

−77

211

703

40

Q0

019

024

015

−00

9−

048

−10

3−

174

−26

2−

367

−48

8−

621

−76

5−

912

−10

50−

1162

−12

19−

1182

−10

02−

625

0

p02

801

1−

002

−01

6−

031

−04

7−

063

−08

0−

096

−11

3−

127

−13

9−

146

−14

5−

129

−09

0−

017

100

269

492

765

~00

7−

014

−03

4−

055

−07

6−

097

−11

9−

140

−16

0−

178

−19

1−

196

−19

1−

170

−12

9−

061

041

184

345

498

648

M

0.8

000

100

400

506

600

4−

001

−01

2−

032

−06

2−

104

−15

9−

226

−30

7−

401

−50

8−

628

−75

2−

870

039

0

Q0

000

022

010

−00

3−

027

−07

6−

151

−24

9−

363

−48

4−

608

−73

7−

872

−10

11−

1142

−12

33−

1235

−10

82−

695

0

p08

200

0−

014

−02

1−

016

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−11

8−

123

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6−

132

−13

8−

138

−11

7−

057

063

257

529

874

~−

067

−08

0−

092

−10

5−

118

−13

1−

144

−15

8−

170

−18

1−

188

−18

8−

177

−15

2−

108

−03

806

420

639

661

082

1

M

0.9

000

200

600

901

000

800

2−

010

−03

0−

060

−10

2−

155

−22

2−

303

−39

8−

506

−62

5−

749

−86

6−

959

0

Q0

042

037

019

−00

5−

038

−08

7−

157

−24

7−

354

−47

4−

603

−73

9−

879

−10

18−

1142

−12

26−

1229

−10

91−

723

0

p08

7−

010

−01

6−

021

−02

7−

040

−05

9−

080

−09

9−

114

−12

5−

133

−13

9−

141

−13

5−

109

−05

205

923

652

095

6

~−

036

−05

1−

066

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1−

097

−11

4−

131

−14

8−

164

−17

8−

189

−19

3−

187

−16

7−

127

−06

203

517

235

759

786

5

M

1.0

000

200

500

901

000

800

1−

012

−03

2−

062

−10

3−

156

−22

2−

304

−40

0−

509

−62

8−

751

−86

7−

960

−10

3

Q0

033

037

024

−00

3−

042

−09

6−

164

−24

8−

349

−46

6−

597

−74

0−

887

−10

28−

1148

−12

24−

1219

−10

81−

723

0

p−

055

−01

5−

005

−02

0−

033

−04

6−

061

−07

6−

092

−10

9−

124

−13

8−

146

−14

6−

134

−10

2−

042

060

230

510

977

~00

4−

015

−03

4−

053

−07

3−

094

−11

5−

136

−15

6−

175

−18

9−

196

−19

4−

177

−14

1−

080

014

148

330

567

864

SAMPLE C

HAPTER

220 Analysis of Structures on Elastic Foundations

tx

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

M

1 032 032 031 028 0.25 021 016 012 007 002 0

2 0.27 0.27 0.27 0.24 0.21 0.19 0.15 0.11 0.06 0.02 0

5 020 020 019 017 0.16 014 011 008 004 002 0

10 012 012 012 012 011 010 009 006 004 001 0

Q

1 0 −009 −019 −028 −035 −043 −049 −052 −049 −036 0

2 0 −008 −016 −022 −030 −036 −042 −045 −046 −032 0

5 0 −004 −010 −015 −020 −024 −030 −033 −033 −023 0

10 0 −0.02 −005 −008 −010 −015 −019 −023 −026 −020 0

p

1 090 090 091 091 092 093 095 100 107 122 153

2 092 092 092 092 093 094 096 098 116 147 198

5 096 095 095 095 095 094 094 093 102 114 141

10 098 097 097 096 093 095 096 096 100 110 134

xY

1 221 220 220 219 218 217 217 216 215 213 212

2 222 222 221 220 220 218 217 214 212 211 208

5 226 226 225 224 223 218 217 213 210 205 201

10 230 230 229 227 224 221 216 212 207 202 196

Table 4.14 Dimensionless coefficients for analysis of beams loaded with a uniformly distributed load

p 0 0.2 0.4 06 0.8 1.0 1.2 1.4 1.6 2.0 2.4 2.8 3.0

p 1.487 1.322 0.981 0.772 0.582 0.418 0.284 0.179 0.100 0.004 −0.034 −0.042 −0.042

p 446 400 294 230 175 125 85 54 30 1.2 −10 −12 −12

Y 2.57 2.32 2.07 1.76 1.44 1.16 0.89 0.67 0.49 0.24 0.12 0.05 0.04

Y 3.86 3.48 3.11 2.64 2.16 1.74 1.34 1.00 0.74 0.36 0.18 0.08 0.06

Table 4.15 Soil pressure and settlements

p 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6

Q 0.00 0.051 0.118 0.200 0.296 0.397 0.500 −0.400 −0.310 −0.231 −0.164 −0.11 −0.067 −0.036

Q 0.0 4.1 9.4 16 24 32 40/−40 −32 −24.8 −18.5 −13.1 −8.8 −5.4 −2.9

Table 4.16 Shear forces

SAMPLE C

HAPTER

Analysis of Beams on Elastic Half-Space 221

β =

0.0

25

ζα

00.

10.

20.

30.

40.

50.

60.

70.

80.

91.

0∞

p

0.0

3.22

72.

747

2.25

91.

828

1.40

21.

101

0.86

70.

723

0.57

00.

434

0.27

30.

761

0.1

2.62

62.

296

1.96

31.

644

1.33

51.

057

0.82

50.

643

0.48

20.

340

0.21

90.

733

0.2

2.10

51.

805

1.68

31.

465

1.24

81.

034

0.82

90.

643

0.48

00.

342

0.22

60.

668

0.3

1.65

81.

538

1.42

11.

291

1.15

91.

106

0.84

60.

680

0.53

70.

395

0.28

40.

602

0.4

1.28

21.

229

1.18

21.

121

1.05

90.

967

0.85

70.

727

0.58

80.

480

0.37

10.

552

0.5

0.96

30.

959

0.96

20.

958

0.94

70.

911

0.85

10.

765

0.66

90.

569

0.47

10.

509

0.6

0.69

80.

730

0.76

80.

802

0.83

10.

839

0.82

50.

775

0.72

30.

649

0.57

30.

456

0.7

0.48

10.

538

0.60

30.

660

0.71

50.

755

0.77

50.

772

o.75

00.

711

0.65

90.

388

0.8

0.30

60.

376

0.45

40.

528

0.60

10.

665

0.71

40.

745

0.75

70.

751

0.72

40.

310

0.9

0.16

50.

246

0.32

80.

413

0.49

40.

575

0.64

50.

703

0.74

20.

759

0.73

60.

233

1.0

0.05

70.

141

0.23

00.

314

0.40

10.

484

0.56

30.

634

0.69

30.

737

0.76

20.

164

1.2

−0.

085

−0.

003

0.07

90.

160

0.24

30.

325

0.40

90.

494

0.57

60.

650

0.71

20.

065

1.4

−0.

152

−0.

083

−0.

012

0.05

70.

127

0.19

90.

275

0.35

30.

432

0.50

70.

583

0.01

3

1.6

−0.

170

−0.

115

−0.

062

−0.

006

0.04

70.

106

0.16

70.

233

0.30

00.

367

0.43

8−

0.00

9

1.8

−0.

158

−0.

121

−0.

083

−0.

040

−0.

003

0.04

30.

089

0.14

10.

192

0.24

30.

301

−0.

016

2.0

−0.

134

−0.

109

−0.

084

−0.

055

−0.

030

0.00

20.

035

0.07

30.

110

0.14

901

92−

0.01

6

2.2

−0.

106

−0.

091

−0.

074

−0.

058

−0.

042

−0.

023

−0.

002

0.02

40.

051

0.07

90.

111

−0.

014

2.4

−0.

076

−0.

068

−0.

059

−0.

051

−0.

043

−0.

034

−0.

019

−0.

008

0.01

30.

028

0.05

0−

0.01

2

2.6

−0.

052

−0.

050

−0.

045

−0.

042

−0.

039

−0.

035

−0.

031

−0.

017

−0.

013

−0.

009

−0.

003

−0.

009

2.8

−0.

032

−0.

033

−0.

033

−0.

033

−0.

035

−0.

035

−0.

034

−0.

032

−0.

031

−0.

031

−0.

031

−0.

007

3.0

−0.

021

−0.

020

−0.

020

−0.

024

−0.

028

−0.

034

−0.

040

−0.

044

−0.

039

−0.

039

−0.

030

−0.

005

Tab

le 4

.17

Dim

ensi

onle

ss c

oeffi

cien

ts f

or a

naly

sis

of lo

ng b

eam

s lo

aded

with

a c

once

ntra

ted

ver

tical

load

P

SAMPLE C

HAPTER

222 Analysis of Structures on Elastic Foundations

β =

0.0

25

ζα

00.

10.

20.

30.

40.

50.

60.

70.

80.

91.

0∞

M

0.0

00

00

00

00

00

00.

176

0.1

0.08

50.

013

0.01

10.

007

0.00

70.

006

0.00

60.

005

0.00

40.

003

0.00

20.

130

0.2

−0.

142

−0.

050

0.04

10.

033

0.02

80.

022

0.01

80.

015

0.01

10.

007

0.00

50.

091

0.3

−0.

181

−0.

094

−0.

011

0.07

20.

059

0.04

80.

039

0.03

10.

022

0.01

70.

011

0.05

8

0.4

−0.

202

−0.

124

−0.

050

0.02

60.

104

0.08

50.

068

0.05

40.

041

0.03

00.

020

0.03

2

0.5

−0.

209

−0.

142

−0.

076

−0.

007

0.05

90.

131

0.10

50.

085

0.06

50.

046

0.03

10.

011

0.6

−0.

207

−0.

150

−0.

092

−0.

035

0.02

20.

085

0.15

20.

122

0.09

20.

070

0.04

8−

0.00

6

0.7

−0.

198

−0.

150

−0.

102

−0.

054

−0.

004

0.05

00.

107

0.16

80.

131

0.10

00.

070

−0.

016

0.8

−0.

185

−0.

144

−0.

105

−0.

067

−0.

024

0.02

00.

068

0.12

00.

176

0.13

70.

100

−0.

023

0.9

−0.

168

−0.

135

−0.

105

−0.

072

−0.

039

−0.

002

0.03

90.

081

0.13

00.

179

0.13

7−

0.02

7

1.0

−0.

150

−0.

124

−0.

100

−0.

074

−0.

048

−0.

018

0.01

50.

050

0.08

90.

135

0.18

1−

0.02

9

1.2

−0.

113

−0.

098

−0.

085

−0.

070

−0.

056

−0.

039

−0.

018

0.00

40.

030

0.05

90.

092

−0.

028

1.4

−0.

080

−0.

072

−0.

067

−0.

061

−0.

054

−0.

044

−0.

033

−0.

020

−0.

007

0.01

10.

031

−0.

024

1.6

−0.

050

−0.

054

−0.

048

−0.

048

−0.

046

−0.

043

−0.

039

−0.

033

−0.

028

−0.

018

−0.

007

−0.

020

1.8

−0.

028

−0.

030

−0.

033

−0.

035

−0.

035

−0.

037

−0.

037

−0.

035

−0.

034

−0.

031

−0.

026

−0.

016

2.0

−0.

013

−0.

017

−0.

020

−0.

024

−0.

026

−0.

031

−0.

031

−0.

032

−0.

035

−0.

035

−0.

035

−0.

012

2.2

−0.

004

−0.

007

−0.

011

−0.

015

−0.

018

−0.

024

−0.

024

−0.

026

−0.

030

−0.

033

−0.

035

−0.

010

2.4

0.00

20.

000

−0.

006

−0.

007

−0.

011

−0.

017

−0.

017

−0.

018

−0.

024

−0.

028

−0.

031

−0.

006

2.6

0.00

50.

002

−0.

002

−0.

003

−0.

005

−0.

009

−0.

009

−0.

013

−0.

017

−0.

020

−0.

024

−0.

006

2.8

0.00

60.

004

0.00

00.

000

−0.

002

−0.

004

−0.

004

−0.

006

−0.

011

−0.

015

−0.

018

−0.

005

3.0

0.00

60.

004

0.00

20.

001

0.00

0−

0.00

1−

0.00

2−

0.00

2−

0.00

5−

0.00

9−

0.01

3−

0.00

4

0.0

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

−0.

500

0.1

−0.

708

0.25

20.

220

0.17

40.

137

0.10

60.

083

0.96

60.

050

0.03

80.

026

−0.

425

0.2

−0.

472

−0.

539

0.39

30.

330

0.26

60.

211

0.16

60.

128

0.09

10.

071

0.04

7−

0.35

5

0.3

−0.

284

−0.

367

−0.

452

0.46

70.

387

0.31

30.

250

0.19

40.

146

0.10

70.

072

−0.

292

0.4

−0.

138

−0.

229

−0.

322

−0.

413

0.49

70.

412

0.33

50.

264

0.20

20.

150

0.10

3−

0.23

4

0.5

−0.

026

−0.

120

−0.

215

−0.

309

−0.

402

0.50

60.

420

0.34

00.

266

0.20

20.

144

−0.

181

0.6

0.05

6−

0.03

6−

0.12

9−

0.22

1−

0.31

4−

0.40

70.

504

0.41

70.

386

0.26

30.

197

−0.

132

0.7

0.11

30.

027

−0.

061

−0.

149

−0.

237

−0.

328

−0.

416

0.49

50.

413

0.33

30.

261

−0.

090

0.8

0.15

30.

073

−0.

009

−0.

089

−0.

171

−0.

256

−0.

340

−0.

427

0.48

80.

407

0.33

0−

0.05

5

0.9

0.17

60.

103

0.03

0−

0.04

2−

0.11

6−

0.19

4−

0.27

2−

0.35

4−

0.43

80.

482

0.40

2−

0.02

8

1.0

0.18

70.

122

0.05

8−

0.00

6−

0.07

2−

0.14

1−

0.21

2−

0.28

8−

0.36

5−

0.44

20.

482

−0.

008

1.2

0.18

30.

135

0.08

80.

040

−0.

008

−0.

060

−0.

114

−0.

174

−0.

237

−0.

303

−0.

373

0.01

4

1.4

0.15

90.

126

0.09

30.

061

0.02

8−

0.00

9−

0.04

7−

0.09

0−

0.13

7−

0.18

8−

0.24

30.

021

1.6

0.14

60.

106

0.08

60.

065

0.04

80.

021

−0.

004

−0.

033

−0.

064

−0.

100

−0.

141

0.02

1

1.8

0.09

20.

091

0.07

00.

060

0.04

90.

036

0.02

2−

0.00

4−

0.01

6−

0.04

0−

0.06

70.

018

2.0

0.06

20.

058

0.05

30.

050

0.04

60.

040

0.03

40.

025

0.01

9−

0.00

1−

0.01

80.

014

2.2

0.03

90.

038

0.03

80.

039

0.03

90.

038

0.03

70.

035

0.02

90.

022

0.01

30.

011

2.4

0.02

00.

022

0.02

40.

027

0.03

00.

032

0.03

40.

036

0.03

50.

032

0.02

80.

009

2.6

0.00

80.

011

0.01

40.

018

0.02

20.

025

0.02

90.

032

0.03

30.

033

0.03

20.

007

2.8

−0.

001

0.00

20.

006

0.01

00.

014

0.01

80.

022

0.02

70.

029

0.02

90.

029

0.00

5

3.0

−0.

005

−0.

003

0.00

00.

005

0.00

80.

012

0.01

60.

020

0.02

20.

022

0.02

20.

004

Tab

le 4

.17

Con

’t

SAMPLE C

HAPTER

Analysis of Beams on Elastic Half-Space 223

β =

0.0

25

ζα

00.

10.

20.

30.

40.

50.

60.

70.

80.

91.

0∞

0.0

00

00

00

00

00

00.

176

0.1

0.08

50.

013

0.01

10.

007

0.00

70.

006

0.00

60.

005

0.00

40.

003

0.00

20.

130

0.2

−0.

142

−0.

050

0.04

10.

033

0.02

80.

022

0.01

80.

015

0.01

10.

007

0.00

50.

091

0.3

−0.

181

−0.

094

−0.

011

0.07

20.

059

0.04

80.

039

0.03

10.

022

0.01

70.

011

0.05

8

0.4

−0.

202

−0.

124

−0.

050

0.02

60.

104

0.08

50.

068

0.05

40.

041

0.03

00.

020

0.03

2

0.5

−0.

209

−0.

142

−0.

076

−0.

007

0.05

90.

131

0.10

50.

085

0.06

50.

046

0.03

10.

011

0.6

−0.

207

−0.

150

−0.

092

−0.

035

0.02

20.

085

0.15

20.

122

0.09

20.

070

0.04

8−

0.00

6

0.7

−0.

198

−0.

150

−0.

102

−0.

054

−0.

004

0.05

00.

107

0.16

80.

131

0.10

00.

070

−0.

016

0.8

−0.

185

−0.

144

−0.

105

−0.

067

−0.

024

0.02

00.

068

0.12

00.

176

0.13

70.

100

−0.

023

0.9

−0.

168

−0.

135

−0.

105

−0.

072

−0.

039

−0.

002

0.03

90.

081

0.13

00.

179

0.13

7−

0.02

7

1.0

−0.

150

−0.

124

−0.

100

−0.

074

−0.

048

−0.

018

0.01

50.

050

0.08

90.

135

0.18

1−

0.02

9

1.2

−0.

113

−0.

098

−0.

085

−0.

070

−0.

056

−0.

039

−0.

018

0.00

40.

030

0.05

90.

092

−0.

028

1.4

−0.

080

−0.

072

−0.

067

−0.

061

−0.

054

−0.

044

−0.

033

−0.

020

−0.

007

0.01

10.

031

−0.

024

1.6

−0.

050

−0.

054

−0.

048

−0.

048

−0.

046

−0.

043

−0.

039

−0.

033

−0.

028

−0.

018

−0.

007

−0.

020

1.8

−0.

028

−0.

030

−0.

033

−0.

035

−0.

035

−0.

037

−0.

037

−0.

035

−0.

034

−0.

031

−0.

026

−0.

016

2.0

−0.

013

−0.

017

−0.

020

−0.

024

−0.

026

−0.

031

−0.

031

−0.

032

−0.

035

−0.

035

−0.

035

−0.

012

2.2

−0.

004

−0.

007

−0.

011

−0.

015

−0.

018

−0.

024

−0.

024

−0.

026

−0.

030

−0.

033

−0.

035

−0.

010

2.4

0.00

20.

000

−0.

006

−0.

007

−0.

011

−0.

017

−0.

017

−0.

018

−0.

024

−0.

028

−0.

031

−0.

006

2.6

0.00

50.

002

−0.

002

−0.

003

−0.

005

−0.

009

−0.

009

−0.

013

−0.

017

−0.

020

−0.

024

−0.

006

2.8

0.00

60.

004

0.00

00.

000

−0.

002

−0.

004

−0.

004

−0.

006

−0.

011

−0.

015

−0.

018

−0.

005

3.0

0.00

60.

004

0.00

20.

001

0.00

0−

0.00

1−

0.00

2−

0.00

2−

0.00

5−

0.00

9−

0.01

3−

0.00

4

Q

0.0

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

−0.

500

0.1

−0.

708

0.25

20.

220

0.17

40.

137

0.10

60.

083

0.96

60.

050

0.03

80.

026

−0.

425

0.2

−0.

472

−0.

539

0.39

30.

330

0.26

60.

211

0.16

60.

128

0.09

10.

071

0.04

7−

0.35

5

0.3

−0.

284

−0.

367

−0.

452

0.46

70.

387

0.31

30.

250

0.19

40.

146

0.10

70.

072

−0.

292

0.4

−0.

138

−0.

229

−0.

322

−0.

413

0.49

70.

412

0.33

50.

264

0.20

20.

150

0.10

3−

0.23

4

0.5

−0.

026

−0.

120

−0.

215

−0.

309

−0.

402

0.50

60.

420

0.34

00.

266

0.20

20.

144

−0.

181

0.6

0.05

6−

0.03

6−

0.12

9−

0.22

1−

0.31

4−

0.40

70.

504

0.41

70.

386

0.26

30.

197

−0.

132

0.7

0.11

30.

027

−0.

061

−0.

149

−0.

237

−0.

328

−0.

416

0.49

50.

413

0.33

30.

261

−0.

090

0.8

0.15

30.

073

−0.

009

−0.

089

−0.

171

−0.

256

−0.

340

−0.

427

0.48

80.

407

0.33

0−

0.05

5

0.9

0.17

60.

103

0.03

0−

0.04

2−

0.11

6−

0.19

4−

0.27

2−

0.35

4−

0.43

80.

482

0.40

2−

0.02

8

1.0

0.18

70.

122

0.05

8−

0.00

6−

0.07

2−

0.14

1−

0.21

2−

0.28

8−

0.36

5−

0.44

20.

482

−0.

008

1.2

0.18

30.

135

0.08

80.

040

−0.

008

−0.

060

−0.

114

−0.

174

−0.

237

−0.

303

−0.

373

0.01

4

1.4

0.15

90.

126

0.09

30.

061

0.02

8−

0.00

9−

0.04

7−

0.09

0−

0.13

7−

0.18

8−

0.24

30.

021

1.6

0.14

60.

106

0.08

60.

065

0.04

80.

021

−0.

004

−0.

033

−0.

064

−0.

100

−0.

141

0.02

1

1.8

0.09

20.

091

0.07

00.

060

0.04

90.

036

0.02

2−

0.00

4−

0.01

6−

0.04

0−

0.06

70.

018

2.0

0.06

20.

058

0.05

30.

050

0.04

60.

040

0.03

40.

025

0.01

9−

0.00

1−

0.01

80.

014

2.2

0.03

90.

038

0.03

80.

039

0.03

90.

038

0.03

70.

035

0.02

90.

022

0.01

30.

011

2.4

0.02

00.

022

0.02

40.

027

0.03

00.

032

0.03

40.

036

0.03

50.

032

0.02

80.

009

2.6

0.00

80.

011

0.01

40.

018

0.02

20.

025

0.02

90.

032

0.03

30.

033

0.03

20.

007

2.8

−0.

001

0.00

20.

006

0.01

00.

014

0.01

80.

022

0.02

70.

029

0.02

90.

029

0.00

5

3.0

−0.

005

−0.

003

0.00

00.

005

0.00

80.

012

0.01

60.

020

0.02

20.

022

0.02

20.

004

SAMPLE C

HAPTER

224 Analysis of Structures on Elastic Foundations

β =

0.0

25

ζα

00.

10.

20.

30.

40.

50.

60.

70.

80.

91.

0∞

Y

06.

926.

005.

124.

283.

522.

842.

201.

721.

240.

880.

602.

14

0.1

6.00

5.32

4.64

3.96

3.32

2.76

2.28

1.80

1.40

1.08

0.80

2.11

0.2

5.12

4.64

4.16

3.68

3.20

2.88

2.34

1.92

1.56

1.24

1.00

2.02

0.3

4.28

3.96

3.68

3.44

3.08

2.72

2.40

2.08

1.76

1.48

1.2

1.90

0.4

3.52

3.32

3.20

3.08

2.88

2.64

2.43

2.16

1.88

1.64

1.40

1.76

0.5

2.84

2.76

2.88

2.72

2.64

2.56

2.44

2.24

2.04

1.80

1.60

1.60

0.6

2.20

2.28

2.34

2.40

2.43

2.44

2.36

2.24

2.12

1.96

1.78

1.44

0.7

1.72

1.80

1.92

2.08

2.16

2.24

2.24

2.24

2.16

2.04

1.92

1.28

0.8

1.24

1.40

1.56

1.76

1.88

2.04

2.12

2.16

2.20

2.12

2.04

1.13

0.9

0.88

1.08

1.24

1.48

1.64

1.80

1.96

2.04

2.12

2.16

2.12

0.99

1.0

0.60

0.80

1.00

1.20

1.40

1.60

1.76

1.92

2.04

2.12

2.16

0.85

1.2

0.16

0.36

0.56

0.80

1.00

1.20

1.40

1.56

1.76

1.52

2.04

0.62

1.4

−0.

080.

080.

280.

480.

680.

881.

041.

241.

401.

601.

760.

43

1.6

−0.

16−

0.04

0.12

0.24

0.44

0.60

0.76

0.92

1.08

1.24

1.40

0.28

1.8

−0.

20−

0.12

0.00

0.16

0.28

0.40

0.52

0.64

0.80

0.96

1.08

0.16

2.0

−0.

20−

0.12

−0.

040.

080.

160.

280.

360.

440.

560.

640.

800.

07

2.2

−0.

12−

0.12

−0.

040.

040.

120.

160.

240.

320.

400.

480.

560.

00

2.4

−0.

08−

0.08

−0.

040.

040.

080.

120.

160.

200.

280.

320.

36−

0.06

2.6

−0.

04−

0.04

0.00

0.04

0.04

0.08

0.12

0.16

0.16

0.20

0.24

−0.

11

2.8

0.00

0.00

0.04

0.04

0.04

0.08

0.08

0.12

0.12

0.12

0.16

−0.

15

3.0

0.04

0.04

0.04

0.04

0.04

0.04

0.04

0.08

0.08

0.08

0.08

−0.

18

Tab

le 4

.17

Con

’t

SAMPLE C

HAPTER

Analysis of Beams on Elastic Half-Space 225

β =

0.0

75

ζα

00.

10.

20.

30.

40.

50.

60.

70.

80.

91.

0∞

p

0.0

2.80

02.

481

2.13

11.

803

1.48

71.

238

1.01

50.

851

0.69

70.

547

0.40

50.

620

0.1

2.29

52.

064

1.81

41.

575

1.33

21.

125

0.92

70.

762

0.61

50.

469

0.34

10.

605

0.2

1.87

61.

716

1.54

91.

387

1.20

81.

041

0.87

30.

726

0.58

90.

641

0.36

10.

572

0.3

1.52

51.

421

1.31

51.

203

1.09

20.

970

0.84

40.

714

0.59

40.

485

0.38

80.

536

0.4

0.22

91.

170

1.11

11.

046

0.98

10.

902

0.81

30.

714

0.61

60.

521

0.43

20.

494

0.5

0.97

90.

945

0.93

00.

903

0.87

50.

832

0.77

90.

712

0.63

80.

561

0.48

40.

460

0.6

0.76

80.

769

0.77

10.

772

0.77

20.

761

0.73

90.

702

0.66

40.

596

0.53

40.

417

0.7

0.58

90.

610

0.63

20.

654

0.67

40.

687

0.68

90.

678

0.65

50.

623

0.58

10.

369

0.8

0.43

90.

474

0.51

00.

546

0.58

20.

612

0.63

50.

646

0.64

80.

637

0.61

40.

318

0.9

0.31

40.

359

0.40

40.

450

0.49

60.

539

0.57

70.

608

0.62

70.

633

0.62

70.

266

1.0

0.20

90.

261

0.31

30.

365

0.41

80.

469

0.51

60.

555

0.59

10.

616

0.62

90.

216

1.2

0.05

50.

112

0.17

00.

233

0.28

40.

341

0.39

70.

452

0.50

50.

553

0.59

20.

132

1.4

−0.

043

0.01

20.

368

0.12

30.

179

0.23

40.

292

0.34

80.

401

0.45

80.

510

0.07

1

1.6

−0.

099

−0.

050

0.00

00.

050

0.10

00.

151

0.20

40.

256

0.30

90.

362

0.41

50.

131

1.8

−0.

126

−0.

084

−0.

042

0.00

10.

043

0.08

80.

132

0.17

90.

225

0.27

20.

320

0.00

8

2.0

−0.

132

−0.

098

−0.

064

−0.

029

0.00

40.

040

0.07

70.

117

0.15

50.

195

0.23

70.

002

2.2

−0.

125

−0.

099

−0.

073

−0.

046

−0.

020

0.00

70.

036

0.06

70.

099

0.13

30.

167

−0.

009

2.4

−0.

111

−0.

092

−0.

072

−0.

061

−0.

034

−0.

013

0.00

60.

025

0.05

40.

081

0.10

9−

0.01

1

2.6

−0.

094

−0.

080

−0.

067

−0.

054

−0.

041

−0.

026

−0.

011

0.00

70.

021

0.03

80.

055

−0.

011

2.8

−0.

076

−0.

068

−0.

059

−0.

051

−0.

042

−0.

034

−0.

025

−0.

016

−0.

006

0.00

30.

014

−0.

010

3.0

−0.

058

−0.

054

−0.

049

−0.

046

−0.

042

−0.

040

−0.

038

−0.

035

−0.

029

−0.

021

−0.

011

−0.

008

Tab

le 4

.18

Dim

ensi

onle

ss c

oeffi

cien

ts f

or a

naly

sis

of lo

ng b

eam

s lo

aded

with

a c

once

ntra

ted

ver

tical

load

P

SAMPLE C

HAPTER

226 Analysis of Structures on Elastic Foundations

β =

0.0

75

ζα

00.

10.

20.

30.

40.

50.

60.

70.

80.

91.

0∞

M

0.0

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

214

0.1

−0.

087

−0.

011

0.00

90.

007

0.00

60.

006

0.00

60.

005

0.00

40.

003

0.00

20.

167

0.2

−0.

150

−0.

057

0.03

90.

033

0.02

80.

022

0.02

00.

018

0.01

50.

011

0.00

70.

127

0.3

−0.

195

−0.

106

−0.

019

0.07

00.

061

0.05

00.

043

0.03

70.

030

0.02

40.

017

0.09

1

0.4

−0.

224

−0.

152

−0.

061

0.02

20.

104

0.08

70.

074

0.06

30.

052

0.03

90.

030

0.06

0

0.5

−0.

242

−0.

169

−0.

093

0.01

70.

057

0.13

80.

113

0.09

60.

078

0.05

40.

046

0.03

6

0.6

−0.

248

−0.

182

−0.

115

−0.

046

0.02

00.

087

0.15

10.

137

0.11

30.

089

0.06

90.

016

0.7

−0.

246

−0.

189

−0.

130

−0.

070

−0.

009

0.05

00.

115

0.18

50.

154

0.12

20.

096

−0.

001

0.8

−0.

241

−0.

191

−0.

139

−0.

085

−0.

033

0.01

90.

078

0.13

90.

202

0.16

30.

130

−0.

013

0.9

−0.

230

−0.

185

−0.

141

−0.

096

−0.

050

−0.

004

0.04

40.

098

0.15

40.

209

0.17

1−

0.02

3

1.0

−0.

217

−0.

180

−0.

141

−0.

119

−0.

063

−0.

024

0.01

90.

065

0.11

30.

163

0.21

5−

0.02

9

1.2

−0.

183

−0.

158

−0.

132

−0.

138

−0.

076

−0.

050

−0.

019

0.01

50.

050

0.08

50.

128

−0.

036

1.4

−0.

148

−0.

135

−0.

109

−0.

096

−0.

080

−0.

061

−0.

041

−0.

017

0.00

70.

031

0.06

1−

0.03

8

1.6

−0.

113

−0.

104

−0.

094

−0.

083

−0.

074

−0.

063

−0.

050

−0.

037

−0.

020

−0.

004

0.01

3−

0.03

6

1.8

−0.

083

−0.

076

−0.

074

−0.

068

−0.

065

−0.

059

−0.

052

−0.

044

−0.

035

−0.

026

−0.

015

−0.

034

2.0

−0.

057

−0.

056

−0.

056

−0.

054

−0.

052

−0.

052

−0.

048

−0.

044

−0.

041

−0.

037

−0.

032

−0.

031

2.2

−0.

037

−0.

037

−0.

039

−0.

041

−0.

041

−0.

043

−0.

043

−0.

041

−0.

039

−0.

039

−0.

039

−0.

027

2.4

−0.

022

−0.

024

−0.

026

−0.

028

−0.

032

−0.

033

−0.

035

−0.

035

−0.

035

−0.

037

−0.

039

−0.

022

2.6

−0.

011

−0.

013

−0.

017

−0.

014

−0.

022

−0.

022

−0.

026

−0.

026

−0.

028

−0.

032

−0.

035

−0.

018

2.8

−0.

004

−0.

006

−0.

009

−0.

011

−0.

013

−0.

017

−0.

017

−0.

019

−0.

020

−0.

024

−0.

028

−0.

015

3.0

0.00

0−

0.00

2−

0.00

2−

0.00

6−

0.00

6−

0.00

7−

0.00

9−

0.01

1−

0.01

1−

0.01

5−

0.02

0−

0.01

1

0.0

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

0−

0.50

0

0.1

−0.

744

0.22

80.

196

0.16

80.

141

0.11

70.

097

0.07

90.

065

0.05

20.

039

−0.

439

0.2

−0.

535

−0.

584

0.36

40.

316

0.27

40.

288

0.18

70.

156

0.12

40.

103

0.07

5−

0.39

1

0.3

−0.

365

−0.

426

−0.

492

0.44

50.

383

0.32

60.

274

0.22

60.

184

0.14

60.

112

−0.

325

0.4

−0.

228

−0.

298

−0.

372

−0.

443

0.48

00.

419

0.35

60.

297

0.24

40.

196

0.14

7−

0.27

4

0.5

−0.

117

−0.

192

−0.

269

−0.

345

−0.

421

0.50

60.

435

0.38

70.

307

0.25

00.

197

−0.

225

0.6

0.03

1−

0.10

7−

0.18

4−

0.26

1−

0.33

8−

0.41

40.

511

0.43

90.

371

0.30

80.

248

−0.

181

0.7

0.03

7−

0.03

5−

0.11

5−

0.19

0−

0.26

6−

0.34

2−

0.41

80.

508

0.43

60.

369

0.30

5−

0.14

2

0.8

0.08

90.

017

−0.

057

−0.

130

−0.

203

−0.

277

−0.

351

−0.

425

0.50

30.

432

0.36

5−

0.10

6

0.9

0.12

50.

058

−0.

012

−0.

080

−0.

150

−0.

219

−0.

289

−0.

362

−0.

433

0.49

60.

426

−0.

079

1.0

0.15

20.

008

0.02

4−

0.04

0−

0.10

4−

0.16

9−

0.23

6−

0.30

4−

0.37

2−

0.44

10.

491

−0.

054

1.2

0.17

70.

125

0.07

10.

019

−0.

034

−0.

088

−0.

145

−0.

203

−0.

262

−0.

317

−0.

388

−0.

020

1.4

0.17

70.

131

0.09

50.

054

0.01

2−

0.03

2−

0.07

6−

0.12

4−

0.17

2−

0.22

4−

0.27

70.

004

1.6

0.16

30.

132

0.10

10.

071

0.03

90.

007

−0.

027

−0.

063

−0.

101

−0.

141

−0.

185

0.01

0

1.8

0.14

00.

118

0.09

60.

075

0.05

50.

031

0.00

6−

0.02

0−

0.04

7−

0.07

8−

0.11

10.

013

2.0

0.11

30.

100

0.08

50.

074

0.05

70.

043

0.02

70.

009

−0.

009

−0.

031

−0.

055

0.01

5

2.2

0.08

80.

079

0.07

20.

065

0.05

60.

048

0.03

90.

028

0.01

6−

0.00

2−

0.01

50.

013

2.4

0.06

40.

061

0.05

70.

054

0.05

00.

047

0.04

20.

037

0.03

10.

027

0.01

20.

011

2.6

0.04

40.

044

0.04

30.

044

0.04

30.

043

0.04

10.

040

0.03

80.

034

0.02

80.

009

2.8

0.02

70.

029

0.03

00.

033

0.03

50.

036

0.03

80.

038

0.03

90.

041

0.03

60.

007

3.0

0.01

30.

017

0.02

00.

023

0.02

60.

030

0.03

20.

033

0.03

60.

037

0.03

60.

004

Tab

le 4

.18

Con

’t

SAMPLE C

HAPTER

Analysis of Beams on Elastic Half-Space 227

β =

0.0

75

ζα

00.

10.

20.

30.

40.

50.

60.

70.

80.

91.

0∞

0.0

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

214

0.1

−0.

087

−0.

011

0.00

90.

007

0.00

60.

006

0.00

60.

005

0.00

40.

003

0.00

20.

167

0.2

−0.

150

−0.

057

0.03

90.

033

0.02

80.

022

0.02

00.

018

0.01

50.

011

0.00

70.

127

0.3

−0.

195

−0.

106

−0.

019

0.07

00.

061

0.05

00.

043

0.03

70.

030

0.02

40.

017

0.09

1

0.4

−0.

224

−0.

152

−0.

061

0.02

20.

104

0.08

70.

074

0.06

30.

052

0.03

90.

030

0.06

0

0.5

−0.

242

−0.

169

−0.

093

0.01

70.

057

0.13

80.

113

0.09

60.

078

0.05

40.

046

0.03

6

0.6

−0.

248

−0.

182

−0.

115

−0.

046

0.02

00.

087

0.15

10.

137

0.11

30.

089

0.06

90.

016

0.7

−0.

246

−0.

189

−0.

130

−0.

070

−0.

009

0.05

00.

115

0.18

50.

154

0.12

20.

096

−0.

001

0.8

−0.

241

−0.

191

−0.

139

−0.

085

−0.

033

0.01

90.

078

0.13

90.

202

0.16

30.

130

−0.

013

0.9

−0.

230

−0.

185

−0.

141

−0.

096

−0.

050

−0.

004

0.04

40.

098

0.15

40.

209

0.17

1−

0.02

3

1.0

−0.

217

−0.

180

−0.

141

−0.

119

−0.

063

−0.

024

0.01

90.

065

0.11

30.

163

0.21

5−

0.02

9

1.2

−0.

183

−0.

158

−0.

132

−0.

138

−0.

076

−0.

050

−0.

019

0.01

50.

050

0.08

50.

128

−0.

036

1.4

−0.

148

−0.

135

−0.

109

−0.

096

−0.

080

−0.

061

−0.

041

−0.

017

0.00

70.

031

0.06

1−

0.03

8

1.6

−0.

113

−0.

104

−0.

094

−0.

083

−0.

074

−0.

063

−0.

050

−0.

037

−0.

020

−0.

004

0.01

3−

0.03

6

1.8

−0.

083

−0.

076

−0.

074

−0.

068

−0.

065

−0.

059

−0.

052

−0.

044

−0.

035

−0.

026

−0.

015

−0.

034

2.0

−0.

057

−0.

056

−0.

056

−0.

054

−0.

052

−0.

052

−0.

048

−0.

044

−0.

041

−0.

037

−0.

032

−0.

031

2.2

−0.

037

−0.

037

−0.

039

−0.

041

−0.

041

−0.

043

−0.

043

−0.

041

−0.

039

−0.

039

−0.

039

−0.

027

2.4

−0.

022

−0.

024

−0.

026

−0.

028

−0.

032

−0.

033

−0.

035

−0.

035

−0.

035

−0.

037

−0.

039

−0.

022

2.6

−0.

011

−0.

013

−0.

017

−0.

014

−0.

022

−0.

022

−0.

026

−0.

026

−0.

028

−0.

032

−0.

035

−0.

018

2.8

−0.

004

−0.

006

−0.

009

−0.

011

−0.

013

−0.

017

−0.

017

−0.

019

−0.

020

−0.

024

−0.

028

−0.

015

3.0

0.00

0−

0.00

2−

0.00

2−

0.00

6−

0.00

6−

0.00

7−

0.00

9−

0.01

1−

0.01

1−

0.01

5−

0.02

0−

0.01

1

Q

0.0

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

0−

0.50

0

0.1

−0.

744

0.22

80.

196

0.16

80.

141

0.11

70.

097

0.07

90.

065

0.05

20.

039

−0.

439

0.2

−0.

535

−0.

584

0.36

40.

316

0.27

40.

288

0.18

70.

156

0.12

40.

103

0.07

5−

0.39

1

0.3

−0.

365

−0.

426

−0.

492

0.44

50.

383

0.32

60.

274

0.22

60.

184

0.14

60.

112

−0.

325

0.4

−0.

228

−0.

298

−0.

372

−0.

443

0.48

00.

419

0.35

60.

297

0.24

40.

196

0.14

7−

0.27

4

0.5

−0.

117

−0.

192

−0.

269

−0.

345

−0.

421

0.50

60.

435

0.38

70.

307

0.25

00.

197

−0.

225

0.6

0.03

1−

0.10

7−

0.18

4−

0.26

1−

0.33

8−

0.41

40.

511

0.43

90.

371

0.30

80.

248

−0.

181

0.7

0.03

7−

0.03

5−

0.11

5−

0.19

0−

0.26

6−

0.34

2−

0.41

80.

508

0.43

60.

369

0.30

5−

0.14

2

0.8

0.08

90.

017

−0.

057

−0.

130

−0.

203

−0.

277

−0.

351

−0.

425

0.50

30.

432

0.36

5−

0.10

6

0.9

0.12

50.

058

−0.

012

−0.

080

−0.

150

−0.

219

−0.

289

−0.

362

−0.

433

0.49

60.

426

−0.

079

1.0

0.15

20.

008

0.02

4−

0.04

0−

0.10

4−

0.16

9−

0.23

6−

0.30

4−

0.37

2−

0.44

10.

491

−0.

054

1.2

0.17

70.

125

0.07

10.

019

−0.

034

−0.

088

−0.

145

−0.

203

−0.

262

−0.

317

−0.

388

−0.

020

1.4

0.17

70.

131

0.09

50.

054

0.01

2−

0.03

2−

0.07

6−

0.12

4−

0.17

2−

0.22

4−

0.27

70.

004

1.6

0.16

30.

132

0.10

10.

071

0.03

90.

007

−0.

027

−0.

063

−0.

101

−0.

141

−0.

185

0.01

0

1.8

0.14

00.

118

0.09

60.

075

0.05

50.

031

0.00

6−

0.02

0−

0.04

7−

0.07

8−

0.11

10.

013

2.0

0.11

30.

100

0.08

50.

074

0.05

70.

043

0.02

70.

009

−0.

009

−0.

031

−0.

055

0.01

5

2.2

0.08

80.

079

0.07

20.

065

0.05

60.

048

0.03

90.

028

0.01

6−

0.00

2−

0.01

50.

013

2.4

0.06

40.

061

0.05

70.

054

0.05

00.

047

0.04

20.

037

0.03

10.

027

0.01

20.

011

2.6

0.04

40.

044

0.04

30.

044

0.04

30.

043

0.04

10.

040

0.03

80.

034

0.02

80.

009

2.8

0.02

70.

029

0.03

00.

033

0.03

50.

036

0.03

80.

038

0.03

90.

041

0.03

60.

007

3.0

0.01

30.

017

0.02

00.

023

0.02

60.

030

0.03

20.

033

0.03

60.

037

0.03

60.

004

SAMPLE C

HAPTER

228 Analysis of Structures on Elastic Foundations

β =

0.0

75

ζα

00.

10.

20.

30.

40.

50.

60.

70.

80.

91.

0∞

Y

04.

313.

843.

402.

972.

572.

21.

851.

551.

271.

040.

851.

37

0.1

3.84

3.47

3.11

2.79

2.44

2.12

1.83

1.54

1.29

1.09

0.93

1.35

0.2

3.40

3.11

2.85

2.59

2.32

2.05

1.80

1.56

1.35

1.16

1.00

1.32

0.3

2.97

2.79

2.59

2.40

2.20

1.99

1.78

1.57

1.40

1.23

1.09

1.27

0.4

2.57

2.44

2.32

2.20

2.07

1.92

1.76

1.59

1.44

1.29

1.16

1.21

0.5

2.20

2.12

2.05

1.99

1.92

1.81

1.71

1.59

1.47

1.35

1.23

1.13

0.6

1.85

1.83

1.80

1.78

1.76

1.71

1.65

1.56

1.49

1.39

1.29

1.05

0.7

1.55

1.54

1.56

1.57

1.59

1.59

1.56

1.52

1.48

1.43

1.35

0.97

0.8

1.27

1.29

1.35

1.40

1.44

1.47

1.49

1.48

1.48

1.45

1.41

0.89

0.9

1.04

1.09

1.16

1.23

1.29

1.35

1.39

1.43

1.45

1.44

1.42

0.80

1.0

0.85

0.93

1.00

1.09

1.16

1.23

1.29

1.35

1.41

1.42

1.44

0.73

1.2

0.49

0.59

0.68

0.80

0.89

0.97

1.08

1.15

1.25

1.31

1.32

0.60

1.4

0.27

0.36

0.45

0.57

0.67

0.76

0.87

0.96

1.07

1.15

1.24

0.48

1.6

0.09

0.20

0.29

0.40

0.49

0.57

0.69

0.77

0.88

0.96

1.08

0.35

1.8

0.00

0.09

0.17

0.27

0.35

0.44

0.53

0.61

0.72

0.80

0.89

0.26

2.0

−0.

050.

030.

090.

170.

240.

320.

400.

470.

570.

640.

730.

17

2.2

−0.

07−

0.01

0.04

0.11

0.17

0.23

0.29

0.36

0.44

0.51

0.57

0.10

2.4

−0.

07−

0.03

0.01

0.07

0.12

0.16

0.21

0.27

0.33

0.39

0.44

0.04

2.6

−0.

06−

0.03

0.00

0.05

0.08

0.11

0.15

0.19

0.24

0.38

0.38

0.00

2.8

−0.

05−

0.01

0.00

0.04

0.05

0.08

0.10

0.13

0.17

0.20

0.23

−0.

03

3.0

−0.

030.

000.

000.

030.

040.

050.

070.

080.

110.

130.

15−

0.07

Tab

le 4

.18

Con

’t

SAMPLE C

HAPTER

Analysis of Beams on Elastic Half-Space 229

β =

0.1

5

ζα

00.

10.

20.

30.

40.

50.

60.

70.

80.

91.

0∞

p

0.0

2.73

22.

435

2.12

81.

841

1.55

21.

316

1.10

70.

946

0.79

40.

640

0.50

70.

555

0.1

2.20

01.

981

1.77

31.

562

1.35

11.

159

0.97

50.

827

0.68

70.

557

0.44

60.

548

0.2

1.78

11.

635

1.48

81.

339

1.19

01.

043

0.89

80.

763

0.63

90.

523

0.42

50.

529

0.3

1.44

41.

349

1.25

81.

154

1.05

50.

949

0.83

80.

727

0.61

90.

520

0.43

20.

503

0.4

1.17

31.

115

1.05

70.

998

0.93

80.

868

0.79

00.

706

0.61

90.

535

0.45

60.

476

0.5

0.94

80.

948

0.89

00.

861

0.83

10.

793

0.74

50.

687

0.62

40.

557

0.49

00.

452

0.6

0.76

10.

753

0.74

60.

741

0.73

30.

720

0.69

80.

666

0.62

40.

575

0.52

20.

426

0.7

0.60

40.

613

0.62

30.

634

0.64

40.

649

0.64

70.

636

0.61

50.

586

0.55

00.

393

0.8

0.47

10.

493

0.51

50.

538

0.56

10.

581

0.59

40.

603

0.59

70.

590

0.57

00.

352

0.9

0.35

70.

389

0.41

90.

453

0.48

40.

516

0.54

30.

565

0.57

80.

582

0.57

60.

302

1.0

0.26

40.

302

0.33

90.

380

0.41

60.

454

0.48

60.

520

0.54

50.

564

0.57

30.

252

1.2

0.11

80.

163

0.20

80.

252

0.29

70.

341

0.38

50.

428

0.46

90.

508

0.53

90.

155

1.4

0.01

90.

065

0.11

10.

157

0.20

30.

249

0.29

90.

340

0.38

50.

433

0.47

10.

081

1.6

−0.

046

−0.

003

0.04

10.

085

0.12

90.

173

0.21

70.

261

0.30

50.

330

0.39

40.

035

1.8

−0.

083

−0.

045

−0.

007

0.03

40.

072

0112

0.15

30.

194

0.23

40.

278

0.31

60.

009

2.0

−0.

102

−0.

069

−0.

037

−0.

002

0.03

10.

065

0.10

10.

137

0.17

30.

210

0.24

6−

0.00

3

2.2

−0.

109

−0.

081

−0.

054

−0.

026

0.00

20.

030

0.05

90.

090

0.12

10.

153

0.18

6−

0.00

9

2.4

−0.

105

−0.

083

−0.

061

−0.

039

−0.

016

0.00

50.

028

0.05

10.

077

0.10

50.

132

−0.

011

2.6

−0.

097

−0.

083

−0.

064

−0.

047

−0.

030

−0.

013

0.00

40.

024

0.04

20.

062

0.08

0−

0.01

1

2.8

−0.

087

−0.

081

−0.

063

−0.

050

−0.

039

−0.

027

−0.

014

−0.

001

0.01

20.

024

0.03

6−

0.01

0

3.0

−0.

076

−0.

075

−0.

061

−0.

053

−0.

046

−0.

040

−0.

032

−0.

025

−0.

017

−0.

007

−0.

003

−0.

008

Tab

le 4

.19

Dim

ensi

onle

ss c

oeffi

cien

ts f

or a

naly

sis

of lo

ng b

eam

s lo

aded

with

a c

once

ntra

ted

ver

tical

load

P

SAMPLE C

HAPTER

230 Analysis of Structures on Elastic Foundations

β =

0.1

5

ζα

00.

10.

20.

30.

40.

50.

60.

70.

80.

91.

0∞

M

0.0

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

230

0.1

−0.

087

0.01

20.

009

0.00

90.

008

0.00

50.

005

0.00

40.

003

0.00

30.

003

0.18

3

0.2

−0.

152

−0.

056

0.03

70.

033

0.02

90.

024

0.02

00.

017

0.01

40.

012

0.01

00.

141

0.3

−0.

200

−0.

110

−0.

021

0.07

00.

061

0.05

20.

044

0.03

70.

031

0.02

60.

020

0.10

5

0.4

−0.

232

−0.

148

−0.

065

0.02

00.

105

0.09

00.

077

0.06

60.

055

0.04

60.

036

0.07

2

0.5

−0.

253

−0.

176

−0.

099

−0.

021

0.05

80.

137

0.11

60.

100

0.08

50.

070

0.05

60.

046

0.6

−0.

265

−0.

194

−0.

124

−0.

053

0.01

90.

091

0.16

50.

139

0.12

00.

100

0.08

10.

024

0.7

−0.

269

−0.

205

−0.

142

−0.

077

−0.

012

−0.

053

0.12

00.

191

0.16

30.

137

0.11

00.

006

0.8

−0.

267

−0.

210

−0.

153

−0.

096

−0.

037

0.02

10.

081

0.14

50.

210

0.17

80.

147

−0.

006

0.9

−0.

259

−0.

209

−0.

158

−0.

108

−0.

057

−0.

005

0.04

90.

106

0.16

40.

225

0.19

0−

0.01

8

1.0

−0.

249

−0.

205

−0.

161

−0.

116

−0.

071

−0.

026

0.02

20.

124

0.12

40.

179

0.23

8−

0.02

5

1.2

−0.

221

−0.

188

−0.

156

−0.

122

−0.

088

−0.

054

−0.

019

0.02

00.

060

0.10

30.

147

−0.

033

1.4

−0.

188

−0.

165

−0.

142

−0.

118

−0.

093

−0.

069

−0.

044

−0.

015

0.01

40.

047

0.08

0−

0.03

5

1.6

−0.

154

−0.

139

−0.

124

−0.

107

−0.

091

−0.

074

−0.

056

−0.

036

−0.

016

0.00

70.

031

−0.

033

1.8

−0.

122

−0.

133

−0.

103

−0.

093

−0.

083

−0.

072

−0.

061

−0.

047

−0.

034

−0.

018

−0.

002

−0.

030

2.0

−0.

093

−0.

088

−0.

083

−0.

077

−0.

072

−0.

066

−0.

059

−0.

051

−0.

043

−0.

033

−0.

022

−0.

026

2.2

−0.

068

−0.

066

−0.

065

−0.

062

−0.

059

−0.

057

−0.

053

−0.

049

−0.

044

−0.

039

−0.

032

−0.

022

2.4

−0.

047

−0.

048

−0.

048

−0.

047

−0.

047

−0.

046

−0.

045

−0.

042

−0.

041

−0.

038

−0.

035

−0.

018

2.6

−0.

031

−0.

033

−0.

034

−0.

034

−0.

035

−0.

036

−0.

035

−0.

031

−0.

034

−0.

033

−0.

032

−0.

015

2.8

−0.

013

−0.

021

−0.

023

−0.

023

−0.

024

−0.

025

−0.

026

−0.

026

−0.

026

−0.

026

−0.

026

−0.

013

3.0

−0.

010

−0.

011

−0.

014

−0.

014

−0.

015

−0.

016

−0.

017

−0.

017

−0.

017

−0.

019

−0.

019

−0.

010

0.0

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

0−

0.50

0

0.1

−0.

755

0.21

90.

196

0.16

80.

144

0.12

40.

104

0.08

90.

074

0.06

20.

046

−0.

445

0.2

−0.

556

−0.

600

0.35

80.

313

0.27

10.

234

0.19

80.

168

0.14

00.

116

0.09

0−

0.39

1

0.3

−0.

395

−0.

454

−0.

505

0.43

90.

384

0.33

40.

285

0.24

20.

203

0.16

70.

133

−0.

339

0.4

−0.

265

−0.

328

−0.

390

−0.

454

0.48

30.

424

0.36

60.

313

0.26

40.

219

0.17

7−

0.29

0

0.5

−0.

159

−0.

226

−0.

293

−0.

293

−0.

428

0.50

70.

443

0.38

20.

326

0.27

30.

224

−0.

244

0.6

−0.

075

−0.

143

−0.

211

−0.

281

−0.

350

−0.

417

0.51

50.

450

0.38

80.

329

0.27

5−

0.20

0

0.7

−0.

007

−0.

075

−0.

143

−0.

213

−0.

280

−0.

349

−0.

418

0.51

50.

451

0.38

80.

329

−0.

159

0.8

0.04

7−

0.02

0−

0.08

6−

0.15

4−

0.22

1−

0.28

7−

0.35

6−

0.42

30.

512

0.47

70.

385

−0.

123

0.9

0.08

80.

024

−0.

040

−0.

104

−0.

169

−0.

232

−0.

299

−0.

364

−0.

429

0.50

60.

442

−0.

090

1.0

0.11

90.

059

−0.

002

−0.

063

−0.

123

−0.

184

−0.

248

−0.

311

−0.

373

−0.

436

0.50

1−

0.06

0

1.2

0.15

70.

105

0.05

20.

000

−0.

053

−0.

105

−0.

160

−0.

215

−0.

272

−0.

330

−0.

389

−0.

021

1.4

0.17

00.

126

0.08

30.

040

−0.

003

−0.

047

−0.

093

−0.

139

−0.

187

−0.

236

−0.

287

0.00

3

1.6

0.16

70.

132

0.09

90.

064

0.03

0−

0.00

5−

0.04

2−

0.07

9−

0.11

8−

0.15

8−

0.20

10.

017

1.8

0.15

30.

128

0.10

20.

075

0.05

00.

023

−0.

005

−0.

034

−0.

064

−0.

096

−0.

130

0.02

0

2.0

0.13

40.

116

0.09

70.

078

0.06

00.

041

0.02

1−

0.00

1−

0.02

3−

0.04

7−

0.07

40.

019

2.2

0.11

40.

102

0.08

80.

076

0.06

30.

061

0.03

70.

022

0.01

6−

0.01

1−

0.03

00.

018

2.4

0.09

20.

084

0.07

60.

069

0.06

10.

054

0.04

50.

036

0.02

60.

014

0.00

10.

015

2.6

0.07

20.

068

0.06

40.

060

0.05

60.

053

0.04

80.

043

0.03

70.

030

0.02

20.

012

2.8

0.05

30.

052

0.05

20.

050

0.04

90.

049

0.04

70.

045

0.04

30.

039

0.03

40.

910

3.0

0.03

70.

037

0.03

90.

040

0.04

10.

042

0.04

30.

043

0.04

20.

040

0.03

80.

009

Tab

le 4

.19

Con

’t

SAMPLE C

HAPTER

Analysis of Beams on Elastic Half-Space 231

β =

0.1

5

ζα

00.

10.

20.

30.

40.

50.

60.

70.

80.

91.

0∞

0.0

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

230

0.1

−0.

087

0.01

20.

009

0.00

90.

008

0.00

50.

005

0.00

40.

003

0.00

30.

003

0.18

3

0.2

−0.

152

−0.

056

0.03

70.

033

0.02

90.

024

0.02

00.

017

0.01

40.

012

0.01

00.

141

0.3

−0.

200

−0.

110

−0.

021

0.07

00.

061

0.05

20.

044

0.03

70.

031

0.02

60.

020

0.10

5

0.4

−0.

232

−0.

148

−0.

065

0.02

00.

105

0.09

00.

077

0.06

60.

055

0.04

60.

036

0.07

2

0.5

−0.

253

−0.

176

−0.

099

−0.

021

0.05

80.

137

0.11

60.

100

0.08

50.

070

0.05

60.

046

0.6

−0.

265

−0.

194

−0.

124

−0.

053

0.01

90.

091

0.16

50.

139

0.12

00.

100

0.08

10.

024

0.7

−0.

269

−0.

205

−0.

142

−0.

077

−0.

012

−0.

053

0.12

00.

191

0.16

30.

137

0.11

00.

006

0.8

−0.

267

−0.

210

−0.

153

−0.

096

−0.

037

0.02

10.

081

0.14

50.

210

0.17

80.

147

−0.

006

0.9

−0.

259

−0.

209

−0.

158

−0.

108

−0.

057

−0.

005

0.04

90.

106

0.16

40.

225

0.19

0−

0.01

8

1.0

−0.

249

−0.

205

−0.

161

−0.

116

−0.

071

−0.

026

0.02

20.

124

0.12

40.

179

0.23

8−

0.02

5

1.2

−0.

221

−0.

188

−0.

156

−0.

122

−0.

088

−0.

054

−0.

019

0.02

00.

060

0.10

30.

147

−0.

033

1.4

−0.

188

−0.

165

−0.

142

−0.

118

−0.

093

−0.

069

−0.

044

−0.

015

0.01

40.

047

0.08

0−

0.03

5

1.6

−0.

154

−0.

139

−0.

124

−0.

107

−0.

091

−0.

074

−0.

056

−0.

036

−0.

016

0.00

70.

031

−0.

033

1.8

−0.

122

−0.

133

−0.

103

−0.

093

−0.

083

−0.

072

−0.

061

−0.

047

−0.

034

−0.

018

−0.

002

−0.

030

2.0

−0.

093

−0.

088

−0.

083

−0.

077

−0.

072

−0.

066

−0.

059

−0.

051

−0.

043

−0.

033

−0.

022

−0.

026

2.2

−0.

068

−0.

066

−0.

065

−0.

062

−0.

059

−0.

057

−0.

053

−0.

049

−0.

044

−0.

039

−0.

032

−0.

022

2.4

−0.

047

−0.

048

−0.

048

−0.

047

−0.

047

−0.

046

−0.

045

−0.

042

−0.

041

−0.

038

−0.

035

−0.

018

2.6

−0.

031

−0.

033

−0.

034

−0.

034

−0.

035

−0.

036

−0.

035

−0.

031

−0.

034

−0.

033

−0.

032

−0.

015

2.8

−0.

013

−0.

021

−0.

023

−0.

023

−0.

024

−0.

025

−0.

026

−0.

026

−0.

026

−0.

026

−0.

026

−0.

013

3.0

−0.

010

−0.

011

−0.

014

−0.

014

−0.

015

−0.

016

−0.

017

−0.

017

−0.

017

−0.

019

−0.

019

−0.

010

Q

0.0

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

0−

0.50

0

0.1

−0.

755

0.21

90.

196

0.16

80.

144

0.12

40.

104

0.08

90.

074

0.06

20.

046

−0.

445

0.2

−0.

556

−0.

600

0.35

80.

313

0.27

10.

234

0.19

80.

168

0.14

00.

116

0.09

0−

0.39

1

0.3

−0.

395

−0.

454

−0.

505

0.43

90.

384

0.33

40.

285

0.24

20.

203

0.16

70.

133

−0.

339

0.4

−0.

265

−0.

328

−0.

390

−0.

454

0.48

30.

424

0.36

60.

313

0.26

40.

219

0.17

7−

0.29

0

0.5

−0.

159

−0.

226

−0.

293

−0.

293

−0.

428

0.50

70.

443

0.38

20.

326

0.27

30.

224

−0.

244

0.6

−0.

075

−0.

143

−0.

211

−0.

281

−0.

350

−0.

417

0.51

50.

450

0.38

80.

329

0.27

5−

0.20

0

0.7

−0.

007

−0.

075

−0.

143

−0.

213

−0.

280

−0.

349

−0.

418

0.51

50.

451

0.38

80.

329

−0.

159

0.8

0.04

7−

0.02

0−

0.08

6−

0.15

4−

0.22

1−

0.28

7−

0.35

6−

0.42

30.

512

0.47

70.

385

−0.

123

0.9

0.08

80.

024

−0.

040

−0.

104

−0.

169

−0.

232

−0.

299

−0.

364

−0.

429

0.50

60.

442

−0.

090

1.0

0.11

90.

059

−0.

002

−0.

063

−0.

123

−0.

184

−0.

248

−0.

311

−0.

373

−0.

436

0.50

1−

0.06

0

1.2

0.15

70.

105

0.05

20.

000

−0.

053

−0.

105

−0.

160

−0.

215

−0.

272

−0.

330

−0.

389

−0.

021

1.4

0.17

00.

126

0.08

30.

040

−0.

003

−0.

047

−0.

093

−0.

139

−0.

187

−0.

236

−0.

287

0.00

3

1.6

0.16

70.

132

0.09

90.

064

0.03

0−

0.00

5−

0.04

2−

0.07

9−

0.11

8−

0.15

8−

0.20

10.

017

1.8

0.15

30.

128

0.10

20.

075

0.05

00.

023

−0.

005

−0.

034

−0.

064

−0.

096

−0.

130

0.02

0

2.0

0.13

40.

116

0.09

70.

078

0.06

00.

041

0.02

1−

0.00

1−

0.02

3−

0.04

7−

0.07

40.

019

2.2

0.11

40.

102

0.08

80.

076

0.06

30.

061

0.03

70.

022

0.01

6−

0.01

1−

0.03

00.

018

2.4

0.09

20.

084

0.07

60.

069

0.06

10.

054

0.04

50.

036

0.02

60.

014

0.00

10.

015

2.6

0.07

20.

068

0.06

40.

060

0.05

60.

053

0.04

80.

043

0.03

70.

030

0.02

20.

012

2.8

0.05

30.

052

0.05

20.

050

0.04

90.

049

0.04

70.

045

0.04

30.

039

0.03

40.

910

3.0

0.03

70.

037

0.03

90.

040

0.04

10.

042

0.04

30.

043

0.04

20.

040

0.03

80.

009

Y

SAMPLE C

HAPTER

232 Analysis of Structures on Elastic Foundations

β =

0.1

5

ζα

00.

10.

20.

30.

40.

50.

60.

70.

80.

91.

0∞

03.

042.

752.

482.

211.

961.

721.

501.

301.

110.

950.

791.

07

0.1

2.75

2.53

2.31

2.09

1.87

1.67

1.47

1.3

1.13

0.98

0.84

1.06

0.2

2.48

2.31

2.12

1.95

1.77

1.60

1.44

1.29

1.14

1.00

0.88

1.04

0.3

2.21

2.09

1.95

1.81

1.67

1.51

1.41

1.27

1.15

1.03

0.92

1.01

0.4

1.96

1.87

1.77

1.67

1.59

1.49

1.38

1.27

1.17

1.06

0.97

0.98

0.5

1.72

1.67

1.60

1.51

1.49

1.41

1.33

1.25

1.17

1.08

1.00

0.94

0.6

1.50

1.47

1.44

1.41

1.38

1.33

1.29

1.24

1.17

1.11

1.06

0.85

0.7

1.30

1.30

1.29

1.27

1.27

1.25

1.24

1.21

1.17

1.11

1.07

0.80

0.8

1.11

1.13

1.14

1.15

1.17

1.17

1.17

1.17

1.14

1.11

1.07

0.80

0.9

0.95

0.98

1.00

1.03

1.06

1.08

1.11

1.11

1.11

1.11

1.09

0.76

1.0

0.79

0.84

0.88

0.92

0.97

1.00

1.03

1.06

1.07

1.09

1.09

0.65

1.2

0.55

0.61

0.66

0.72

0.78

0.83

0.89

0.93

0.97

1.01

1.05

0.55

1.4

0.36

0.43

0.49

0.55

0.62

0.68

0.75

0.81

0.86

0.91

0.97

0.46

1.6

0.22

0.29

0.35

0.41

0.49

0.55

0.62

0.68

0.74

0.80

0.86

0.36

1.8

0.12

0.18

0.24

0.31

0.37

0.43

0.50

0.56

0.62

0.68

0.75

0.28

2.0

0.05

0.11

0.17

0.22

0.28

0.34

0.40

0.46

0.51

0.57

0.63

0.15

2.2

0.01

0.06

0.11

0.16

0.21

0.26

0.32

0.37

0.42

0.47

0.53

0.15

2.4

−0.

010.

030.

070.

110.

160.

200.

250.

290.

330.

370.

430.

10

2.6

−0.

020.

010.

050.

080.

110.

140.

190.

230.

260.

290.

340.

05

2.8

−0.

020.

000.

030.

050.

090.

110.

140.

170.

190.

220.

260.

02

3.0

−0.

01−

0.01

0.02

0.04

0.06

0.08

0.11

0.12

0.13

0.15

0.18

−0.

01

Tab

le 4

.19

Con

’t

SAMPLE C

HAPTER

Analysis of Beams on Elastic Half-Space 233

β =

0.3

ζα

00.

10.

20.

30.

40.

50.

60.

70.

80.

91.

0∞

p

0.0

2.25

1.81

1.34

0.95

0.54

0.34

0.24

0.28

0.31

0.28

0.13

0.49

0.2

1.65

1.55

1.14

0.89

0.64

0.44

0.29

0.20

0.13

0.07

0.03

0.47

0.4

1.18

1.06

0.94

0.80

0.67

0.52

0.37

0.23

0.11

0.03

0.02

0.43

0.6

0.82

0.78

0.75

0.70

0.65

0.56

0.45

0.30

0.18

0.09

0.02

0.36

0.8

0.54

0.56

0.58

0.59

0.60

0.57

0.50

0.39

0.28

0.19

0.11

0.30

1.0

0.34

0.37

0.44

0.49

0.53

0.54

0.52

0.45

0.38

0.30

0.22

0.23

1.2

0.19

0.25

0.32

0.39

0.47

0.49

0.51

0.49

0.45

0.39

0.33

0.17

1.4

0.08

0.15

0.22

0.30

0.37

0.43

0.48

0.50

0.49

0.46

0.42

0.13

1.6

0.00

0.07

0.14

0.22

0.29

0.36

0.43

0.47

0.50

0.50

0.48

0.09

1.8

−0.

040.

020.

080.

150.

220.

300.

370.

430.

490.

500.

510.

06

2.0

−0.

08−

0.02

0.04

0.10

0.16

0.23

0.30

0.37

0.43

0.48

0.51

0.04

2.2

−0.

09−

0.03

0.01

0.06

0.12

0.18

0.24

0.31

0.38

0.44

0.48

0.02

2.4

−0.

09−

0.05

−0.

020.

030.

080.

130.

180.

250.

310.

380.

440.

01

2.6

−0.

09−

0.06

−0.

030.

010.

050.

090.

140.

190.

250.

320.

380.

00

2.8

−0.

09−

0.06

−0.

030.

000.

030.

060.

100.

140.

200.

250.

320.

00

3.0

−0.

08−

.005

−0.

04−

0.01

0.01

0.04

0.07

0.10

0.15

0.20

0.25

0.00

3.2

−0.

07−

0.05

−0.

04−

0.02

0.00

0.02

0.04

0.07

0.11

0.15

0.19

−0.

01

3.4

−0.

06−

0.05

−0.

03−

0.02

−0.

010.

010.

030.

050.

080.

100.

14−

0.01

3.6

−0.

05−

0.04

−0.

03−

0.02

−0.

010.

000.

010.

030.

050.

070.

10−

0.01

3.8

−0.

04−

0.03

−0.

03−

0.02

−0.

02−

0.01

0.00

0.02

0.03

0.05

0.07

−0.

01

4.0

−0.

03−

0.03

−0.

03−

0.02

−0.

02−

0.01

0.00

0.01

0.02

0.03

0.05

−0.

01

Tab

le 4

.20

Dim

ensi

onle

ss c

oeffi

cien

ts f

or a

naly

sis

of lo

ng b

eam

s lo

aded

with

a c

once

ntra

ted

ver

tical

load

P

SAMPLE C

HAPTER

234 Analysis of Structures on Elastic Foundations

β =

0.3

ζα

00.

10.

20.

30.

40.

50.

60.

70.

80.

91.

0∞

M

0.0

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

282

0.2

−0.

159

0.03

40.

026

0.01

90.

012

0.00

80.

005

0.00

50.

005

0.00

40.

003

0.19

2

0.4

−0.

252

−0.

077

0.09

60.

073

0.04

90.

028

0.02

20.

019

0.01

60.

012

0.00

70.

120

0.6

−0.

296

−0.

145

0.00

50.

159

0.11

20.

079

0.05

30.

041

0.03

10.

021

0.01

20.

066

0.8

−0.

310

−0.

182

−0.

057

0.07

30.

202

0.14

60.

103

0.07

50.

053

0.03

30.

017

0.02

6

1.0

−0.

300

−0.

196

−0.

095

0.01

10.

115

0.23

70.

172

0.12

50.

087

0.05

40.

026

−0.

002

1.2

−0.

280

−0.

194

−0.

115

−0.

032

0.05

00.

149

0.26

20.

193

0.13

60.

086

0.04

4−

0.02

2

1.4

−0.

246

−0.

183

−0.

123

−0.

067

0.00

30.

081

0.17

20.

281

0.20

20.

133

0.07

7−

0.03

3

1.6

−0.

212

−0.

166

−0.

120

−0.

075

−0.

036

0.03

00.

101

0.18

90.

289

0.20

00.

124

−0.

039

1.8

−0.

178

−0.

145

−0.

114

−0.

081

−0.

049

−0.

006

0.04

70.

115

0.19

50.

288

0.19

1−

0.04

1

2.0

−0.

146

−0.

124

−0.

110

−0.

82−

0.06

0−

0.03

10.

010

0.05

80.

120

0.19

10.

277

−0.

041

2.2

−0.

117

−0.

104

−0.

091

−0.

078

−0.

065

−0.

045

−0.

020

0.01

80.

062

0.11

60.

189

−0.

039

2.4

−0.

090

−0.

084

−0.

078

−0.

072

−0.

065

−0.

054

−0.

037

−0.

012

0.02

00.

059

0.11

0−

0.03

7

2.6

−0.

068

−0.

067

−0.

065

−0.

064

−0.

062

−0.

058

−0.

047

−0.

031

−0.

010

0.01

70.

055

−0.

033

2.8

−0.

050

−0.

051

−0.

054

−0.

055

−0.

057

−0.

056

−0.

052

−0.

043

−0.

031

−0.

013

0.01

3−

0.03

0

3.0

−0.

035

−0.

039

−0.

043

−0.

047

−0.

051

−0.

053

−0.

053

−0.

049

−0.

043

−0.

032

−0.

015

−0.

027

3.2

−0.

023

−0.

027

−0.

033

−0.

039

−0.

044

−0.

048

−0.

050

−0.

051

−0.

049

−0.

044

−0.

034

−0.

023

3.4

−0.

013

−0.

019

−0.

025

−0.

031

−0.

037

−0.

042

−0.

047

−0.

050

−0.

050

−0.

049

−0.

044

−0.

020

3.6

−0.

007

−0.

013

−0.

019

−0.

025

−0.

031

−0.

036

−0.

041

−0.

046

−0.

049

−0.

050

−0.

049

−0.

018

3.8

−0.

002

−0.

008

−0.

013

−0.

019

−0.

025

−0.

030

−0.

036

−0.

041

−0.

045

−0.

049

−0.

050

−0.

015

4.0

0.00

1−

0.00

4−

0.00

9−

0.01

5−

0.02

0−

0.02

5−

0.03

0−

0.03

5−

0.04

0−

0.04

5−

0.04

8−

0.01

3

0.0

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

0−

0.50

0

0.2

−0.

612

0.32

10.

246

0.18

40.

119

0.07

80.

051

0.04

60.

040

0.03

20.

020

−0.

403

0.4

−0.

331

−0.

435

0.45

40.

353

0.22

40.

174

0.11

80.

099

0.06

20.

041

0.02

1−

0.31

3

0.6

−0.

133

−0.

253

−0.

377

0.50

30.

384

0.25

60.

200

0.14

80.

091

0.05

20.

020

−0.

234

0.8

−0.

001

−0.

120

−0.

244

−0.

368

0.51

00.

397

0.29

60.

210

0.13

70.

079

0.03

2−

0.16

8

1.0

0.08

8−

0.02

6−

0.14

2−

0.26

0−

0.37

60.

508

0.39

70.

294

0.20

30.

128

0.06

5−

0.11

5

1.2

0.13

90.

036

−0.

066

−0.

173

−0.

277

−0.

388

0.50

00.

390

0.28

30.

197

0.12

1−

0.07

4

1.4

0.16

50.

076

−0.

013

−0.

105

−0.

195

−0.

295

−0.

400

0.48

90.

381

0.28

40.

195

−0.

043

1.6

0.17

30.

097

0.02

3−

0.05

3−

0.12

9−

0.21

5−

0.31

0−

0.41

10.

481

0.38

00.

285

−0.

025

1.8

0.16

20.

105

0.04

6−

0.01

6−

0.07

8−

0.14

9−

0.23

1−

0.32

3−

0.42

10.

481

0.38

40.

006

2.0

0.15

60.

106

0.05

70.

009

−0.

039

−0.

096

−0.

164

−0.

243

−0.

330

−0.

421

0.38

60.

006

2.2

0.14

00.

101

0.06

40.

026

−0.

011

−0.

056

−0.

110

−0.

174

−0.

248

−0.

329

−0.

417

0.01

2

2.4

0.12

00.

092

0.06

40.

037

0.00

9−

0.02

5−

0.06

7−

0.11

8−

0.18

1−

0.24

6−

0.32

30.

016

2.6

0.10

10.

082

0.06

20.

040

0.02

2−

0.00

4−

0.03

6−

0.07

6−

0.12

5−

0.17

7−

0.24

10.

017

2.8

0.08

40.

070

0.05

70.

042

0.02

90.

012

−0.

042

−0.

042

−0.

078

−0.

121

−0.

176

0.01

7

3.0

0.06

60.

058

0.05

10.

040

0.03

30.

022

0.00

5−

0.01

6−

0.04

3−

0.07

6−

0.11

60.

016

3.2

0.05

20.

047

0.04

40.

037

0.03

40.

027

0.01

60.

001

−0.

014

−0.

041

−0.

071

0.01

5

3.4

0.03

90.

038

0.03

60.

033

0.03

30.

030

0.02

30.

014

−0.

001

0.01

5−

0.03

70.

014

3.6

0.02

90.

029

0.02

90.

029

0.03

10.

030

0.02

70.

022

0.01

30.

002

−0.

014

0.01

3

3.8

0.02

00.

022

0.02

30.

025

0.02

70.

030

0.02

90.

028

0.02

20.

014

0.00

30.

011

4.0

0.01

50.

016

0.01

70.

021

0.02

40.

028

0.02

90.

030

0.02

60.

022

0.01

40.

010

Tab

le 4

.20

Con

’t

SAMPLE C

HAPTER

Analysis of Beams on Elastic Half-Space 235

β =

0.3

ζα

00.

10.

20.

30.

40.

50.

60.

70.

80.

91.

0∞

0.0

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

282

0.2

−0.

159

0.03

40.

026

0.01

90.

012

0.00

80.

005

0.00

50.

005

0.00

40.

003

0.19

2

0.4

−0.

252

−0.

077

0.09

60.

073

0.04

90.

028

0.02

20.

019

0.01

60.

012

0.00

70.

120

0.6

−0.

296

−0.

145

0.00

50.

159

0.11

20.

079

0.05

30.

041

0.03

10.

021

0.01

20.

066

0.8

−0.

310

−0.

182

−0.

057

0.07

30.

202

0.14

60.

103

0.07

50.

053

0.03

30.

017

0.02

6

1.0

−0.

300

−0.

196

−0.

095

0.01

10.

115

0.23

70.

172

0.12

50.

087

0.05

40.

026

−0.

002

1.2

−0.

280

−0.

194

−0.

115

−0.

032

0.05

00.

149

0.26

20.

193

0.13

60.

086

0.04

4−

0.02

2

1.4

−0.

246

−0.

183

−0.

123

−0.

067

0.00

30.

081

0.17

20.

281

0.20

20.

133

0.07

7−

0.03

3

1.6

−0.

212

−0.

166

−0.

120

−0.

075

−0.

036

0.03

00.

101

0.18

90.

289

0.20

00.

124

−0.

039

1.8

−0.

178

−0.

145

−0.

114

−0.

081

−0.

049

−0.

006

0.04

70.

115

0.19

50.

288

0.19

1−

0.04

1

2.0

−0.

146

−0.

124

−0.

110

−0.

82−

0.06

0−

0.03

10.

010

0.05

80.

120

0.19

10.

277

−0.

041

2.2

−0.

117

−0.

104

−0.

091

−0.

078

−0.

065

−0.

045

−0.

020

0.01

80.

062

0.11

60.

189

−0.

039

2.4

−0.

090

−0.

084

−0.

078

−0.

072

−0.

065

−0.

054

−0.

037

−0.

012

0.02

00.

059

0.11

0−

0.03

7

2.6

−0.

068

−0.

067

−0.

065

−0.

064

−0.

062

−0.

058

−0.

047

−0.

031

−0.

010

0.01

70.

055

−0.

033

2.8

−0.

050

−0.

051

−0.

054

−0.

055

−0.

057

−0.

056

−0.

052

−0.

043

−0.

031

−0.

013

0.01

3−

0.03

0

3.0

−0.

035

−0.

039

−0.

043

−0.

047

−0.

051

−0.

053

−0.

053

−0.

049

−0.

043

−0.

032

−0.

015

−0.

027

3.2

−0.

023

−0.

027

−0.

033

−0.

039

−0.

044

−0.

048

−0.

050

−0.

051

−0.

049

−0.

044

−0.

034

−0.

023

3.4

−0.

013

−0.

019

−0.

025

−0.

031

−0.

037

−0.

042

−0.

047

−0.

050

−0.

050

−0.

049

−0.

044

−0.

020

3.6

−0.

007

−0.

013

−0.

019

−0.

025

−0.

031

−0.

036

−0.

041

−0.

046

−0.

049

−0.

050

−0.

049

−0.

018

3.8

−0.

002

−0.

008

−0.

013

−0.

019

−0.

025

−0.

030

−0.

036

−0.

041

−0.

045

−0.

049

−0.

050

−0.

015

4.0

0.00

1−

0.00

4−

0.00

9−

0.01

5−

0.02

0−

0.02

5−

0.03

0−

0.03

5−

0.04

0−

0.04

5−

0.04

8−

0.01

3

Q

0.0

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

0−

0.50

0

0.2

−0.

612

0.32

10.

246

0.18

40.

119

0.07

80.

051

0.04

60.

040

0.03

20.

020

−0.

403

0.4

−0.

331

−0.

435

0.45

40.

353

0.22

40.

174

0.11

80.

099

0.06

20.

041

0.02

1−

0.31

3

0.6

−0.

133

−0.

253

−0.

377

0.50

30.

384

0.25

60.

200

0.14

80.

091

0.05

20.

020

−0.

234

0.8

−0.

001

−0.

120

−0.

244

−0.

368

0.51

00.

397

0.29

60.

210

0.13

70.

079

0.03

2−

0.16

8

1.0

0.08

8−

0.02

6−

0.14

2−

0.26

0−

0.37

60.

508

0.39

70.

294

0.20

30.

128

0.06

5−

0.11

5

1.2

0.13

90.

036

−0.

066

−0.

173

−0.

277

−0.

388

0.50

00.

390

0.28

30.

197

0.12

1−

0.07

4

1.4

0.16

50.

076

−0.

013

−0.

105

−0.

195

−0.

295

−0.

400

0.48

90.

381

0.28

40.

195

−0.

043

1.6

0.17

30.

097

0.02

3−

0.05

3−

0.12

9−

0.21

5−

0.31

0−

0.41

10.

481

0.38

00.

285

−0.

025

1.8

0.16

20.

105

0.04

6−

0.01

6−

0.07

8−

0.14

9−

0.23

1−

0.32

3−

0.42

10.

481

0.38

40.

006

2.0

0.15

60.

106

0.05

70.

009

−0.

039

−0.

096

−0.

164

−0.

243

−0.

330

−0.

421

0.38

60.

006

2.2

0.14

00.

101

0.06

40.

026

−0.

011

−0.

056

−0.

110

−0.

174

−0.

248

−0.

329

−0.

417

0.01

2

2.4

0.12

00.

092

0.06

40.

037

0.00

9−

0.02

5−

0.06

7−

0.11

8−

0.18

1−

0.24

6−

0.32

30.

016

2.6

0.10

10.

082

0.06

20.

040

0.02

2−

0.00

4−

0.03

6−

0.07

6−

0.12

5−

0.17

7−

0.24

10.

017

2.8

0.08

40.

070

0.05

70.

042

0.02

90.

012

−0.

042

−0.

042

−0.

078

−0.

121

−0.

176

0.01

7

3.0

0.06

60.

058

0.05

10.

040

0.03

30.

022

0.00

5−

0.01

6−

0.04

3−

0.07

6−

0.11

60.

016

3.2

0.05

20.

047

0.04

40.

037

0.03

40.

027

0.01

60.

001

−0.

014

−0.

041

−0.

071

0.01

5

3.4

0.03

90.

038

0.03

60.

033

0.03

30.

030

0.02

30.

014

−0.

001

0.01

5−

0.03

70.

014

3.6

0.02

90.

029

0.02

90.

029

0.03

10.

030

0.02

70.

022

0.01

30.

002

−0.

014

0.01

3

3.8

0.02

00.

022

0.02

30.

025

0.02

70.

030

0.02

90.

028

0.02

20.

014

0.00

30.

011

4.0

0.01

50.

016

0.01

70.

021

0.02

40.

028

0.02

90.

030

0.02

60.

022

0.01

40.

010

SAMPLE C

HAPTER

236 Analysis of Structures on Elastic Foundations

β =

0.3

ζα

00.

10.

20.

30.

40.

50.

60.

70.

80.

91.

0∞

Y

0.0

2.29

1.92

1.56

1.25

1.56

1.25

1.01

0.78

0.57

0.40

0.12

0.78

0.2

1.92

1.66

1.40

1.14

0.99

0.80

0.62

0.46

0.35

0.27

0.22

0.77

0.4

1.56

1.40

1.24

1.09

0.97

0.82

0.66

0.52

0.40

0.33

0.29

0.72

0.6

1.25

1.14

1.09

1.03

0.95

0.84

0.72

0.61

0.50

0.43

0.38

0.66

0.8

1.01

0.99

0.97

0.95

0.92

0.86

0.77

0.67

0.58

0.43

0.38

0.66

1.0

0.78

0.80

0.82

0.84

0.86

0.84

0.80

0.73

0.65

0.59

0.54

0.53

1.2

0.57

0.62

0.66

0.72

0.77

0.80

0.79

0.76

0.71

0.65

0.61

0.45

1.4

0.40

0.46

0.52

0.61

0.67

0.73

0.76

0.77

0.75

0.71

0.67

0.39

1.6

0.28

0.35

0.40

0.50

0.58

0.65

0.71

0.75

0.76

0.75

0.72

0.32

1.8

0.19

0.27

0.33

0.43

0.51

0.59

0.65

0.71

0.75

0.78

0.77

0.26

2.0

0.12

0.22

0.29

0.38

0.46

0.54

0.61

0.67

0.72

0.77

0.79

0.21

2.2

0.07

0.16

0.22

0.31

0.38

0.45

0.52

0.59

0.66

0.72

0.77

0.16

2.4

0.03

0.12

0.17

0.25

0.31

0.38

0.45

0.52

0.59

0.66

0.72

0.12

2.6

0.00

0.08

0.14

0.21

0.26

0.33

0.39

0.46

0.53

0.60

0.66

0.08

2.8

−0.

010.

060.

110.

170.

220.

270.

330.

400.

460.

530.

600.

05

3.0

−0.

010.

040.

090.

140.

180.

230.

280.

340.

400.

460.

530.

02

3.2

−0.

010.

020.

070.

120.

150.

200.

240.

290.

340.

400.

460.

00

3.4

−0.

010.

010.

050.

100.

130.

170.

200.

250.

290.

340.

39−

0.02

3.6

−0.

010.

000.

040.

090.

110.

140.

170.

210.

250.

290.

34−

0.04

3.8

0.00

0.00

0.03

0.07

0.10

0.12

0.15

0.18

0.21

0.25

0.28

−0.

05

4.0

0.00

0.00

0.02

0.06

0.09

0.11

0.13

0.16

0.18

0.21

0.24

−0.

07

Tab

le 4

.20

Con

’t

SAMPLE C

HAPTER

Analysis of Beams on Elastic Half-Space 237

β =

0.5

ζα

00.

10.

20.

30.

40.

50.

60.

70.

80.

91.

0∞

p

0.0

2.27

1.86

1.43

1.04

0.64

0.39

0.24

0.22

0.21

0.18

0.14

0.44

0.2

1.63

1.39

1.15

0.92

0.68

0.48

0.31

0.19

0.11

0.05

0.01

0.43

0.4

1.15

0.96

0.92

0.80

0.67

0.53

0.39

0.25

0.14

0.05

0.00

0.40

0.6

0.79

0.75

0.72

0.68

0.64

0.56

0.46

0.33

0.22

0.12

0.05

0.36

0.8

0.53

0.54

0.55

0.56

0.57

0.55

0.50

0.41

0.32

0.23

0.14

0.31

1.0

0.32

0.37

0.42

0.46

0.50

0.52

0.51

0.47

0.40

0.33

0.25

0.25

1.2

0.20

0.25

0.30

0.37

0.42

0.47

0.49

0.49

0.46

0.41

0.34

0.20

1.4

0.09

0.15

0.22

0.28

0.35

0.41

0.46

0.48

0.49

0.46

0.42

0.14

1.6

0.02

0.09

0.15

0.21

0.28

0.34

0.41

0.46

0.48

0.49

0.47

0.10

1.8

−0.

020.

040.

090.

150.

220.

280.

350.

410.

460.

480.

490.

07

2.0

−0.

050.

000.

060.

110.

160.

220.

290.

350.

400.

460.

490.

04

2.2

−0.

07−

0.03

0.03

0.07

0.12

0.17

0.23

0.29

0.35

0.41

0.46

0.02

2.4

−0.

08−

0.04

0.01

0.04

0.09

0.14

0.18

0.23

0.29

0.36

0.41

0.00

2.6

−0.

08−

0.05

−0.

010.

020.

060.

100.

140.

190.

250.

300.

350.

00

2.8

−0.

07−

0.05

−0.

020.

010.

040.

060.

100.

140.

190.

240.

300.

00

3.0

−0.

07−

0.05

−0.

020.

000.

020.

050.

070.

110.

150.

190.

24−

0.01

3.2

−0.

07−

0.05

−0.

03−

0.01

0.01

0.03

0.05

0.08

0.11

0.15

0.19

−0.

01

3.4

−0.

06−

0.04

−0.

03−

0.01

0.00

0.02

0.04

0.06

0.08

0.10

0.14

−0.

01

3.6

−0.

05−

0.04

−0.

03−

0.02

−0.

010.

010.

020.

040.

060.

080.

10−

0.01

3.8

−0.

04−

0.03

−0.

03−

0.02

−0.

010.

000.

010.

030.

040.

060.

08−

0.01

4.0

−0.

03−

0.03

−0.

03−

0.02

−0.

010.

000.

010.

020.

030.

040.

060.

00

Tab

le 4

.21

Dim

ensi

onle

ss c

oeffi

cien

ts f

or a

naly

sis

of lo

ng b

eam

s lo

aded

with

a c

once

ntra

ted

ver

tical

load

P

SAMPLE C

HAPTER

238 Analysis of Structures on Elastic Foundations

β =

0.5

ζα

00.

10.

20.

30.

40.

50.

60.

70.

80.

91.

0∞

M

0.0

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

300

0.2

−0.

159

0.03

40.

026

0.02

00.

013

0.00

90.

005

0.00

40.

003

0.00

20.

001

0.20

7

0.4

−0.

253

−0.

076

0.09

90.

077

0.05

30.

036

0.02

30.

017

0.01

10.

008

0.00

40.

135

0.6

−0.

300

−0.

144

0.00

90.

165

0.12

10.

085

0.05

70.

039

0.02

50.

019

0.00

60.

075

0.8

−0.

316

−0.

183

−0.

052

0.08

10.

217

0.15

60.

108

0.07

40.

048

0.02

70.

012

0.03

3

1.0

−0.

310

−0.

199

−0.

091

0.01

90.

128

0.25

00.

180

0.12

60.

083

0.04

90.

023

0.00

1

1.2

−0.

291

−0.

201

−0.

113

−0.

025

0.06

40.

162

0.27

20.

197

0.13

40.

084

0.04

3−

0.02

0

1.4

−0.

264

−0.

192

−0.

123

−0.

053

0.01

60.

094

0.18

30.

287

0.20

40.

135

0.07

8−

0.03

3

1.6

−0.

232

−0.

177

−0.

124

−0.

071

−0.

018

0.04

20.

113

0.19

70.

293

0.20

40.

129

−0.

041

1.8

−0.

200

−0.

159

−0.

119

−0.

080

−0.

041

0.00

50.

058

0.12

40.

201

0.29

30.

198

−0.

044

2.0

−0.

169

−0.

139

−0.

111

−0.

082

−0.

055

−0.

022

0.01

80.

068

0.12

80.

201

0.28

8−

0.04

5

2.2

−0.

140

−0.

120

−0.

101

−0.

081

−0.

062

−0.

039

−0.

011

0.02

60.

071

0.12

70.

196

−0.

044

2.4

−0.

114

−0.

101

−0.

089

−0.

077

−0.

065

−0.

050

−0.

030

−0.

006

0.02

80.

070

0.12

6−

0.04

3

2.6

−0.

091

−0.

083

−0.

077

−0.

070

−0.

064

−0.

055

−0.

043

−0.

025

−0.

003

0.02

80.

067

−0.

029

2.8

−0.

070

−0.

068

−0.

066

−0.

063

−0.

061

−0.

057

−0.

051

−0.

040

−0.

024

−0.

004

0.02

4−

0.01

7

3.0

−0.

053

−0.

054

−0.

055

−0.

055

−0.

056

−0.

056

−0.

053

−0.

047

−0.

039

−0.

025

−0.

006

−0.

012

3.2

−0.

039

−0.

042

−0.

045

−0.

048

−0.

051

−0.

053

−0.

054

−0.

051

−0.

046

−0.

039

−0.

027

−0.

009

3.4

−0.

028

−0.

032

−0.

036

−0.

041

−0.

045

−0.

048

−0.

051

−0.

051

−0.

050

−0.

046

−0.

039

−0.

009

3.6

−0.

019

−0.

024

−0.

029

−0.

034

−0.

038

−0.

043

−0.

047

−0.

019

−0.

050

−0.

046

−0.

008

−0.

008

3.8

−0.

012

−0.

017

−0.

022

−0.

028

−0.

033

−0.

038

−0.

042

−0.

046

−0.

049

−0.

050

−0.

049

−0.

008

4.0

−0.

007

−0.

012

−0.

017

−0.

022

−0.

027

−0.

032

−0.

037

−0.

041

−0.

045

−0.

048

−0.

049

−0.

008

0.0

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

0−

0.50

0

0.2

−0.

614

0.32

30.

258

0.19

60.

133

0.08

80.

055

0.03

90.

029

0.02

00.

014

−0.

411

0.4

−0.

339

−0.

436

0.48

40.

367

0.26

90.

189

0.12

60.

083

0.05

30.

030

0.01

3−

0.32

9

0.6

−0.

147

−0.

258

−0.

372

0.51

40.

400

0.29

90.

211

0.14

20.

088

0.08

20.

036

−0.

186

0.8

−0.

017

−0.

127

−0.

245

−0.

362

0.52

20.

410

0.30

70.

218

0.14

20.

082

0.03

6−

0.18

6

1.0

0.06

8−

0.04

0−

0.14

9−

0.26

0−

0.37

10.

516

0.40

80.

305

0.21

40.

137

0.07

5−

0.13

0

1.2

0.12

00.

021

−0.

075

−0.

178

−0.

278

−0.

384

0.50

90.

401

0.30

20.

211

0.13

4−

0.08

5

1.4

0.14

90.

061

−0.

026

−0.

113

−0.

202

−0.

296

−0.

396

0.49

90.

397

0.29

90.

211

−0.

050

1.6

0.16

00.

084

0.01

0−

0.06

4−

0.14

0−

0.22

1−

0.31

0−

0.40

60.

494

0.39

50.

300

−0.

027

1.8

0.16

00.

096

0.03

4−

0.02

8−

0.09

0−

0.15

9−

0.23

5−

0.32

0−

0.41

20.

493

0.39

7−

0.01

1

2.0

0.15

20.

100

0.04

8−

0.00

1−

0.05

3−

0.10

8−

0.17

1−

0,24

3−

0.32

5−

0.41

30.

495

0.00

0

2.2

0.13

90.

097

0.05

60.

016

−0.

025

−0.

069

−0.

119

−0.

179

−0.

247

−0.

325

−0.

410

0.00

6

2.4

0.12

40.

092

0.05

90.

028

−0.

004

−0.

039

−0.

080

−0.

126

−0.

182

−0.

248

−0.

322

0.00

9

2.6

0.10

90.

083

0.06

00.

035

0.01

1−

0.01

6−

0.04

7−

0.08

6−

0.13

0−

0.18

3−

0.24

40.

010

2.8

0.09

20.

074

0.05

60.

038

0.02

10.

001

−0.

023

−0.

051

−0.

087

−0.

129

−0.

178

0.00

9

3.0

0.07

70.

064

0.05

20.

039

0.02

70.

012

−0.

006

−0.

028

0.05

3−

0.08

6−

0.12

50.

008

3.2

0.06

30.

055

0.04

70.

038

0.03

10.

019

0.00

7−

0.00

9−

0.02

8−

0.05

2−

0.08

20.

007

3.4

0.05

00.

045

0.04

10.

035

0.03

10.

024

0.01

60.

004

−0.

009

−0.

027

−0.

049

0.00

5

3.6

0.04

00.

037

0.03

50.

032

0.03

00.

026

0.02

10.

014

0.00

4−

0.00

8−

0.02

50.

004

3.8

0.03

10.

030

0.02

90.

029

0.02

80.

027

0.02

50.

021

0.01

40.

005

−0.

007

0.00

3

4.0

0.02

30.

023

0.02

30.

025

0.02

50.

027

0.02

70.

025

0.02

10.

014

0.00

60.

002

Tab

le 4

.21

Con

’t

SAMPLE C

HAPTER

Analysis of Beams on Elastic Half-Space 239

β =

0.5

ζα

00.

10.

20.

30.

40.

50.

60.

70.

80.

91.

0∞

0.0

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

300

0.2

−0.

159

0.03

40.

026

0.02

00.

013

0.00

90.

005

0.00

40.

003

0.00

20.

001

0.20

7

0.4

−0.

253

−0.

076

0.09

90.

077

0.05

30.

036

0.02

30.

017

0.01

10.

008

0.00

40.

135

0.6

−0.

300

−0.

144

0.00

90.

165

0.12

10.

085

0.05

70.

039

0.02

50.

019

0.00

60.

075

0.8

−0.

316

−0.

183

−0.

052

0.08

10.

217

0.15

60.

108

0.07

40.

048

0.02

70.

012

0.03

3

1.0

−0.

310

−0.

199

−0.

091

0.01

90.

128

0.25

00.

180

0.12

60.

083

0.04

90.

023

0.00

1

1.2

−0.

291

−0.

201

−0.

113

−0.

025

0.06

40.

162

0.27

20.

197

0.13

40.

084

0.04

3−

0.02

0

1.4

−0.

264

−0.

192

−0.

123

−0.

053

0.01

60.

094

0.18

30.

287

0.20

40.

135

0.07

8−

0.03

3

1.6

−0.

232

−0.

177

−0.

124

−0.

071

−0.

018

0.04

20.

113

0.19

70.

293

0.20

40.

129

−0.

041

1.8

−0.

200

−0.

159

−0.

119

−0.

080

−0.

041

0.00

50.

058

0.12

40.

201

0.29

30.

198

−0.

044

2.0

−0.

169

−0.

139

−0.

111

−0.

082

−0.

055

−0.

022

0.01

80.

068

0.12

80.

201

0.28

8−

0.04

5

2.2

−0.

140

−0.

120

−0.

101

−0.

081

−0.

062

−0.

039

−0.

011

0.02

60.

071

0.12

70.

196

−0.

044

2.4

−0.

114

−0.

101

−0.

089

−0.

077

−0.

065

−0.

050

−0.

030

−0.

006

0.02

80.

070

0.12

6−

0.04

3

2.6

−0.

091

−0.

083

−0.

077

−0.

070

−0.

064

−0.

055

−0.

043

−0.

025

−0.

003

0.02

80.

067

−0.

029

2.8

−0.

070

−0.

068

−0.

066

−0.

063

−0.

061

−0.

057

−0.

051

−0.

040

−0.

024

−0.

004

0.02

4−

0.01

7

3.0

−0.

053

−0.

054

−0.

055

−0.

055

−0.

056

−0.

056

−0.

053

−0.

047

−0.

039

−0.

025

−0.

006

−0.

012

3.2

−0.

039

−0.

042

−0.

045

−0.

048

−0.

051

−0.

053

−0.

054

−0.

051

−0.

046

−0.

039

−0.

027

−0.

009

3.4

−0.

028

−0.

032

−0.

036

−0.

041

−0.

045

−0.

048

−0.

051

−0.

051

−0.

050

−0.

046

−0.

039

−0.

009

3.6

−0.

019

−0.

024

−0.

029

−0.

034

−0.

038

−0.

043

−0.

047

−0.

019

−0.

050

−0.

046

−0.

008

−0.

008

3.8

−0.

012

−0.

017

−0.

022

−0.

028

−0.

033

−0.

038

−0.

042

−0.

046

−0.

049

−0.

050

−0.

049

−0.

008

4.0

−0.

007

−0.

012

−0.

017

−0.

022

−0.

027

−0.

032

−0.

037

−0.

041

−0.

045

−0.

048

−0.

049

−0.

008

Q

0.0

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

0−

0.50

0

0.2

−0.

614

0.32

30.

258

0.19

60.

133

0.08

80.

055

0.03

90.

029

0.02

00.

014

−0.

411

0.4

−0.

339

−0.

436

0.48

40.

367

0.26

90.

189

0.12

60.

083

0.05

30.

030

0.01

3−

0.32

9

0.6

−0.

147

−0.

258

−0.

372

0.51

40.

400

0.29

90.

211

0.14

20.

088

0.08

20.

036

−0.

186

0.8

−0.

017

−0.

127

−0.

245

−0.

362

0.52

20.

410

0.30

70.

218

0.14

20.

082

0.03

6−

0.18

6

1.0

0.06

8−

0.04

0−

0.14

9−

0.26

0−

0.37

10.

516

0.40

80.

305

0.21

40.

137

0.07

5−

0.13

0

1.2

0.12

00.

021

−0.

075

−0.

178

−0.

278

−0.

384

0.50

90.

401

0.30

20.

211

0.13

4−

0.08

5

1.4

0.14

90.

061

−0.

026

−0.

113

−0.

202

−0.

296

−0.

396

0.49

90.

397

0.29

90.

211

−0.

050

1.6

0.16

00.

084

0.01

0−

0.06

4−

0.14

0−

0.22

1−

0.31

0−

0.40

60.

494

0.39

50.

300

−0.

027

1.8

0.16

00.

096

0.03

4−

0.02

8−

0.09

0−

0.15

9−

0.23

5−

0.32

0−

0.41

20.

493

0.39

7−

0.01

1

2.0

0.15

20.

100

0.04

8−

0.00

1−

0.05

3−

0.10

8−

0.17

1−

0,24

3−

0.32

5−

0.41

30.

495

0.00

0

2.2

0.13

90.

097

0.05

60.

016

−0.

025

−0.

069

−0.

119

−0.

179

−0.

247

−0.

325

−0.

410

0.00

6

2.4

0.12

40.

092

0.05

90.

028

−0.

004

−0.

039

−0.

080

−0.

126

−0.

182

−0.

248

−0.

322

0.00

9

2.6

0.10

90.

083

0.06

00.

035

0.01

1−

0.01

6−

0.04

7−

0.08

6−

0.13

0−

0.18

3−

0.24

40.

010

2.8

0.09

20.

074

0.05

60.

038

0.02

10.

001

−0.

023

−0.

051

−0.

087

−0.

129

−0.

178

0.00

9

3.0

0.07

70.

064

0.05

20.

039

0.02

70.

012

−0.

006

−0.

028

0.05

3−

0.08

6−

0.12

50.

008

3.2

0.06

30.

055

0.04

70.

038

0.03

10.

019

0.00

7−

0.00

9−

0.02

8−

0.05

2−

0.08

20.

007

3.4

0.05

00.

045

0.04

10.

035

0.03

10.

024

0.01

60.

004

−0.

009

−0.

027

−0.

049

0.00

5

3.6

0.04

00.

037

0.03

50.

032

0.03

00.

026

0.02

10.

014

0.00

4−

0.00

8−

0.02

50.

004

3.8

0.03

10.

030

0.02

90.

029

0.02

80.

027

0.02

50.

021

0.01

40.

005

−0.

007

0.00

3

4.0

0.02

30.

023

0.02

30.

025

0.02

50.

027

0.02

70.

025

0.02

10.

014

0.00

60.

002

SAMPLE C

HAPTER

240 Analysis of Structures on Elastic Foundations

β =

0.5

ζα

00.

10.

20.

30.

40.

50.

60.

70.

80.

91.

0∞

Y

0.0

1.69

1.43

1.21

1.01

0.82

0.66

0.52

0.40

0.31

0.24

0.20

0.60

0.2

1.43

1.25

1.08

0.94

0.78

0.66

0.54

0.44

0.36

0.29

0.24

0.59

0.4

1.21

1.08

0.97

0.87

0.76

0.66

0.56

0.48

0.40

0.34

0.28

0.56

0.6

1.01

0.94

0.87

0.80

0.73

0.66

0.59

0.51

0.45

0.39

0.32

0.52

0.8

0.82

0.78

0.76

0.73

0.71

0.66

0.61

0.54

0.50

0.44

0.39

0.47

1.0

0.66

0.66

0.66

0.66

0.66

0.65

0.62

0.57

0.54

0.49

0.44

0.43

1.2

0.52

0.54

0.56

0.59

0.61

0.62

0.62

0.60

0.57

0.53

0.48

0.38

1.4

0.40

0.44

0.48

0.51

0.54

0.57

0.60

0.60

0.59

0.56

0.53

0.33

1.6

0.31

0.36

0.40

0.45

0.50

0.54

0.57

0.59

0.60

0.59

0.57

0.28

1.8

0.24

0.29

0.34

0.39

0.44

0.49

0.53

0.56

0.59

0.61

0.60

0.24

2.0

0.20

0.24

0.28

0.32

0.39

0.44

0.48

0.53

0.57

0.60

0.61

0.20

2.2

0.16

0.19

0.24

0.29

0.34

0.39

0.43

0.48

0.52

0.57

0.60

0.16

2.4

0.13

0.15

0.20

0.24

0.29

0.34

0.39

0.43

0.48

0.52

0.57

0.12

2.6

0.10

0.13

0.16

0.21

0.25

0.30

0.34

0.39

0.44

0.48

0.53

0.09

2.8

0.08

0.11

0.14

0.18

0.22

0.26

0.30

0.34

0.39

0.44

0.49

0.05

3.0

0.07

0.09

0.12

0.15

0.19

0.23

0.26

0.30

0.34

0.39

0.44

0.03

3.2

0.06

0.08

0.10

0.13

0.16

0.20

0.23

0.27

0.30

0.35

0.39

0.00

3.4

0.05

0.07

0.09

0.12

0.14

0.17

0.20

0.23

0.27

0.31

0.35

−0.

03

3.6

0.04

0.06

0.08

0.10

0.12

0.15

0.18

0.20

0.23

0.27

0.30

−0.

05

3.8

0.03

0.05

0.07

0.09

0.11

0.13

0.15

0.18

0.20

0.23

0.27

−0.

07

4.0

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

0.23

−0.

09

Tab

le 4

.21

Con

’t

SAMPLE C

HAPTER

Analysis of Beams on Elastic Half-Space 241

References1. Borowicka, H. 1938. The distribution of pressure under a uniformly loaded strip resting on

elastic isotropic ground. 2nd Congress International Association for Bridge and Structural En-gineering (Berlin), Final Report, VIII, 3.

2. Borowicka, H. 1939. Druckverteilung unterelastischen Platten. ING. Arch. (Berlin) 10(2): 113–125.

3. Gorbunov-Posadov, M. I. 1940. Analysis of beams and plates on elastic half-space. Applied Mechanics and Mathematics 4(3): 60–80.

4. Gorbunov-Posadov, M. I. 1949. Beams and slabs on elastic foundation. Moscow: Mashstroiizdat.5. Gorbunov-Posadov, M. I. 1953. Analysis of structures on elastic foundation. Moscow:

Gosstroiizdat.6. Gorbunov-Posadov, M. I. 1984. Analysis of structures on elastic foundation. 3rd ed. Moscow:

Stroiizdat, pp. 252–298.7. Selvadurai, A.P.S. 1979. Elastic analysis of soil-foundation interaction. Amsterdam/Oxford/

New York: Elsevier Scientific, pp. 11–121.8. Tsudik E. A. 2006. Analysis of beams and frames on elastic foundation. Canada/UK: Trafford,

pp. 29–47.

SAMPLE C

HAPTER