CC BY
14Magazine of Civil Engineering. 2020. 95(3). Pp. 31-41
Magazine of Civil Engineering issn
2071-0305
journal homepage: http://engstroy.spbstu.ru/
DOI: 10.18720/MCE.95.3
Stability of fixed-fixed shallow arches under arbitrary radial
and vertical forces
L.P. Kiss*
Institute of Applied Mechanics, University of Miskolc, Miskolc, Hungary * E-mail: [email protected]
Keywords: buckling, stiffness, strength, design, composite materials
Abstract. In the present paper the planar stability of fixed-fixed shallow circular arches is investigated. The arches are made of linearly elastic, functionally graded material and are subject to a concentrated radial or vertical dead force at an arbitrary position. To describe the behaviour, the one-dimensional Euler-Bernoulli kinematic hypothesis is used. The effect of the bending moment on the membrane strain is incorporated into the model. The related coupled differential equations of the problem are derived from the principle of virtual work. Exact solutions are found both for the pre- and post-bucking displacements. Closed-form analytical solution is given for the buckling load when the load is radial while for vertical force, the solution is numerical. It is found that for fixed-fixed members, only limit point buckling is possible. Such shallow arches are not sensitive to small imperfections in the load position or in the load direction. It turns out that the material composition and geometry have significant effects on the behaviour and buckling load. If the load is placed far enough from the crown point, the load bearing abilities become better than for crown-load. Comparisons with an analytical literature model and commercial finite element software confirm the validity of the new findings.
1. Introduction
Curved members (arches, shells) are often used in engineering applications, e.g., in aerospace structures or in civil engineering structures [1, 2]. These elements are preferred because of their advantageous mechanical behaviour under various loading conditions and partially because of their favourable design.
Compared to classical homogeneous materials, composites have such advantageous properties like the increased durability or reduced weight [3-5]. Functionally graded materials (FGM) have a further notable benefit, that is the smooth stress distribution because of the continuously changing material properties. For these previous reasons, such composite materials are being used more and more frequently [6-9] by civil engineers. Thus, it is important to be able to predict the mechanical behaviour of such members.
Buckling is a well-known phenomenon, which can occur for various structural members [10-12]. Thus, the in-plane stability of arches is not a new topic as there is a bunch of available works, like [13-15] in the open literature. It is well-known that their behaviour is nonlinear since the pre-buckling deformations are significant. There are many arch problems that have been solved either analytically or by means of a numerical method [16-22]. These issues involve the effect of the foundation stiffness, remote unconnected equilibria, inclusion of geometric imperfections, elastic horizontal and rotational end restraints. The preferred numerical methods include the analog equation method, the modified Riks method and the Galerkin method as well.
Studies [23-27] report about analytical solutions on the stability of shallow arches with multiple stiffness regions along the centroidal axis. The Euler-Bernoulli hypothesis is considered and various load and support conditions are investigated. Articles [28-30] focus on FGM arches analytically using the classical shallow arch theory.
It is found that simple crown-point loads are preferred in the literature. However, in practice, other load positions are also possible. Early articles [31, 32] investigate the effect of the load position on arches but the later one neglects the effect of the axial compressive force. There is a recent article series about the in-plane stability of arches under arbitrary concentrated radial load [33-35] by using the same hypotheses as in [3637]. These works present analytical investigations for homogeneous material but the effect of the bending
Kiss, L.P. Stability of fixed-fixed shallow arches under arbitrary radial and vertical forces. Magazine of Civil Engineering. 2020. 95(3). Pp. 31-41. DOI: 10.18720/MCE.95.3
This work is licensed under a CC BY-NC 4.0
moment on the membrane strain is neglected. An improved kinematical hypothesis is presented and evaluated in the likes of [38-40].
According to the literature review, it is an open question how fixed-fixed FGM shallow arches behave under a radial or vertical dead load at an arbitrary position. The aim of the current article is to find the answer to this issue. The related tasks, therefore, are:
1. to introduce a new geometrically nonlinear beam model, incorporating the effect of the bending moment on the membrane strain;
2. to find the related static equilibrium equations using a suitable variational principle and to solve these coupled equations in closed-form;
3. to derive the buckling equations and investigate the possibility and conditions of bifurcation and limit point buckling;
4. to show the effect of various geometrical and material parameters on the buckling load;
5. to verify the outcomes, i.e., compare the new results with the literature and finite element computations.
2. Methods
2.1. Kinematics
A fixed-fixed shallow FG circular arch is considered. Its (E -weighted) centroidal axis is shown in Fig. 1. The axis £ coincides with the circumferential direction, while the cross-sectional coordinates are r;£ . The
cross-section is symmetric to £ , i.e., r is a principal axis. The initial radius of the centroidal axis is p and the included angle of the arch is 23 with ) being the angle coordinate. The external load Q is a dead concentrated force at ) = a and is either radial (y = 0) or vertical (y = a). The FG material composition is a mixture of metallic (subscript m) and ceramic (subscript c) phases. The material constituents - like the Young modulus E are assumed to follow a power-law distribution (e.g., the Voigt rule of mixture) along the thickness as
E(£) = (Ec - Em ) £ + Em
with b being the height of the cross-section and k is the volume fraction index [42].
(1)
Figure 1. A fixed-fixed shallow arch with radius p and included angle 2 3 under a concentrated radial/vertical load Q.
Using the Euler-Bernoulli hypothesis, the longitudinal strain at an arbitrary point is [41] expressed by
P
p+C
du w
-+-+C
ds p
(
du
dw
pds ds1
1
+ —
2
u vP
dw ^ ds
(2)
with the nonlinear membrane strain - £ = 0 - and the cross-sectional rotation being
du w 1 2 u dw
s =--1---h — y ; y =---.
ds p 2 p ds
The circumferential and lateral displacements are denoted by u and w .
According to the Hooke law, the axial force and bending moment are as follows [41]:
(3)
E3
N = J jJsdA = E1s —3
M = J ^¿dA = - E3
P
1 du d 2w ^
P ds ds2
' 2 \
d2w w ^ M ^ N +— = E1s.
P
j 2 + 2 ds p
(4a) (4b)
Thus, clearly, the effect of the bending moment on the membrane strain is accounted, unlike in some previous models. In the expressions of the inner forces the various stiffnesses are defined by
Ej = JaEC]-X dA.
(5)
At the points of the centroidal axis E2 = 0.
2.1.1 Equilibrium equations
The principle of virtual work is used to find the nonlinear pre-buckling equilibrium configuration. It requires that equation
JyO-^Ss^ dV - Q -Sw
= a
= 0
(6)
to be satisfied by any kinematically admissible virtual field S(). Due to the shallowness of the arch, only the virtual work done by the external force on the radial displacement field is accounted. From the above principle, the following coupled differential equation (DE) system can be found [40]:
dN ds 1 + — P dM f.r --N + ds ^
d " dM U M1 N + — 1 P )
ds ds
M
\
P ) ¥
¥
= 0,
- N = 0. P
(7)
(8)
These equations can be given in terms of the dimensionless displacement W = w / p and the notation
r
d()/ds = () as
; rttr / \ tt r\ r\
(9)
W'+[z2 + i)w"+X2W = z2 -1,
^ = 0,
with the strain- and geometrical-material parameters being defined as
2 y EiP P 2 E3
x =1 m = = -r; r = T3"
E
E
(10)
So, the effect of the material composition is incorporated into the model through these quantities. Furthermore, it is found that the membrane strain is constant. Equilibrium equations are comparable to, e.g., [28, 43].
Focusing now on the post-buckling equilibrium state, the typical increments - noted by a hat symbol () - are [38]
P
P + Ç
du + w + dy
ds
ds P
u dw + yy, y =---
P ds
du w s =— + — + ds p
NN = Es■
u dw p ds
u dw
p ds
(12)
ML p
M = — E,
f 72 ~ ~ \
d w w T + -j
ds p1
y
From the principle of virtual work for the buckled equilibrium configuration, equation
jv&gSs* dV — Q* • Sw*
<j) = a
= 0
(13)
(14)
is to be satisfied. Here () = () + () is the sum of the pre- and post-buckling components of any quantity. The following DEs are found from this principle [41]:
dNN 1 dM 1f ■ +
ds p ds p
d2M d_
ds2 p ds
M
\
N + ■ v P
V-
1
f
Li
^
Y
* M
N +
p
p
^ r V +
NN + v P y
N M^
NN + —
V = 0,
p
V
= 0,
which are actually
N = 0,
(15a) (15b)
(16)
^ ffff o ^ n O ^
W + (x + 1)W +%W = ßs
1-
(w " + W )
W = w / p.
These results are again comparable to previous models, like [28, 44].
3. Results and Discussion
3.1. Solution - pre-buckling state The following closed-form general solution was found to satisfy Eq. (9),
C
C
7 — 1 C C
Wi $ = ^—r + CH cos $ + C2isin $--7 cos Xi$ —y sin 7$
7i Xi 7i
with Cii — C4i being integration constants, 7. being the membrane strain parameter defined by
X =V 1-
and furthermore, here and now on, subscript i = {r,l} if ^ is {>a; <a}.
(17)
(18)
(19)
When the load is radial, the strain is constant on the whole centroidal axis while for vertical load, the strain has a jump at $ = a . The jump (discontinuity) magnitude is found from the variational principle as
2 2 2Psinr Qp23
x =xr +-- ; P = ■
3
2E
3
with P being a dimensionless load. Thus, the average membrane strain - recalling (19) and (20) - is
1 (Xt— l)09 + a) + (xr2— 1 ^^^^a)
s = —■
(20)
(21)
U 25
with xl = xr for radial load. As a consequence, the solution for the displacement field is sought separately on
the two sides of the external load. Based on the virtual work principle, the following boundary, continuity and discontinuity conditions are valid for the arch:
W U= W
-3
= W = W
r 3 r
= 0,
w,\ = wj w
l la r la l
= W„
W " = W " r - W + Wr
a a a
2P cos y
3
(22)
(23)
(24)
All these above fields are continuous except the third derivative, which is proportional to the shear force. It has a jump for both radial and vertical load. Substituting here solutions (18), the system of linear equations is found:
cos& - sin3 cos x¡3 xx sin x3 xX
-X sin& -X cos3 sin X3 cos X3
cos a sin a cos Xa x2 sin Xa x2
X sin a -X cosa - sin Xa cos Xa
- cosa - sin a cos Xa sin Xa
- sin a cos a X sin Xa -X cos Xa
0 0
0 0
0 0
0 0
- cosa
~xr sin a cosa sin a
cos3
0 Xr sin 3
r X2 -i
2
X
Cil ' Xr2 -1 X2 -
C21 2 2 Xr X
C31 0
C4l 0
Cir 0
C2r -2 P cosy
C3r 3
C4r _ Xr2 -i
2
Xr
0
0 0
- sin a
Xr cosa sin a
- cosa
sin& -Xr cos &
0 0
0 0
cos xa sin x__
2 2
Xr Xr
sin xa - cos Xra
- cos xa - sin xra
-Xr sin Xra Xrcos Xra
cos Xr3 sin Xr3
2 2
Xr Xr
- sin Xr3 cos Xr3
The coefficients c can now be calculated in closed-form.
Because of equilibrium Eq. (9)^, the membrane strain is constant on the centroidal axis, so it is equal
to the average membrane strain given by:
_ 1
1
s = s =
\a_3¡l ()d / + --\aer r)d / = constant.
(25)
Recalling Eq. (3) and assuming w >> u due to the shallowness, which is generally accepted in the literature [33], a quadratic equation in P is established for fixed-fixed shallow arches under arbitrary radial
( xi=xr ; 7 = 0) load as:
a
a
a
G0 + GlP + G2 P2 = 0.
(26)
The constants G0-G2 are functions of ju,a,3, % ■ This previous equation can be solved analytically for any given strain-, geometrical and material parameter.
When the load is vertical - % ^ %r; 7 = a - the above equation becomes strongly nonlinear in the dimensionless load because of Eq. (20) and
P, %, %) = 0 (27)
can only be tackled numerically.
3.2. Buckled equilibrium The general solution to the inhomogeneous post-buckling equilibrium equation (17) is
W ($) = L\i cos $ + D2i sin $ + D3i sin xi$ + D4i cos xi$ -
f
2zf v
A
— + C3$ sin - C4$ cos X
(28)
with the unknown constants D -D ■ The related boundary and continuity conditions are
li 4i
= Wl
= Wr = Wr
-3 r 3 r
= 0,
W,
= W„
W,
=wr
(29)
(30)
(31)
with all the typical fields being continuous at ( = a as there is no load increment (Q = 0). These conditions determine an inomogeneous linear equation system which can be solved in closed-form. According to (16), the strain increment is constant, thus, with (12) and the inequality W >> U used on (16) it is found that
« ttt
Wt II , W
a a a
* = + W + W W') d(
It is again a quadratic equation in the dimensionless load when the force is radial:
Jn + JP + JiP2 = 0
(32)
(33)
with Jj;J2;J3 being functions of /u,a,3,P,% and it leads to a strongly nonlinear relationship for vertical load:
Ko(^,a,7,3, P, %, Xr) = 0. (34)
Solution for limit point (of snap-through) buckling can be evaluated for any geometry and material distribution by solving simultaneously nonlinear Eqs. (26); (33) for radial load and (27); (34) for vertical load. Or, alternatively, since limit points are local maxima on the equilibrium path [42], the very same results can be found by differentiating Eq. (26) or (27) with respect to the strain parameter.
In theory, it is also possible for fixed-fixed shallow arches under concentrated load to undergo bifurcation buckling [38]. During bifurcation, the load and membrane strain remain constant [36] and the arch buckles to an infinitesimally close bifurcation equilibrium configuration. In this case, solution to DE (17) with * = 0 is
() = Sji COS ( + S2i Sin ( + S3i Sin %i( + S4i COS (35)
This solution and the conditions (29)-(31) determine a homogeneous linear system of equations. Nontrivial solution exists if the determinant of the coefficient matrix vanishes, therefore
a
a
a
cos 3 - sin 3 - sin x3 cos x3 0
sin 3 cos3 xi cos Xi3 Xi sin x3 0
cos a sin a sin Xia
- sin a cos a Xi cos xa
cos Xia Xisin Xia
0 0
- cos a - sin a sin a - cos a
0 0
sin Xra
0 0
cos Xra
- cos a - sin a sin a - cos a
,2 .
2
0 0
0 0
Xi sin Xia -X cos Xia cos a sin a
Xx cos Xia X/3sin Xia - sin a cos a
0 cos3 sin3
0 - sin 3 cos 3
Xr cos Xra Xr sin Xra
x;2 sin Xra xX cos Xra
Xr cos Xra X2r sin Xr3 xl cos Xr3
Xr sin Xra x;?cos Xr3 -sin Xr3
= 0
This determinant, in fact, yields the critical strain for bifurcation buckling. However, recalling the solutions of [33] and [38], it is clear that, for fixed-fixed shallow arches under a concentrated load, bifurcation buckling is possible only when a = 0 and in this case, the bifurcation point is always located on the unstable equilibrium branch. It means that only limit point buckling can occur.
3.3. Analytical and numerical solutions
P(a * 0) 1.50 P(a * 0) 1.4
P(a = 0)
1.40 1.30 1.201.10 1.00 0.90
-9 = 20°
- » = 30"
------,9 = 40'
■ ■ 9 > 50"
« 0 = 20° / /
■ 9 - 40° y / m
/ ^ " / /
,—__
------ Sir = 75
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
S/r - 200
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 b
Figure 2. The critical load ratio for non-crown and crown loads in terms of the relative load position parameter a/ 3 for arches with: a - S/r = 75; b - S/r = 200.
First, the effect of the load position a on the lowest critical load is investigated for multiple semi-vertex angles 3. Results are plotted in Fig. 2a and 2b for two S/r ratios - S is the length of the centroidal axis and r is the E-weighted radius of gyration. P(a = 0) is the critical dimensionless load at the crown and P(a ^ 0) is the critical load when the force is placed elsewhere. Different line types are drawn for radial load while markers are used to show the findings for vertical force. When the load is in the small vicinity of the crown point a << 23 the buckling load changes (decreases) slowly for all the selected geometries. Thus, fixed-fixed arches, unlike pinned-pinned members [39], are not sensitive to small imperfections in the load position. In general, replacing the load from the crown makes the buckling load decrease for a while. Then it is always followed by an increase and if the load is exerted sufficiently far from the crown point, the load bearing abilities become better than for a = 0. Overall, smaller included angle means smaller change in the buckling load. When S / r = 75, the semi-vertex angle and load position have greater effects compared to S / r = 200. Above a certain angle, the behaviour seems to become independent of the angle as well.
It is also found that in the investigated range it does not really matter if the load is vertical or radial, i.e., fixed-fixed shallow arches are not sensitive to small imperfections in the load direction in relation to the radial one.
200 15.0 10.0 5.0 00
-- a - 0 II - [1.38 a = 0. Of
, -*
f^à--
Sir = 200 S = 0.26
P 160
12.0
0.5
1.5
S
cslr
a b
Figure 3. Dimensionless load vs membrane strain parameter to depict the effect of load position a on the behaviour of arches with S/r=200: a - 3 =0.26; b -3=0.87.
Next, it is investigated how the load position can affect the behaviour of arches as shown in Fig. 3a and 3b. Here, sstr is the buckling strain of a straight column under compression with the same material
composition, length and cross-section as the arch. Independently of the angle, it is common in the diagrams that as the strain is increased from zero, there is a possible stability loss where the tangent is horizontal. Furthermore, increasing a makes the strain ratio decrease at the moment of buckling and at the same time the dimensionless buckling load gradually becomes greater. One branch always starts from the origin and the other one from a different level. The later one is greater in the dimensionless load P if a is greater and 3 is smaller. These diagrams are plotted for radial load but can be considered valid for vertical force also with a good accuracy.
Fig. 4-5 give information about the effect of the ratio S/r on the in-plane behaviour when the load is radial. The dimensionless displacement at the application point of the force is Wa= W(p = a). As S / r
decreases, the loop always becomes smaller with lower critical load and strain ratio as well. The different branches start from a same point, independently of S / r and intersect each other again at a same point as shown in Fig. 4a and 4b. This later intersection point is close to the strain ratio 1. In Fig. 5a and 5b, there are three equilibrium branches: a primary stable, a remote stable and an unstable between them. (When there is no possibility of buckling, there is only one stable branch.) It is somewhat different to pinned-pinned members [39, 40] as then, the number of equilibrium branches showed an increases with a and could reach a number quite more than three, but for fixed-fixed members the number of equilibrium branches seems to be always three. It is set down to the fact that fixed supports are stiffer. When a = 0, the unstable branches are very flat but still there is limit point buckling at smaller dimensionless displacement compared to a = 0.23.
10.0
4.0 2.0 0.0
— Sir - 200 Sir - 100
— Sir - 75
3 - 0. 35 a = 0
P 12.0 10.0 8.0 6.0 4.0 2.0 0.0
-Sir = 200
...........Sir = 100 — — Sir = 75
3- = 0.35 a = 0.23
0.5 1 1.5
£
0.4 0.8 1.2 1.6 Sstr
b
Figure 4. Dimensionless load vs membrane strain parameter to depict the effect of ratio S/r on the behaviour of arches with 3=0.35 when: a -a =0; b - a =0.23.
Figure 5. Dimensionless load vs dimensionless displacement at p = a to depict th effect of ratio S/r on the behaviour of arches with 3=0.35 when: a -a =0; b - a =0.23.
In what follows, the lowest buckling loads are plotted in terms of the included angle as shown in Fig. 6a and 6b. The various lines represent radial load and markers are used for vertical force. Clearly, the buckling load increases with S / r. It makes no real sense if the load is radial or vertical - the maximum difference between them is only a few percent at most. Increasing S/r makes the typical curves shift to the left - the possibility of buckling appears for smaller included angles.
Figure 6. Lowest critical loads for multiple load positions in terms of the semi-vertex angle:
a - S/r=75; b - S/r=200.
3.4. Verification of the results
For radial load, comparisons were carried out with [33] assuming homogeneous material. Results for arches with the same geometrical and material parameters are gathered in Table 1. The correlation is really good. Differences are greatest when 3 and a are greatest. Since the new model of this article accounts for the effect of the bending moment on the membrane strain, it is assumed to be more reliable.
The new results for vertical load are also verified by the commercial finite element software Abaqus 6.13. A geometrically nonlinear model was used with the Riks method to find the limit points on the equilibrium path. A one-dimensional model was created with eighty B21 beam elements. A doubly symmetric /-section was selected [34]. The related geometrical data were the following: depth 256 mm, flange width 146 mm, flange thickness 10.9 mm, web thickness 6 mm. The moment of inertia is 54,255,292 mm4 and the modulus of section is 423,869.469 mm3 to the bending axis ^ while these values are 5,657,929 mm4 and 77,505.875 mm3 to the other principal axis £ . So the radius of gyration r to the bending axis is 108 mm.
Since S/r = 200 was selected, the centerline legth is 21,600 mm. The Young modulus was 210 GPa. Comparisons of the models with some further geometrical data are gathered in Table 2, indicating that the results of the current model are reliable.
Table 1. Comparison of the lowest radial dimensionless buckling loads with [33] for arches with S/r = 75 and 200.
s P [m] span [m] altitude [m] P P in [33]
S/r = 75, a = {0/0.2/0.4} s
0.38 10.65 7.90 0.76 5.20/5.61/7.62 5.15/5.56/7.35
0.69 5.87 7.47 1.34 6.67/6.35/8.14 6.57/6.26/7.89
1.05 3.85 6.69 1.94 7.16/6.68/8.67 7.06/6.41/7.99
S/r = 200, a = {0/0.2/0.4} s
0.38 28.42 21.08 2.03 6.92/6.38/8.05 6.92/6.37/7.97
0.69 15.65 19.92 3.58 7.29/6.54/8.42 7.28/6.49/8.05
1.05 10.28 17.84 5.17 7.39/6.76/8.74 7.34/6.51/8.07
Table 2. Comparison of the lowest dimensionless buckling loads for vertical load with Abaqus for arches with S/r=200.
s P P, Abaqus
S /r = 200, a = {0.1/0.2/0.3} s
0.38 6.42/6.40/6.88 6.30/6.29/6.78
0.69 6.61/6.54/7.09 6.72/6.62/7.04
1.05 6.76/6.77/7.30 7.07/6.93/6.84
4. Conclusion
1. In this article a new geometrically nonlinear model was established to investigate the in-plane behaviour and stability of fixed-fixed shallow arches under the effect of a concentrated radial or vertical dead load. The arches are made of functionally graded material. The Euler-Bernoulli kinematical hypothesis was applied. Account for the effect of the bending moment on the membrane strain was made.
2. The related pre-buckling and buckled equilibrium equations were derived from the principle of virtual work. Analytical evaluations were carried our for radial load and numerical ones for vertical force.
3. It was found that only limit point buckling is possible with three equilibrium branches: two stable and one unstable for any geometry and material composition.
4. Unlike pinned-pinned arches, fixed-fixed members are not sensitive to small imperfections in the load position and in the load direction.
5. At the same time, the load position can have a significant effect on the buckling load. Generally, the buckling load decreases for a while when the force is moved from the crown point but as it is placed close enough to the supports, the load bearing abilities become even better.
6. In limit case, the presented model is also capable of predicting the buckling load for homogeneous materials.
7. The new results were confirmed by comparisons with literature and finite element computations.
5. Acknowledgements
The described article was carried out as part of the EFOP-3.6.1-16-2016-00011 „Younger and Renewing University - Innovative Knowledge City - institutional development of the University of Miskolc aiming at intelligent specialisation" project implemented in the framework of the Szechenyi 2020 program. The realization of this project is supported by the European Union, co-financed by the European Social Fund.
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Contacts:
Laszlo Peter Kiss, [email protected]
© Kiss L.P., 2020
