Beam on Elastic Foundations

8/13/2019 Beam on Elastic Foundations

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Loadings on locks

1. Water level = max

2. Water level= min

3. Empty lock - repairs

N N

N N

N N

T TT T

T T

T T

M M

M M

M M


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N N

T TM MPa

S

gr

p

N* N*

M* M*

Equivalent loadings on the floor


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Beams on elastic foundations

Foundation slab&

Foundation terrain

Plane strain conditionst=ct

q=ct

All the sections - same displacements analyze on a 1 m slice

Plane strainconditions

B= (theoretic)

B> 3 x Lg (practice)

( end sections)

Soil reactions

NN

M

M

Lg


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Methods to calculate BoEF

Mathematical equations(soil structure interaction)

Behavior of the soil

Deformed average fiber of the slab

= + ()

= k

= + 1 + 22 + 33 + +

Winkler

Gurbonov-Posadov


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Winkler Model

=

=

The reacting forces of the foundation are proportional at every point to the deflection of thebeam at that point

Introduced by E. Winkler in 1867 The application of this modelon soil foundations should be regarded only as a practical

approximation

includes the flexural rigidity of thebeam and also the elasticity of thesupporting medium

= 4

Characteristic of the system

N

mfoundation modulus


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Stiffness of the beam

Characterizes the relative stiffness of the beam on elastic foundation

< 0.5 ( 4 )

> 5 ()

0.5 4 < < 5 ()

Short beam

Medium beam

Long beam

The beam deformation can beneglected beam absolutely rigid computed using statics

Loads applied at one end have afinite and not negligible effect onthe other end no approximationsare possible

The effect of a Load applied at oneend can be neglected at the otherend computations are greatlysimplified


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Mathematical expressions for y M T

= 1() +1

2()

1

3()

1

()

1

3(( )) +

1

(( )) +

1

(( ))

= 1 1 2() 1

3() 4() 1 2(( )) + 1

3(( )) + 1

3(( ))

= 1 +1

2 +

3() +

() +1(( ))

1

2(( )) -

1

2(( ))

= 1 +

2 +

3() 4() 4(( )) 1(( )) -

1(( ))

1 = cosh() cos()

2() =1

2 cosh sin + sinh() cos()

3 =1

2 sinh() sin()

() = 12 cosh sin sinh() cos()

= = 2

2 =


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Winkler model advantages and disadvantages

Disadvantages:

k has no physical sense. it is determined from in situ tests. a wide variety of formulas for k different values for the same E and

you cant determine k from those formulas.

k depends on : physical properties of the foundation

the shape and the length of the loading plate

the concentrated force value the model doesnt consider the lateral loadings

for an uniform loading (weight) there is no moment or shear force diagrams (false)

Advantages:

for sand foundations correct results (if k is determined accurately)

very easy and simple model

For floor locks Winkler model can beused for thin compressible

foundations (


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Winkler Example

k=50000 K=kPaLg=15mhg=2m

a = 1.5 mc = 3 mconcrete beam

P=500 kNM = 150 kNmq = 10 kN/m

s = - 20 kN/m

Ansys beam54 elements


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Gurbonov Posadov Method

it is based on Boussinesq/ Flamandmodels (elastic half-space/ half-plane model)

the soil is replaced by a solid body with the following characteristics:

extends laterally and down elastic linear deformable homogeneous and isotropic

Disadvantages:

overestimate the settlements

settlements at the ends tend to infinite

overestimates the moments

the model doesnt consider the lateral loadings

Boussinesq spatial problem

Flamand - plane problem


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Gurbonov Posadov Method

= + ()

= + 1 + 22 + 33 + + Exponential series

polynomial equation (n degree)

, 1, 2, - unknowns

n = 10 (practical reasons)

In order to calculate y, M, T G.P. realized tables for unitary loadings

The tables are realized for half of the slab foundation (because the loadingsare symmetrical).

This half is divided in 10 parts where the efforts, moments and shear forceare calculated.


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N NT TM MPa

S

gr

p

N* N*

M* M*

Equivalent loadings on the floor


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flexibility index

t < 1 short beam (absolutely rigid)1


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Gurbonov Posadov Example


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Conclusions

Winkler model is a very simple model but it should be regarded as a practical

approximation when it is used on soil foundations.

For a uniform loading (weight) the Winkler model doesntcompute shear force

diagrams or moment diagrams which in reality is false.

Gurbonov Posadov model take into consideration the displacement of the soil

outside the beam.

For predimensioning purposes of a lock floor we can use Gurbonov Posadov

Method.


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Thank you for your attention!

Beam on Elastic Foundations

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