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Large deflections of eccentrically loaded arches

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Zusammenfassung

Auf Grund der Hypothesen von Ebenbleiben und Normalität der Querschnitte werden die Differentialgleichungen der nichtlinearen Theorie der Bogenträger abgeleitet und im Falle des schlanken, durch Einzellasten belasteten Kreisbogenträgers mit undehnbarer Mittellinie auf die Form der Pendelgleichung gebracht. Diese Gleichung wird dann benutzt, um die grossen Durchbiegungen und die Spannungsresultierenden eines Zweigelenkkreisbogens, der durch eine lotrechte exzentrische Einzellast belastet wird, zu berechnen. In der Nähe der kritischen Last bewirken kleine Exzentrizitäten bedeutende Grössenänderungen der Spannungsresultierenden und der Durchbiegungen.

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Abbreviations

A :

cross-sectional area of curved beam

a :

radius of centroidal circle

E :

modulus of elasticity

e :

eccentricity of the load (Fig. 2)

F :

an arbitrary function

H :

horizontal component of the internal forceR acting on a cross section of the arch rib (Fig. 2)

h P :

horizontal displacement of the loadP (Fig. 2)

I :

moment of inertia of the cross-sectional area

k 2 :

=4p 2/(1+4p 2 sin2ψ0)

L :

span (distance between supports),L=2a sin α

M :

internal bending couple (Figs. 1 and 2)

N :

internal normal tensile force (Figs. 1 and 2)

n :

distributed tangential load (Fig. 1)

P :

downward point load (Fig. 2)

p 2 :

−R a 2 /E I

Q :

internal shearing force (Figs. 1 and 2)

q :

distributed normal load (Fig. 1)

R :

internal resultant force (Fig. 2);R 2=H 2+V 2=N 2+Q 2

ν:

radius of curvature of the undeformed centroidal curve

s :

length along the unextended centroidal curve measured from the left support

\(\bar s\) :

length along the unextended centroidal curve measured from the right support

u :

tangential displacement component of the centroidal curve (Fig. 1)

V :

vertical component ofR (Fig. 2)

v P :

vertical displacement of the loadP (Fig. 2)

w :

normal displacement component (Fig. 1)

x, y :

rectangular coordinates of the deformed left portion of the centroidal curve (Fig. 2)

Z :

\( = \{ \iint\limits_A {\left[ {z^2 /(\nu - z)} \right]dA}\} /\nu A\)

z :

normal distance (positive inward) from centroidal curve (Fig. 1)

α:

half subtending angle of the arch (Fig. 2)

β:

angle of rotation of the centroidal curve (Fig. 1)

ε:

extensional strain of the centroidal curve

εz :

extensional strain of the linez=constant

η:

y cosμ−x sinμ

θ:

angle between the tangent to the formed left portion of the centroidal curve and the horizontal (Fig. 2)

λ:

(u′−w)/r, whereu′=du/dø

μ:

angle betweenH andR

ξ:

x cosμ+y sinμ

σ:

normal stress along the centroidal curve

σz :

normal stress along the linez=constant

ϕ:

angle measured from the radius at the left support of the undeformed arch

ψ:

(θ−μ)/2 (Fig. 2)

ω:

(ω′+u)/r, where ω′=dω/dø

References

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Author information

Authors and Affiliations

  1. Dept. of Civil Engineering, University of Detroit, Michigan, USA

    Robert Schmidt

  2. Dept. of Civil Engineering and Engineering Mechanics, University of Arizona, Tucson, Arizona, USA

    Donald A. DaDeppo

Authors
  1. Robert Schmidt
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  2. Donald A. DaDeppo
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Additional information

A bar over a letter indicates that the entity pertains to the right portion of the arch. Asterisk indicates the deformed configuration. Primes indicate derivatives with respect to ø.

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Schmidt, R., DaDeppo, D.A. Large deflections of eccentrically loaded arches. Journal of Applied Mathematics and Physics (ZAMP) 21, 991–1004 (1970). https://doi.org/10.1007/BF01594857

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  • DOI: https://doi.org/10.1007/BF01594857

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