Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

3, 39, 2001

CONTACT OF A RIGID FLAT PUNCH WITH A WEDGE

SUPPORTED BY THE WINKLER FOUNDATION

Joanna Marzęda

Volodymyr Pauk

Margaret Woźniak

Department of Geotechnics and Engineering Structure, Technical University of Łódź

e-mail: pauk@ck-sg.p.lodz.pl, mwozniak@ck-sg.p.lodz.pl

The contribution deals with the new class of contact problems relatedwith
an elastic wedge. It is supposed that the wedge rests on the Winkler foun-
dation. The wedge is in the plane frictionless contact with a rigid flat plate
(punch). The problem is solvedusing theMellin integral transformsmethod
and is reduced to an integral equation for unknown contact pressure, which
was solved numerically. The results concerning the contact pressure distri-
bution and the punch displacement and slope are presented for different
values of mechanical and geometrical parameters.

Keywords: contact problem, elasticwedge, rigid punch,Winkler foundation

1. Introduction

Solutions to contact problems involving a deformable subgrade and a rigid
plate (punch) havemany applications, particularly in soil mechanics, geotech-
nical engineering and foundation design. Deformable subgrades are generally
considered as an elastic half-space or a layer, see for exampleGladwell (1980).
But many geotechnical applications prove that the subgrade soil has the sha-
pe of a wedge. Previous investigations of contact problems related with the
elastic wedge, see e.g. Aleksandrov (1967), Aleksandrov and Pozharski (1988)
were done on the assumption that the wedge rests without friction on a rigid
base. In this paperwe propose new formulation of the contact problem for the
elastic wedge assuming that the wedge is underlain by a deformable base of
theWinkler type.



564 J.Marzęda et al.

Weinvestigate thecontact problem for anelastic, homogeneous, and isotro-
pic wedge supported by theWinkler foundation (Fig.1). The wedge is planar
and cuts out an infinite sector of θ0. The upper surface of the wedge is in
tensionless smooth contact with the rigid flat punch. The problem is assumed
to be planar and stationary.

Fig. 1. Geometry of contact

Mathematically, the above formulated contact problem is reduced to so-
lving the elasticity equations in the wedge (Timoshenko and Goodier, 1951)

∂σr

∂r
+
1

r

∂τrθ

∂θ
+
σr−σθ
r
=0

(1.1)

1

r

∂σθ

∂θ
+
∂τrθ

∂r
+
2

r
τrθ =0

with the following boundary conditions on the wedge surfaces

τrθ(r,0)= 0 r­ 0

σθ(r,0)= 0 0¬ r <a r> b

uθ(r,0)= g0+rg1 a¬ r¬ b

σθ(r,θ0)= r
−1kθuθ(r,θ0) r­ 0

τrθ(r,θ0)= r
−1krur(r,θ0) r­ 0

(1.2)

where ur, uθ and σr, σθ, τrθ are displacements and stresses in the polar coor-
dinates system 0rθ, respectively; kr, kθ are theWinklermedium stiffnesses in
the radial and angular directions; (a,b) is the contact area, which is given for
the flat punch. The unknown parameters g0, g1 define the rigid displacement
and slope of the punch, respectively.



Contact of a rigid flat punch... 565

2. General solutions

Differential equations (1.1) in the polar coordinate system can be solved
by the method of Mellin’s integral transforms (Sneddon, 1951). The fields of
stresses and displacements in the wedge have the forms of contour integrals

σr(r,θ) = −
1

2πi

c+i∞
∫

c−i∞

r−s−1s
{

(s−1)[Asin(s−1)θ+Bcos(s−1)θ]+

(2.1)

+ (s+3)[C sin(s+1)θ+Dcos(s+1)θ]
}

ds

σθ(r,θ) =
1

2πi

c+i∞
∫

c−i∞

r−s−1s(s−1)
{

[Asin(s−1)θ+Bcos(s−1)θ]+

(2.2)

+ [C sin(s+1)θ+Dcos(s+1)θ]
}

ds

τrθ(r,θ) =
1

2πi

c+i∞
∫

c−i∞

r−s−1s
{

(s−1)[Acos(s−1)θ−B sin(s−1)θ]+

(2.3)

+ (s+1)[C cos(s+1)θ−Dsin(s+1)θ]
}

ds

uθ(r,θ) = −
1+ν

2πiE

c+i∞
∫

c−i∞

r−s
{

(s−1)[Acos(s−1)θ−B sin(s−1)θ]+

(2.4)

+ (s−κ)[C cos(s+1)θ−Dsin(s+1)θ]
}

ds

ur(r,θ) =
1+ν

2πiE

c+i∞
∫

c−i∞

r−s
{

(s−1)[Asin(s−1)θ+Bcos(s−1)θ]+

(2.5)

+ (s+κ)[C sin(s+1)θ+Dcos(s+1)θ]
}

ds

where A,B,C,D are the unknown functions of s and c is the real number
which makes the integrands in (2.1)-(2.5) regular. Moreover, ν and E are
Poisson’s ratio andYoung’smodulus, respectively, and κ=3−4ν isKolosov’s
constant.



566 J.Marzęda et al.

3. Point load solution

Fig. 2. Scheme of point load solution

First, we consider the point load problem for a wedge as shown in Fig.2.
Satisfying the following point load boundary conditions by equations (2.1)-
(2.5)

τrθ(r,0)= 0 r­ 0

σθ(r,0)=Pδ(r−a) r­ 0

σθ(r,θ0)= r
−1kθuθ(r,θ0) r­ 0

τrθ(r,θ0)= r
−1krur(r,θ0) r­ 0

(3.1)

we obtain a system of four algebraic equations for A,B,C,D, which has the
solutions

A(s)=−
s+1

s−1
C(s)

B(s)=−D(s)−P
as

s(s−1)
(3.2)

C(s)=P
U0(s)+αrU1(s)+αθU2(s)+αrαθU3(s)

∆0(s)+αr∆1(s)+αθ∆2(s)+αrαθ∆3(s)

as

s

D(s)=P
V0(s)+αrV1(s)+αθV2(s)+αrαθV3(s)

∆0(s)+αr∆1(s)+αθ∆2(s)+αrαθ∆3(s)

as

s

where

∆0(s)= 2s
2(s2−1+cos2sθ0−s

2cos2θ0)

∆1(s)= s(κ+1)(ssin2θ0− sin2sθ0)



Contact of a rigid flat punch... 567

∆2(s)=−s(κ+1)(ssin2θ0+sin2sθ0)

∆3(s)= 2s
2−1−κ2−2s2cos2θ0−2κcos2sθ0

U0(s)= s
2(ssin2θ0+sin2sθ0)

U1(s)=0.5s(κ+1)(cos2θ0+cos2sθ0)

U2(s)=−0.5s(κ+1)(cos2θ0− cos2sθ0)

U3(s)= ssin2θ0−κsin2sθ0

V0(s)= s
2(scos2θ0+cos2sθ0−s−1)

V1(s)= 0.5s(κ+1)(sin2θ0+sin2sθ0)

V2(s)=−s(κ+1)(sin2θ0+sin2sθ0)

V3(s)= scos2θ0−κcos2sθ0−s−1

and

αr =
1+ν

E
kr αθ =

1+ν

E
kθ

are dimensionless stiffnesses of theWinkler medium.
To satisfy contact boundary condition (1.2)3 we need a normal deflection

of the wedge upper surface. Substituting solutions (3.2) into formula (2.4) we
obtain

uθ(r,0)=
2(1−ν2)

πiE
P

c+i∞
∫

c−i∞

(a

r

)

−sL(s)

s
ds (3.3)

where the kernel of this equation has the form

L(s)=
U0(s)+αrU1(s)+αθU2(s)+αrαθU3(s)

∆0(s)+αr∆1(s)+αθ∆2(s)+αrαθ∆3(s)
(3.4)

Let us observe the following properties of the kernel L(s)
(i) L(−s)=−L(s)

(ii) L(s)∼
a0s
3+αra1s+αθa2s+αrαθa3s

b0s
4+αrb1s

2+αθb2s
2+αrαθb3

for s→ 0

where ai, bi, i=0,1,2,3 are some known constants.
Taking c = 0 in integral (3.3) and using methods of contour integration

the normal deflection of the wedge upper surface can be obtained in the form

uθ(r,0)=

∞
∫

0

L∗(t)

t
cos(tR) dt (3.5)



568 J.Marzęda et al.

where

δ=
2(1−ν2)

E
L∗(t)=

2L(it)

it
R= ln

a

r

Assuming now that the loading p(r) is distributed over the region (a,b)
we obtain from (3.5) the normal deflection

uθ(r,0)=
δ

π

b
∫

a

p(ρ)K
(

ln
ρ

r

)

dρ r­ 0 (3.6)

where the kernel K(·) has the form of the integral

K(R)=

∞
∫

0

L∗(t)

t
cos(tR) dt (3.7)

Using the value of the integral (Gradshteyn and Ryzhik, 1965)

∞
∫

0

1− e−t

t
cos(tR) dt=− ln |R| (3.8)

we can present the kernel K(·) in the following form

K(R)=− ln |R|+Φ(R) (3.9)

where

Φ(R)=

∞
∫

0

L∗(t)−1+e−t

t
cos(tR) dt (3.10)

is a regular function.
Let us note here that the well known result for the elastic wedge resting

on a rigid base (see Aleksandrov, 1967), can be obtained directly from (3.6),
(3.4) for αr,αθ →∞.

4. Integral equation of the contact problem

Satisfying boundary condition (1.2)3 by formula (3.6) we arrive at the
integral equation of the considered contact problem

δ

π

b
∫

a

p(ρ)K
(

ln
ρ

r

)

dρ= g0+rg1 r∈ (a,b) (4.1)



Contact of a rigid flat punch... 569

This equation must be solved together with the two equilibrium conditions

b
∫

a

p(r) dr=P

b
∫

a

rp(r) dr=eP (4.2)

The distribution of the contact pressure p(r), rigid displacement g0 and
slope g1 of the punch are unknown in the system of integral equations (4.1)
and (4.2).

Introducing dimensionless variables and functions

τ =λ ln
ρ

a
−1 t=λ ln

r

a
−1 r= aexp

(t+1

λ

)

q(τ)=
ρ

λP
p(ρ) λ=2

(

ln b
a

)

−1
(4.3)

the system of integral equations (4.1) and (4.2) can be rewritten into the new
form

1

π

1
∫

−1

q(τ)K
(τ− t

λ

)

dτ =G0+(L−1)G1 exp
(t+1

λ

)

t∈ (−1,1)

(4.4)
1
∫

−1

q(t) dt=1

1
∫

−1

exp
(t+1

λ

)

q(t) dt= ε

where

G0 =
g0

δP
G1 =

g1c

δP
ε=
e

a

L=
l

c
l=
b+a

2
c=
b−a

2

5. Numerical solutions to the system of integral equations

Introducing collocation points

τi =−1+(i−1)dt i=1, ...,n+1

ti =−1+
(

i−
1

2

)

dt i=1, ...,n dt=
2

n

(5.1)



570 J.Marzęda et al.

and using rectangular quadratic formulae we obtain the discretized form of
the system of integral equations (4.4)

1

π

n
∑

1=1

q(τi)Aim−G0− (L−1)G1 exp
(tm+1

λ

)

=0 m=1, ...,n

(5.2)

dt

n
∑

1=1

q(ti)= 1 λ
n
∑

1=1

q(ti)
[

exp
(ti+1+1

λ

)

− exp
(ti+1

λ

)]

= ε

where the matrices {Aim} have the forms

Aim =

τi+1
∫

τi

K
(τ − tm
λ

)

dτ (5.3)

and using the formulae (3.9), (3.10) can be calculated as

Aim = −

τi+1
∫

τi

ln
∣

∣

∣

τ− tm
λ

∣

∣

∣
dτ+

τi+1
∫

τi

Φ
(τ− tm
λ

)

dτ =

(5.4)

= λ[Z2 ln |Z2|−Z1 ln |Z1|+Φ(Z2)−Φ(Z1)]

where

Z1 =
τi− tm
λ

Z2 =
τi+1− tm
λ

i,m=1, ...,n (5.5)

and

Φ1(Z)=

∞
∫

0

L∗(t)−1+e−t

t2
sin(Zt) dt (5.6)

is the regular integral which was calculated numerically.
The set of n+2 linear algebraic equations (5.2) is sufficient to find n+2

unknowns: thedimensionless rigiddisplacement G0 and slope G1 of thepunch
and the distribution of the dimensionless contact pressure q(ti), i=1, ...,n.

6. Numerical results

The system of algebraic equations (5.2) was solved numerically. The input
parameters for the calculations were: ν – Poisson’s ratio, θ0 – wedge angle,



Contact of a rigid flat punch... 571

Fig. 3. Distribution of dimensionless contact pressure for L=2 (a) and L=5 (b)



572 J.Marzęda et al.

Fig. 4. Dimensionless rigid vertical displacement (a) and slope (b) of the punch
versus the stiffness αθ



Contact of a rigid flat punch... 573

Fig. 5. Dimensionless rigid vertical displacement (a) and slope (b) of the punch
versus the stiffness αr



574 J.Marzęda et al.

ε> 1 – dimensionless eccentricity, L> 1 – dimensionless location of the plate
center, αr,αθ – dimensionless stiffnesses of the Winkler medium. The para-

meter λ can be calculated as λ = 2
(

ln L+1
L−1

)

−1

. For numerical calculations

we put ν = 0.3 and ε= L, which means that the load P is applied to the
center of the rigid plate.

The calculations were performed to display complex effects of theWinkler
medium, wedge angle θ0 and distance L to the punch on the distribution of
the contact pressure q(t), rigid vertical displacement G0 and slope G1 of the
punch.

The distributions of the dimensionless contact pressure q(t) are presented
in Fig.3 for some values of the wedge angle θ0. The curves in Fig.3a were
found for the punch situated near to thewedge corner (L=2) but the results
presented in Fig.3b were obtained for a larger distance (L = 5). The effect
of the angle θ0 is greater for small values of L. The distributions of the
contact pressure shown in Fig.3b are almost symmetrical. These results were
obtained for αr =αθ =1.0, and our investigation displayed that theWinkler
mediumhad small effect on the contact pressure. This does notmean that the
boundary conditions on the lower surface of the wedge have no effect on the
solution to the contact problem.The stiffnesses αr,αθ have great effect on the
values of the rigid displacement G0 and slope G1 of the punch. These effects
are shown in Fig.4 and Fig.5 for some values of the wedge angle and for the
fixed distance L=2.The diagrams presented inFig.4 are found for αr =1.0
and those in Fig.5 for αθ = 1.0. The parameters G0 and G1 decrease with
the growth of stiffnesses and tend to constant values for αθ or αr equal to 5.
The comparison of the results presented in Fig.4 and Fig.5 displays that the
angular stiffness αθ plays a greater role than the radial one αr. The main
effects are observed for small values of the stiffnesses and the large values
correspond to the problem of the wedge resting on the rigid base.

References

1. Aleksandrov V.M., 1967, Contact Problems for an Elastic Wedge, Izvestia
AN SSSR, MTT, 2, 120-131

2. Aleksandrov V.M., Pozharski D.A., 1988, On the Contact Problem for
an ElasticWedge,PMM J. Appl. Math. Mech., 52, 4, 651-656

3. GladwellG.M.L., 1980,Contact Problems in the Classical Theory of Elasti-
city, Sijthoff andNoordhoff, Alphen aan den Rijn



Contact of a rigid flat punch... 575

4. Gradshteyn I.S., Ryzhik I.M., 1965, Tables of Integrals, Series and Pro-
ducts, Academic Press, NewYork

5. Sneddon I., 1951,Fourier Transforms, McGrawHill, NewYork

6. Timoshenko S., Goodier J.N., 1951, Theory of Elasticity, McGraw Hill,
NewYork

Współpraca sztywnego płaskiego stempla z klinem opartym na podłożu

Winklera

Streszczenie

Praca dotyczy nowej klasy zagadnień kontaktowychdla sprężystego klina spoczy-
wającegona podłożuWinklera.Klin ten znajduje się w płaskimkontakcie ze sztywną
płytą (stemplem). Używając transformacji całkowych Mellina, zagadnienie sprowa-
dzono do równania całkowego względem funkcji nacisków kontaktowych, które roz-
wiązywano numerycznie. Przedstawionowyniki dla ciśnienia kontaktowego, osiadania
i przechylenia stempla w zależności od różnychmechanicznych i geometrycznych pa-
rametrów zagadnienia.

Manuscript received November 21, 2000; accepted for print December 27, 2000