Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

52, 1, pp. 247-255, Warsaw 2014

THE INFLUENCE FUNCTION IN ANALYSIS OF BENDING CURVE

AND REACTIONS OF ELASTIC SUPPORTS OF BEAM WITH

VARIABLE PARAMETERS

Jerzy Jaroszewicz, Krzysztof K. Żur, Łukasz Dragun
Bialystok University of Technology, Faculty of Management, Kleosin, Poland

e-mail: j.jaroszewicz@pb.edu.pl; k.zur@pb.edu.pl; l.dragun@doktoranci.pb.edu.pl

In the paper, theoretical knowledge about base solution of common differential equations
with variable parameters is presented. The solutions are applied to mechanical problems of
discrete, distributed and discrete-distributed homogeneous elastic models of structures and
structural components. In this paper, the influence function and its properties are presented.
The influence function is applied to analysis of the bending curve of a beamwith constants
and variable parameters. The presented method of the influence function is based on the
mathematical similarity of differential equations describing free vibrations and deflection of
beams, which are fourth-order equations with variable coefficients. In this paper, examples
of calculation of support reactions as function of stiffness of the beam and elastic supports
are presented.

Key words: influence function, bending curve, reactions of elastic supports

1. Introduction

In innovation technology, solution methods of static and dynamic problems have application to
design of drive assemblies of machines which can be modeled as elastic systems with variable
parameters. Solutions to problems such as an increase of rotor velocity and acceleration or
loading of bearings is very important (Lazopoulos, 2010; Jaroszewicza nd Żur, 2012a,b,c).
In a previous paper (Jaroszewiczet et al., 2008), the authors analyzed the simplest lower

estimator to calculate the basic frequency of axi-symmetrical vibration of plates with variable
thickness of a circular diaphragm type. The existence of the simplest estimator of the actual
value of the parameter depending on the frequency rate of change characterized by a thick plate
was analyzed. The accuracy of the method differed from the FEM, and in order to improve the
accuracy of the estimators, it was decided to use a higher order, in that case double. Using the
bilateral estimator, the similar problem arose in calculation of the exact solution in a paper by
Conway (1958).
The aim of this paper is to present theoretical knowledge about the base solution (influence

function) to common differential equations with variable parameters. The advantage of the
influence function is the possibility of omission of the boundary of conjugation in the solution
to the boundary value problem. The solutions are applied to mechanical problems of discrete,
distributed and discrete-distributed homogeneous elastic models of structures and structural
components like beams, shafts and plates (Timoshenko, 1940; Solecki and Szymkiewicz, 1964).

2. The solution by means influence function

Inmany works (Zoryj, 1987c) an important property of the influence function whoose base was
constructed by solutions to the boundary value problem of the bending curve, vibrations and
stability was presented. It is known that the base solution to the equation



248 J. Jaroszewicz et al.

L[y(x)] = δ(x−α) (2.1)

is the influence function φ(x,α) in which the linear operator have analytical parameters

L[y(x)] = p0(x)y(x)
n+p1(x)y(x)

n−1+ . . .+pn(x)y(x) (2.2)

which are finite in intervals (a,b), where a<x<b, a<α<b, x is the independent coordinate,
α – parameter, δ(x) – Dirac’s delta, pi(x)> 0 – variable coefficients.
The fundamental solution (influence function) to Eq. (2.1) is defined in the following form

φ(x,α) =K(x,α)Θ(x−α) (2.3)

where Θ(x) is the Heaviside step function, K(x,α) – Cauchy function which is the solution to
homogeneous Euler’s equation L[y(x)] = 0 which satisfies following conditions

K(x,α) =K′(x,α) = . . .=Kn−2(x,α) = 0 Kn−1(x,α) =
1
p0(x)

(2.4)

In the case of equations

L[y(x)] = δj(x−α) j=1,2,3 . . . (2.5)

the solutions to them are partial derivativeswith respect to the parameter α of the influence
function

yj(x)= (−1)j
∂jφ(x,α)
∂αj

(2.6)

In a general equation with the variable right part

L[y(x)] = g(x) (2.7)

we have the following solution

y=
n−1∑

k=0

Ak
∂kK

∂αk
+y∗(x,α) (2.8)

where Ak are arbitrary constants, y∗(x,α) – particular solution to Eq. (2.7) well known by the
Cauchy formula

y∗(x,α) =
x∫

α

K(x,s)g(s) ds (2.9)

We can concludde that the partial derivative of the Cauchy function relative α parameter
(arbitrary steps) satisfies the equation L[y(x)].
Below, in Table 1, we present six formulas for the Cauchy function corresponding to chosen

differential operators which have practical applications to mechanical engineering problems.

3. Influence function in the analysis of the bending curve and reactions of elastic

supports of a beam with constant and variable cross-sections

3.1. Example of a general equation of the bending curve of the beam

The differential equation for deflection of an elastic beam in the static case has form (Jaro-
szewicz and Zoryj, 1997)

(fy′′)′′ =G(x) (3.1)



The influence function in analysis of bending curve... 249

Table 1. Formulas of Cauchy function (Zoryj, 1987)

L[y(x)] K(x,α)

(fy′)′
x∫
α

1
f(s)
ds

(fy′′)′′
x∫
α

(x−s)(s−α)
f(s)

ds

y′′+ 1
x
y′ α ln x

α

y′′+ p
f(x)
y,

p= const

∞∑
k=0
(−p)kuk(x,α)≡U(x,α)

u0(x,α)=x−α

uk(x,α)=
x∫
α

x−s
f(s)
u(k−1)(s,α) ds, k=1,2, . . .

1
p
[x−α−U(x,α)]≡

x∫
α

x−s
f(s)
U(s,α) ds

(f(x)y′′)′′+py′′
∞∑
k=0
(−p)kuk(x,α)≡U(x,α)

p= const u0(x,α)=x−α

uk(x,α)=
x∫
α

x−s
f(s)
u(k−1)(s,α) ds, k=1,2, . . .

where f =EJ(x) is the bending rigidity of the beam (Fig. 1), G(x) – transversal loading takin
into account all possible spatialy discrete compositions

G(x) =
n∑

i=1

qi(x)[θ(x−xi)−θ(x−xi− li)]+Piδ(x−xi)+Miδ
′(x−xi) (3.2)

where qi(x) is the distributed load acting on the segment xi < x < xi + li, Pi, Mi – discrete
force and moment located in section x= xi and 0<x1 <x2 < ... < xn < l, l – length of the
beam.

Fig. 1. Bending of elastic beam (Zoryj, 1987)

On the base of (2.3) and (2.6), the particular solution y∗ equation (3.1) has the following
form

y∗(x)=
n∑

i=1

[θ(x−xi)Fi(x,xi)−θ(x−xi− li)Fi(x,xi+ li)]

+
n∑

i=1

[
Piφ(x,xi)−Mi

∂φ

∂α

∣∣∣
α=xi

] (3.3)

where



250 J. Jaroszewicz et al.

φ(x,α) = θ(x−α)
x∫

α

x∫

a

1
f(s)
(x−s)(s−α) ds

Fi(x,α) =
x∫

α

φ(x,τ)qi(τ) dτ

(3.4)

The particular solution y∗(0) for x=0 satisfies the boundary condition

y∗(0)= y
′

∗
(0)= y′′

∗
(0)= y′′′

∗
(0)= 0 (3.5)

These expressions allow one to describe general solution (3.1) in the following form

y(x)= y(0)+y′(0)x−M(0)
x∫

0

x−s

f(s)
ds+Q(0)φ(x,0)+y∗(x) (3.6)

Expression (3.6) is a general equation of the bending curve of the beam with a variable cross-
section. In (3.6), the initial parameters: y(0), y′(0),M(0),Q(0), i.e. deflection, angle of rotation,
bending moment and transversal force on the left end of the beam (x = 0), respectively, may
take arbitrary values.
If f =EJ = const we have

φ(x,α) =
1
6f
(x−α)3

∂φ(x,α)
∂α

=−
1
2f
(x−α)2 (3.7)

Considering (3.7), we can obtain from (3.6) the well known equation of an elastic beam of
constant rigidity. General equation (3.6) gives a new way of the solution of a few problems of
static multi-supported beams with rigidity distributed by steps.

3.2. Example of application of the influence function in the analysis of bending curve of

the beam

Figure 2 gives a model of an elastically supported beam (Jaroszewicz and Zoryj, 1994)
subject to a transverse load G. The rigidity of the support at x= l1 is c. The flexural rigidity
is f =EJ(x), where theYoungmodulus of elasticity E = const, as thematerial is assumed to
be homogeneous and the planemoment of inertia J(x) is variable. The function 1/f(x) should
be continuous, positive definite and should have a finite value and integral in [0, l].

Fig. 2. Model of an elastically supported beam

When we consider the beam shown in Fig. 2, the deflection is defined as follows

L[y] =−Rδ(x− l1)+Gδ(x− l2) (3.8)

where L[y] = (fy′′)′′, δ is Dirac’s function.



The influence function in analysis of bending curve... 251

The boundary conditions have the following form

y(0)= 0 fy′′(0)= 0

fy′′(l)= 0 cy(l1)=R
(3.9)

The following solution to equation (3.8) can be proposed (Jaroszewicz and Zoryj, 2000)

y=C0+C1(x−α)+C2Kx0+C3K
′

x0−Rφx1+Gφx2 (3.10)

where K =K(x,α) is the Cauchy function for L[y(x)] = 0 which can be defined

K(x,α) =
x∫

α

1
f(s)
(x−s)U(s,α) ds U(s,α) = s−α (3.11)

Ci arre arbitrary constants and

Kx0 =K(x,0) K
′

x0 =
∂K

∂α

∣∣∣∣
α=0

φx1 =φ(x,x1)=K(x,l1)θ(x− l1)

The fundamental solution to equation (3.8) has form (2.3).
After calculation of constants of integration Ci, we have a particular solution for the deflec-

tion of the beam at the end x= l in the following form

y(l)=
G

l1−α
[−(l−α)(D0 +Kl1)+(l− l1)Klα (3.12)

where

Kl1(l, l1)=
l∫

l1

1
f(s)
(x−s)(s− l1) ds Klα(l,α)=

l∫

α

1
f(s)
(x−s)(s−α) ds

D0 =
1
l1−α

[
(l− l1)K1α−

l−α

c

]
K1α =(l1,α)

(3.13)

When we assume the constant cross-section f(x)= f0 = const

K(x,α) =
1
6f0
(x−α)3 (3.14)

we have the expression for deflection as follows

x(l)=G
[ l2

cl21
+
1
3f0
l(l− l1)

2
]

α=0 (3.15)

In the case of the rigid support of this beam c→∞, it has been obtained

y(l)→
1
3f0
Gl(l− l1)

2 (3.16)

3.3. Influence function in the determination ofe elastic support reactions of the

multi-supported beam

We consider an elastic beam with free ends supported on elastic supports. we examine the
displacement in points with coordinates aj (j = 1, . . . ,n), so 0< a1 <a2 < ... < an < l. We
apply the principle of release from constrains, and considering Hook’s law, we have

y(aj)= c
−1
j Rj (3.17)



252 J. Jaroszewicz et al.

where Rj is themodulus of j-th support reactions, cj – j-th support rigidity, y(aj) – deflection
of the beam in j-th support. Boundary conditions for the considered beam are defined by
equaling to zero the bendingmoments and forces at the beam end

(fy′′)
∣∣∣
x=0
=M(0)= 0 (fy′′)′

∣∣∣
x=0
=Q(0)= 0

(fy′′)
∣∣∣
x=l
=M(l)= 0 (fy′′)′

∣∣∣
x=l
=Q(l)= 0

(3.18)

From (3.6), by taking into consideration (3.18)1, we have

y(x)= y(0)+y′(0)x+ ỹ∗(x) (3.19)

We put into formula (3.3) the support reaction Rj

ỹ∗(x)= y∗(x)−
n∑

j=1

Rjϕ(x,aj) (3.20)

Substituting (3.19) to conditions (3.18)2, we have received the following equations

n∑

j=1

Rj(l−aj)=
τ∑

i=1

[
Pi(l−xi)+

xi+li∫

xi

(l− τ)qi(τ) dτ+Mi

]

n∑

j=1

Rj =
τ∑

i=1

(
Pi+

xi+li∫

xi

qi(τ) dτ

(3.21)

Equation (3.21)1 presents the condition of equilibrium of moments at the point x= l (left end
of the beam), the second condition – equilibrium of transverse.
Substituting (3.19) to (3.17), after transformation, we receive the follows equations

y0+y
′

0a1−
R1
c1
=−y∗(a1)

y0+y
′

0a2−R1(a2,a1)−
R2
c2
=−y∗(a2)

y0+y
′

0an−
n−1∑

j=1

Rn(an,aj)−
Rn
cn
=−y∗(an)

(3.22)

This way we determine two constants y0 = y(0) and y′0 = y
′(0) and we look for n reactions Ri

of the elastic supports through the system n+2 linear algebraic equations (3.21), (3.22). If
any arbitrary support of them is rigid (e.g. first) then in (3.22) c1 →∞ we obtain the system
considered in this case.
Nowwe consider the determinants ∆n got from the system of equations n=2,3,4, and we

find

∆2 =(a2−a1)
2 ∆3 =ϕ21(a3−a2)

2+ϕ32(a2−a1)
2

∆4 =ϕ21ϕ32(a4−a3)
2+ϕ21ϕ43(a3−a2)

2+ϕ32ϕ43(a2−a1)
2

(3.23)

Next, we proove that ∆n > 0 for n=2,3,4, which implies that the consider problem has only
one solution in all cases (rigid, elastic supports).



The influence function in analysis of bending curve... 253

3.4. Examples of calculation of the reaction Ri(β,ci) from the elastic supports

We take into consideration the beam presented in Fig. 3 which has three elastic supports
and constant stiffness (EJ = const). The systems of equations (Zoryj, 1987; Jaroszewicz and
Zoryj, 1994) can be expressed as follows (a1 → 0, a2 = l, a3 =2l)

2R1+R2 =2ql R1+R2+R3 =2ql

y0−
R1
c1
=0 y0+y′0l−

R1l
3

6EJ
−
R2
c2
=− ql

4

24EJ

y0+y′02l−
R1(2l)3

6EJ
−

R2l
3

6EJ
−

R3
c3
=
q(2l)2

24EJ

(3.24)

Fig. 3. Beamwith three elastic supports (Zoryj, 1987)

From the first equation (3.24) it follows that R1 =R3. Next, reducing the constants y0, y′0,
we obtain

R1(β,ci)
ql

=
R3(β,ci)
ql

=
1
4
+
4c−1
2

β

c
−1

1

β
+
4c
−1

2

β
+
c
−1

3

β
+ 2
3

R2(β,ci)
ql

=2
c
−1

1

β
+
c
−1

3

β
+ 5
12

c
−1

1

β
+
4c−1
2

β
+
c
−1

3

β
+ 2
3

(3.25)

where β= l3(EJ)−1. Examples of calculations are presented in Table 2.

Table 2. Examples of calculation of reaction forces in the elastic supports of constant stiffness
beam

β→∞
Variants

I II III IV

c1 1 100 1000 ∞

c2 2 1 1000 ∞

c3 1 100 1000 ∞
R1(β,ci)
ql
= R3(β,ci)

ql
0.375112 0.375375 0.375 0.375

R2(β,ci)
ql

1.249775 1.249251 1.24999 1.25

In particular cases, we find from (3.25) well known values. If all supports are rigid (ci →∞),
we have the well known results (Niezgodziński and Niezgodziński, 2012)

R1 =R3 =
3
8
ql R2 =

5
4
ql (3.26)



254 J. Jaroszewicz et al.

For the absolutely rigid bar on elastic supports, in the case EJ →∞, β=0, we receive

R1 =R3
4c−12 ql

c−11 +4c
−1
2 +c

−1
3

R2 =2ql
c−11 + c

−1
3

c−11 +4c
−1
2 + c

−1
3

(3.27)

For EJ →∞ and c2 →∞, R1 =R3 → 0, R2 =2ql and for EJ →∞, c1 = c3 →∞, we have
R1 = R3 → ql, R2 → 0, which can be concluded on the base of equations (3.21) and (3.22).
We can also formulate conditions for applying simpler equations, e.g. c1 = c3 =∞, c2 = c. We
present the corresponding values found from (3.25)1 in Table 3.

Table 3.Results of calculation

cl3/(EJ) 0 0.1 0.5 1 5 1000 ∞
R1/(ql) 1 0.99 0.95 0.9 0.55 0.38 0.775
R2/(ql) 0 0.02 0.1 0.2 0.9 1.24 1.25

We can notice that for 0 < cl3(EJ)−1 < 0.1 the influence of rigidity of the supports on
reactions at the end support is very small, and for cl3(EJ)−1 > 103 we can consider the beam
as absolutely rigid and calculate the reactions bymeans of formulas (3.27).

4. Conclusions

In this work, properties of the influence function applied to solution of mechanical problems
of discrete, distributed and discrete-distributed homogeneous components were presented. The
advantages of the influence function is the possibility of omission of the boundary of conju-
gation in the solution to the boundary value problem. The presented method gives possibility
of solving the problem of bending curve deflection of a multi-supported beam with a variable
cross-section in a closed analytical form. The presented method enables calculation of the force
reaction Ri(β,ci) for elastic supports of beams with constant and variable stiffness β(x).

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Manuscript received January 29, 2013; accepted for print September 9, 2013