BEAMPASK

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BEAMPASK

Finite Length Beam on a Continuous Pasternak Support

Version 11.11

Author: Ren FREUND

ENGINEERING CONSULTANTS LTD

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GRANTS & RESTRICTIONS

ATLANTIS Engineering Consultants Ltd. (ATLANTIS) grants to you (the Customer) the non-exclusiverights with respect to the utilisation of the program BEAMPASK (the Software) and its User Documentation.One copy of the Software may be made for backup or archival purposes. This Software and Documentation mayotherwise not be used, modified, reproduced, sold, transferred or distributed without ATLANTIS prior written

permission.No attempt shall be made to de-assemble or de-compile the executable.

WARRANTY

ATLANTIS warrants that the Software and User Documentation are free from defects in material andworkmanship under normal use for a period of two months following its acquisition.The liability of ATLANTIS under the warranty set forth above shall be limited to the amount paid by theCustomer for the product.

DISCLAIMER

ATLANTIS does not make any warranty expressed or implied as to the use, or the results of use, of the Softwareand User Documentation in terms of correctness, accuracy and reliability. The Customer is expected to make thefinal evaluation as to the usefulness of the Software in his own environment and shall assume the entire risks as toits results and performance.

ATLANTIS shall not be liable for any direct, indirect, consequential or incidental damages (including damagesfor loss of business profits, business interruption and the like) arising from the use or inability to use the Softwareeven if ATLANTIS has been advised of the possibility of such damages.

REVISION HISTORY

11.11 Finite-Length Beam09.09 First Issue

Version Description

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BEAMPASK Finite Length Beam on a Continuous Pasternak Support Version 11.11

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CONTENTS

1 SCOPE 12 ASSUMPTIONS AND METHODOLOGY 2

2.1 Model 22.1.1 Beam 22.1.2 Foundation 2

2.1.2.1 Winkler Spring Layer 22.1.2.2 Pasternak Shear Layer 3

2.2 Governing Differential Equation 42.2.1 Subcritical Deformed Shape 42.2.2 Critical Deformed Shape 42.2.3 Supercritical Deformed Shape 42.2.4 Unitization of Deformed Shape Functions 5

2.3 Forces and Deformations Along an Infinitely Long Strip 62.3.1 Concentrated Force 62.3.2 Uniformly Distributed Load 7

2.3.3 Triangular Load 82.3.4 Concentrated Moment 92.4 Beams of Finite Length 10

2.4.1 Soil Layer Deformations Outside the Beam 102.4.2 Boundary Conditions 112.4.3 Elements of Reduction at Beam Extremities 122.4.4 Boundary System 12

2.4.4.1 General Solution 123 VARIABLES AND UNITS 14

3.1 Input Data 143.2 Default Values 153.3 Output Data 15

4 OPERATING LIMITS AND WARNINGS 16

4.1 Recommended Values of the Shear Stiffness 164.2 Residual Shear at the Free Edges of a Finite Beam 164.3 Infinite Hinged Beam 174.4 Finite Length Beam Equivalent to an Infinite Strip 18

4.4.1 Semi-Infinite Beam 184.4.1.1 Concentrated Force 184.4.1.2 Concentrated Moment 18

4.4.2 Equivalent Stiffness Matrix 184.5 Beam in Tension Resting on a Winkler Foundation 19

5 ASSOCIATED ROUTINES 205.1 Upstream 205.2 Downstream 205.3 Related Software 20

6 REFERENCES & BIBLIOGRAPHY 216.1 References 216.2 Bibliography 22

7 EXAMPLES 247.1 Infinite Beam 24

7.1.1 Concentrated Load 247.1.1.1 Winkler Model 257.1.1.2 Sub-Critical Pasternak Model 26

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7.1.1.3 Critical Pasternak Model 277.1.1.4 Super-Critical Pasternak Model 28

7.1.2 Uniform Load 327.2 Semi-Infinite Beam 34

7.2.1 Winkler Foundation 357.2.2 0% Deflection Ratio 367.2.3 50% Deflection Ratio 377.2.4 100% Deflection Ratio 38

7.3 Finite Beam 397.3.1 Centrally Loaded Beam 397.3.2 Beam Loaded at its Edges 39

7.4 Finite Beam Equivalent to an Infinite Strip 437.4.1 Concentrated Load at 1.0m from Right Support 437.4.2 Concentrated Load at 5.0m from Right Support 447.4.3 Concentrated Load at 7.0m from Right Support 44

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1 SCOPEThe program computes the forces and deformations along an infinite Euler-Bernoulli beam subjected to a verticalconcentrated load and resting horizontally on an elastic two-parameter Pasternak foundation.

KEYWORDS

two-parameter foundation, Pasternak layer, Winkler layer, modulus of subgrade reaction, shear modulus

Figure 1-1 Projected Freeway Extension Outside Los Angeles, CA

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2 ASSUMPTIONS AND METHODOLOGY

2.1 Model

2.1.1 Beam

The beam is characterized by a constant widthB and flexural rigidityEIover its entire lengthL.The system attached to the beam is such that the origin is set at the left extremity of the beam, thex-axis corresponds to the beam centerline, directed from left to right, they-axis is positive upwards.

2.1.2 Foundation

The soil is regarded as an elastic medium represented by a two-parameter Winkler-Pasternak model.

Figure 2-1 Infinite Beam on a Pasternak Foundation [from Ref./3/]

2.1.2.1 Winkler Spring Layer

This representation assumes that the vertical reaction force exerted by the foundation at any point is proportionalto the beam deflection at that point alone.

ykBp =where

k = modulus of subgrade reaction of the soil

The combined flexural rigidity of the beam and spring stiffness of the soil can be merged into the foundationparameter which has dimension L-1.

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4

4EI

kB=

2.1.2.2 Pasternak Shear Layer

The shear force Q in the Pasternak layer is proportional to the rotation of the beam at this section

dx

dyGBQ =

The ratio of flexural to shear stiffness is expressed by the parameter which has dimension L-1.

G

k=

Figure 2-2 Hard Landing of an Antonov 124 Under Strong Lateral Winds

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2.2 Governing Differential Equation

The expression of the deflection in any section of an infinitely long strip resting on a continuous elastic supportand supporting the load w(x) is the solution of the 4 th order differential equation

)(2

2

4

4xwkBy

dx

ydGBdx

ydEI =+ [1]

The normalized associated homogeneous equation

044 42

22

4

4

=+ ydx

yd

dx

yd where

kEI

BG

2

2

=

=

is transformed to the characteristic equation 044 4224 =+ rr where rxey =

which admits the roots

= 12 24,3,2,1 r

These roots are all either complex conjugate or real, depending on the magnitude of with respect to unity.The integration constants CandD factoring the positive exponential term in the solution of the homogeneousequation are identically equal to zero on the conditions that the deflections of the beam must remain finite faraway from the loaded area.

2.2.1 Subcritical Deformed Shape

When the flexural stiffness of the beam prevails over the shear stiffness of the soil, then < 1 and

ir =4,3,2,1 where

=

+=

1

1

The solution of the homogeneous equation becomes

[ ] [ ]xDxCexBxAeyxx

sincossincos +++=

2.2.2 Critical Deformed Shape

When the flexural stiffness of the beam is about equivalent to the shear stiffness of the soil, then = 1 and

=2,1r where 2=


[ ] [ ]DxCeBxAey xx +++=

2.2.3 Supercritical Deformed Shape

When the shear stiffness of the soil prevails over the flexural stiffness of the beam, then > 1 and

=4,3,2,1r where1

1

=

+=


[ ] [ ]xDxCexBxAey xx sinhcoshsinhcosh +++=

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2.2.4 Unitization of Deformed Shape Functions

The above equation can be expressed analytically in terms of hyperbolic, polynomials or circular functionsdepending on the rigidity ratio , i.e. whether or not the shear rigidity of the Pasternak layer prevails over theflexural stiffness of the beam combined with the spring stiffness of the Winkler layer.

function 0 < 1 = 1 > 1

)(0 xf xcos 1 xcosh

)(1 xf

xsinx

xsinh

)(2 xf 2cos1

x

2

2x2

1cosh

x

)(3 xf 3sin

xx

6

3x3

sinh

xx

)(4 xf 4

2 cos2/)(1

xx 24

4x4

2 2/)(1cosh

xx

Those functions are more conveniently represented here by their series development

)(xfn

( )

= +

0

2

)!2(i

in

ni

xx

!n

x n ( )( )

= +

0

2

)!2(1

i

iin

ni

xx

Figure 2-3 High Speed Track of the East TGV

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2.3 Forces and Deformations Along an Infinitely Long Strip

The forces and deformations along the continuous beam (x 0) at the point of load application are

( )

1

1

1

if

cosh

cos

)( 20>

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