A theoretical and experimental study on geometric nonlinearity of initially curved cantilever beams

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A theoretical and experimental study on geometric nonlinearity of initially curved cantilever beams Sushanta Ghuku, Kashi Nath Saha * Mechanical Engineering Department, Jadavpur University, Kolkata 700032, India

A R T I C L E

I N F O

Article history: Received 5 June 2015 Received in revised form 9 July 2015 Accepted 10 July 2015 Available online Keywords: Cantilever beam Leaf spring Large deflection Geometric nonlinearity Numerical solution

A B S T R A C T

This paper presents a theoretical and experimental study on large deflection behavior of initially curved cantilever beams subjected to various types of loadings. The physical system as a straight cantilever beam subjected to a tip concentrated load is considered in this study. Nonlinear differential equations are obtained for large deflection analysis of such a straight cantilever beam, and this problem is known to involve geometrical nonlinearity. The equations are solved numerically with the help of MATLAB® computational platform to get deflection profiles of the concerned problem. These results are imposed subsequently on the center line of an initially curved beam to get theoretical load-deflection behavior of curved beam problems. To verify the theoretical model, experiment is carried out with the master leaf of a leaf spring bundle by modeling it as an initially curved cantilever beam. The effects of initial clamping and geometry variations in the eye-region are observed from experimental investigation which is commonly neglected in the mathematical formulation. Comparisons of the theoretical results with the experimental results are quite good, but the avenues for further improvement are also reported. The proposed approach is further extended to study large deflection behavior of an initially curved cantilever beam subjected to distributed and combined load. These results are successfully validated with existing results for straight beams and some new results are furnished for initially curved cantilever beams. Copyright © 2015 The Authors. Production and hosting by Elsevier B.V. on behalf of Karabuk University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction In structural analysis, two types of nonlinearities are most commonly encountered: geometric and material. Material nonlinearity is associated with nonlinear stress–strain relations whereas nonlinear curvature–slope and strain–displacement relations give rise to geometric nonlinearity. Depending on the nature of the problem any one or both of the nonlinearities are included in the analysis. In the earlier years, studies of deflection behavior of a cantilever beam under different loadings were based on linear models in order to simplify the analysis. Several researchers [1–3] pointed out that for better characterization of such beams, analysis should be carried out through geometric non-linear model. Geometrically nonlinear large deflection problem of elastic cantilever beam under tip concentrated vertical load had been solved classically by Bisshopp and Drucker [1], and afterward many researchers have extended the theory. Wang [2,3] proposed a simple numerical method for analyzing nonlinear bending of beam under

* Corresponding author. Tel.: +91 3324146908, fax: +91 3324146890. E-mail address: [email protected] (K. Nath Saha). Peer review under responsibility of Karabuk University.

tip concentrated and uniformly distributed loads respectively. Beléndez et al. [4,5] also studied the same problem, both theoretically and experimentally. Kumar et al. [6] suggested genetic algorithm based search strategies in the context of direct numerical solution of governing differential equation and the principle of stationarity of the energy functional in the equilibrium state. Dado and Al-sadder [7] developed an approach that approximates the angle of rotation by a polynomial function and applied this method effectively for complex load on non-prismatic beam with very large deflection. Banerjee et al. [8] proposed non-linear shooting and Adomian decomposition methods to determine the large deflection of a cantilever beam under arbitrary loading conditions. Chen [9] proposed an integral approach for large deflection study of a cantilever beam with complex load and varying beam properties. Roy and Saha [10] applied a geometrically updating technique by using variational method to find out deflection profiles of non-uniform beams under various loading conditions. Large deflection of beams made of functionally graded material had been studied by Almeida et al. [11] using a tailored Lagrangian formulation and also by several other researchers [12,13]. Xiao-Ting He et al. [14] proposed a new perturbation method with two small parameters, describing the effect of load and geometry of the problem, to solve nonlinear large deflection problem of initially curved beams under two different boundary conditions. Large deflection problem of initially straight

http://dx.doi.org/10.1016/j.jestch.2015.07.006 2215-0986/Copyright © 2015 The Authors. Production and hosting by Elsevier B.V. on behalf of Karabuk University. This is an open access article under the CC BY-NCND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Please cite this article in press as: Sushanta Ghuku, Kashi Nath Saha, A theoretical and experimental study on geometric nonlinearity of initially curved cantilever beams, Engineering Science and Technology, an International Journal (2015), doi: 10.1016/j.jestch.2015.07.006

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(b)

(a)

L x

x

L-x

x y

s

A(x, y)

F

y

x

s y

NL

L-x NL

i

A(x, y)

tip

x y

N

L

F

tip N

N s = L (x tip , ytip )

l Large deformation Bending moment at A

Small deformation Bending moment at A M(x) = F(L - x)

M(s) = F(l - x)

Fig. 1. (a) Small deformation and (b) large deformation of a cantilever beam.

cantilever beam under follower type loading have been solved numerically by several researchers [15,16]. Shvartsman [17] studied large deflection of a curved cantilever beam under follower force by direct numerical method, whereas Nallathambi et al. [18] studied the same problem for a constant curvature cantilever beam by fourth order R-K method. Design and manufacture of automotive leaf spring using functionally graded and composite materials have been addressed by several researchers [19–23]. Shenhua et al. [24] carried out experimental work on precision roll-forging taper-leaf spring of vehicle, and results have been used in the design of roll-forging process and dies for the forming of taper-leaf springs. Osipenko et al. [25] introduced a contact problem in the theory of leaf spring bending. Sugiyama et al. [26] reported development of nonlinear elastic leaf spring model for multi-body vehicle system. Rahman et al. [27] carried out nonlinear geometric analysis of parabolic leaf spring. Charde et al. [28] used strain gauge technique to evaluate the stress field in the master leaf of a leaf spring and compared the results with finite element method. Large deflection study of an initially straight cantilever beam under different loading is ever interesting and a huge number of studies are reported in the literature. However, geometric nonlinear analyses of an initially curved cantilever beam under different loading conditions are few. The present paper focuses on both theoretically and experimentally geometric nonlinear behavior of an initially curved cantilever beam under different loading conditions. For the purpose of experimentation, the master leaf of a leaf spring bundle is considered as a cantilever beam with initial curvature.

Large deflection problem of cantilever beams is generally analyzed in curvilinear coordinate system. Euler Bernoulli beam theory in curvilinear coordinate system ( s, n) is 1 ρ = M EI [1], where curvature 1 ρ = dϕ ds . So Euler Bernoulli bending moment–curvature relationship is given as follows,

dϕ =M ds

(1)

In equation (1), ϕ is the slope dy dx at location s, and it is also the measure of normal direction n. For the purpose of computation ϕ is designated as ϕ ij , where i ( = 1, … , N L ) is the measure of load and j ( = 1, … , N g N f ) corresponds to the location, where ϕ is measured in x s coordinate system. When large deflection analysis is carried out in Cartesian coordinate system ( x, y ), the curvature is given by

1 d2 y = ρ dx 2

EI

d 2ϕ dM = ds2 ds

(2)

The bending moment M at location s is,

M ( s ) = F (l − x ) .

(3)

Differentiating equation (3) with respect to s, and comparing with equation (2) the following non-linear differential equation is obdx and tained, taking into account the geometrical relations cos ϕ = ds

sin ϕ =

EI

dy . ds

d 2ϕ + F cos ϕ = 0 ds2

(4)

dϕ dϕ d 2ϕ dϕ + F cos ϕ =0 to yield EI ds ds ds2 ds and after carrying out some mathematical manipulations, it is expressed as Equation (4) is multiplied by

2. Mathematical formulation

EI

1 d2 y = , and ρ dx 2 as a consequence the domain of x becomes 0 ≤ x ≤ L , i.e., the beam stretches with increase in loading as shown in Fig. 1(a). On the other hand, in large deflection bending analysis of cantilever beams, it is assumed that the length of the beam does not change with loading. Hence the domain of s remains unchanged and spans from 0 to L ( 0 ≤ s ≤ L ). To maintain constancy in beam length, the domain of x changes with loading, spanning from 0 to the projected length l of the beam, as shown in Fig. 1(b). The first derivative of equation (1) with respect to s, yields, deflection problems, the curvature is approximated as

⎛ 3⎞ ⎜ ⎟

⎡ ⎛ dy ⎞ 2 ⎤⎝ 2⎠ ⎢1 + ⎜⎝ ⎟⎠ ⎥ . However in the analysis of small dx ⎦ ⎣

2 ⎤ d ⎡ EI ⎛ dϕ ⎞ ⎢ ⎜⎝ ⎟⎠ + F sin ϕ ⎥ = 0 ds ⎣ 2 ds ⎦

(5)

Equation (5) is integrated and the associated constant of inteNL gration is evaluated by using boundary conditions (i) ϕ = ϕ tip and dϕ dy N L = 0 at s = L . ϕ tip represents the slope corresponding to (ii) ds dx load F at load step number N L . Hence equation (5) becomes 2

2F ⎛ dϕ ⎞ (sinϕtipNL − sinϕ ) ⎜⎝ ⎟⎠ = ds EI

(6)

⎛ FL2 ⎞ Using a normalized load parameter α ⎜ = , the above equa⎝ 2EI ⎟⎠ tion is expressed as

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NL NL The coordinate ( xtip ) at the free end of the beam provides , ytip NL NL 0 0 ) or stretching ( δ x = xtip ) correbeam shortening ( δ x = xtip − xtip − xtip sponding to load F at load step N L . In addition the tip coordinate NL 0 , which may be in veralso provides the tip deflection, δ y = ytip − ytip tically upward or downward direction.

2.1. The deflection profile As mentioned earlier, the master leaf of leaf spring is modeled as a cantilever beam following large deflection theory. The equations of deflection profile of a cantilever beam of length L which is subjected to a vertical concentrated load F at the free end is reNL as a function ported in the previous section. In order to obtain ϕ tip

NL Fig. 2. Inter relationship for ϕ tip –vs.– α .

NL of α , equation (9) is integrated between 0 and ϕ tip

⎛ dϕ ⎞ L⎜ ⎟ = 2 α ⎝ ds ⎠

(sinϕtipN − sinϕ ) L

(7)

Upon integration, the equation provides arc length s as a function of ϕ through the relation

s 1 = L 2 α

ϕ

∫ (sinϕ 0

dϕ NL tip

(8)

− sin ϕ )

Noting that s L = 1 at the free end of the beam, equation (8) solves NL for the unknown slope ϕ tip corresponding to load parameter α , from iterative solution of the following relation. NL ϕtip

2 α=

∫ (sinϕ 0

dϕ NL tip

(9)

− sin ϕ )

An appropriate transformation of equations (7) and (8), obdϕ dϕ dϕ dϕ = sinϕ = cosϕ and , yields ds dy ds dx the ( x, y ) coordinate at any location s, as given below. tained by using the relations

1 ⎡ x NL = sin ϕ tip − L α⎣

(sinϕtipN − sinϕ ) ⎤⎦ L

(10)

y 1 = L 2 α

ϕ

sin ϕ dϕ

∫ (sinϕ 0

NL tip

− sin ϕ )

(11)

for different

NL tip

2.2. Cantilever beam with initial curvature The deflection profile of a curved beam may be represented in x, y as well as in s, n coordinate system, as highlighted for point A in Fig. 1(b). The correlation between these two systems is estabs x ⎡ ⎛ dy ⎞ 2 ⎤ lished by the relation s = ∫ ds = ∫ ⎢1 + ⎜ ⎟ ⎥dx . The Cartesian ⎣ ⎝ dx ⎠ ⎦ 0

and

max

value of ϕ . The results provide the relationship ϕ vs. α , as shown in Fig. 2. The deflection profiles are now obtained from equations (10) and (11) and their plots are shown in Fig. 3 for different values of load parameter. The deflection profile, as reported in this section, 0 = 0 ). It is reported pertains to an initially straight beam (i.e., ϕ tip earlier that for a given value of α , axial stretching (or shortening) NL , but for other of the tip δ x is determined from tip co-ordinate xtip coordinate values of x , axial stretching has some other value δ x ( s ) . Similarly δ y is the particular value at s = L and in general it is also a field variable in s. Coordinate x ( s ) is readily obtained from equation (10), but evaluation of y ( s ) from equation (11) requires evaluation of another elliptic integral, called as incomplete integral of a second kind. However a leaf spring has an initial curvature 0 = 0 at the free boundary is and hence the boundary condition ϕ tip not valid here. The deflection behavior of such an initially curved cantilever beam is analyzed by a tricky method as presented in the next section. NL tip

0

coordinates of the leaf spring under study is noted in its unloaded condition and a best fit polynomial equation of the curved profile dy and y = f ( x ) is established. This equation provides slope dx hence arc length s is obtained as a function of x . By using the reverse

Fig. 3. Deflection profile for different values of load parameter α .

Please cite this article in press as: Sushanta Ghuku, Kashi Nath Saha, A theoretical and experimental study on geometric nonlinearity of initially curved cantilever beams, Engineering Science and Technology, an International Journal (2015), doi: 10.1016/j.jestch.2015.07.006

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Fig. 4. Deflection profile for different values of load parameter α of a cantilever beam with initial curvature.

relation x ( s ), the x coordinates are determined for a number of equidistant points N f along the arc length. Obviously for these points, si = i ( L N f ) , where i = 1, … , N f . The load deflection behavior of an initially straight beam is known analytically as described in the previous section. Now N f number of points are taken on the cantilever beam with initial straight profile having coordinates ( xiNL , 0) , i = 1, … , N f . The distance of these points from origin are known and they are a measure of its arc length as well. At each of these points axial shortening δ x ( s ) and vertical deflection δ y ( s ) are calculated for a load step N L . The same N f number of points is also located on the initially curved leaf spring and x, y coordinates of these points are calculated. To get the elastic curve of the leaf spring at the current load step NL , δ x (s) is added to the x coordinate and δ y (s) is subtracted from the y coordinate of the

elastic curve in its previous configuration. Considering a cantilever beam with known initial curvature, deflection profiles are computed for different α values and the results are shown graphically in Fig. 4. 3. Experimental setup and observation The experiment is carried out on a master leaf of an automobile leaf spring, made from spring steel, which is a cantilever beam with initial curvature. Photograph of the experimental setup is shown in Fig. 5(a), and the detail components are shown through a schematic diagram in Fig. 5(b). The dimensions of the spring crosssection (in mm) are width =38.5 and thickness =6.25. The span, camber and arc-length along periphery of the leaf spring are

Fig. 5. (a) Photograph and (b) schematic diagram of the experimental setup.

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Fig. 6. Photographs of deflection profiles under different loading.

measured in its free state, and these measurements (in mm) are 864, 133 and 921.8 respectively. The spring is clamped centrally with the help of a hydraulic cylinder at a pressure of one ton. It is observed that the clamping produces an initial deflection of the spring and divides the spring into two halves. Thereafter the spring is loaded symmetrically by placing equal weights on the weight pans at both ends. Experimental observation is made in one half of the spring only, which is modeled as an initially curved cantilever beam in the theoretical analysis. The engagement of ram of the hydraulic cylinder with the leaf spring changes its span, and the effective span is found to be 433.3 mm for each cantilever. Loading is increased gradually until the limit load of the experiment is reached. The limit load is calculated from the theoretical bending stress equation of curved beam and taking 75% of yield stress value of spring steel material. In each step of loading deflection profile is captured and recorded by using a digital camera, and it is observed that under the maximum load beam has become almost horizontal. 3.1. Experimental deflection profiles The photographs of deflection profiles under different loading are taken for the left side of the spring only and shown in Fig. 6. Deflections of the spring under each loading condition are post processed from the photographs, and for this purpose a graph paper is placed immediately behind the spring. Each photograph is taken as background in the editor of a graph handling software (AutoCAD) and a curvature line is drawn along the center line of the loaded beam. The length of the curvature line is measured, and the drawing is scaled to equate this length with the initial beam length. Now the projected length of this line is divided into equal ten divisions

and ( x, y ) coordinates are measured at each of the division points. Some of these curvature lines are presented in Fig. 7 and in addition, the profile of the spring in its free state is also appended to this figure. The ( x, y ) coordinates of the ten division points are also shown in Fig. 7 for each of the curvature lines. The best fit deflection curve and its analytical equation is obtained by using MSExcel software and shown in Fig. 8 for all the six loading conditions of Fig. 7. 3.2. Post processing of experimental load-deflection behavior Deflections of the leaf spring due to applied loads are observed at the tip as indicated in Table 1. It is obvious from Fig. 7 and Table 1 that the spring has deflected at clamped position, although no external load in the form of dead weights has been applied. Deflection of the tip at clamped position with respect to the initial no load configuration is 4.7 mm. This deflection is due to bending effect of clamping force acting at the contact surface of hydraulic cylinder head and the leaf spring. The clamping effect is modeled through an equivalent force at the tip, which is unknown at this stage, but need to be calculated for obtaining the actual experimental load-deflection behavior. The best fit linear load-deflection curve is obtained from data points of Table 1, as shown in Fig. 9, and it does not pass through origin. Using MATLAB® software, the load-deflection curve is shifted so as to pass through origin and the equation of the best fit line is y = 0.2242x , which is also shown in Fig. 9 by solid line with dots on it. Now corresponding to tip-deflection 4.7 mm, clamping force is calculated as 20.9634 N. Hence, this additional tip load is considered to capture the effect of initial clamping, although in actual case the spring has

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Clamped condition

No load condition ((438.5, 122.6)

10

9

7

8

(219.25, 29.2) ( (175.4, 18.8) (131.55, 10.5)

((394.65, 98.3) (350.8, 76.3) (306.95, 57.5) ((263.1, 41.8)

4

3

1

2

9

((402.3, 71.51) ((357.6, 55.11) ((312.9, 41.12)

8

7

6

5

0

6

7

8

9

10

9

8

7

4

3

2

1

0

2

1

0

((225.65, 16.7) ((180.52, 10.6) ((135.39, 5.9) ((90.26, 2.8) ((45.13, 1) (0, 0))

10

9

8

7

6

5

4

3

2

1

0

((183.76, -2.1)

((181.88, 6.6) ((136.41, 3.9)

((459.4, 19.6)

((272.82, 16.7)

((90.94, 2.2)

((227.35, 11)

((45.47, 0.8)

5

3

460.089 N

((318.29, 24.2)

6

4

((315.91, 33.5) ((270.78, 23.9)

((367.52, 4.5)

4

3

2

1

0

((137.82, -1.3) ((91.88, -0.3)

((321.58, 0.4)

((413.46, 10.7)

((45.94, 0.1)

((275.64, -1.3) ((229.7, -2.8)

(0, 0)) 10

5

((406.17, 59.7) ((361.04, 45.2)

((451.3, 77.5)

(223.5, 21.16) ((178.8, 13.84) ((134.1, 8.27) ((89.4, 4.31) (44.7, 1.1) (0,0))

((363.76, 33.7)

((409.23, 45.5))

((44.01, 1.5) (0, 0)

287.433 N ((454.7, 60.4)

((88.02, 4.4)

211.896 N

((268.2, 30.89)

10

((132.03, 10.1)

((220.05, 28.1))

138.321 N ((447, 91.62)

((176.04, 17.7)

((264.06, 40.4)

(87.7, 4.7) ( (43.85, 1.6) (0, 0)

5

6

((352.08, 73.1) ((308.07, 55.1)

((440.01, 117.9) ( (396.09, 94.2)

10

9

8

7

6

5

4

(0, 0)) 3

2

1

0

Fig. 7. Curvature lines of the spring for some applied loads.

a locked up moment. It should also be noted that the magnitude and direction of this locked up moment gets changed with the application of external load at different load levels. However following the present proposition, the corrected loaddeflection behavior of the tip is given in Table 2. Actual experimental loads (in N) are calculated by adding the clamping force with every applied load and their new values are given in Table 2. Similarly the tip deflection due to clamping is also added with the observed tip deflections during experiment.

3.3. Comparison between experimental and theoretical results Load parameters α are calculated from the actual experimenNL is calculated tal load and the slope at the free end of the beam ϕ tip NL for each of these load parameters from α vs. ϕ tip correlation. When NL α and ϕ tip are known, it is easy to obtain deflection profiles of the loaded spring as shown in Fig. 10. The figure also shows comparison between the theoretical and experimental deflection profiles for loads 159.2844, 232.8594 and 308.3964 (in N). Deflections at the tip of the beam are obtained from deflection profiles, and this theoretical load-deflection behavior is shown in Fig. 11. The figure also shows comparison between the experimental and theoretical load-deflection behavior of the tip. The theoretical and experimental results match quite well, and the slight difference in the progressive and the digressive nature between them may be due to the following reasons.

(1) The physical system is modeled as a cantilever beam of uniform cross-section throughout the length, but leaf spring has a geometry variation at the tip portion. (2) Theoretical analysis is carried out for tip concentrated loading, but actual load application point has an eccentricity with respect to the center line of the beam, as may be seen in Fig. 7. Similarly, the length of the spring is assumed to be constant, but due to the eccentricity, effective length of the spring is changing at each loading condition. (3) The ideal clamping requires a line load, but in actual experimental setup the contact at hydraulic cylinder head is of finite size. Due to this clamping deficiency, the profile of the deflected spring shows a point of inflection at higher values of applied load. Present analysis is not done with due consideration for actual locked up moment, the magnitude of which is changing in the course of experiment.

4. Curved beam under distributed and combined load To establish the robustness of proposed theoretical method further analysis is carried out on large deflection analysis of cantilever beam under distributed and combined loading. These problems are not readily solvable by using elliptic integrals and hence the iterative method of solution in Cartesian coordinate system, as proposed by Chen [9] has been used. The large deflection

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Fig. 8. Best fit deflection curves with their equations. Table 1 Observed load-deflection behavior of the tip. Applied load (dead weight) (N)

Tip-deflection (mm)

62.784 138.321 211.896 287.433 361.989 438.507 460.089

8.8 26.28 40.4 57.5 74.6 93 98.3

behavior of an initially straight beam is solved first and after appropriate validation, new results are obtained for initially curved beams. The schematic diagram of the present problem is shown in Figs. 12(a) and (b) for uniform and combined loading. A brief description of the direct integration method is furnished here for ready reference. To carry out large deflection analysis in Cartesian coordinate system ( x, y ), the slope curvature relation

Fig. 9. Experimental and modified best fit linear load deflection of the tip.

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Table 2 Actual load-deflection behavior of the tip.

1 dϕ = ρ dx

Load (N)

Tip-deflection (mm)

0 20.9634 83.7474 159.2844 232.8594 308.3964 382.9542 459.4704 481.0524

0 4.7 13.5 30.98 45.1 62.2 79.3 97.7 103

⎛ 3⎞

[1 + ϕ 2 ]⎝⎜ 2⎠⎟ , is used, where

ϕ is the slope

From the moment curvature relation tion is established,

dϕ ⎛ 3⎞ 2 ⎜⎝ 2 ⎟⎠

[1 + ϕ ]

=

dy at location s. dx

1 = M EI , the following relaρ

M (x) dx. EI

(12)

Integration of the above differential relation over the domain ‘0 to s’, i.e., from clamped end to the point A of Fig. 1(b), yields ϕ

∫ 0

M (x) dx EI 0 x

dϕ ⎛ 3⎞

[1 + ϕ 2 ]⎜⎝ 2⎟⎠

=∫

(13)

ϕ

and 1+ ϕ 2 dy dy for ϕ , the LHS is finally evaluated as . The RHS substituting dx ds of the equation is designated as G ( x, l ) following the notation of Chen [9]. Finally some mathematical manipulations lead to the following two differential equations as given in equations (14) and (15).

)

ds = 1 1 − G 2 dx

(

dy = G

4.1. Analysis for uniformly distributed and combined load When the cantilever beam is under uniformly distributed load, P 2 the bending moment at location s is M ( s ) = (l − x ) 2 . It should l be noted that the magnitude of uniformly distributed load is a function of the current beam configuration, because the total transverse load P = q ( s ) l is conserved at all times and for all configurations. Thus G ( x, l ) is found to be [9]

G ( x, l ) =

q 2 P x3 (l x − x 2l + x 3 3) = ⎛⎜⎝ lx − x2 + ⎞⎟⎠ . 2EI 2EI 3l

(16)

Using equation (16), equations (14) and (15) are converted into

It is easy to evaluate the left hand side integral as

(

Fig. 11. Theoretical and experimental load-deflection behavior of the tip.

)

1 − G 2 dx

(14) (15)

Equation (14) is evaluated iteratively with assumed values of l until the condition ∫ ds = L is satisfied, and subsequently equation (15) is solved with known value of l .

⎛ ds = ⎜ 1 ⎜⎝

2 2 P ⎞ ⎛ x3 ⎞ ⎞ 2 1 − ⎛⎜ lx − x + ⎟ dx ⎟ ⎝ 2EI ⎠ ⎜⎝ 3l ⎟⎠ ⎟⎠

3 ⎛ P ⎛ ⎞ lx − x 2 + x ⎞ dy = ⎜ ⎛⎜ ⎟ ⎜ ⎜⎝ ⎝ 2EI ⎠ ⎝ 3l ⎟⎠

2 2 P ⎞ ⎛ x3 ⎞ ⎞ lx − x 2 + ⎟ ⎟ dx 1 − ⎛⎜ ⎟ ⎜ ⎝ 2EI ⎠ ⎝ 3l ⎠ ⎟⎠

(17)

(18)

Equation (17) is solved iteratively to get projected length l and once the appropriate value of l is obtained, deflection profile of the beam is obtained by solving equation (18), as mentioned in the previous section. When the cantilever beam is under combined uniform and tip concentrated loading, the bending moment at location s is P 2 M (s) = F (l − x ) + (l − x ) 2 . Thus G ( x, l ) is found to be l

Fig. 10. Theoretical and experimental deflection profiles.

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The solution can easily be done in this case also, following the above-mentioned numerical method.

4.2. Validation of results To carry out validation with the available results, intensity of the uniformly distributed load is defined in a non-dimensional form as

q=

Fig. 12. Cantilever beam under uniform and combined loading.

G ( x, l ) =

P ⎛ x3 ⎞ F ⎛ x2 ⎞ lx − x 2 + ⎟ + ⎜ lx − ⎟ ⎜ ⎝ ⎠ ⎝ 2EI 3l EI 2⎠

(19)

Equations (14) and (15) are now converted into

⎛ ds = ⎜ 1 ⎜⎝

x3 ⎞ F ⎛ x2 ⎞ ⎤ ⎡ P ⎛ 1− ⎢ lx − x 2 + ⎟ + ⎜ lx − ⎟ ⎥ ⎜ 3l ⎠ EI ⎝ 2 ⎠⎦ ⎣ 2EI ⎝

2

⎞ ⎟ dx ⎟⎠

⎛ P x3 ⎞ F ⎛ x2 ⎞ ⎤ ⎡ ⎛ dy = ⎜ ⎢ lx − x 2 + ⎟ + ⎜ lx − ⎟ ⎥ ⎜ 3l ⎠ EI ⎝ 2 ⎠⎦ ⎜⎝ ⎣ 2EI ⎝ x3 ⎞ F ⎛ x2 ⎞ ⎤ ⎡ P ⎛ 1− ⎢ lx − x 2 + ⎟ + ⎜ lx − ⎟ ⎥ ⎜ ⎝ ⎠ ⎝ 3l 2 ⎠⎦ EI ⎣ 2EI

2

⎞ ⎟ dx ⎟⎠

(20)

(21)

qL3 EI

(22)

However, in case of combined loading, no such non-dimensional load parameter can be prescribed and hence results are presented in dimensional plane. The problem of a cantilever beam bending under simultaneous action of a concentrated load and gravity is validated with Chen [9]. In this comparative study, the values of the system parameters are L = 0.4 m , E = 194.3 GPa and I = 1.333 × 10−13 m 4 . The weight of the beam P (=0.3032 N) produces uniformly distributed load and in addition, the beam is acted upon by concentrated load F at tip. Three different cases for F = 0 , 0.098 and 0.196 N are taken up, and Fig. 13(a) shows the deflection profiles for these loading conditions in dimensional plane. It is observed that comparison with the results of Chen [9] is matching quite well. To validate the proposed method when the beam is under uniformly distributed load only, numerical results presented by Dado

Fig. 13. Numerical simulation of the results of (a) Chen [9] and (b) Dado et al. [7].

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Fig. 14. Transverse load vs. error plot.

et al. [7] for a prismatic slender cantilever beam bending problem is simulated. For any prescribed value of q we can find out the diEI mensional value q0 ( = P L ) by using the relation, q0 = ql 4 , where L q0 is the initial value of uniformly distributed load at straight configuration of the beam. Numerical computation is carried out for the aforesaid beam geometry and the computational results are shown by solid lines in Fig. 13(b) for load intensities q = 4 , 10 and 20. It is obvious from the figures that results are not matching with the results of Dado [fig. 5 of Ref. [7]] appropriately. This discrepancy is coming from the assumption that load intensity q ( s ) = P l remains constant along x axis for the beam configuration under consideration. However the intensity of the distributed load is constant along the arc length, and one must consider the net vertical component of such a loading condition. The expressions of bending moment M ( x ) and the function G ( x, l ) are evaluated following the proposed change in loading condition, and they are reported in the next section. Hence Fig. 13(b) contains another set of results which are indicated by solid lines with dots on them and observation on those lines is also reported in the next section.

x

G ( x, l ) = ∫ 0

P l is constant along the arc length, but it is not constant along the projected length. At location x ( 0 ≤ x ≤ l ) the vertical component of q is As mentioned earlier, the intensity of the distributed load q =

qn ( x ) =

q cosϕ

(23)

For a known distribution of loading qn , shear force V ( x ) and bending moment M ( x ) are given by the following relations, for the point A of Fig. 12. l

V ( x ) = ∫ qn ( x ) dx

(24)

x

l

M ( x ) = ∫ V ( x ) dx x

Thus G ( x, l ) is found to be

(25)

(26)

It is observed from equations (23)–(26) that determination of G ( x, l ) is a stepwise procedure starting from a known distribution of loading qn . To determine qn one must know two field variables ϕ and q a priori, where ϕ is a function of q . The coupled problem is solved numerically by using an iterative method, in which the final load value is reached with increment Δq . At load step i , a load increment Δq is given on the current load value of iΔq , i = 1, 2, … , N L . The corresponding dimensional load intensity q0 is EI obtained from the relation q0i = ql i 4 , where l i is the current proL jected length. At this load step the incremented value of total transverse load P is calculated as P i = q0i L , which is uniformly distributed over the current projected length l i . So intensity of the Pi EI distributed load is given by qi = i = q 3 , i = 1, 2, … , N L . Net vertil L cal component field of the load intensities are calculated as

(q)i

, i = 1, 2, … , N L and j = 1, … , N f . As mentioned cos (ϕ ij−1 ) earlier the search procedure begins with an assumed projected length l i for load step i . N f number of points are taken between ‘0 to l i ’ and at each of these points shear force, bending moment and the

(qn )ij =

4.3. Non-uniformly distributed load

M (x) dx EI

function (G ( x, l i )) j , i = 1, … , N L and j = 1, … , N f , are calculated from i

equations (24), (25) and (26) respectively. Once the function is dei

termined, ⎛⎜ ds ⎞⎟ is calculated from equation (14), and this is ⎝ dx ⎠ j numerically integrated between ‘0 to l i ’ to find arc length si , i = 1, … , N L . l i value is adjusted until the condition {( L − si ) L} < ε err is satisfied. Once l i is calculated from the search procedure, deflections at N f points are obtained from equations (26) and (15). At load step i , with known value of l i one can check the total i by numerically integrating (qn )ij over the domain transverse load Pcal i 0 to l , i = 1, … , N L and j = 1, … , N f , which ideally should be equal to P i . In the computation scheme this is a source of error, and this is accounted for in each load step through post-processing. This error i is calculated as {( P i − Pcal ) P i } and plotted against transverse load P i which is shown in Fig. 14. This is clearly seen from the figure that error is bounded between −8.89% and 3.34% and in general error increases with load. This indicates that selection of a proper error limit value ε err is a function of the loading. However in the present

Please cite this article in press as: Sushanta Ghuku, Kashi Nath Saha, A theoretical and experimental study on geometric nonlinearity of initially curved cantilever beams, Engineering Science and Technology, an International Journal (2015), doi: 10.1016/j.jestch.2015.07.006

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Fig. 15. Load-deflection behavior of the master leaf under uniformly distributed load and combined load.

work computation is carried out within 0.01% error. Fig. 13(b) shows the deflection profiles for non-dimensional load intensities q = 4, 10, 20 , and these results are successfully validated with the results of Dado [fig. 5 of Ref. [7]]. 4.4. New results for cantilever beam with initial curvature New results are furnished for an initially curved beam considering the geometry of leaf spring mentioned in experimental work. The load deflection behavior is observed under two different loading conditions: uniform and combined. Under uniform loading four different distributed loads, totaling 232, 308, 383 and 460 N are considered. In case of combined loading a base distributed load 232 N is considered and on top of that three different tip concentrated loads 76, 151 and 228 N are imposed. Load deflection behavior under above said loading conditions is determined by using the method of superposition mentioned in section 2.1.1 and the results are shown in Fig. 15. It is clearly seen from Fig. 15 that deflections under uniformly distributed loads are less compared to those under combined loads when the magnitude of total vertical load is the same. As a particular case, it may be noted that for the cases of uniform and combined loading as shown by curves 8 and 4, the magnitudes of total transverse loads are the same. 5. Conclusions Large deflection behavior of an initially curved cantilever beam subjected to various loading conditions has been studied both theoretically and experimentally. Large deflection behaviors of straight and initially curved cantilever beams under tip concentrated loads have been studied theoretically. Solutions of such geometric nonlinear problem are obtained iteratively with the help of MATLAB® computational simulation. Experiment has been carried out with the master leaf of a leaf spring bundle by modeling it as an initially curved cantilever beam. Theoretical results are compared successfully with experimental results in general. From the slight difference in trends of the comparison study, some relevant parameters of the physical system are identified. Further theoretical study on the large deflection behavior of a cantilever beam under distributed and combined load reveals that analytical solution based on elliptic integral is insufficient to predict the correct result. An iterative method with incremental loading has been introduced additionally to study such problems. Here also results of other

researchers have been compared successfully, and new results have been furnished for initially curved cantilever beams. Nomenclature

b F h I l L M N f , Ng NL P q ( s) , q ( s) qn ( s ) s, n x, y 0 0 xtip , ytip NL NL xtip , ytip α δ x, δ ξ

δ y, δ η ε err ξ, η ϕ NL 0 ϕ tip ,ϕ tip ϕ 00, ϕ 0NL ρ

Width of the beam Vertical conservative load applied at the tip of the beam Thickness of the beam Moment of area of the beam Projected length of the beam ( L − δ x ) Length of the beam Bending moment Number of precision points used to represent physical and computational domain in the solution algorithm Number of load steps A constant transverse load coming from distributed loading ( = qL ) Intensity of distributed load in dimensional and nondimensional form Vertical component of q Curvilinear coordinate system, in normalized form Cartesian coordinate system x, y coordinates of the tip at no load condition x, y coordinates of the tip at load step N L Normalized load parameter corresponding to F Shortening/stretching of the beam in dimensional and normalized form Beam deflection in dimensional and normalized form Error limit in calculation of arc length Cartesian coordinate system in normalized form The slope dy dx at location s The slope dy dx at the free end of the beam at location s = L at initial configuration (i.e., at no load condition) and for load step N L Slope at the fixed end ( s = 0 ) corresponding to no load and load step N L (It should be noted that ϕ 00 = ϕ 0NL = 0 ) Radius of curvature

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Please cite this article in press as: Sushanta Ghuku, Kashi Nath Saha, A theoretical and experimental study on geometric nonlinearity of initially curved cantilever beams, Engineering Science and Technology, an International Journal (2015), doi: 10.1016/j.jestch.2015.07.006