Analytical formulation and solution of arches defined in global coordinates

Engineering Structures 60 (2014) 189–198

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Analytical formulation and solution of arches defined in global coordinates F.N. Gimena, P. Gonzaga, L. Gimena ⇑ Department of Engineering Projects, Public University of Navarre, Campus Arrosadia C.P. 31006, Pamplona, Navarre, Spain

a r t i c l e

i n f o

Article history: Received 13 May 2013 Revised 26 November 2013 Accepted 9 December 2013 Available online 23 January 2014 Keywords: Structural analysis Curved beam Arches Global coordinates

a b s t r a c t This work deals with the arch defined in global coordinates. It shows the procedure followed to obtain its formulation from the differential system of a curved beam. This procedure consists mainly in applying in the differential system, besides the habitual assumptions of the Strength of Materials, the simplifications of geometric conditions of the planar curves. The resulting model includes the analysis of arches of variable section under any action force, moment, rotation or displacement in its plane. The successive integration of the equations of the system permits obtaining the solution that represents the structural behavior of the arch under any type of support. Applying the boundary conditions of each problem it is obtained directly the exact analytical solution. Examples of calculus of parabolic arches are given to show the practical viability of the procedure followed. Results presented in graphs and tables are comparable with those obtained in the literature. The method is suitable for educational purposes. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction The mechanical behavior of a curved beam, applying the theories of Euler–Bernuolli or Timoshenko, is expressed usually through the equations of equilibrium, constitutive relationships and compatibility equations [1–4] or through compact equations of energy [5–12]. These two ways to annotate the structural behavior have permitted offering results, analytical and/or numerical, only for certain type of pieces. The planar pieces [13–16] more widely studied are with circular axis-line [17,18], parabolic [19– 21] and elliptical [22]. The twisted beam more studied is with circular helix axis-line [23–26]. Also the analysis of a curved beam could be concreted using a system of twelve linear ordinary differential equations [27–29]. This joint statement has permitted finding new resolution procedures and thereby broadening the types of pieces to study. But still analytical procedures are limited depending on the complexity of the shape of the axis-line and section of the piece, the characteristics of the material, the action system and the type of support. Both in the case of approaching the study of the pieces by a single differential system or through separate equations (equilibrium and compatibility), functions are expressed in natural coordinates using the Frenet reference system and the variable independent used is the arc length. Authors that subscribe this investigation, published a general formulation of curved beam elements expressed in the Frenet ⇑ Corresponding author. Tel.: +34 948 169225; fax: +34 948 169644. E-mail addresses: [email protected] (F.N. Gimena), lazaro.gimena@unavarra. es, [email protected] (L. Gimena). 0141-0296/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.engstruct.2013.12.004

frame and with independent variable the arc length [30] or with another variable [31], taken into account shearing deformations, variable section, asymmetrical section, generalized loads and any support condition. Later, this general formulation was expressed under a system of global coordinates and with the arc length as independent variable [32], with the objective of representing and interpreting more efficiently, results of internal forces and displacements. In this later document there were shown the advantages of this new formulation compared with the formerly. The principal advantage is the lower triangular nature of the system annotated, that permits the obtaining of analytical results through successive integrations row by row. As a novelty, the present paper focuses on solving the problem of the arch, because it is a less common type of piece, but of high interest in the design and in analysis of structures. From the differential model in global coordinates, we obtain the exact analytical solution. To do this, first a particular formulation is provided from the general system, representing the arch of symmetrical crosssection loaded into its plane taken into account the axial and shearing deformation and with whatever independent variable. Sometimes, the length of the arch that defines the axis-line is not the fundamental variable of the design. Is remarkable in the formulation and in the examples of this article, the employment of a generic parameter that can be particularized in function of the geometry of the piece element. The formulation is particularized and solved analytically in two cases of arches which its axis-line is parabolic without taken into account the shearing deformation. In the first example the section is of hyperbolic variation. Data have been chosen to be compared

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with those obtained in the cited literature. In the second example the section is constant and shows how the procedure permits to parameterize and tabulate the obtaining of results. In both cases graphs and tables are offered, useful to specify dimensions, testing and comparison of results.

with unit vectors i, j and k instead of unit vectors tangent t, normal n and binormal b, through the direction cosines:

2 3 2 ttx t 6 7 6 4 n 5 ¼ 4 tnx

tbx

b 2. Curved beam formula defined in Global Coordinates A curved beam is generated by a plane cross section whose centroid sweeps through all the points of an axis curve. The vector ra-

32 3

tty ttz i 76 7 tny tnz 54 j 5 tby tbz k

ð3Þ

Fig. 1 represents the axis-line of a curved beam and its systems of reference, Frenet–Serret and global, associated. The differential system that governs the structural behavior of a curved beam in global coordinates is expressed as [32]:

DV x

þqx ¼ 0 DV y

þqy ¼ 0 DV z

þqz ¼ 0

ttz V y þ tty V z þ DM x

þmx ¼ 0

þttz V x

ttx V z

tty V x þ ttx V y

þmy ¼ 0

þDM y þDMz

þmz ¼ 0

cxx Mx  cyx My  czx M z þ Dhx

Hx ¼ 0

cxy Mx  cyy My  czy M z

Hy ¼ 0

þDhy

cxz Mx  cyz M y  czz M z

Hz ¼ 0

þDhz

exx V x  eyx V y  ezx V z

ttz hy þ tty hz þ Ddx

exy V x  eyy V y  ezy V z exz V x  eyz V y  ezz V z

þttz hx tty hx þ ttx hy

dius r(s) expresses this curved line, where s length of the arch, is the independent variable. The reference coordinate system used here to represent the intervening known and unknown functions of the problem is the Frenet–Serret frame Ptnb. Its unit vectors tangent t, normal n and binormal b are:

ttx hz

 Dx ¼ 0 þDdy

Dy ¼ 0 þDdz Dz ¼ 0

The first six rows of the system (4) represent the equilibrium equations. The functions involved in the equilibrium equations are represented in Fig. 2 and its expressions are: Internal forces

V ¼ V xi þ V yj þ V zk ¼ ðt; n; bÞ ¼ ðDr; D2 r=jD2 rj; t ^ nÞ

ð4Þ

Z

r dAt þ

A

ð1Þ

Z

sn dAn þ

A

Z

sb dAb

ð5Þ

A

Internal moments where D is the derivative with respect to the parameter s. The Frenet–Serret equations [33] describe the movement of the frame system along the axis line. They are obtained with the vectors tangent, normal and binormal derivates with respect to the arch length. Its matricial expression is:

2 3 2 t 0 vðsÞ 6 7 6 D4 n 5 ¼ 4 vðsÞ 0 b 0 sðsÞ

32 3 t 76 7 sðsÞ 54 n 5 b 0

M ¼ Mx i þ My j þ Mz k Z Z ¼ ðsb n  sn bÞdAt þ A

A

rb dAn 

Z

rn dAb A

0

ð2Þ

where v(s) and s(s) are the flexure and torsion curvatures respectively. Assuming the habitual principles and hypotheses of the strength of materials [3] and considering the stresses associated with the normal cross-section (r, sn, sb), the geometric characteristics of the section are: area A(s), shearing coefficients an(s), anb(s), abn(s), ab(s) and moments of inertia It(s), In(s), Ib(s), Inb(s). E(s) and G(s) are the longitudinal and transversal elastic moduli which give the elastic properties of the material. Equilibrium and kinematics equations compose a system of twelve linear ordinary differential equations of a curved beam element [30]. It is possible to apply a change of basis in the referenced equations and express the functions in a global coordinate system Pxyz

Fig. 1. Curved beam and its reference systems.

ð6Þ

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Fig. 2. Functions of forces and moment.

Fig. 3. Functions of rotations and displacements.

Load force

3. Formula of arches defined in global coordinates

q ¼ qx i þ qy j þ qz k

ð7Þ

Load moment

m ¼ mx i þ my j þ mz k

ð8Þ

The last six rows of the system (4) represent the kinematics equations. The functions involved in the kinematics equations are represented in Fig. 3 and its expressions are: Rotations

h ¼ hx i þ hy j þ hz k

ð9Þ

Displacements

d ¼ dx i þ dy j þ dz k

ð10Þ

Load rotation

H ¼ Hx i þ Hy j þ Hz k

ð11Þ

Load displacement

D ¼ Dx i þ Dy j þ Dz k

ð12Þ

Besides, in formulation (4), the components that multiplied by the internal moments generate rotations could be expressed by: 2 3 2 3

cxx cxy cxz

ttx tnx tbx tny tby 7 5 ttz tnz tbz

6c c c 7 6 4 yx yy yz 5 ¼ 4 tty 2

czx czy czz

3 2 0 0 1=GIt     6 7 ttx 2 2 6 0 6 Ib =E In Ib  Inb Inb =E In Ib  Inb 7 6 74 tnx 4    5 tbx 0 Inb =E In Ib  I2nb In =E In Ib  I2nb

3

tty ttz tny tnz 7 5 tby tbz

ð13Þ

In this section it is exposed the procedure to follow and the formulation to analyze arches. Particularizing the differential system (4) for plane curves of symmetrical section loaded in its plane (arches): DV x DV y tty V x þ ttx V y þ DM z exx V x  eyx V y exy V x  eyy V y

þqx ¼ 0 þqy ¼ 0 þmz ¼ 0 z þ Dh  Hz ¼ 0 M z EIz þtty hz þ Ddx Dx ¼ 0 ttx hz þDdy Dy ¼ 0 ð15Þ

In the case of arches, the relation between the reference systems of Frenet–Serret and global could be annotated as:

2 3 2 t ttx 6 7 6 4 n 5 ¼ 4 tnx b 0

32 3

tty 0 i 6 7 tny 0 7 54 j 5 0

1

It is observed that binormal and z axis coincide. The Fig. 4 represents a generic arch loaded onto his plane. In matricial notation, the system (15) is obtained:

DeðsÞ ¼ ½TD ðsÞeðsÞ þ qD ðsÞ

exx 6 4 eyx ezx 2 ttx 6 4 tnx tbx

exy eyy ezy tty tny tby

3

2

32

eðsÞ ¼ fV x ; V y ; Mz ; hz ; dx ; dy gT

3

exz ttx tnx tbx 1=EA 0 0 6 76 7 eyz 7 an =GA anb =GA 5 5 ¼ 4 tty tny tby 54 0 ezz 0 anb =GA ab =GA ttz tnz tbz 3 ttz tnz 7 5 tbz ð14Þ

This new general expression (4), which simulates the structural behavior of the linear element, has a lower-triangular form. This important property permits to solve analytically the differential equation system using successive integrations row by row.

ð17Þ

Vectors involved in the above expression of the arch are: effect vector

Also, the components that multiplied by internal forces produce displacements are:

2

ð16Þ

k

Fig. 4. Generic arch with punctual and distributed load.

ð18Þ

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F.N. Gimena et al. / Engineering Structures 60 (2014) 189–198

load vector

qD ðsÞ ¼ fqx ; qy ; mz ; Hz ; Dx ; Dy g

T

differential transfer matrix

2

0

6 0 6 6 6 tty ½TD ðsÞ ¼ 6 6 0 6 6 4 exx

exy

ð19Þ

0

0

0 0

0

0

0

ttx 0

0

0 0

0 07 7 7 0 07 7 0 07 7 7 0 05

eyx eyy

1 EIz

0

tty

0

ttx

4.1. Parabolic arch with variable cross-section under concentrated force

3

0

ð20Þ

2

Integrating the above system directly, row by row, the general solution can be written:

ð21Þ

where the vector of arbitrary coefficients is:

C ¼ fC 1 ; C 2 ; C 3 ; C 4 ; C 5 ; C 6 gT

ð22Þ

With the proper change of variable of the arch length s by the parameter k, Eq. (17), yields:

Dk eðkÞ ¼ ½TD ðkÞDk seðkÞ þ qD ðkÞDk s

ð23Þ

In the same manner, integrating the above system (7), general solution is as follows:

eðkÞ ¼ ½TT ðkÞC þ qT ðkÞ

ð24Þ

Let suppose a punctual load applied at a generic point A (see Fig. 4); the equilibrium and kinematics relate the effects (forces and displacements) at this point:

eðkA;I Þ þ eðkA;II Þ þ Q A ¼ 0

The equation that represents the geometry of a parabolic arch in terms of the height f and span l is given by:

y ¼ 4fx =l

0 0

eðsÞ ¼ ½TT ðsÞC þ qT ðsÞ

parabolic shaped arches, with variable cross-section and different heights and spans.

ð25Þ

ð26Þ

Solution given in Eq. (24) is particularized in the extremes of both parts: for the first part kI P k P kA 1

eðkA;I Þ ¼ ½TT ðkA Þ½TT ðkI Þ ðeðkI Þ  qT ðkI ÞÞ þ qT ðkA Þ ¼ ¼ ½TT ðkA ; kI ÞðeðkI Þ  qT ðkI ÞÞ þ qT ðkA Þ

ð27Þ

for the second part kA P k P kII

eðkA;II Þ ¼ ½TT ðkA Þ½TT ðkII Þ1 ðeðkII Þ  qT ðkII ÞÞ þ qT ðkA Þ ¼ ¼ ½TT ðkA ; kII ÞðeðkII Þ  qT ðkII ÞÞ þ qT ðkA Þ

ð32Þ

In parametric equations:

xðkÞ ¼ pk; yðkÞ ¼ pk2 =2; zðkÞ ¼ 0

ð33Þ

being p = l2/8f and k = 8f x/l2, in this example for 1 = kI P k P kII = 1. A punctual load Q0 = {0,Qy0,0,0,0,0}T is applied at the point kA = k0 = 0. pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Direction cosines of the curve are ttx ¼ 1= k2 þ 1 and pffiffiffiffiffiffiffiffiffiffiffiffiffiffi tty ¼ k= k2 þ 1. The derivative of the arch length s with respect to the paramepffiffiffiffiffiffiffiffiffiffiffiffiffiffi ter k is Dk s ¼ p k2 þ 1. Properties of the variable cross-section (with height h(k) = h and pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi width bðkÞ ¼ b0 k2 þ 1) are: A0 = b0h, AðkÞ ¼ A0 k2 þ 1, an = 0, pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Iz0 = b0h3/12, Iz ðkÞ ¼ Iz0 k2 þ 1 and i0 ¼ Iz0 =A0 . Fig. 5 show the parabolic arch which is studied in this example. The differential system (15) with this geometry will be:

Dk V x

¼0

In this equation the vector of punctual action is annotated as:

Q A ¼ fQ xA ; Q yA ; M zA ; 0; 0; 0gT

2

Dk V y pkV x

¼0

þpV y

þDk M z  EIpz0 M z

p pk  ðk2 þ1ÞEA V x  ðk2 þ1ÞEA Vy 0

0

2

pk pk  ðk2 þ1ÞEA V x  ðk2 þ1ÞEA Vy 0

0

¼0 þDk hz

¼0

þpkhz þDk dx

¼0

phz

þDk dy ¼ 0

Its direct integration, row by row, gives:

ð28Þ

Substituting former values (27) and (28) in Eq. (25), it is obtained:

 ½TT ðkA ; kI ÞðeðkI Þ  qT ðkI ÞÞ þ ½TT ðkA ; kII ÞðeðkII Þ  qT ðkII ÞÞ þ QA ¼ 0

ð29Þ

The former expression represents an algebraic system of six equations with twelve unknowns, six from both extremes of the arch. Six support conditions are introduced in Eq. (29) to obtain values at both extremes initial e(kI) and final e(kII). Once knowing these values, the exact solution in both parts is written:

eðkÞ ¼ ½TT ðk; kI ÞðeðkI Þ  qT ðkI ÞÞ þ qT ðkÞ for kI P k P kA

ð30Þ

eðkÞ ¼ ½TT ðk; kII ÞðeðkII Þ  qT ðkII ÞÞ þ qT ðkÞ for kA P k P kII

ð31Þ

4. Examples In the previous section, a general procedure has been presented for analyzing arches with distributed and a punctual load. In the following examples, the geometry of these arches is restrained to

Fig. 5. Pinned parabolic arch with variable cross-section.

ð34Þ

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F.N. Gimena et al. / Engineering Structures 60 (2014) 189–198

V x ðkÞ ¼

C1

V y ðkÞ ¼

C2

M z ðkÞ ¼

pk2 2

hz ðkÞ ¼

p2 k3 6EIz0

C1 C1

p½30i20 arctan kp2 k5 

dx ðkÞ ¼

30EIz0 p½

dy ðkÞ ¼

12i20

C1

lnðk2 þ1Þþp2 k4  24EIz0

pkC 2

þC 3

p2 k2  2EI z0

þ EIpkz0 C 3

þ

C1

þ

C2

p½4i20 lnðk2 þ1Þþp2 k4  8EIz0

p2 k3

p½



2 2

p k þ 2EI C 3 þ pkC 4 z0

C2

6EIz0

2

 3EIz0 C 3  pk2 C 4 þ C 5

C2

6i20 ðkarctan kÞp2 k3

Following the procedure explained in Section 3, considering the parabolic arch pinned in both extremes, values at both ends of the first part of the curve are given:

eðkI Þ ¼



eðk0;I Þ ¼ 0; 0;

5½12i20 ln 25p2 

Q y0 ; 12 Q y0 ; 0;

4½15i20 pþ8p2 



5½12i20 ln 25p2  4½15i20 pþ8p2 

p2 ½15i20 ð4 ln 2þ3pÞp2  12EIz0 ½15i20 pþ8p2 

Q y0 ; 12 Q y0 ; 

p½60i20 ðln 2þpÞþ7p2  8½15i20 pþ8p2 

p½p4 þ8p2 i20 ð6pþ25 ln 2þ16Þ60i40 ðp2 4pþ4ðln 2Þ2 Þ 32EIz0 15i20

½

pþ8p2 

Q y0

Q y0 ; 0; 0

Q y0 ;

¼0 ¼0 þDk M z

þpV y

¼0

 EIpz0 Mz

ð37Þ

¼0

þDk hz þpkhz þ Dk dx

¼0

þ Dk dy

phz

¼0

Its direct integration, row by row, gives:

V x ðkÞ ¼

C1

V y ðkÞ ¼

C2

M z ðkÞ ¼

pk2 2

hz ðkÞ ¼

p2 k3 6EIz0

C1

pkC 2

þC 3

C1

p2 k2  2EI z0

þ EIpkz0 C 3

p3 k5

p3 k4

C2

p2 k3

dx ðkÞ ¼

 30EIz0 C 1

þ 8EIz0 C 2

 3EIz0 C 3 

dy ðkÞ ¼

p3 k4 24EIz0

p3 k3  6EI z0

p2 k2 þ 2EI z0

C1

C2

ð38Þ

þ C4 2

pk 2



C4 þ C5

C 3 þ pkC 4



T 25 1 p2 Q y0 ; Q y0 ; 0; Q y0 ; 0; 0 32 2 96EIz0

ð39Þ

 T 25 1 7p p3 eðk0;I Þ ¼  Q y0 ; Q y0 ;  Q y0 ; 0; 0; Q y0 32 2 64 256EIz0

Neglecting the axial deformation, the differential system Eq. (34), yields:

pkV x

In the same way, values at both ends of the first part of the curve are obtained:

eðkI Þ ¼

ð36Þ

Dk V y

þ C6

T

T

Dk V x

ð35Þ

þ C4

In Fig. 6, results of forces, moment, rotation and displacements, particularized for data given in Benedetti and Tralli [20], l = 42 m; f = 10.5 m; b0 = 4 m; h = 1 m; E = 10,000 MPa; Qy0 = 200 kN, are plotted with and without axial deformation. This graph shows how the hypothesis of neglecting axial deformation is acceptable, and results can be compared with those obtained in the literature. In the Table 1, are represented for different values of the parameter k, the functions graphed in the above Fig. 6. Taking as initial values the data of the Fig. 6 and maintaining the length l = 42 m, it is presented in Table 2, results of the effects Vx, Mz(0) and dy(0) for different values of slenderness and shallowness, with and without axial deformation. Knowing that the slenpffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi ffi derness is K ¼ l= Iz0 =A0 ¼ 2 3l=h it has been developed the Table 2 for five values of height: h = 0.5, h = 0.707, h = 1, h = 1.414 and h = 2, and five values of shallowness: f/l = 0.125, f/l = 0.177, f/l = 0.25, f/l = 0.354 and f/l = 0.5. This Table 2 also shows how the hypothesis of neglecting axial deformation is acceptable. 4.2. Parabolic arch with constant cross-section under distributed force The equation that represents the geometry of a parabolic arch in terms of the height f and span l is given by:

þ C6

2 2

y ¼ f =l ½l  4x2 

ð40Þ

Table 1 Forces, moment, rotation and displacements of the parabolic arch with variable cross-section. k

Mz (kN m)

hz (103 rad)

dx (103 m)

dy (103 m)

Mz (kN m)

hz (103 rad)

dx (103 m)

dy (103 m)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

461.71 268.10 107.25 20.84 116.16 178.71 208.50 205.53 169.78 101.27 0.00

0.00 0.23 0.34 0.37 0.33 0.23 0.11 0.03 0.15 0.23 0.27

0.00 0.01 0.09 0.28 0.53 0.78 0.97 1.01 0.87 0.51 0.00

2.34 2.08 1.46 0.69 0.05 0.65 1.01 1.10 0.92 0.53 0.00

459.37 265.78 105.00 22.97 118.13 180.47 210.00 206.72 170.63 101.72 0.00

0.00 0.23 0.34 0.37 0.32 0.22 0.10 0.03 0.15 0.24 0.28

0.00 0.02 0.11 0.30 0.56 0.81 1.00 1.04 0.89 0.53 0.00

2.17 1.91 1.30 0.54 0.19 0.77 1.11 1.18 0.98 0.55 0.00

Vx(kN) Vy(kN)

With axial deformation 156.03 100.00

Without axial deformation 156.25 100.00

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F.N. Gimena et al. / Engineering Structures 60 (2014) 189–198

Table 2 Values of the effect in a parabolic arch with variable cross-section for different data of slenderness and shallowness (f/l). Vx(kN)

With axial deformation

Without axial deformation

Mz(0) (kN m) dy(0) (103 m)

h = 0.50 K = 290.98

f/l = 0.125

312.06 461.69 18.47

312.50 459.37 17.36

311.62 464.01 6.92

f/l = 0.177

220.81 460.54 17.96

220.97 459.37 17.36

f/l = 0.25

156.19 459.96 17.70

f/l = 0.354

f/l = 0.5

h = 0.707 K = 205.76

h=1 K = 145.49

h=1 K = 145.49

h = 1.414 K = 102.88

h=2 K = 72.75

312.50 459.37 6.14

310.74 468.61 2.72

312.50 459.37 2.17

309.00 477.76 1.15

312.50 459.37 0.77

305.57 495.76 0.54

312.50 459.37 0.27

220.66 461.70 6.56

220.97 459.37 6.14

220.35 464.01 2.47

220.97 459.37 2.17

219.72 468.63 0.98

220.97 459.37 0.77

218.49 477.80 0.42

220.97 459.37 0.27

156.25 459.37 17.36

156.14 460.55 6.38

156.25 459.37 6.14

156.03 461.71 2.34

156.25 459.37 2.17

155.80 464.05 0.89

156.25 459.37 0.77

155.36 468.70 0.36

156.25 459.37 0.27

110.47 459.67 17.58

110.49 459.37 17.36

110.45 459.97 6.29

110.49 459.37 6.14

110.41 460.57 2.28

110.49 459.37 2.17

110.32 461.76 0.84

110.49 459.37 0.77

110.16 464.14 0.32

110.49 459.37 0.27

78.12 459.53 17.51

78.13 459.37 17.36

78.11 459.69 6.24

78.13 459.37 6.14

78.10 460.00 2.24

78.13 459.37 2.17

78.07 460.62 0.82

78.13 459.37 0.77

78.01 461.86 0.31

78.13 459.37 0.27

Fig. 6. Forces, moment, rotation and displacements of the parabolic arch with variable cross-section.

2

In parametric equations:

xðkÞ ¼ pk;

2

yðkÞ ¼ f  pk =2;

zðkÞ ¼ 0

ð41Þ

being p = l2/8f and k ¼ 8fx=l . Direction cosines of the curve are pffiffiffiffiffiffiffiffiffiffiffiffiffiffi tty ¼ k= k2 þ 1.

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ttx ¼ 1= k2 þ 1 and

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F.N. Gimena et al. / Engineering Structures 60 (2014) 189–198

where 

 qffiffiffiffiffiffiffiffiffiffiffiffiffi



qffiffiffiffiffiffiffiffiffiffiffiffiffi

a1 ðkÞ ¼ 2k2 þ 1 k k2 þ 1  ln k þ k2 þ 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  3 k2 þ 1  qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi a3 ðkÞ ¼ k k2 þ 1 þ ln k þ k2 þ 1

a2 ðkÞ ¼

 qffiffiffiffiffiffiffiffiffiffiffiffiffi









qffiffiffiffiffiffiffiffiffiffiffiffiffi

a4 ðkÞ ¼  4k4 þ 4k2 þ 3 k k2 þ 1 þ 3 2k2 þ 64i2z =p2 þ 1 ln k þ k2 þ 1

qffiffiffiffiffiffiffiffiffiffiffiffiffi k2 þ 1   qffiffiffiffiffiffiffiffiffiffiffiffiffi    qffiffiffiffiffiffiffiffiffiffiffiffiffi a6 ðkÞ ¼ 2k2  1 k k2 þ 1 þ 4k2 þ 1 ln k þ k2 þ 1  qffiffiffiffiffiffiffiffiffiffiffiffiffi   qffiffiffiffiffiffiffiffiffiffiffiffiffi k2 þ 1  15kln k þ k2 þ 1 a7 ðkÞ ¼ 6k4 þ 7k2 þ 16 15i2z =p2 þ 1   qffiffiffiffiffiffiffiffiffiffiffiffiffi    qffiffiffiffiffiffiffiffiffiffiffiffiffi a8 ðkÞ ¼ 2k2 þ 12i2z =p2  5 k k2 þ 1  12i2z =p2 þ 3 ln k þ k2 þ 1  qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffi a9 ðkÞ ¼ k2  2 k2 þ 1 þ 3kln k þ k2 þ 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    3 a10 ðkÞ ¼ 3k2  2 k2 þ 1  qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffi a11 ðkÞ ¼ 6k4  2k2  315i2z =p2 k  8 k2 þ 1 þ 315i2z =p2 ln k þ k2 þ 1  qffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffi a12 ðkÞ ¼ 8k5 þ 6k3  240i2z =p2 k2  17k þ 480i2z =p2 k2 þ 1  15ln k þ k2 þ 1 

a5 ðkÞ ¼ k4 þ 2k2 þ 15i2z =p2 þ 1

Fig. 7. Parabolic arch with constant cross-section vertically loaded.

2

with iz ¼ Iz =A. The general solution in function of the arbitrary coefficients is obtained integrating row by row the differential system (42). Then, support conditions could be applied to determine the particular solution. Neglecting the axial deformation, the differential system Eq. (42), yields: Dk V x

¼0 pqy ¼ 0 Dk V y ¼0 pkV x pV y þDk M z pffiffiffiffiffiffiffiffi p k2 þ1  EIz M z þDk hz ¼0 ¼0 pkhz þ Dk dx þDk dy ¼0 phz

Fig. 8. Bi-fixed parabolic arch with constant cross-section.

The derivative of the pffiffiffiffiffiffiffiffiffiffiffiffiffi ffi arch length s with respect to the parameter k is Dk s ¼ p k2 þ 1. A distributed load is applied qy in the y direction and the section remains constant. Fig. 7 show the parabolic arch which is studied in this section. The differential system (15) in this example will be: Dk V x Dk V y pV y

pkV x

pqy þDk M z pffiffiffiffiffiffiffiffi 2  p EIkz þ1 M z þ Dk hz

p Vx  pffiffiffiffiffiffiffiffi 2

pk þ pffiffiffiffiffiffiffiffi Vy 2

pk þ pffiffiffiffiffiffiffiffi Vx 2

pk  pffiffiffiffiffiffiffiffi Vy 2

k þ1EA

2

k þ1EA

¼0

pkhz þ Dk dx

k þ1EA

phz

k þ1EA

¼0 ¼0 ¼0

¼0 ¼0

þDk dy

ð42Þ

Its direct integration, row by row, gives: V x ðkÞ ¼

C1

V y ðkÞ ¼

C2

þpkqy

M z ðkÞ ¼

2  pk2

hz ðkÞ ¼

a1 ðkÞ  p16EI C1 z

a2 ðkÞ  p 3EI C2 z

dx ðkÞ ¼

p3 a4 ðkÞ C1 192EIz

a5 ðkÞ  p15EI C2 z

dy ðkÞ ¼

a7 ðkÞ  p240EI C1 z

C1

2

3

pkC 2

2 3

þ p 3k qy

þC 3

2

a3 ðkÞ þ p2EI C3 z

3

a6 ðkÞ þ p16EI C 3 þ pk2 C 4 þ C 5 z

3

a9 ðkÞ þ p 6EI C 3 þ pkC 4 z

a8 ðkÞ þ p24EI C2 z

2

2

p3 a10 ðkÞ qy 45EIz

þ C4 2

þ C6

4

a11 ðkÞ þ p630EI qy z 4

a12 ðkÞ  p720EI qy z

ð43Þ

ð44Þ

Its direct integration, row by row, gives: V x ðkÞ ¼ C 1 V y ðkÞ ¼

C2

þpkqy

M z ðkÞ ¼

2  pk2 C 1

pkC 2

þC 3

þ p 3k qy

hz ðkÞ ¼

2 a ðkÞ 1  p16EI C1 z

2 a ðkÞ 2  p 3EI C2 z

a3 ðkÞ þ p2EI C3 z

p3 a10 ðkÞ qy 45EIz

p3 b2 ðkÞ

p2 a6 ðkÞ

dx ðkÞ ¼ dy ðkÞ ¼

p3 b1 ðkÞ 192EIz

C1

3 b ðkÞ 3  p240EI C1 z



15EIz

C2 þ

3 b ðkÞ 4 þ p24EI C2 z

16EIz

2 3

þC 4 2

C 3 þ pk2 C 4 þ C 5

2 a ðkÞ 9 þ p 6EI C3 z

þpkC 4

ð45Þ

p4 b5 ðkÞ

þ 630EIz qy 4

b6 ðkÞ þ C 6  p720EI qy z

where  qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi



b1 ðkÞ ¼  4k4 þ 4k2 þ 3 k k2 þ 1 þ 3 2k2 þ 1 ln k þ k2 þ 1 qffiffiffiffiffiffiffiffiffiffiffiffiffi

b2 ðkÞ ¼ k4 þ 2k2 þ 1 k2 þ 1  qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi

b3 ðkÞ ¼ 6k4 þ 7k2 þ 16 k2 þ 1  15kln k þ k2 þ 1  qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi

b4 ðkÞ ¼  2k2 þ 5 k k2 þ 1  3ln k þ k2 þ 1 qffiffiffiffiffiffiffiffiffiffiffiffiffi

b5 ðkÞ ¼ 6k4  2k2  8 k2 þ 1  qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi

b6 ðkÞ ¼ 8k5 þ 6k3  17k k2 þ 1  15ln k þ k2 þ 1

In Fig. 8 a particular case of a bi-fixed parabolic arch is represented with next data, l = 40 m; f = 10 m; b = 2 m; h = 1 m; E = 20,000 MPa; qy = 5 kN/m. Results of forces, moment, rotation and displacements are plotted in Fig. 9 with and without axial deformation.

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Fig. 9. Forces, moment, rotation and displacements of the parabolic arch with constant cross-section.

Table 3 Forces, moment, rotation and displacements of the parabolic arch with constant cross-section. k

Vy (kN)

Mz (kN m)

hz (103 rad)

dx (103 m)

dy (103 m)

Vy (kN)

Mz (kN m)

hz (103 rad)

dx (103 m)

dy (103 m)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 10 20 30 40 50 60 70 80 90 100

3.23 3.15 2.88 2.44 1.82 1.03 0.05 1.10 2.42 3.92 5.60

0.000 0.002 0.004 0.005 0.007 0.008 0.008 0.008 0.006 0.004 0.000

0.000 0.005 0.009 0.012 0.013 0.012 0.008 0.004 0.001 0.003 0.000

0.135 0.131 0.120 0.103 0.082 0.060 0.038 0.019 0.005 0.002 0.000

0 10 20 30 40 50 60 70 80 90 100

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

Vx(kN)

With axial deformation 99.12

This graph shows how the hypothesis of neglecting axial deformation is acceptable to obtain the values of the internal forces Vx and Vy. In the Table 3, are represented for different values of the parameter k, the functions graphed in the above Fig. 9. Taking as initial values the data of the Fig. 9 and maintaining the length l ¼ 40 m, it is presented in Table 4, results of the effects Vx, Mz(0) and dy(0) for different values of slenderness and shallowness, with and without axial deformation. Knowing that

Without axial deformation 100.00

pffiffiffiffiffi pffiffiffiffiffiffiffiffiffi the slenderness is K ¼ l= Iz =A ¼ 2 3l=h it has been developed the Table 4 for five values of height: h = 0.5, h = 0.707, h = 1, h = 1.414 and h = 2, and five values of shallowness: f/l = 0.125, f/l = 0.177, f/l = 0.25, f/l = 0.354 and f/l = 0.5. 5. Conclusions Normally, authors use the mobile Frenet system of reference with flexure and torsion curvatures to approach the structural

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F.N. Gimena et al. / Engineering Structures 60 (2014) 189–198 Table 4 Values of the effect in a parabolic arch with constant cross-section for different data of slenderness and shallowness (f/l). Vx(kN)

With axial deformation

Without axial deformation

Mz(0) (kN m) dy(0) (103 m)

h=0.5 K = 277.13

f/l = 0.125

198.18 3.13 0.82

200.00 0.00 0.00

196.39 6.21 0.58

200.00 0.00 0.00

192.89 12.21 0.40

200.00 0.00 0.00

186.23 23.64 0.27

200.00 0.00 0.00

174.12 44.46 0.18

200.00 0.00 0.00

f/l = 0.177

140.78 1.59 0.45

141.42 0.00 0.00

140.15 3.17 0.32

141.42 0.00 0.00

138.90 6.28 0.22

141.42 0.00 0.00

136.45 12.37 0.16

141.42 0.00 0.00

131.78 24.02 0.11

141.42 0.00 0.00

f/l = 0.25

99.78 0.81 0.27

100.00 0.00 0.00

99.56 1.62 0.19

100.00 0.00 0.00

99.12 3.23 0.14

100.00 0.00 0.00

98.24 6.42 0.09

100.00 0.00 0.00

96.54 12.68 0.07

100.00 0.00 0.00

f/l = 0.354

70.63 0.42 0.19

70.71 0.00 0.00

70.56 0.84 0.13

70.71 0.00 0.00

70.40 1.68 0.09

70.71 0.00 0.00

70.10 3.36 0.07

70.71 0.00 0.00

69.49 6.68 0.05

70.71 0.00 0.00

f/l = 0.5

49.97 0.22 0.15

50.00 0.00 0.00

49.95 0.45 0.11

50.00 0.00 0.00

49.89 0.89 0.08

50.00 0.00 0.00

49.78 1.78 0.05

50.00 0.00 0.00

49.57 3.56 0.04

50.00 0.00 0.00

h = 0.707 K = 195.96

h=1 K = 138.56

problem of curved beams elements, when trying to reach exact analytical results. Otherwise, they can use different numerical approximations or simplifications in geometry (considering the curved composed of straight beams) to obtain acceptable results. In this article, this problem is approached analytically with differential equations and solved using the global reference system. The system of equations to determine the internal forces and displacements of curved elements is presented. The method considers in general, a twisted element with varying cross-sectional area with generalized loads and different boundary conditions. This article applies the general differential formulation to the case of the arch, both for its high interest in structural analysis as for to be covered its calculus from an analytical point of view. Because of the employment of a system in global coordinates, it is obtained a differential system lower triangular of order six. This system can be solved through the simple successive integration of its equations. Once integrated, the notation of the general solution is important, since if you eliminate the integration constants and apply the support conditions for each case, you get a six-order algebraic system, which provides the solution to the problem. On one hand, the application over the annotated system of the developed procedure of calculus permits to approach the analysis of any type of arches of symmetrical cross-section expressed in its parametric equations. – It has not been necessary neither for the approach nor for the calculus, the use of energetic theorems or procedures derivatives thereof. – Normally, methods of calculus distinguish between arches statically determinate or not. The solution procedure developed is independent of the support conditions of the element to be calculated. – In the same manner, is possible to take into account or not, through the same analytical procedure, the deformation produced in the arch by the axial and, or shearing effort. – In global coordinates, the results of displacement obtained are referred to the horizontal and vertical axes of the design, been directly suitable to the structural testing. On the other hand, from the two examples developed in this article (arch of axis-line parabolic with cross-section variable or constant, and under a punctual or uniform force load) next conclusions are deduced:

h = 1.414 K = 97.98

h=2 K = 69.28

– In the first example (parabolic arch of hyperbolic variable section with punctual load) results of internal forces and displacements showed are coincident with those cited in the literature. – In both examples, been offered the exact analytical solution, is possible to represent graphically the results as continuous functions in all the points of the axis-line of the arch without discretizations. – Tables of results, permit the designer to consider or not the deformations produced by the efforts. Presented formulation, and the solution method developed and applied in the examples, can be a useful tool in the design and structural testing of arches, since is possible to parameterize and tabulate its calculus in function of the conditions of shape, material, type of load and support conditions.

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