optimal arch shape solution under static vertical loads

Acta Mech 225, 679–686 (2014)DOI 10.1007/s00707-013-0985-0

Giuseppe Carlo Marano · Francesco Trentadue · Floriana Petrone

Optimal arch shape solution under static vertical loads

Received: 1 May 2013 / Published online: 20 September 2013© Springer-Verlag Wien 2013

Abstract In this paper, a new analytical solution for the optimal shape of a plane-statically determined archsubjected to uniform vertical loads is presented. The classical problem of a catenary subjected to the self-weightis extended to an inverted catenary subjected to the self-weight and to a constant vertical load distribution.In this condition, the authors demonstrate that a class of analytical solutions exists and that unlike previouslyproposed solutions it corresponds to the minimum ratio of the self-weight of the arch to the total applied load.Finally, existence conditions for such a solution are derived.

1 Introduction

Arches play a fundamental role in structural engineering as well as in architecture, as structural elements pro-viding esthetics and strength by means of their geometry. They have been widely used in structural engineeringfrom the early Roman Empire to nowadays, since they are still one of the most handsome forms of bridgedesign and at the same time offer resistance and safety.

Because of their importance, many researches about this topic have been developed from the beginning ofcontinuum mechanics to define optimal arches shapes under different load conditions and referring to differentmaterials.

Many analytical methods for arches have been developed starting from statically determinate two- and three-hinged arches to extensive investigations presented in [1] and [2] where equations for determining bendingand axial stresses in given geometry arches are provided. A fundamental reference is Timoshenko and Young’swork [3] where the advantages of funicular design in which an arch geometry is defined so that under deadloads the arch remains in a momentless state are pointed out. It is important to underline that momentless archsolutions depend on load conditions. A momentless arch solution under live loads presents many advantages,and about this topic, Tadjbakhsh [4] proposed a momentless solution with a uniform compression in anysections, considering also buckling effects. The solution is given for symmetric configurations of archessubjected to vertically uniform loads. More recently, Serra [5] proposed an analytical solution obtained bymeans of a simplified approach, considering also the arch self-weight as vertical load. Moreover, the solutiondepends on the axial forces acting horizontally at the top of the arch and on the area of the cross section where

G. C. Marano · F. TrentadueDepartment of Civil Engineering and Architectural Science, Technical University of Bari,via Orabona, 10, 70126 Bari, Italy

F. Petrone (B)University of California, Davis, One Shields Avenue, Davis, CA 95616, USAE-mail: [email protected]

F. PetroneSapienza, University of Rome, via A. Gramsci 53, 00197 Rome, Italy

680 G. C. Marano et al.

the subject matter axial force is applied. Under this condition, not finite solutions are given. All the solutionsdepend on this value that can be selected without any restrictions (has only to be a positive value) so that thefinal arch shape is a function of this parameter.

In this paper, the optimal arch shape is completely defined by minimizing the total structural volume. Theproblem is laid down so that it is a generalization of the dual problem of finding equal strength catenariessubjected to self-weight that was solved by Gilbert [6]. In this paper, this situation has been extended to the caseof a uniform compressed inverted catenary subjected to external constant loads in addition to the self-weight.

The problem consists of finding the optimal arch shape and the optimal cross section—variable along thearch—to have a momentless configuration with a uniform compression in any section. As stated previously,vertical dead loads and self-weight are taken into account, so that the final configuration is obtained byminimizing the ratio between the arch self-weight, which is related to the volume, and the total applied load.When existence conditions for such a solution are satisfied, a single solution is found.

2 Problem formulation

The problem consists of determining the optimal profile y(x) and the cross-sectional area A(x) of a staticallydeterminate arch (Fig. 1) subjected to the self-weight and to an external load, defined for unit horizontal lengthp(x). The aim is finding the arch shape of equal strength such that all the cross sections are subjected tothe constant stress σ , and the ratio between the arch self-weight and the global external load

∫ L0 p(x)dx is

minimum.Figure 2 shows an arch section having a horizontal distance x from the origin. Generally speaking, in arches

subjected only to vertical loads, the x section is subjected to a horizontal internal force that, for equilibriumreasons, is equal to H(x), a vertical internal force V (x), and a bending moment m(x).

The arch is made up of a homogeneous material with constant specific gravity γ , so the arch self-weightfor unit horizontal length q(x) is

q(x) = γ A(x)

cos θ(x)= γ A(x)

(1 + y,x (x)2) 1

2 (1)

where cos θ (x) is the horizontal projection of an arch element of unit length. In an arch having all sectionsuniformly compressed, the bending moment is zero in each section and the following condition holds:

⎧⎪⎪⎨

⎪⎪⎩

m (x) = −⎡

⎣H y − VAx +x∫

0

[

p (τ ) + γ A (τ )(1 + y,τ (τ )2) 1

2

]

(x − τ)dτ

⎤

⎦ = 0

x ∈ [0, L]

. (2)

Fig. 1 Arch structure with a variable section subject to a distributed vertical load p = p(x)

Optimal arch shape solution 681

Fig. 2 Internal forces acting on a generic section

The differentiation of Eq. (2) with respect to x gives the following expression:

m,x (x) = −⎡

⎣H y,x (x) − VA +x∫

0

p (τ )dτ +x∫

0

γ A (τ )(1 + y,τ (τ )2) 1

2 dτ

⎤

⎦ = 0, (3)

and a further differentiation gives the following:

m,xx (x) = −[

H y,xx (x) + p (x) + γ A (x)(1 + y,x (x)2) 1

2

]

= 0. (4)

This last equation together with the following boundary conditions:{

m (0) = 0m (L) = 0 (5)

allows determining the arch shape, which gives zero bending moment and zero shear stress in each crosssection under the given external load p (x).

Further, it must be imposed that each cross section is subject to the constant axial compressive stress σ , sothat

σ = N (x)

A (x)= H

A (x) cos θ (x)= H

(1 + y,x (x)2) 1

2

A (x), (6)

Then, the cross-sectional area A (x) is determined by the condition

A (x) = H

σ

(1 + y,x (x)2) 1

2 . (7)

By substituting Eq. (7) into Eq. (6), the following expression holds:

q (x) = γ

σH

(1 + y2) = H

h

(1 + y2) (8)

where the constant h = σ /γ is the height of a column, subject to its self- weight, and made up of the samematerial of the arch, in which the compressive stress σ is reached at the base section.

By substituting Eq. (8) in Eq. (4), the following second-order nonlinear differential equation is obtained:{

y,xx (x) + p(x)H + 1

h

(1 + y,x (x)2) = 0

m (0) = m (L) = 0(9)

Approximate solutions for Eq. (9) have been proposed by Tadjbakhsh and Serra, see [4] and [5].


In the following, an analytical solution of this equation is presented and evaluated referring to the particularcase of constant value of the applied external load p (x) = p. To this aim, the solution of Eq. (9) has beencalculated in the form

y (x) = h log (z (x)) , z (x) > 0. (10)

By differentiating Eq. (10) with respect to x , the following expressions hold:

y,x (x) = hz,x (x)

z (x); y,xx (x) = h

(z,xx (x)

z (x)− z,2

x (x)

z2 (x)

)

. (11)

By substituting Eq. (11) and Eq. (10) in Eq. (9):

hz,xx (x)

z (x)− h

(z,x (x)

z (x)

)2

+ p (x)

H+ 1

h

[

1 + h2(

z,x (x)

z (x)

)2]

= 0, (12)

and after some easy algebra:

z,xx (x) + ω2 (x) z (x) = 0

ω (x) =√

p(x)

Hh+ 1

h2

m (0) = m (L) = 0

(13)

The parameter ω is function of x by means of the external load p(x). However, note that Eq. (13) has a closedsolution in case of external constant load p(x) = p,

z,xx (x) + ω2z (x) = 0. (14)

Moreover, the hypothesis of uniform constant external load appears reasonable if in a first design phase onlyquasi permanent conditions are considered, in order to furnish a preliminary arch modeling.

Then, the solution of Eq. (14) is

z (x) = C cos ω (x − xc) (15)

where the parameters C and xc depend on the boundary conditions.The analyses presented in the following refer to symmetric shape arches; therefore, only the case xc = L/2

will be considered. With reference to Fig. 1 and to Eq. (2), note that the condition y (0) = 0 implies that theboundary condition m (0) = 0 holds. Furthermore, in the symmetric case, because of the symmetry of the loadcondition and the arch shape, the boundary condition m (0) = 0 implies that also the condition m (L) = 0holds.

The condition y(0) = 0 gives

0 = h log (z (0)) ⇒ z (0) = 1, (16)

so:

C = 1

cos(

ωL2

) . (17)

Then, the geometrical arch profile and the cross-sectional area can be evaluated as follows:

y (x) = h log

[cos

(ω

(x − L

2

))

cos(

ωL2

)

]

, (18)

A (x) = H

σ

(1 + h2ω2 tan [ω (x − L/2)]2) 1

2 . (19)

In order that the solution Eq. (18) exists, the following condition must be fulfilled:

cos(ω

(x − L

2

))

cos(

ωL2

) ≥ 0 x ∈ [0, L] . (20)


Equation (20) allows defining

ωL = L

√p

Hh+ 1

h2≤ π (21)

and therefore

L ≤ π h√

phH + 1

. (22)

In the particular case of p = 0, Eq. (22) gives L ≤ π h. Then, in the particular case of null external load, it ispossible to find an optimum profile only if in the arch span L it is not greater than π h. In the following, it willbe shown that this condition holds also in the case of nonnull applied load p.

It must be noted that the horizontal reaction H is indeterminate also after imposing the boundary conditionsm (0) = m (L) = 0. Then, the optimal value of the horizontal reaction H for which the uniform compressedarch reaches the minimum ratio between its self-weight and the global external load must be found.

For this purpose, two different cases must be considered. In the particular case of null external loadp = 0, Eq. (13) gives ω = 1/h, and then the optimum arch geometrical profile Eq. (18) does not dependon the horizontal reaction H , while the cross-sectional area Eq. (19) is proportional to H . Note that, in thisparticular case, the minimum weight arch is the virtual one characterized by zero horizontal reaction H , nullcross-sectional area, and null weight.

Differently, in the general case of not null external load p, the arch geometrical profile depends on thehorizontal reaction H , and the optimal arch profile corresponding to the minimum ratio between the archweight W and the applied external load pL has a not null weight.

Referring to Eq. (19), the specific arch weight per unit horizontal length is as follows:

q(x) = H

h

(1 + h2ω2 tan (ωx − L/2)2) (23)

and then, the ratio � between the arch weight W and the external load pL is

� = W

pL= 1

pL

L∫

0

q(x)dx = H

phL

L∫

0

[1 + h2ω2 tan (ωx − L/2)2]dx

= H

phL

(

L − h2ω

(

Lω − 2 tanLω

2

))

. (24)

For the following developments, it is useful introducing the two dimensionless parameters α = h/ω andη = L/h.

Then, starting from the Eq. (13) and Eq. (24), the following expressions hold:

H = h p

h2ω2 − 1= h p

α2 − 1, (25)

� = 2αtan

(αη2

)

η(α2 − 1

) − 1, (26)

and the existence condition Eq. (21) becomes

α ≤ π

η. (27)

It should be noted that when the above existence condition is strictly satisfied, both the self-weight and theheight of the arch tend to an infinite value, and the following conditions hold:

limα→ π

η− � = +∞,

limα→ π

η− y(L/2) = +∞,

limα→ π

η− H = ph

(πη

)2 − 1.

(28)


Fig. 3 Graph of the function �(α, η)

Furthermore, it should be noted that the analysis is limited only to finite positive values of the horizontalreaction (0 ≤ H ≤ +∞), so that Eq. (25) gives

1 ≤ α, (29)

and when the above condition is strictly satisfied

limα→1+ � = +∞,

limα→1+ y(L/2) = −h log

[cos

(η

2

)], (30)

limα→1+ H = +∞.

The conditions Eq. (27) and Eq. (29) define the existence domain of the function �(α, η), whose graph isshown in Fig. 3.

As stated in the previous and as it is shown in Fig. 3, the function tends to an infinite value on the boundariesα = 1 and α = π/η of the existence domain. For the following analyses, the parameter αopt (η) has beenintroduced: It refers to the optimal value of α that, for a given value η, produces the minimum value of thefunction �η (α) = �(α, η).

In Fig. 4, the graphs of the function �η (α) are shown, for different values of η.The function αopt (η) is implicitly defined by the condition:

�,α(αopt (η) , η

) = 0

which can be written as

sin(ηαopt (η)

) = ηαopt (η)(αopt (η)2 − 1

)

αopt (η)2 + 11 ≤ α ≤ π

η(31)

and can be easily solved numerically.In Fig. 5, the green lines represent the boundaries α = 1 and α = π/η of the existence domain, and the

blue dotted line indicates the curve(η, αopt (η)

)of the minimum points of the function �η (α).

Figure 5 clearly shows that also in the case of an arch subjected to a uniform load the existence condition1 ≤ α ≤ π

ηimplies that η = L/h ≤ π , so that the following theorem holds:


Fig. 4 Graphs of the functions �η (α)

Fig. 5 Function αopt(η) and boundaries of the existence domain

Theorem The span L of an arch of equal strength subjected to a constant vertical load cannot be larger thanπ h, where h is height of a column subject to its self-weight and made up of the same material of the arch andin which, at the base section, the compressive stress σ acting in the arch is reached.

Finally, it should be remarked that the proposed solution refers only to axial strains, while bending curvatureand shear strains are null in each sections. Then, for slender statically indeterminate arches, the proposedsolution can be assumed as an approximate (pre-design) solution also in case of a simply supported or clampedarch.

3 Numerical example

In the following, the results of a numerical example are shown. The case refers to an arch with variablesquare cross section, made up of a material whose specific gravity is γ = 25 KN/m3. The working stress isσ = 9.75 N/mm2, and the arch span is equal to L = 2h = 780 m.

In Fig. 6, two optimum profiles are shown, corresponding to two different values of the applied load:p = 1,000 KN/m (blue profile) and p = 10,000 KN/m (green profile).


Fig. 6 Optimum arch profiles

It is interesting to remark that for an assigned value of η = L/h, only the cross-sectional area of theoptimal equal strength arch depends on the applied load p, while the curve of the centers of section is thesame.

4 Conclusions

In the present work, an analytical solution for the optimal shape of a plane-statically determined arch subjectedto a constant vertical load is presented. The classical problem of finding the profile of an equal strength catenarysubjected to its self-weight has been extended to the case of an inverted catenary of equal strength subjectedto both its self-weight and an external applied constant load. The solution corresponding to the minimum ratioof the arch self-weight on the total applied load is found. Finally, a theorem on the existence of such a solutionis provided.

References

1. Withney, C.J.: Design of symmetrical concrete arches. Trans. ASCE 88, 931–1029 (1925)2. Leontiev, V.: Frames and Arches. McGraw Hill, New York (1969)3. Timoshenko, S., Young, D.H.: Theory of Structures. McGraw-Hill, New York (1965)4. Tadjbakhsh, I.G.: Stability and optimum design of arch-type structures. Int. J. Solids Struct. 17, 565–574 (1981)5. Serra, M.: Optimal arch: approximate analytical and numerical solutions. Comput. Struct. 52, 1213–1220 (1994)6. Routh, E.J.: Analytical Statics, vol. 1. Cambridge University press, Cambridge (1891)

optimal arch shape solution under static vertical loads

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