Nonlinear Beam Deflection - transverse force

Method of derivation:Curvature = Moment / Flexural RigidityElliptic IntegralsPrinciple of Elastic Similarity

End result:x, & y as a function of FF = K1q + K2*q3 => F=F(q)

Identification of terms

L -

dy

dx

Lx

y

F

2

1 ds

EI

s = distance along the beam from node 1 to node 2r(s) = radius of curvature at s(s) = angle of the beam at s with respect to the horizontalM(s) = moment at sE = modulus of elasticityI = moment of inertia about the z-axis (out of plane)x and y are the in-plane horizontal and vertical axes with the origin at node 1.Fo = external force at node 2

L = beam length = projected beam shortening

Fig 1

Beam Curvature due to M, F

The curvature at each point s along the beam is defined to be

EI

sM

ds

sd

dx

dy

dx

yd

srcurvature

)()(1

)(

1 3

22

2

2

If an external transverse force is applied at the node 2, the curvature at s is

EI

xLF

EI

sM o )()( (1)

Change of variables

Equation (1) contains two independent variables, s and x. We eliminate x by differentiating (1) with respect to s, then re-integrate it.

Noting dx/ds=cos() , differentiating (1) gives

Applying the identity , the integration of (2) leads to

cos

2

2

EI

F

ds

dx

EI

F

ds

d oo

2

2

2

2

1

ds

dd

ds

d

CEI

F

ds

d o

sin2

12

(2)

(3)

Boundary conditions

To solve for the integration constant in (3) we apply the following boundary condition at the end node 2

since the moment vanishes at node 2. Therefore, equation (3) becomes

Now, we want to end up with a solution of the form

0Ls

ods

d

00

00 sinsin2

sinsin2

oo

oo

EI

F

dLds

EI

F

ds

d

LEIFo ,,00

(4)

(5)

(6)

Transformation to Elliptic IntegralsElliptic integrals have been well characterized (see M. Abramowitz, A. Stegun, Handbook of Mathematical Functions). Transforming equation (6) to an elliptic integral form provides us with their benefits.

Introducing two new variables p and defined as

We seek substitutions for d, sin(), and sin() in equation (6).

We find the differential element d by the differentiation of equation (8) with respect to and then using the trigonomic identity

This leads to

And from (7) & (8) we see that

op sin12

12 o

sin1

sin1sin 2

222 sin1sin2sin1cos pp

dpp

pd

22

2

sin1sin2

cossin4

1sin2sin 22 p12sin 2 po

(7) (8)and

(9)

(10)

(11) (12)

The Elliptic Integral Form

Substitution of equations (10-12) into equation (6) results in

where the lower and upper limits of the integral are determined from equation (11)

Associating equation (13) with the complete and incomplete elliptic integrals of the first kind, we have

)(

)(22

20

2

1 sin1

p

p p

d

EI

LF

p2

1sin0 1

1220

1

20 ,)( pFpKEI

LF

(13)

(14)

(15)

Realization of Force and Displacement, SIMO

0Given , it’s required force is found by plugging (7) & (14) into (15).

Equation (15) is the nondimensionalized external force at node 2.

After finding the nondimensional force, the relationship gives us the nondimensional transverse deflection, LL

sindsdy

0

0 02

0 )sin()sin(

sin

2

1

d

LF

EI

L

Equation (1) with the BCs , and

gives the nondimensional projected beam shortening as

00

x )(

sin20

000

0x

x

LFEI

F

EI

M

ds

d

)12(21 22

0

pLF

EI

L

(16)

(17)

Nonlinear Stiffness

The outputs of equations (15-17) as functions of the input 0 are plotted as “O’s” below. The solid curves are 3rd order, piecewise continuous, polynomial fits. To obtain nonlinear stiffness, we first assume that the curves can be approximated by a third order polynomial of the form

were q stands for , x, y, etc. Seeing that the solution has odd symmetry, we only need to keep constants B and D, which also eases iterative computations. Absorbing the material and geometric terms into B and D, respectively K1 and K2, we find that

The coefficients of these polynomial curves are associated with the linear stiffnesses K1,i and the cubic nonlinearities K2,i.

In order to maintain accuracy, (19) is applied in a continuous piecewise fashion by dividing the total physical range into, say, 4 intervals, were i(q)=[1,4]. (see plot)

3220

L

qD

L

qC

L

qBA

EI

LF

3,2,10 qKqKF ii

(18)

(19)

Plot of Nonlinear Stiffness vs Theory

EI

LF 20 L

L

2/0

Fig 2

Principle of Elastic Similarity

Using the geometric properties of the above elliptic integrals together with the principle of elastic similarity we find an intuitive relationship. This type of analysis was first reported by [R. Fay, “A new approach to the analysis of the deflection of thin cantilevers,” Journal of Applied Mechanics 28, Trans. ASME, 83, Ser. E (1961) 87].

From similarity:

From Fig 1, using your imagination, assume that the beam, which remains anchored at node 1, continues arching to the left of node 1 until it reaches a new node 0. Node 0 is such that it is oriented normal to the horizontal axis (see Fig 3). The length of the section from node 0 to 1 is L2. Also imagine that the shape of L2 is the shape that would be required if L1 is to remain unaltered when node 1 is no

longer fixed.

Geometric Properties

From geometry: From the geometric properties of the elliptic functions, we make the following identifications.

bkpKpEpFpE

hL

kph

pFpKk

LLL

p

d

kL

p

p

kpKLLL

BB

B

B

BB

B

/)]}()(2[)],(),(2{[

)cos(

/2

),()(1

)(sin1

1

2/

)2/sin(

)sin()2/sin(

/)(

1

21

0 222

0

21 EI

Fk 0

0

(x,y)

B=/2

L2

L1

L1-B

F0

1

2

0

b

h

Identification of terms

Fig 3

Nonlinear Beam Deflection - Fx, Fy, & M

Method of derivation:By intuition, jumping straight to elastic similarity and the geometric properties of elliptic integrals.

End result:x, & y as a function of Fx, Fy, and MF = K1q + K2*q3 => F=F(q)

Here we attempt answer the question:Given the angles of node2, determine x, y, Fx, Fy, and M.

A New Approach to Analyzing Nonlinear Beams with Coupled Fx, Fy, and M Forces.

The method presented above will now be extending to a much more general situation.The geometric quantities listed below are self-described in a new geometric representation shown in Fig 4.

)sin(

)cos(

),(),(

2sin)sin(/)2/sin(

)cos(

0

0

03

220

02

0

03

By

Bx

B

BBB

FF

FF

FLMEI

LFpFpFkL

p

hL

Given 0 and B, the following holds

The Modified Geometric Representation

1

0 2

F0

B

0

h=2p/k

B

B

B

L2 L1

L3

0

0

Fig 4