Axially Loaded Generalized Beam Element on a Two-Parameter Elastic Foundation with Semi-Rigid...

8/13/2019 Axially Loaded Generalized Beam Element on a Two-Parameter Elastic Foundation with Semi-Rigid Connections an…

http://slidepdf.com/reader/full/axially-loaded-generalized-beam-element-on-a-two-parameter-elastic-foundation 1/10

— 1 —

COMPUTATIONAL MECHANICS

WCCM VI in conjunction with APCOM’04, Sept. 5-10, 2004, Beijing, China

2004 Tsinghua University Press & Springer-Verlag

Axially Loaded Generalized Beam Element on a Two-Parameter

Elastic Foundation with Semi-Rigid Connections and Rigid Offsets

E. I.Avramidis *, K. Morfidis

Department of Civil Engineering, Aristotle University of Thessaloniki, 54006 Thessaloniki, Greece

e-mail: [email protected], [email protected]

Abstract A geometric nonlinear Timoshenko beam element on a two-parameter elastic foundation is

presented. This new element is based on the analytical solution of the differential equation of the problem

and has the ability to optionally taking into account the shear deformations of the beam, semi-rigid

connections and rigid offsets at its nodes. The gonerning equations of the element are formulated on the

basis of its deformed axis, a fact that enables its use in stability analyses or in analyses where second-order

effects must be taken into account. Apart from the stiffness matrix of the element, the load vector in the

case of uniform transverse loading is also presented. The usefulness and reliability of the proposed element

are documented by a numerical application.

Keywords: beams on elastic foundation, two-parameter elastic foundation, Timoshenko beam, finite

element method, geometric non-linearity.

INTRODUCTION

The problem of an axially and transversely loaded beam resting on elastic foundation is encountered in the

analysis of building structures, ships, aircrafts and various machines. Such analyses require the use ofspecial modeling techniques. Focusing on building structures, the modeling of the foundation-soil

interaction constitutes one of the most serious problems. Since many decades, the Winkler model [1],

distinguished for its simplicity, is widely used for the modeling of the soil. However, its obvious

inadequacy led to the formulation of two-parameter models [2-3] which, unlike the Winkler model,

account for the continuity of the deformations between the loaded and the free surface of the soil.

Furthermore, when analyzing a foundation structure (most often consisted of column footings

interconnected by foundation beams), it is necessary to model footings as rigid elements resting on elastic

foundation. Their modeling is usually achieved through the use of classical beam elements with large (in

effect: infinite) values attributed to their cross-section parameters, so that they behave as absolutely rigid.

Also, the elastic soil is usually modeled by discrete translational and/or rotational springs. Another

category of modeling problems relates to the analysis of steel structures. Two typical problems are worthmentioning: The modeling of the semi-rigid connections of the beams at their nodes and the consideration

of the deformations (second order theory) in calculating the stresses. The former is dealt with by placing

rotational springs at the ends of the beams [4], as well as rigid offsets for the modeling of the practically

rigid node area. To confront the latter, the use of beam elements governed by differential equations

formulated on the basis of the deformed axis is indispensable.

In order to achieve a combined solution to all the aforementioned problems that is both correct and simple,

a geometric nonlinear Timoshenko beam element resting on a two-parameter elastic foundation is

proposed. For the connection between the rigid offsets incorporated in this element and its internal flexible

segment rotational springs are used. In this paper, the analytical formulation of the stiffness matrix and the

load matrix for uniform transverse load are presented. The usefulness and reliability of the proposed

element are documented by a numerical application.



— 2 —

FORMULATION OF THE STIFFNESS MATRIX

The new generalized beam element is shown in Fig. 1a. It consists of: Two rigid segments between nodes

1,2 and 3,4 respectively (rigid offsets). The flexible median segment between nodes 2 and 3, which is a

Timoshenko beam element. For the connection of the median segment to the rigid offsets rotational springs

are used. The element rests on a two-parameter elastic foundation and is loaded by a static axial force.

When laying out the differential equations of the element, its deformed shape is taken into account. The

derivation of the stiffness matrix [K] is divided into two stages:1st. The formulation of the median segment's stiffness matrix [Kint] based on the analytical solution of its

differential equations.

2nd. The formulation of the relations between the coefficients of the stiffness matrix of the median

segment [Kint] and those of the generalized element [K].

P

Gk

1

1

d

R Α

2

K

L

PEI, G(A/n)

ks

d2

KRB

43

(a)

Stiffness Matrices

[K] :Element dGeneralize

][K :Segment Median int

1w

1

2

∆φ2

4

3

d1 L

Y

Xφ

4φ

2w w3 w4

d2

3φ

4φ

φ2

φ1

φ1

φ1 ∆ (b)

Fig. 1 (a) New generalized geometric nonlinear beam element (b) Deformed configuration of the element

First stage

The bending of an axially loaded Timoshenko beam element on a two-parameter elastic foundation is

governed by the following differential equations:

( ) 012

2

2

2

4

4

=−++

−−+

−− q

dx

qd

Φ

EI wk

dx

wd

Φ

EIk k P

dx

wd

Φ

k P EI S

S

G

G (1a)

( ) 012

2

4

4

=−+

−−+

−−

dx

dqφk

dx

φd

Φ

EIk k P

dx

φd

Φ

k P EI S

S G

G , (1b)

where EI is the flexural stiffness of the beam, Φ=AG/n, φ is the flexural rotation of cross- section, A is the

cross-section area, G is the shear modulus of elasticity, n is the shear factor, P is the axial load, q is the

lateral external load, w is the lateral deflection, k S (kN/m2) is the modulus of the subgrade reaction, and k G

(kN) is the second parameter of the elastic foundation.The structures of the two homogenous equations of (1a) and (1b) are identical, the six different forms of

their solutions being dependent on the values of the coefficients of the beam and the soil (Fig. 2). From all

six cases of solution, only cases 1 and 3 (Fig. 2) are of practical interest with regard to analyses according

to the second order theory [5]. The process of the analytical formation of the stiffness matrix [K int] is

already known. The stiffness coefficients for the two typical forms of solution of (1a) and (1b) are given in

reference [5].

Second stage

In order to formulate the relations between the coefficients of the stiffness matrix [K int] and the respective

coefficients of [K], the following three-step procedure must be adopted:a. Formulation of the relation between the displacements of nodes 2,3 and 1,4 respectively.

From Fig. 1b, the following relation is obtained:



— 3 —

−

−

+

=

−

−

3

3

2

2

1

1

4

4

1

1

2

1

3

3

2

2

0000

000

0000

000

100

0100

001

0001

V

M

V

M

)(K

)(K

w

φ

w

φ

-d

d

w

φ

w

φ

R Β

RA

(2)

where M2 and M3 are the bending moments at nodes 2 and 3 respectively, V2 and V3 are the shear forces,

and K RA and K RB are the constants of the rotational springs. Eq. (2) can be expressed in a “symbolic” form

as:

[ ] [ ][ ] [ ] [ ]intint S T uΤ u KR+= (3)

b. Formulation of the relation between the stresses at nodes 2,3 and 1,4 respectively.

From Fig. 3, the following relation is obtained:

( )

−+

−

−

+

=

4

4

1

1

2

1

222

22

32

12

1

21

31

3

3

2

2

2

1

4

4

1

1

0000

000

0000

000

2

100

21

3100

002

1

002

1

3

1

1000

100

0010

001

w

φ

w

φ

-d

-d

k P

d k d k

d k d k

d k d k

d k d k

V

M

V

M

-d

d

V

M

V

M

G

S S

S S

S S

S S

G

G

G

G (4)

The terms k Gd1 and k Gd2 are moments resulting from the assumptions on which the two-parameter elastic

foundation model is based, and are necessary for fulfilling the equilibrium conditions [6]. VGi is the

“generalized shear force” [3], which takes into account the effects of shearing stresses in the soil medium

as well as the shearing stresses in the beam (Fig. 3). Eq. (4) can be expressed in a “symbolic” form as:

[ ] [ ] [ ] [ ] ( )[ ]{ }[ ]uT k P Κ S T S P GW

T −++= int

(5)

c. Formulation of the stiffness matrix [K] as a function of the stiffness matrix [K int].A number of matrix transformations of (3) and (5) are performed, in combination with the familiar

relations:

[ ] [ ] [ ]intintint u K S = (6a)

[ ] [ ] [ ]u K S = (6b)

which express the force-deformation relation of the internal segment and of the whole element

respectively. The required transformations ultimately lead to the following relation:

[ ] [ ] [ ] [ ] [ ]{ } [ ] [ ] [ ] ( )[ ] P GW KR

T T k P Κ nΤ Κ Τ Κ m Ι Τ K −++−=

−

int

1

int (7)

where [I] is the 4x4 identity matrix.

The “switches” added in (7) determine whether the rotational springs, the elastic support or/and the effect

of the axial force will be considered or not. The indices must be set to 1, if the corresponding effect is to be

taken into account. If not, they must be set to 0. If m=0, then Eq. (7) describes the geometric nonlinear

stiffness matrix of a beam element without semi-rigid connections, supported by a one or two parameter

elastic support. If n=k G=0, then Eq. (7) expresses the geometric nonlinear stiffness matrix of a beam

element without elastic support. If m=P=0 and n=1, then Eq. (7) represents the linear stiffness matrix of a

beam element with one or two parameter elastic support. Finally, if k G=P=0 and n=1, then Eq. (7) describes

the linear stiffness matrix of a beam element with one parameter elastic support [7]. The coefficients of the

stiffness matrix [Κ ] are given in Appendix 1.



— 4 —

sin(Qx)

xsin(Qx)

-Rx

sin(Qx)

xe

-Qx

e sin(Qx)

Sections of curve D =0: (0-4) and (1-2-3): Case 6

-Rxe cos(Qx)Rxe sin(Qx)Rxe cos(Qx)D <0c>0b>0Case 3

Rx

Rx

Rx

eD =0c>0b<0Case 6 ch

b<0

b<0

b<0

b>0

Case 5

Case 4

c>0 echD >0

c<0 D >0ch e

ch

e-RxRxxe

e-Rx

-Rxe

eQx e

cos(Qx)

cos(Qx)

cos(Rx)

Section of curve D =0: (0-1): Case 2

b>0

b>0

Case 2

Case 1

c>0 chD =0

c>0 D >0ch

Case 5 4

D =0ch

f (x)f (x)f (x)

xcos(Qx)

1

sin(Rx)

2

sin(Qx)

cos(Qx)

3

ch

ch

k

(c/4)+(b/4) (c/4)-(b/4)

(1/2)(-b- D )

(1/2)(b+ D )

--b/2-Rx

(1/2)(-b+ D )

(1/2)(-b+ D )

ch

ch

2b -4cchD =

θ=1-

ch

ch

c=

GP-k

Φ

ΕΙθS

(1/2)(b+ D )

QRf (x)

(1/2)(b- D )

-

4

b/2

ch

(P-k )-1

b=ΕΙθ

ch

ΕΙkΦG

S

Case 5

chD =0

Case 3

C a s e 3

P<k

0

G

Φ /(ΕΙ)2

b>0

Case 1

Case 4

P>k G

Φ

b<0

1 b>0

2

Case 3

4Φ /(ΕΙ)2

Case 4

D <0ch

D >0ch

c<0

c>0

b = 0

G(P-k ) b<0

Sk

3

c

Fig. 2 The possible forms of solution of the governing differential equations and the corresponding ranges

of the values of the parameters k S , k G , P

shear layer

Winkler forces

shear layer

1M

VG1

w1

2

1

M

Sk w

VG2

2

G2V

2M

wφ

2

Υ

φ

Winkler forces

shear layer

3

Χ

M

VG3

3

M3

VG3

w

V

φ4

M4

w4

G4

P

P

P

P

P

P

S 1k w

S 3k w

Sk w4

S 2k w

Sk w3

Winkler forces

dx

dw)k(P

dx

φdEIVVV G2

2

soilbeamG −−−=+=

Fig. 3 Relationships between the forces at the internal joints and the element end forces

FORMULATION OF THE LOAD VECTOR FOR LATERAL UNIFORM LOADING

The load vector [p] (Fig. 4) is derived in two stages: The first stage comprises the formation of the load

vector of the median segment of the element [pint], with rotational springs being part of the process. At the

second stage, the presence of rigid offsets is taken into consideration as well.

First stage

The analytical solution of the homogenous forms of Eqs (1a) and (1b) and the application of the

well-known initial parameter method (e.g. [5], [9]) lead to the formulation of the load vector [p int]. The

expressions of its coefficients for the common solution cases of (1a) and (1b) are given in Appendix 2.



— 5 —

(a) (b)

2dd1

P

L

P

L

q

1 2 43

KRA

2 3

RBK

w =02

2φ =Μ /Κ2 RA RB

w =0

3φ =Μ /Κ3

3

Internal Boundary Conditions[ ]

[ ]G33G22int

G44G11

VMVM][p

VMVM[p]

=

=

Fig. 4 Transverse uniform loading (a), boundary conditions of the internal nodes

Second stage

At the second stage, the relations between the stresses at nodes 2, 3 and 1, 4 are formulated. Application of

a method similar to the one employed in the case of the stiffness matrix (Eq. 5) gives:

][T ][p[T][p] q

T += int (9)

The matrix [T] is given in Eq. 2, while [Tq] is the load vector for the rigid offsets. When the rigid offsets are

loaded by a uniform load q, [Tq] is obtained from the following relation:

[ ]2221

21 22

2d d d d

q ][T

T

q −−−= (10)

NUMERICAL EXAMPLE

The steel frame given in Fig. 5a rests on the soil through a reinforced concrete foundation. The soil consists

of a layer of loose sand 20m in thickness, resting on a rigid base. Young's modulus and Poisson's ratio of

the sand are Es=17500kN/m2 and νs=0.28 respectively.

(a)

Width of beam b =0.3m

Width of footings b =1.2m

6.1m

IPE400

7.180m

24 kN/m

1.0m

0.7m

K

IPB320

11 kN/m

R Α

25 kN

0.160m

B

f

0.7m

1.0m

0.35m

RΒK

A=0.21mI=0.0085m

2

4

4.450m

0.160m

0.20mν =0.3S

E =2.0E08 kN/m

E =2.9E07 kN/m

ν =0.2

S

Β

Β

2

2

500 kN 500 kN 2

w

φu

3

1 4

N=3

N=6

N=12

N=20

(b)

N= The Number of Elements between Rigid Offsets

rigid

offsets

rigid

offsetsrigid

offsets

transnational springs

rigid

offsets

Fig. 5 (a) Steel frame on reinforced concrete foundation, (b) Discretization of foundation beam and rigid

footings by conventional beam elements

With regard to the given steel frame, two kinds of second order analyses were carried out:

1) The objective of the first kind of analyses was the investigation of the reliability of the new element inmodeling of beams on elastic foundation. Initially, the frame was analyzed using the modified Vlasov soil



— 6 —

model, proposed by Vallabhan and Das [8], which enables the estimation of the two soil parameters. Their

values were found to be k S=3321,3kN/m2 and k G=11143,19kN (assuming plane stress conditions).

A number of analyses were then carried out using the Winkler model, in which the value of k S was the

same as in the case of the modified Vlasov model, i.e. k S=3321,3kN/m2. Finally, an additional analysis

based on the soil constants, which resulted from the analysis using the modified Vlasov model, was carried

out using the Pasternak model.

Solutions based on the Winkler model were achieved both using the new element, and using four different

conventional models of the foundation beam (Fig. 5b). The main objective of the Winkler-model-analyses

was to prove that the new element can, in fact, yield results that may only be attained using a large number

of simple (classical) beam elements in case of conventional modeling. In addition, conventional modeling

of continuous support requires additional calculations in order to determine the stiffnesses of the discrete

springs required.

-10%

-5%

0%

5%

10%

15%

20%

25%

30%

35%

40%

Ν=3 Ν=6 Ν=12 Ν=20

φ1

φ4

w1

w4

u2

(a)

-50%

-40%

-30%

-20%

-10%

0%

10%

20%

30%

Ν=3 Ν=6 Ν=12 Ν=20

M4

M1

V1

V4

(b)

2φ

w

u

1

3

4

Fig 6. Deviations of the displacements (a) and stresses (b) of the four models relative to the reference

solution using the proposed new element

A careful study of Fig. 6 leads to the following conclusions:

1.1) The model with N=3 simple elements fails to approximate the “exact” values of the stresses, as it

displays deviations of the order of 25% for bending moments and 40% for shear forces.

1.2) The model with N=6 simple elements, although better in comparison with the previous one, still

exhibits considerable deviations, especially for the shear forces (20%). However, divergences are

relatively small when it comes to bending moments (below 10%).

1.3) The model with N=12 simple elements is immensly improved in comparison with the previous one

with deviations remaining below 15%.

1.4) The model with N=20 elements consists of the minimum number of elements necessary for an

absolutely satisfactory approximation of all stress values. In this particular case, the deviations do notexceed 7%.

The analyses in which the two-parameter models were employed aimed at investigating the influence of

the second parameter on the stress values of the foundation beam. Apart from that, in the case of the

solution based on the Vlasov model, the influence of the soil on either side of the frame was also taken into

account by placing transnational springs with constantsGSk k [3] at the ends of the foundation beam

(nodes 5 and 6, Fig. 7). Fig. 7a indicates that the difference between the Winkler and Pasternak models

with reference to the maximum bending moment of the beam is of the order of 20%, whereas the respective

difference between the Winkler and Vlasov models approaches 3400%! From Fig. 7b, it is evident that the

difference between the Winkler and Pasternak models in terms of shear forces does not exceed 10% at any

given point of the beam. The respective differences between the Winkler and Vlasov models reach atcertain points 80-90%. Therefore, the consideration of the soil on either side of the frame substantially

modifies its stress condition, since it alters the way in which it rests on the soil.



— 7 —

-270

-170

-70

30

130

230

330 Bending Moments MKNm

4

1 -7,30kNm

-205,6kNm

-253,1kNm

1

2

w4

φu

3

56

26 3 5

(a)

-500

-400

-300

-200

-100

0

100

200

300

400

500 Winkler

Pasternak

Vlasov

Generalized Shear ForcesKN

53

6

2 1

4

388kN

248kN

241,7kN (Winkler)

-341,1kN (Winkler)

-335,4kN

-196kN

-263kN (Winkler)

-255,7kN

-402kN

194,6kN

334,3kN (Winkler) 341kN

-250kN

(b)

Fig. 7 Diagrams of the bending moments and shear forces of the foundation beam (analysis using the

Winkler, Pasternak and Vlasov models).

2) The objective of the second kind of analyses was to provide evidence for the usefulness and practicality

of the new element in modeling of semi-rigid connections in steel structures. In the conventional solution

using the well-known structural analysis program SAP2000 [10], the modeling of semi-rigid connectionswas based on two different connection models (Fig. 8a).

The solutions were carried out using three different values for the rotational stiffness of the semi-rigid

connections [4]. The comparisons of rotations and bending moments at the semi-rigid connections are

illustrated in Figs 8a and 8b. From the these diagrams, it is concluded that the conventional modeling of

semi-rigid connections (Fig. 8a) leads to values of stresses and displacements that are very close to those

derived from the analysis using the new element. However, this does not diminish the usefulness of the

new element since the deviation of results requires: (a) extensive preparatory work in order to determine

the appropriate lengths of the auxiliary elements used, as well as the geometric properties of their sections

(Fig. 8a), and (b) the introduction of additional auxiliary nodes and elements in the model. It is clear, that a

more complicated frame, with more bays and stories, would demand multiple repetitions of this

conventional modeling procedure, thus dramatically increasing pre- and postprocessing efforts. Incontrast, using the proposed new element simplifies the modeling of steel structures with semi-rigid

connections to a remarkably large extent.

(only torsional deformations)

Κ LG

Auxiliary Element

Κ R ∆φ=1 rad

BL

Α, I J =tR

1 rad

B

Model A Model B

(only flexural deformations)

Κ R

BL

I= Α, J BRΚ L

Et

Auxiliary Element

Κ R Κ R

Connection (FP)Connection (ΗΡ) Connection (ΕΡ)

K =2081kNm/radΦ K =4510kNm/rad K =48226kNm/radΦ Φ

-4,0%

-3,0%

-2,0%

-1,0%

0,0%

1,0%

Model A Model B Model A Model B Model A Model B

HP FP EP

φ2

φ3

Μ2

Μ3

2 3

1 4

φu

w

(a) (b)

Fig. 8 Conventional models and semi-rigid connection types (a), Deviations of the analyses using the

conventional models from the analyses using the proposed generalized element (b)

CONCLUSIONSA geometric nonlinear Timoshenko beam element, with rigid offsets at its ends, resting on two-parameter

elastic foundation was presented. The connection of the flexible median segment to the rigid offsets is



— 8 —

achieved by means of rotational springs. The final form of the analytically formulated stiffness matrix of

the element features incorporated “switches”, through which it is possible to activate or de-activate one or

more of its structural characteristics.

The usefulness of the new element was demonstrated by a numerical example. The initial purpose of this

application was to assess the effectiveness and ease with which the modeling of beams on Winkler-type

elastic foundation is achieved. The main conclusion is that the discretization of the foundation beam with

at least 20 simple beam elements is necessary, if the results obtained from the use of conventional models

are to converge to the results from the use of the new element. The second purpose of the example was to

make a comparison between the Winkler model, on the one hand, and the two-parameter Pasternak and

Vlasov models, on the other hand. The chief conclusion reached, in this case, is that the consideration of

the influence of the soil on either side of the frame, by means of the Vlasov model, influences its stress

values to a remarkably large extent. Finally, the effectiveness of the new element in modeling of semi-rigid

connections was tested as well. This led to the conclusion that the new element succeeds in modeling steel

elements nodes in a way that is both simple and reliable.

REFERENCES

[1] E. Winkler, Die lehre von der elasticitaet und festigkeit, Prag, Dominicus, (1867).

[2] P.L.Pasternak, On a new method of analysis of an elastic foundation by means of two constants, (in

Russian), Gosudarstvennoe Izdatelstvo Literaturi po Stroitelstvui Arkhitekture, Moscow, (1954).

[3] V.Z. Vlasov, U.N. Leontiev, beams, plates, and shells on elastic foundation (Translated from Russian),

Israel Program for Scientific Translations, Jerusalem, (1966).

[4] Semi-Rigid connections in steel frames, Council on tall buildings and urban habitat , Committee 43,

Mc Graw-Hill, New York, (1993).

[5] Morfidis K., Research and development of methods for the modeling of foundation structural elements

and soil , PhD Thesis, Department of Civ. Eng., Faculty of Engineering, Aristotle University of

Thessaloniki, Greece, Oct., (2003).

[6] K. Morfidis, I.E. Avramidis, Formulation of a generalized beam element on a two – parameter elastic

foundation with semi – rigid connections and rigid offsets, Computers and Structures, 80, (2002),

1919 – 1934.

[7] I.E. Avramidis, K. Morfidis, Generalized finite beam element for equivalent frame modeling of

reinforced concrete building structures, (in Greek). Proc. 13th Greek Concrete Conference,

Rethymno – Crete, (1999), pp. 23-32.

[8] C.V.G. Vallabhan, Y.C. Das, Modified vlasov model for beams on elastic foundation, Journal of

Geotechnical Engineering, 117(6), (1991), 956-966.

[9] A.P.S. Selvadurai, Elastic analysis of soil – foundation interaction, Elsevier scientific publishing

company, Amsterdam, (1979).

[10] SAP2000 NonLinear Version 7.42, Integrated finite element analysis and design of structures.

Berkeley, California, USA: Computers and Structures, Inc., (1995).



— 9 —

APPENDIX 1: ELEMENTS OF STIFFNESS MATRIX [K]

1

3

1

int

21

int

22113

int

231

2

int

21111

3

11 )d k (P d k ) K K (d d

D

D

K

K d

D

D )

K

K d ( K GS

(a)

M

RB

(a)

M

RA

−−+++−−=

) K d (K K

K ) K d )(K

K

K ( D

RB RB

(a)

M

int

321

int

31

int

13int

121

int

11

int

332 1 +−++= ) K d (K

K

K ) K d )(K

K

K ( D

RA RA

(a)

M

int

121

int

11

int

31int

321

int

31

int

113 1 +−++=

2

1

int

221

3

int

23

1

2

int

21

112 2

1

1 d k K d D

D

K

K

d D

D

) K

K

d ( K S

(b)

M

RB

(b)

M

RA ++−−= ]d k [K D

D

K

K

D

D

K

K

K S

(b)

M

RB

(b)

M

RA1

int

22

3

int

232

int

21

22 ++−−=

int

32

int

13int

12

int

332 1 K

K

K )K

K

K ( D

RB RB

(b)

M −+= int

12

int

31int

32

int

113 1 K

K

K )K

K

K ( D

RA RA

(b)

M −+=

) K d (K d D

D

K

K d

D

D )

K

K d ( K

(c)

M

RB

(c)

M

RA

int

242

int

2313

int

231

2

int

21113 1 −+−−= ) K d (K

D

D

K

K

D

D

K

K K

(c)

M

RB

(c)

M

RA

int

242

int

233

int

232

int

2123 −+−−=

2

3

2

int

442

int

4323

int

432

2

int

41233

3

11 )d k (P d k ) K d K ( d

D

D )

K

K d (

D

D

K

K d K

GS

(c)

M

RB

(c)

M

RA

−−++−+++=

) K d (K K

K ) K d )(K

K

K ( D

RB RB

(c)

M

int

342

int

33

int

13int

142

int

13

int

332 1 −−−+= ) K d (K

K

K ) K d )(K

K

K ( D

RA RA

(c)

M

int

142

int

13

int

31int

342

int

33

int

113 1 −−−+=

int241

3

int

231

2

int

21114 1 K d

D D

K K d

D D )

K K d ( K

(d)

M

RB

(d)

M

RA

+−−= int24

3

int

232

int

2124 K

D D

K K

D D

K K K

(d)

M

RB

(d)

M

RA

+−−=

]d k [K d D

D )

K

K d (

D

D

K

K d K S

(d)

M

RB

(d)

M

RA

2

int

4423

int

432

2

int

41234

2

11 +−++=

2

int

443

int

432

int

4144 d k K

D

D

K

K

D

D

K

K K S

(d)

M

RB

(d)

M

RA

++−−=

RB RB

(d)

M K

K K )

K

K ( K D

int

13int

34

int

33int

142 1 −+= RA RA

(d)

M K

K K )

K

K ( K D

int

31int

14

int

11int

343 1 −+= RB RA RB RA K K

)(K )

K

K )(

K

K ( D

2int

13

int

33

int

11 11 −++=

where K ijint are the coefficients of stiffness matrix of median segment.

APPENDIX 2: LOAD VECTOR FOR UNIFORM LOAD

Solution Case 3 (Fig. 2):( )[ ] S S G

ΕΙ k / Φ EIk )k (P 42<+−

−+′−′+−+−= m)ωn( ω )mnQ)( ω R( ω

K

EI Qm) )(Rnω( ω

Dk

EIq M

RBQS

2112

2

2

2

12

4

where n=sin(QL), n΄=cos(QL), m=sinh(RL) and m΄=cosh(RL).

The bending moment M3 results from the aforementioned equation by substituting K RA for K RB.

+−+−+−+−+= ) F F

K

EI F

K

EI F

K K

(EI) )( k (P ) F F

K

EI F

K

EI F

K K

(EI) )( ω EI( ω

Dk

qV GG

RB

G

RA

G

RB RA

G

RB RA RB RAQS

G 4321

2

4321

22

2

2

12

4

Qm) )(RnmnQ)( ω R )( ωQ(R F +′−′++= 12

22

1

)m RωnQ )( ωmn RQ( Rn)m)(Qmωn )( ωQ(R F ′−′′−′+−−−= 2121

22

2 2

[ ] )mn )( n RωmQ( ωQn)m)(Rmωn( ω RQ F ′−′′−′++−= 21213 2

m)ωn )( ωmn RQ( F 214 2 −′−′= m)ωnQ)( ωω R )( ωmn )( Q(R F G 2112

22

1 −+′−′+=

m)ωnn)( ωωm RQ( ω )nQωm )(RωmnQ)( ω R( ωm)ωQnωm)(Rωn( ω F G 212112122

2

1

2

212 ++−′+′′−′+−−−=

[ ] )n RωmQ )( ωmn( m)ωnQn)( ω(RmQ)ω R( ω F G ′−′′−′+−+−= 2121123

m)ωn )( ωmnQ)( ω R( ω F G 21124 −′−′−=

)nωm( ω )nnωmmQ)( ωω R )( ω K

EI

K

EI ( )n(mQ)ω R( ω

K K

EI)( D

RB RA RB RA

Q

22

1

22

21212

222

12

2

442

−+′−′+++++=

RQθ ΑΦ

)Qθ (R AQ R

R ΑQAω

Q R

Q Α RAω ,

k , , S

22

22

122

21222

211 =−−

+

+−

+

+===

The shear force VG3 results from the above equations by substituting K RA for K RB and vice-versa.



— 10 —

Solution Case 1 (Fig. 2): ( )[ ] S S G ΕΙ k / Φ EIk )k (P 42>+−

[ ] [ ] [ ] )ν D(D )µ D(D )ν µ( Q) R( Q)-D R( D K

EI Qν µ R

Dk

D EIqD= M

R ΒQS

−−++′−′−+−− 212121

212

2

where: ν=sin(R L), ν΄=cos(R L), µ=sin(QL), µ΄=cos(QL)

The bending moment M3 results from the above equation by substituting K RA for K RB.

++−−

++= 65421321212 U )U U D(D K

EI )k (P U )U (U K

EI D EID Dk

qV

RA

G

RAQS

G

[ ] [ ][ ]

′+

′−−′++−+′+−−−= µQν µν R )νQ R(

Κ

ΕΙ µνQ) R( DQ) R( D µνQ) R( DQ) R( DU

RB

222

1

2

2

2

1

2

21 2

[ ] [ ][ ]ν µQ) R( DQ) R( D )ν µ( Q) R( DQ) R( DO )OQ R( K

EI U

RB

−++−′′−−−+

−−= 212112

22

2 12

[ ] [ ] [ ][ ]

−++−′−+′′−−++′−′+−−−= µνQ) R( DQ) R( D )ν µν µ( Q) R( DQ) R( D Κ

ΕΙ )ν µ( )µ D(D )ν D(DQ RU

RB

21

22

2121213 4

[ ]222222

2

22

4 424 µνQν µ Rν µQ R )OQ R( K

EI µνU

RB

′+′+

′′−−=

[ ] [ ]{ } [ ] [ ]

−−

−++−−′′+−−−++−′′−−−+=

3

22

2121

21215 1 )OQ R(

Κ

ΕΙ

µνQ) R( DQ) R( Dν µQ) R( DQ) R( D µνQ) R( DQ) R( D )ν µ( Q) R( DQ) R( DU

RB

[ ] [ ] [ ]

[ ]

−++−

−′−+′′−−++′−′+−−+−−=

µνQ) R( DQ) R( D

)ν µν µ( Q) R( DQ) R( D

Κ

ΕΙ )ν µ( )µ D(D )ν D(DQ) R( DQ) R( DU

RB 21

22

21

2121216

ν µQ R )µµQ R( O ′′++= 222

1 ν µ R µQνO ′−′=2

ν )µ D(D µ )ν D(DΟ ′++′−= 21213

[ ] [ ][ ]

[ ]22

21

22

21

21212121

2

2121

2 11

2

µ ) D(Dν ) D(D

µ ) D(Dν ) D(D )Q D(D R ) D(D K K

EI )Q D(D R ) D(D

K K

(EI) D

RB RA RB RA

Q

+−−+

++−−++−

++++−−=

++

+=

Φ

k Q) Rθ (

Q) R( D S 2

1

1

+−

−=

Φ

k Q) Rθ (

Q) R( D S 2

2

1

L) R( ν 2sin= L) R( µ 2cos=

The shear force VG3 results from the above equations by substituting K RA for K RB and vice-versa.

Axially Loaded Generalized Beam Element on a Two-Parameter Elastic Foundation with Semi-Rigid...

Documents

Transcript of Axially Loaded Generalized Beam Element on a Two-Parameter Elastic Foundation with Semi-Rigid...