Computers & Structures. Vol. 14, No. 5-6, pp. 357-360. 1981

Printed in Great Britain.

c@45-7949/9/81/110357-4#&?.ln3/0

@ I981 Pergamon Press Ltd.

LARGEDEFLECTIONSOFCANTILEVERBEAMS OFNONLINEARMATERIALS

GILBERT L~wr.9 and FRANK ~ONASA~

Michigan Technological University, Houghton, MI 49931, U.S.A.

(Received I8 August 1980; in revised form 12 January 1981)

Abstract-This paper deals with the large deflections (finite) of thin cantilever beams of nonlinear materials, subjected to a concentrated load at the free end. The stress-strain relationships of the materials are represented by the Ludwick relation. Because of the large detections, geometrical nonlinearity arises and, therefore, the analysis is formulated according to the nonlinear bending theory. Consequently, the exact expression of the curvature is used in the moment-curvature relationship. The resulting second-order nonlinear differential equation is solved numerically using fourth-order Runge-Kutta method. For comparison purposes, the differential equation is solved for linear material and the results are compared to the exact solution which uses elliptic integrals. Deflections and rotations along the central axis of beams of nonlinear materials are obtained. The numerical algorithm was performed on the UNIVAC I 110.

width of rectangular cross section, in. coefficient of material property; Young’s modulus of elas-

ticity for linear elastic materials, psi Young’s modulus of elasticity, psi height of rectangular cross section, in. moment of inertia. in4

a”b~h~“+t~”

= Er for linear elastic materials =2”+‘(1 t2nl”P” P

length of the beam along the central axis, in. bending moment, lb-in. coefficient of material property point load acting at the free end of the beam, Ibs arc length along the central axis of the beam measured

from the origin horizontal and vertical coordinates, respectively dummy variables stress, psi strain in/in slope of the central axis at s slope of the central axis at the free end horizontal and vertical displacements at the free end,

respectively, in.

INTRODUCTION

The subject of the large deflections of thin beams is one of great historic interest and has received renewed attention during the past two decades. Thin beams, being flexible, exhibit large deflections and slopes when subjected to loads. Therefore, due to these large deformations, geometrical nonlinearity arises while strains remain small. As a result, problems involving large deflections of beams must be formulated according to the nonlinear bending theory. In the past, applications of the nonlinear bending theory to thin beams have been con&red to linear elastic materials. Solutions to much of the previous work can be found in the book by Frisch-Fay fll. Barten 121 and Bisshopp and Drucker 131 provided a solution for the large deflection of cantilever beams of linear elastic material subjected to a concentrated load at the free end. However, solutions to problems of large deflections of thin beams of nonlinear materials have been limited. Prathap and Varadan [4] examined the inelastic large

tAssistant Professor, Department of Mathematical and Com- puter Sciences.

SAssociate Professor, Department of Civil Engineering.

deformation of a uniform cantilever of rectangular cross section with a vertical point load at the free end, where the material of the beam has a stress-strain law of the Ramberg-Osgood type. Lo and Gupta [5] examined the bending problem of a nonlinear rectangular beam with large deflections where linear elastic behavior is con- sidered for sections of the beam that deformed elastically, and a logarithmic function of strain is used for regions stressed beyond the elastic limit. The logarithmic function was approximated by a semilogarithmic relation which is only applicable for special cases.

In this paper, the problem of finite deflections of cantilever beams of nonlinear elastic materials and sub- jected to a concentrated load at the free end is considered. This problem involves both material and geometrical nonlinearities. To test the validity of the method of solution used in this study the results are compared with previously published results for thin beams of linear elastic materials. Deflections and rotations of nonlinear cantilever beams are presented in a ~bulated form.

PROBLEMSTATEMENT

Thin cantilever beam of rectangular cross section sub- jected to a concentrated load, P, at the free end as shown in Fig. 1 is considered for this study. A point m(x, y) on the deflected central axis of the beam is identi~ed by the x and y coordinates, the arc length s, and the angle of rotation $. The vertical and the horizontal displacements and the angle of rotation at the free end are denoted by S,, S,,, and (Go, respectively. The beam is constructed of a nonlinear material where the experimental stress-strain curve is represented by the Ludwick relation [6], i.e.

B

Fig. 1. Deflected form of a cantilever beam.

CAS Vol. 14, No. 563 3.57

358 G. LEWIS and F. MONASA

where (T and e represent the stress and strain, respec- METHODOFSOLUTION

tively, and the constants B and n represent material In rectangular coordinates the curvature is expressed properties. This relation applies primarily to metals as: which work harden.

Experimental data of annealed commercially pure dll, d*y/dx’ copper [7], and N.P.8 aluminum alloy 181 show that the

Y”(X)

stress-strain relationships can be represented, respec- ds = [l t (dy/dx)*]“* = [ 1 t y’*(x)]“*’

tively, as follows: Therefore, from eqn (6)

u = 66500 e”.463 (2a) and

(L-x-&)” K” . (7)

o = 66100 l o.*O9 (2b)

where D is the stress in psi. The above representation of Equation (7) is a second order non-linear differential

the stress-strain curve is a purely empirical curve-fitting equation for the vertical deflection, y.

technique. It is assumed that the weight of the beam is To solve eqn (7), let u = y’, then

negligible, strains remain small, and the axis of the beam is inextensible. u’ = (1 t u2)3’2 (L - x - &)

4, * 03)

MOMENT-CURVATURE RELATIONSHIPS After rearranging eqn (8) and integrating, one obtains

The analysis of beam deflections is based on the Bernoulli-Euler bending moment-curvature relation. In

du

the elementary bending theory the expression of the I (1 t U*))‘* = I $ (L - x - S,,)” dx. ”

curvature is linearized by neglecting the square of the slope in comparison with unity. This approximation is

Therefore,

valid only for beams where the deflections are small compared with the length of the beam. However, thin

u I (L-x-S,)“+‘tC

beams, being flexible, deform under the action of loads in m=-- K (ntl) ‘.

a configuration with large deflections, curvatures, and slones. Therefore. the exact exnression for the cur- The condition y’ (0) = 0, implies that

vature, d$/ds, of the central axis of the deflected beam must be considered in the moment-curvature relation. c = J_ (L - &I)“+’

Using the general stress-strain relation of eqn (1). the ’ K, (ntl)

moment-curvature relationship for a rectangular cross section can be written as follows: Thus,

d$ ds=

2”+’ (1 t 2n)"M" n”b”h2”+1p ,

I (L - &Jn+’ (3) *=K,

-(L-x-&)“+‘=f(x) (n + 1)

where M is the bending moment, and b and h are the and

width and depth of the cross section, respectively. Figure 1 shows that the bending moment M at any point m(x, y) f(x) along the central axis of the beam is

y’(x) = u(x) = X41 -f’(x))’

(9)

M = P(L-x- 6,). (4)

Substituting the expression of M from eqn (4) into eqn (3), the curvature becomes

drC, 2”+‘(1 t2n)"(P(L-x-S,,))" ds= n"b",,2"+,g" 9 (9

and, letting

K,, = ,,"b"h*"+lfj"

,“+I(1 t 2n)“P”

then

(6)

Integrating eqn (9) and setting the constant of in- tegration equal to zero (y(O) = 0), results in the following:

In eqn (IO) z is used as a dummy variable. Because of the initially unknown displacement, &, of the free end of the cantilever beam, it is not possible to solve Eqn (10) directly for the vertical deflection, y. However, the ver- tical and horizontal displacements can be obtained by using both eqn (10) and the equation for arc length,

I L--6)#

L= d/(1 t y"(t)) dt. (11) 0

For a linear elastic material, K. is equal to the flexural rigidity, EZ, divided by the load, P.

Because of the transcendental nature of eqn (lo), one must resort to numerical integration procedures to evaluate the displacements 8, and S,,, and the angle of rotation tJo at the free end of the cantilever beam.

Large deflections of cantilever beams of nonlinear materials 359

NUMERICAL METHOD

By using numerical integration, eqns (10) and (11) can be solved for the deflections along the central axis of the cantilever beam. However, care must be taken in choos- ing an appropriate value for Sh in order that the quantity under the square root sign in eqn (10) always remains positive. Because of this complication, we prefer to solve eqn (7) directly, using a fourth-order Runge-Kutta method. The procedure is as follows: (1) Assume a small load, P, and zero (or some other educated guess) horizontal displacement, S,, at the free end; (2) Use Runge-Kutta to solve eqn (7). This yields the vertical deflection along the central axis of the beam, and eqn (11) gives a new value of &; (3) Iterate on eqns (7) and (11) until the value of &, does not change by more than a given tolerance from one iteration step to the next, at which a solution for the deflection of the beam is said to be reached. To obtain the displacement for higher loads, the computed displacement 8,, for the previous load is used with a slightly increased load, and steps (l)-(3) are repeated until a solution is achieved. This procedure may by used for any desired load.

During the creation of the above algorithm there arose several difficulties, which had to be resolved. The prob- lem with eqn (10) has already been mentioned. The fact that & is unknown while solving eqn (7) seemed to cause the most problems. Since 6, is just the y value at the end of the beam (x = L - &Jr it is natural to evaluate S, by calculating y(L - S,,) using Runge-Kutta. However, since S,, is not known, we do not even known how to break down the interval into subintervals of the correct size. Since Sh is estimated (not exact), then invariably the numerical integration of eqn (7) did not end at exactly x = L - S,,. Thus, a correction in the S,, S,, and $,, values had to be made. This is done by using the equation for arc length:

s = I

x v(l t y’*(t))dt (12) D

and Simpson’s Rule to calculate the total length of the deflected beam. The difference between the exact value of the length and this calculated value gives a way of determining the corrections required for S,, S,, and I&.

Another problem that occurred involved slow con- vergence or divergence of the algorithm. If two suc- cessive calculated estimates of 8,, bound the exact value (determined by the fact that high (low) estimate for S,, at one step leads to a decrease (increase) in & at the next step), then a certain weighted average of them was used as a new estimate of S,,. This weighted average was found to improve performance and, in certain cases, it created convergence for an otherwise divergent scheme.

It is interesting to note that if the algorithm is con- vergent, then it is self-correcting. The S,, value calculated for one load (even though not exact) is used as input for the next higher load, and the S,, value for this higher load can be calculated to any desired degree of accuracy.

The above numerical algorithm was performed on the UNIVAC 1110.

DISCUSSION OF RESULTS

For the case that n = 1, the material is linear elastic, and the integral in eqn (10) reduces to an elliptic integral. Elliptic integrals of the first and second kind have been used in [9] to obtain the vertical and horizontal dis- placements and the angle of rotation at the free end of a cantilever beam of linear elastic material subjected to a concentrated load at the free end.

To test the accuracy of our algorithm, we have used eqns (7) and (11) and the above-mentioned numerical method, with n = 1, to find the displacements 8, and Sh and the angle $, at the free end of a cantilever beam with concentrated load at the free end. These results are in agreement with the ones obtained from the exact solu- tion using elliptic integrals [9]. The maximum dis- crepancy between the exact solution and the numerical algorithm is 0.099%.

In Table 1 the nondimensional ratios of 1,&/?r/2, 8,/L. and &/Z_. are given for two nonlinear materials, respec- tively. It is shown that for the same values of Z,“+‘/K”, the above nondimensional ratios decrease as the exponent of the strain in eqn (1) decreases. It is also observed that for the same values of L”+‘/K,, the vertical and horizontal deflections and the slopes along the central axis of the beam are the same for anv h/L ratio. Therefore. deflections and slopes are not dependent on the h/L values.

Table 1. End deflections and end rotation vs &“+‘I&) for beams of annealed copper and aluminum alloy

Annealed Copper 1+1

N.P.8 Aluminum Alloy

" = 66100 co*209

5l $Jo/42 0 = b%;o,o, CO.463

" $)/L $ofrr/2 6&)/L 6h/L

0 0 0 0 0 0 0

0.25 0.05009 0.05973 0.00203 0.02740 0.03669 0.00073

0.50 0.09864 0.11741 0.00766 0.05419 0.07251 0.00284

0.75 0.14443 0.17142 0.01685 0.07984 0.10672 0.00617

1.00 0.18675 0.22085 0.02814 0.10401 0.13884 0.01046

2.00 0.32039 0.37235 0.08241 0.18414 0.24407 0.03270

3.00 0.41049 0.46910 0.13468 0.24190 0.31822 0.05629

4.00 0.47437 0.53433 0.17925 0.28484 0.37211 0.07785

5.00 0.52222 0.58115 0.21668 0.31816 0.41308 0.09692

6.00 0.55967 0.61646 0: 24839 0.34501 0.44548 0.11374

7.00 0.58995 0.64414 0.27562 0.36725 0.47190 0.12868

8.00 0.61511 0.66652 0.29931 0.38613 0.49398 0.14204

9.00 0.63642 0.68503 0.32017 0.40246 0.51282 0.15409

10.00 0.65477 0.70065 0.33871 0.41677 0.52913 0.16504

360 G. LEWIS and F. MONASA

CONCLUSIONS REFERENCES

In this study the finite vertical and horizontal deflections and rotations along the central axis of can- tilever beams of nonlinear elastic materials of the Lud- wick type and subjected tb concentrated load at the free end are obtained. The problem involved material and geometrical nonlinearities and solution to such problems can be obtained bv using numerical methods only. Fourth-order Runge-Kutta method proved to be an excellent technique to solve the resulting second-order non-linear diEerentia1 equation in this problem. It is shown that for the same values of L”+‘/K,,, the non- dimensional ratios of 1&/7r/2, &IL and &,/L decrease as the exponent of ihe strain in the Ludwick relation decreases. The computer program provides the solution for the deflections and rotations along the central axis of nonlinear cantilever beams constructed of materials exhibiting a Ludwick stress-strain relationship.

1. R. Frisch-Fay, Flexible Bars, Butterworths, London (1%2). 2. H. J. Barten, On the deflection of cantilever beam. Q. Appl.

Math. 2, 16&171 (1944) and ibid. 3, 275-276 (1945). (NOTE: The second article corrects a mistake in the first one).

3. K. E. Bisshopp and D. C. Drucker, Large deflections of cantilever bea& Q. Appl. Math. 3,2?2-275(1945).

4. G. Prathan. and T. K. Varadan. The inelastic laree defor- mation of’deams. J. Appl. Mech.’ ASME 43, 689-660 (1976).

5. C. C. Lo and S. D. Gupta, Bending of a non-linear rectangular beam in large deflection. I. Appl. Mech. ASME 45, 213-215 (1978).

6. Alfred M. Freudenthal, The Inelastic Behavior of Engineering Materials and Sfructures, p. 203. Wiley, New York (1950).

7. A. A. Denton, Plane strain bending with work hardening. J. Strain Anal. l(5), 196-203 (1966).

8. M. B. Bassett and W. Johnson, The bending of plate using a three-roll pyramid type plate bending machine. J. Strain Anal. l(5), 39Ul4 (1%6).

9. S. P. Timoshenko and J. M. Gere, Mechanics of Materials, pp. 208-211. D. Van Nostrand, New York (1972).

Acknowledgemen&The authors would like to acknowledge the assistance of Dr. Linda Ottenstein who helped with the computer programming.