论文研究-用Winkler路基的两个反应系数将弹性梁上的基础梁弯曲资源-CSDN文库

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Open Journal of Civil Engineering, 2019, 9, 123-134

http://www.scirp.org/journal/ojce

ISSN Online: 2164-3172

ISSN Print: 2164-3164

DOI:

10.4236/ojce.2019.92009 Jun. 5, 2019 123 Open Journal of Civil Engineering

Bending the Foundation Beam on Elastic Base

by Two Reaction Coefficient of Winkler’s

Subgrade

Mirko Balabušić

, Boris Folić

, Slobodan Ćorić

Independent Designer of Structure, Herceg Novi, Montenegro

Innovation Center, Faculty of Mechanical Engineering, University of Belgrade, Belgrade, Serbia

Faculty of Mining and Geology, University of Belgrade, Belgrade, Serbia

Abstract

A new method for analysis of counter beams is presented in the paper. The

analysis has taken into account their stiffness EI, Winkler’s space with mod-

ulus of subgrade reaction

and equality deformities of the foundation beam

with the ground. The solution is found by using the numerical analysis of the

Winkler’s model, with variation of different moduli of the subgrade reaction

outside the force zone

, while under the force

exists the modulus of the

subgrade reaction

, up to the definition of minimum bending moments. The

exponential function

(

), as the geometric position of the minimum mo-

ments is approximately assumed. From the potential energy conditions of the

reciprocity of displacement and reaction, the width of the zone

and the

modulus of the subgrade reaction

are

explicitly determined, introducing in

the calculation initial and calculation soil displacement

successively. At

the end of the paper, it presented numerical example in which the influence

and

values on bending moments of the counter beam is analyzed. The

essential idea of this paper is to decrease the quantity of the reinforcement in

the foundations, beams,

i.e

. to obtain a cost-efficient foundation construction.

Keywords

Foundation Beam, Winkler’s Model, Coefficient of Subgrade Reaction

Modulus

and

, Zone

under the Force

, Soil Displacement

1. Introduction

When calculating a beam on a continuous deformable base, it is important to

provide a modeling of the foundation beam base as realistic as possible,

i.e

. its

How to cite this paper:

Balabušić, M.,

Folić, B

. and Ćorić, S. (2019)

Bending the

Foundation Beam on Elastic Base by Two

Reaction Coefficient of Winkler’s Subgrade.

Open Journal of Civil Engineering

, 123-134.

https:

//doi.org/10.4236/ojce.2019.92009

Received:

February 26, 2019

Accepted:

June 2, 2019

Published:

June 5, 2019

9 by author(s) and

Scientific

Research Publishing Inc.

This work is licensed under the Creative

Commons Attribution International

License (CC BY

4.0).

http://creativecommons.org/licenses/by/4.0/

Open Access

M. Balabušić et al.

DOI:

10.4236/ojce.2019.92009 124 Open Journal of Civil Engineering

approximation with the actual properties of soil beneath the foundations. Yet,

the calculation methods should be kept simplified so that they could be widely

implemented in practical applications.

Beam on elastic foundation has been analysed, most usually, based on the

Winkler’s model in which the soil is replaced by a bed of elastic springs. The

compressive resistance of soil against the beam deflection is quantified in terms

of spring constant

[force/length

/length], which is a frequent occurrence in the

Euler-Bernoulli beam theory. Shear deformations are neglected and plane

cross-section is assumed to remain plane and normal to the longitudinal axis

deformation. Many researchers [1] [2] [3] [4] [5] have investigated the modulus

of subgrade reaction and found that the geometry, the foundation dimensions

and soil layering below the foundation structure are the most important para-

meters to define the value of this modulus. In the Winkler foundation model, the

soil reactive pressure at any arbitrary point

is proportional to the deflection

and can be expressed as,

( ) ( )

qx kwx= ⋅

(1)

where

is the Winkler’s coefficient of ground reaction at point

. Winkler

foundation is a single parameter model,

is used to describe the soil reaction.

The subgrade reaction modulus

is dependent on some parameters like soil

type, size and shape of foundations, depth and stress level. The foundation

represented by the Winkler model [6] cannot sustain shear stress, and hence a

discontinuity of adjacent spring displacement can occur. A different model may

result in significant inaccuracies in the evaluated structural response. In order to

overcome this problem, many researches have introduced a different mechanical

model [7]-[14]. Among them is the class of two parameters foundation. The

second parameter introduced the interaction between adjacent springs, in addi-

tion to the first parameter from the ordinary Winkler’s model [15]. This proce-

dure is proposed in [13] for homogenous elastic semi space.

In the [13] the elastic base is represented with a layer having thickness

, ex-

posed to pressure, lying on top on an infinitely stiff horizontal base. One dimen-

sion of the compressed layer is large, and the load invariable in this direction;

the supporting conditions and values of the elastic characteristic are constant,

too. An in-plane stress and strain state is considered. The proposed model of soil

has two soil characteristics,

(characterising displacement of the elastic base

under pressure) and

(and describing behaviour of the subgrade during sliding

and base “distribution properties”). Pasternak proposed that both soil characte-

ristics (

and

) are named the subgrade coefficients,

i.e

. the model of soil of two

subgrade coefficients (as cited in [9]).

In [16] a closed-form analytical solution of the problem of bending of a beam

on elastic foundation is proposed. The solution based on the total potential

energy functional. In order to eliminated the bearing soil reaction as a variable

in the problem solution of beam on elastic foundation, the simplified continuum

M. Balabušić et al.

DOI:

10.4236/ojce.2019.92009 125 Open Journal of Civil Engineering

approach, with a numerical research, is presented in [17]. Study the behaviour of

a math foundation on subsoil from the plate theory taking into account the

soil-structure interaction, and several model have been described, presented in

[18]. Very important work related to subgrade reaction and analysis of beams on

elastic foundation is [4] [5] [19].

The equations available for estimating the soil spring constant

are mostly

developed empirically [5], which is limitation of Winkler’s model. In some in-

stances, plate-load test are used to estimate

, but that estimations are not free

from errors because the results depend on size, thickness and stiffness of the

plate.

Two-parameter foundation models provide the displacement continuity of the

soli medium by adding of a second spring which interacts with the first spring of

the Winkler’s model. Displacement continuity is provided for by the introduc-

tion of a virtual shear layer which integrates the vertical spring elements and the

second foundation parameter

, is the shear modulus

of the shear layer [16]

[17]. The soil reaction

(

) for two parameter foundation model is given in gen-

eral by:

( ) ( )

( )

q x kwx k wx x= −

(2)

where

and

are two foundation parameters.

In the paper will be demonstrated that in a case of

and

model, bending

moments in the counter beam are smaller than in a case of a

model. The

change of

values does not significantly affect this difference.

2. Theoretical Bases

2.1. Bernoulli Beam Theory

The main assumption of Bernoulli beam theory is that cross-sectional rotations

are the same as the rotations of the beam centroidal line. The Bernoulli beam

element can be obtained by defining the total potential energy and applying the

variational principle to it.

• Ground as linearly deformable described by Winkler’s model.

• Ground is inhomogeneous and acting non-linearly under load

In all solutions, it is referred to the basic differential equation for Bernoulli

beam resting on Winkler soil model [11] of the elastic beam, where soil replaced

with an elastic spring [7] and [10] that displacements of points on the beam axis

and the corresponding soil displacements are equal.

( )

( ) ( )

ddEI w x qx px B=−−





(3)

The dependence between

(

) and soil displacement

(

) of basic ground

surface is defined according to one of proposed models.

2.2. Coefficient of Subgrade Modulus Reaction, K

Winkler proposed a model that assumes the soil stiffness that is considered as

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