Free vibrations of continuous horizontally curved beams

Journal of Sound and Vibration (1992) 157(2), 345-355

FREE

VIBRATIONS

OF CONTINUOUS CURVED BEAMS

J. M. SNYDER

HORIZONTALLY

AND J. F. WILSON

Department of Civil Engineering, Clemson University, Clemson, South Carolina 29634-0911, U.S.A. (Received 8 June 1990, and in final form 10 June 199 1) This paper presents a closed form solution for the out-of-plane free vibration frequencies of a horizontally curved thin walled beam which is continuous over multiple supports. The bending and twisting of this beam is coupled in the mathematical model used. This coupled bending/twisting behavior causes two free vibration frequencies to be associated with each vibration mode shape. The solution is used to calculate the first six free vibration frequencies and the corresponding mode shapes for prismatic beams with three equal spans over a range of non-dimensional parameters representing variations in warping stiffness, torsional stiffness, radius of curvature, included angle of the curve and polar mass moment of inertia. The mathematical model and numerical results are presented in non-dimensional form. The results are applicable to the design of highway, rail, rapid transit and guideway structures.

1. INTRODUCTION

In this paper a method is presented for calculating the coupled bending/twisting out-ofplane free vibration frequencies of a horizontally curved thin walled beam continuous over multiple supports. The differential equations used to model the static elastic behavior of the beam, derived by Vlasov [I], are based on the assumption that the centroid and shear center of the beam are coincident. The corresponding dynamic theory [2,3] involves two coupled fourth order partial differential equations for bending and twisting (an eighth order boundary value problem). State-of-the-art papers [4-61 summarize much of the available literature dealing with the static and dynamic behavior of horizontally curved beams. The free vibration frequencies of a simple-span curved beam of this type were found by Culver [2] using a closed form solution of the equations of motion. The dynamic response of a simple-span curved beam to moving loads was calculated by Tan and Shore [3,7]. The free vibration frequencies of a simple-span curved beam with an asymmetrical cross-section, with the centroid and shear center not coinciding, was calculated by Yoo and Fehrenbach [8] using the finite element method. Studies of the dynamic behavior of continuous multiple span curved beams are less common. The first free vibration frequency of a multiple span curved beam was calculated by Heins and Sahin [9] using a discretized, lumped mass beam model and the finite difference method; and experimental investigations of multiple span curved beams were performed by Joseph and Wilson [lo]. In this paper the free vibration frequencies of a continuous multiple span curved beam are obtained from a non-explicit closed form solution of the partial differential equations of motion. This method of solution eliminates discretization errors and allows the eigenfunctions describing the vibration mode shapes to be determined. This solution is applied to a uniform prismatic beam with three equal length spans. The first six free vibration 345 0022-460X/92/1 70345+ 11 %08.00/O

Q 1992 Academic

Press Limited

346

J. M. SNYDER AND J. E. WILSON

frequencies and the corresponding mode shapes are calculated over a range of nondimensional parameters representing variations in horizontal radius of curvature, included angle of the curve, warping constant, torsion constant, and polar mass moment of inertia. To enhance their generality, the mathematical model and the numerical results are presented in non-dimensional form.

2. MATHEMATICAL

MODEL

The left-hand co-ordinate system used by the model is shown in Figure 1. The x and y axes shown in this figure are the principal centroidal axes of the beam cross-section; the x-axis is in the horizontal plane of curvature and the z-axis coincides with the centroid. The horizontal radius of curvature R is constant. The assumptions of the model follow Vlasov [ 1] and are as follows. The member is prismatic, the shear center and centroid of the member coincide, the member has a thin walled cross-section with an inflexible contour, and the cross-section dimensions are small compared with both the length of the beam and the horizontal radius of curvature.

Figure 1. Co-ordinate system and sign convention for a horizontally curved beam segment.

For out-or-plane

free vibrations the equations of motion are

lJ4q C a*q D a"p EI+Cl@+ma'tl=O a,2+2 az”R a2 at* ’

g-2

a*p EI D a4f7 EI+C a*q a4p --__ -+Dg-Cs+$/?+mr,p=O, R az4 R a2

2a*p

(2)

where C is the torsional stiffness, D is the warping stiffness, E is the modulus of elasticity, I is the second moment of area about the x-axis, m is the mass per unit length, r is the polar radius of gyration, R is the horizontal radius of curvature, t is time, /I is the torsional rotation about the z-axis, q is the vertical deflection, and 4 is the horizontal deflection. Other notations are as defined in Figure 1. Equations (1) and (2) must be solved simultaneously to find n(z, t) and p(z, t). To find the corresponding free vibration frequencies the following solutions are assumed:

rl(z, 0 = W$

sin(pt + P),

(3)

P(z, 0 = V.4

sin(pt + P),

(4)

CURVED

347

BEAMS

wherep is the circular frequency and p is the phase. Using equations (3) and (4) equations (1) and (2) become

V” - mp2 W=O,

EI+C R

(5)

Wt’+DViv-CV”+($-mr$‘)V=O,

(6)

where the prime denotes the operator d/dz. Differentiating equations (5) and (6) twice by z, dividing equation (6) by R, and subtracting the two equations, the following equation is obtained :

(7) Equation (5) may be solved for Vi” to obtain viv =

E1+c

Wiv+s

D

W”+gmp2W.

(8)

Substituting equation (8) into equation (7) and solving for V”, the following is obtained : Vrf=p-~W”+!-[-$-~(E~+$)]Wiv+~(mp2+~)W”+$mp2W,

(9)

where

a=~(mr2p’_5)_;~.

(10)

When equation (9) is substituted into equation (7), the result is the eighth order differential equation F

j,fpiii,

E

Er)

+[g(

1 +$)mp2-F]

[“R~-S?$!~(~+~+!!$)mp2]

We+

W”+[~m2p4-$mp2]

W”

W=O.

(11)

Defining the parameters L= R80, D = D/( EIR’),

c = r/R,

z= z/L,

W(Z) = W(ZL),

c= C/(EZ), jj = (mL4/( n4E1))“2p,

(12)

348

J. M. SNYDER

AND J.E. WILSON

(a)

-mm NtI

N

j+l

j

3

2

I

( b) Figure 2. Horizontally curved beam with N continuous spans: (a) plan; (b) elevation.

where (I,-,is the included angle of curve of the beam between supports (see Figure 2). Equation (11) may be written in non-dimensional form as

(13) Note that the frequency parameter ~5 is the ratio of the actual span frequency to the fundamental frequency of its simple-span straight counterpart. Recall that equation (13) was derived by eliminating V(z) from equations (5) and (6). If, instead, W(z) was eliminated from equations (5) and (6) the same equation, with Y replacing IV, would result. This is a consequence of the coupling of bending and twisting in the beam model. The boundary and continuity conditions for equation (13) are considered next.

3. BOUNDARY

CONDITIONS

To solve equation (13) for the case of a beam that is continuous over multiple supports, it is necessary to define the boundary conditions at a simple end support and the boundary and continuity conditions at an interior support. At a simple end support the boundary conditions are zero vertical deflection, zero torsional rotation, zero flexural moment, and zero bimoment :

(14)

CURVED

BEAMS

349

Substituting equations (3) and (4) into equations (14) the boundary conditions become w=o,

v= 0,

W” = 0,

V”= 0.

(15)

Equations (15) are rewritten exclusively in terms of W by using equations (5) (6) and (9). In non-dimensional form, the boundary conditions are Ij?f = 0,

w= 0,

Ij+,=(),

ljP’=().

(16)

To present the boundary and continuity conditions for interior simple supports, quantities pertaining to the portion of the beam to the right of the support are subscripted with an Y,those to the left with an 1. The boundary conditions are: zero vertical deflections and zero torsional rotations. The continuity conditions pertain to the slope, warping (twist), flexural moment, and bimoment. In mathematical form, the boundary and continuity conditions are

(17) Substituting equations (3) and (4) into equations (17) the conditions become w,=o, Wi=

w,=o,

W:,

w;=

v,=o,

w:,

v,= VI,

Vi= V:,

v;=

v:‘.

(18)

Equations (18) are rewritten exclusively in terms of W by using equations (5) (6) and (9). The resulting boundary conditions in non-dimensional form are W,=o, Wi= W:,

Wr=o,

WY= WY,

(Lo W, = 0,

I@= jj?‘:,

(N),=

Wyi = @r,

WN),,

(19)

where 4=Ds+

d6

- i2 (D-C)e:+D-Pr~ [ 0:

+[D(c’-

l)7r4p2-e;(2C+D+

L,=

$ (Lo).

1

4-Z d4 -$

l)] 2,

(20) (21)

Alternatively, an equivalent set of boundary conditions may be derived in terms of V rather than W. Free vibration solutions of equation (13) using the boundary and continuity conditions of equations (16) and (19), are presented next.

350

J. M. SNYDER

4. SOLUTION

AND

J. E. WILSON

FOR FREQUENCIES

To solve equation (13) for w(Z), the vertical deflection mode shape of the beam, the solution is assumed to be of the form

P(Z)= K ez.

(22)

The equation obtained by substituting the assumed solution into equation (13) is ~a8+(2~-~)e~a~+[(~-22c)e~-(~++~2+~~)~4~~]a4

+[~ef(l+r’)a”~“-Ce~]~2+r21t8~4-e~n4~2=0.

(23)

Equation (23) defines the relationship between the free vibration frequencies p and the eigenvalues 1’. For any value of 6, equation (23) may be solved for fi, which determines the eigenfunctions of the solution m(P). The solution of equation (23) gives three classes of eigenvalues and eigenfunctions, as follows. When fi is real and positive, tt
cash (&)+K2

sinh (a&

(24)

where a = $? and K, and K2 are arbitrary constants. When fi is real and negative, fV(F)=K, cos (aZ)+K2 sin (az),

(25)

where a = p and K, and K2 are arbitrary constants. When a2 is complex, there will be two complex conjugate roots, a2 = a + bm a - bfi. The eigenfunctions for these two roots are

and a2 =

~(~)=KKIcosh(a,i)cos(a2~)+K2cosh(a1~)sin(a2~) +K, sinh (a@) cos (a2z) + K4 sinh (a,Z) sin (a2i),

(26)

where al = [f(J2Ti7+a)]“2,

a2 = IdI [2(J2S+

a)]-‘12,

(27,28)

and K,,K2, K3 and K4 are arbitrary constants. For a curved beam with N spans (Figure 2), there will be N functions, Pj(?j), i= 1 to N, constituting one mode shape. Define & as the average angular support spacing and define the span ratio fi such that eOj =f;.

00

9

(29)

and consider the jth span in Figure 2. At support Rj,fj=Oand at support Rj+t, i;=fi. The resulting N functions will be of the form w,(z,)=

i

j&&i;),

(30)

k-l

with the form of the eigenfunction gk determined by equations (24)-(26). There are 8N constants K,kto be determined using the 8N boundary and continuity conditions given by equations (16) and (19). All of the boundary conditions are homogeneous, and the matrix of coefficients associated with the constants &k vanishes if fi is a free vibration frequency.

351

CURVED BEAMS

For multiple-span beams, a numerical procedure may be used to find the values of p for which the determinant of this matrix vanishes. When the warping stiffness of the beam is zero, application of this solution is simplified. Equation (23) becomes a sixth order equation, since iT=O, and the boundary and continuity conditions for warping and bimoment become unnecessary. Thus there will only be three eigenvalues a* to be calculated, and the range of the index k in equation (30) will be 1 to 6, since only 6N constants K/k must be determined. 5. NUMERICAL RESULTS The mathematical model and solution developed in the previous sections of this paper were used to investigate the behavior of a continuous curved beam with three equal spans. c was taken as 0.05 and 2.00 and c was taken as 0.005 and 0.020. These values were selected to match upper and lower limits for elevated guideways reported by Wilson et al. [6]. The parameter D was allowed to vary from 0 to 0.1. A computer program was used to calculate the free vibration frequencies to within six decimal places, using Gaussian elimination with row and column pivoting to search for sign changes in the frequency determinant. The constants Kjk were then calculated and developed elevations of the mode shapes w(Z) were plotted. The results are given in Tables l-4 for angular support spacings of 15”, 30”, 45” and 60”, respectively. For each case considered, these tables show the first six non-dimensional free vibration frequencies, denoted by J%-&, and the corresponding mode shapes. The mode shapes are designated by the capital letters A-F, printed adjacent to the frequencies. These letters refer to the developed elevations shown in Figure 3. Note that the actual frequencies p are given by equation (12), using the p values in these tables. The data in Tables l-4 show that when (?=2.00, variations in D have little effect on the first six free vibration frequencies. This value of c is representative of tubular and box TABLE 1

First six non-dimensional free vibration frequencies for & = 15” (mode shapes are designated by the adjacent capital; see Figure 3)

0.005 0.005 0.005 0.005 0.005

0

0.001 0.01 0.1

0.020 0.020 0.020 0.020 0.020

O-926A 0~0001 0.941 A

A

0.975 A 0.991 A 0.993 A

l-217 l-235 1.265 1.274 1.276

B B B B B

1.815 1.839 1.861 1.866 1.867

C C c c C

3.898 3.955 3.987 3.992 3.993

D D D D D

3.995 4.465 4.548 4.552 4.552

A E E E

4.461 4.520 5.584 5.587 5 587

E E F F F

0 0~0001 0.001 o-01 0.1

0.794 0.855 0.966 O-990 0.993

A A A A A

0.909 I.069 1.257 1.274 1,275

B B B B B

0.969 1.228 1.856 1.865 1.866

C A C C c

1.166 1.382 l-886 3.991 3.992

A C A D D

1.333 1.399 2.286 4.551 4.551

B B B E E

1.784 1.904 3.164 5.101 5.586

C c C A F

0.005 0.005 0*005 0.005

0 0.001 0.01 o* 1

O-991 A 0.991 A 0.992 A 0.993 A

1.274 1.274 1,275 1.276

B B B B

1.865 1.866 1.866 l-867

C c c C

3.991 3.992 3.993 3.993

D D D D

4.551 4.551 4.552 4.552

E E E E

5.586 5.586 5.587 5.587

F F F F

0*020 0,020 0.020 0.020

0 0.001 0.01 0.1

O-991 A 0.991 A 0.992 A O-993 A

1,274 l-274 1.275 1.275

B B B B

1.865 1.865 1.866 l-866

C C c c

3.989 3.990 3.992 3.992

D D D D

4.548 4.549 4.551 4.551

E E E E

5.582 5.584 5,585 5.586

F F F F

352

J. M. SNYDER AND J. E. WILSON TABLE 2

First six non-dimensionalfree vibration frequencies for 80 = 30” (mode shapes are designated by the adjacent capital letter; see Figure 3)

0.05 0.05

O-778A 0.787 A 0.844 A

o-005

l-093 1.106 1.166 1.240 1.256

B B B B B

1.716 1.735 1.798 1.844 1.852

C c C C C

3.706 D 3.763 D

4.228 E 4.351 E

5.340 F

3.901 D 3.963 D 3.971 D

4.477 E 4.526 E 4.533 E

5.415 5.529 5.565 5.570

F F F F

0.05 0.05 0.05

0.005 0.005 0.005 0.005

0.05 0.05 0.05 0.05 0.05

0.020 0.020 0.020 0.020 0.020

0.761 0.829 0.939 0.968

A A A A

1.048 1.065 1.148 1.238 1.256

B B B B B

1.610 1.653 1.780 1.843 1.852

C C C C C

2,417 2.464 2,886 3.961 3.970

A A A D D

2.424 2.546 3.176 4.524 4.532

B B B E E

2.465 2.740 3.866 5.526 5.568

C c D A F

2.00 2-00 2.00 2.00

0.005 0.005 0.005 0.005

0.966 0.966 0.967 0.970

A A A A

1.253 1.253 1.254 1.257

B B B B

1.848 1.849 l-850 1.852

C C C C

3.965 3.966 3.968 3.971

D D D D

4.527 4.528 4.530 4.533

E E E E

5.564 5.565 5.567 5.570

F F F F

2.00 2.00 2.00 2.00

0.020 0.020 0.020 O-020

0.965 A 0.965 A O-966 A 0.970 A

1.252 1.252 1.253 1.256

B B B B

1.848 1.848 1.849 1.852

C C C C

3.964 3.964 3.967 3.970

D D D D

4.525 4.526 4.529 4.532

E E E E

5.562 5.563 5.566 5.568

F F F F

0.941 A 0.969 A

0,750 A

TABLE 3

First six non-dimensional free vibration frequencies for & = 45” (mode shapes are designated by the adjacent capital letter; see Figure 3)

c

c

0.05 0.05 0.05 0.05 0.05

0.005 0.005 0.005 0.005 0.005

0

0.05 0.05 0.05 0.05 0.05

0.020 0.020 0.020 0,020 0.020

2.00 2.00 2.00 2.00 2.00 2.00 2.00 2-00

B

81 O-624A 0.630 A

P2

84

!h

p6

l-600 1.618 l-684 1.793 1.826

c c C c C

3.426 D

4.038 E

5.112 F

0.671 A O-823 A 0.920 A

0.962 B 0.973B 1,023 B 1.157 B 1.219 B

3.475 3.689 3.893 3.933

D D D D

4.092E 4,299 E 4.468 E 4,498 E

5.179 5.388 5.518 5.539

F F F F

0 0~0001 0.001 0.01 0.1

0.614 0.620 0.663 0.820 0.920

A A A A A

0.945 0.956 1.010 1.154 1.219

B B B B B

1.566 1.585 1.665 1.791 1.825

C C C c C

3.223 3.302 3.629 3.889 3.932

D D D D D

3.764 E 3.875 E 4.238 E 4.463 E

4-267 4.295 4.543 5.513 5.538

A A A F F

0.005 0.005 0.005 0.005

0 0.001 0.01 0.1

0.923 0.923 0.924 0.929

A A A A

1.217 l-217 1.218 1,224

B B B B

1.821 1.821 1,822 1.827

C C C C

3.922 3.923 3.926 3.934

D D D D

5.529 5.530 5.534 5.540

F F F F

0.020 0.020 0.020 0.020

0 0.001 0.01 0.1

0.923 A O-923 A 0.924 A 0.929 A

1.217 1.217 1.218 1.223

B B B B

1.820 1.820 1.822 1.827

C C C C

3.920 3.921 3.924 3.933

D D D D

5.527 5.528 5.532 5.539

F F F F

0*0001 0.001 0.01 0.1

83

4.497 E

4.488 E 4,489 E 4.492 E 4,499 E

4.486 E 4.487 E 4.490 E 4.498 E

353

CURVED BEAMS TABLE 4

First six non-dimensional free vibration frequencies for tIO= 60” (mode shapes are designated by the adjacent capital; see Figure 3)

c

r

b

0.05 0.05 0.05 0.05 0.05

O-005 0.005 0.005 0.005 0.005

0 0.0001 0.001 0.01 0.1

0.495 A O-498 A 0.523 A 0.664 A 0.841 A

0.845 0.859 0.899 I.036 1.162

B B B B B

1.483 1.507 1.568 1.711 1.787

C c C c C

3.111 3.149 3.377 3.763 3.874

D D D D D

3.755 3.800 4.030 4,360 4.447

E E E E E

4.847 4,905 5.157 5,434 5.495

F F F F F

0.05 0.05 0.05 0.05 0.05

0.020 0.020 0.020 0.020 0.020

0 0~0001 0.001 0.01 0.1

0.491 0.494 0.519 0.661 0.840

A A A A A

0.838 0.851 0.892 1.033 1.161

B B B B B

1467 1.491 1.555 1.707 1.786

C c c c C

2.998 3.044 3.315 3.755 3.873

D D D D D

3.615 3.672 3.964 4.353 4.445

E E E E E

4.635 4.717 5.082 5.426 5.493

F F F F F

2.00 2.00 2.00 2.00

0.005 0.005 0.005 0.005

0 0.001 0.01 0.1

0.865 0.865 0.866 0.872

A A A A

1.169 1.170 1.171 1.177

B B B B

1.784 1.784 1.785 1.792

C C C C

3.862 3.863 3.866 3.879

D D D D

4.434 4.434 4.438 4,450

E E E E

5.481 5,481 5.486 5-497

F F F F

2.00 2.00 2.00 2.00

0.020 0.020 0.020 0.020

0 0.001 0.01 0.1

0.865 0.865 0.866 0.872

A A A A

1.169 1.169 1.170 1.177

B B B B

1.783 1.783 1.785 1.791

C C C c

3,860 3.861 3.865 3.878

D D D D

4.432 4.433 4.437 4.449

E E E E

5.478 5479 5.484 5.496

F F F F

PI

P4

P3

62

Is6

P5

----___ ----v % ,

(0)

(b)

(cl

(d)

-w * (e)

(f)

Figure 3. Typical developed elevations of deflection mode shapes for first six free vibration frequencies: (a) mode A; (b) mode B; (c) mode C; (d) mode D; (e) mode E; (f) mode F.

354

J. M. SNYDER

AND

J. E. WILSON

shapes. It is common design practice to assume D=O when working with these types of cross-sections. When c=O*OS, variations in b do have a significant effect on the first six frequencies and mode shapes. This lower value of (? is representative of open thin walled cross-sections. In general, as the torsion constant of a beam cross-section becomes smaller, the warping stiffness of the cross-section becomes more significant. Culver [2] and Tan and Shore [3] have shown that two free vibration frequencies correspond to each mode shape of a single span horizontally curved beam. This is a result of the coupling of flexural and torsional vibrations in a horizontally curved beam. Let F, be the ratio of the higher frequency associated with a mode shape to the lower frequency. F, will vary with the parameters of the beam. F, may be decreased by increasing c, which represents the polar mass moment of inertia, or by decreasing c or D, which represent the torsional and warping stiffnesses. Also, if the angular support spacing 8,, is increased, twisting motions are resisted to a greater extent by flexural action, causing F, to increase. If the polar radius of gyration of the beam is taken as zero, torsional vibrations do not occur and there will be only one free vibration frequency for each mode shape. Furthermore, Ayre and Jacobsen [I 1] observed that the free vibration frequencies of straight multiple span continuous beams occur in clusters, with N frequencies in each cluster for a beam with N spans. The data in Tables l-3 show that both of the above phenomena occur in horizontally curved, multiple span continuous beams. The free vibration frequencies occur in clusters with N mode shapes and 2N frequencies in each cluster. Also, the clusters may overlap. For both straight and curved continuous beams with equal spans, the first frequency of the nth cluster corresponds to the nth frequency of the simple span beam with a length equal to one span length of the continuous beam. For example, consider Table 2 with c= 0.05, c = 0.020 and d = 0 or O*OOOl. In both of these cases the six lowest free vibration frequencies are members of the first cluster; there are only three different mode shapes in the cluster. When D = O+OOl, the five lowest frequencies are members of the first cluster, but the sixth frequency is the first frequency of the second cluster. When D = 0.1, the first three frequencies are members of the first cluster and the next three frequencies are members of the second cluster. Also, whenever c= 2.0 or (, = 0.005, the first three frequencies are always members of the first cluster and the next three frequencies are always members of the second cluster. This behavior occurs because the ratio F, becomes larger as c and d increase and also increases as ( decreases. Tables 1 and 3 exhibit data similar to that shown by Table 2. For every case considered in Table 4, where &=60”, the first three frequencies are members of the first cluster and the next three frequencies are members of the second cluster. This occurs because, as discussed earlier, the ratio F, increases with increasing f3,,.

6. CONCLUSIONS

Our analysis affords a method for computing the free vibration frequencies and mode shapes of continuous multiple span horizontally curved thin walled beams. Numerical results are provided for the case of a continuous beam on simple supports with three equal spans; however, the method can accommodate different numbers of spans, unequal span lengths and various support types. The numerical results show that the mode shapes of multiple span curved beams occur in clusters and that each mode shape possesses two frequencies because of the coupling of torsional and flexural vibrations.

CURVED

BEAMS

355

REFERENCES 1. V. Z. VLASOV 1961 Thin-walled Beam Theory. Washington, D.C. : National Science Foundation. 2. C. G. CULVER 1967 Journal of the Structural Dioision, ASCE 93, 189-203. Natural frequencies of horizontally curved beams. 3. C. P. TAN and S. SHORE 1968 Journal ofthe Structural Division, ASCE 94, 761-781. Dynamic response of a horizontally curved bridge. 4. P. F. MCMANUS, G. A. NASIR and C. G. CULVER 1969 Journal of the Structural Division, of the art. ASCE 95, 853-870. Horizontally curved girders-state 5. TASK COMMIII-EE ON CURVED Box GIRDERS 1978 Journal of the Structural Division. ASCE 104, 1719-1739. Curved steel box-girder bridges: state-of-the-art. 6. J. F. WILSON, C. BIRNSTIAL, C. W. DOLAN, P. E. POTTER and P. R. SPENCER 1985 Journal of Structural Engineering 111, 1873-1898. Dynamics of elevated steel guideways. 7. C. P. TAN and S. SHORE 1968 Journal of the Structural Division, ASCE 94,2135-2151, Response of a horizontally curved bridge to moving load. 8. C. H. Yoo and J. P. FEHRENBACH 1981 Journal of the Engineering Mechanics Division, ASCE 107, 339-354. Natural frequencies of curved girders. 9. C. P. HEINS and M. A. SAHIN 1979 Journal of the Structural Division, ASCE 105, 2591-2600. Natural frequency of curved box girder bridges. 10. T. P. JOSEPHand J. F. WILSON 1980 Journal of the Engineering Mechanics Diuision, AXE 106, 255-272. Vibrations of curved spans for mass transit. 11. R. S. AYRE and L. S. JACOBSEN 1950 Journal of Applied Mechanics 17, 391-395. Natural frequencies of continuous beams of uniform span length.