BEAMS ON ELASTIC FOUNDATION

Under the guidance

Of

Dr.M.V.RENUKA DEVI

Associate Professor

Department of Civil Engineering, RVCE

By

M.PRASANNA KUMAR

(1RV13CSE05)

INTRODUCTION• Beneath the foundation soil mass is considered as

identical, independent, closely spaced, linearly elastic springs which is known as Winkler foundation.

• Bending of beams on an elastic foundation is developed on the assumption that the reaction forces of the foundation are proportional at every point to the deflection of the beam at that point .

• One of the most important deficiencies of the Winkler model is that a displacement discontinuity appears between the loaded and the unloaded part of the foundation surface. In reality, the soil surface does not show any discontinuity.

Classification of beams:

The beams on elastic foundation can be classified into three types • Short beams for which • Medium beams or semi-infinite beams for

which

0.6• Long beams or infinite beams

L= length of the beam

Infinite beam subjected to point load

• As we know , = −

= −

= −

• Where the distributed reaction force q is positive when acting upward

• For linearly elastic foundation, the distributed force q is linearly proportional to the deflection y . Thus,

Where k is the elastic coefficient, is the elastic foundation modulus, and b is the width of the foundation.

= − = −• To solve this homogeneous, fourth order, linear

differential equation.

we will assume that ,then

• By using method of differential equation the solution

of above equation is

Since the deflection , z then the term and we obtain • By applying boundary conditions

1. at

+

we get ;

2. 2 Then equation for deflection is

=

• Defining,

= = = ) =

Then, we have

• A rail road uses steel rails (E = 200.0 GPa ) with a depth of 184 mm . The distance from the top of the rail to its centroid is 99.1 mm , and the moment of inertia of the rail is . The rail is supported by ties, ballast, and a road bed that together are assumed to act as an elastic foundation with modulus of subgrade reaction .length is 6m.

(a) Determine the maximum deflection, maximum bending moment, and maximum flexural stress in the rail for a single wheel load of 170 KN as shown in Fig

(b) If a locomotive has 3 wheels per truck equally spaced at 1.70 m, determine the maximum deflection, maximum bending moment and maximum flexural stress in rail when the load on each wheel is 170 kN.

Solution:

(a) The maximum deflection and the maximum bending moment occur under the load Where z =0

therefore

(b) Case (i):

• For case 1, let the origin of the coordinate be located under one of the end wheel. The distance to the first wheel , we have

The distance from the origin to the next wheel is

The distance from the origin to the next wheel is

• we get the maximum deflection and the maximum bending moment equal to

+Case(ii): let the origin of the coordinate be located under one of the end wheel. The distance to the first wheel , we have

The distance from the origin to either of the end wheel is is

• + we obtain, ,

Beam Supported on Equally Spaced Separated Elastic Supports

Each spring has same spring constant K . The reaction force R exerted on the beam is directly proportional to the deflection y

the load R can be idealized as uniformly distributed over a total span L

k is the elastic coefficient

However it has to satisfy the condition

Check for spacing of the spring

To obtain a reasonable approximate solution

Check for length of the beam is

• Problem2: An aluminium alloy I-beam (depth , E = 72.0GPa) as shown in figure has a length of 7m and is supported by 7 springs . Spaced at a distance of 1.1m centre to centre along the beam. A load P = 12.0KN is applied at the centre of the beam. Determine the maximum deflection of the beam, the maximum bending moment, and the maximum bending stress in the beam.

Solution: The elastic coefficient

the value of

Check the spacing of the spring.

Check the length of the beam

The maximum deflection and the maximum bending moment of the beam occur under the load where,

Infinite Beam Subjected to a Distributed Load Segment

From the displacement solution of the beam subjected to concentrated load

By using the principle of superposition, the total deflection due to the distributed load is

where,

• Problem 3: A long wood beam (E = 10.0 GPa) has a rectangular cross section with a depth of 200 mm and a width of 100 mm . It rests on an earth foundation having spring constant of and is subjected to a uniformly distributed load 35 N/mm extending over a length. Taking the origin of the coordinate at the centre of the segment . determine the maximum deflection, the maximum bending stress in the beam, and the maximum pressure between the beam and the foundation. The moment of inertia of the beam about x -axis is.

Solution: The elastic coefficient k = b

value of The maximum deflection occurs at the centre of segment since ,

• The maximum pressure between the beam and the foundation occurs at the point of the maximum deflection

Maximum bending occurs at where shear force is zero

(= a+b)

KN-m Bending stress is MPa

Semi-infinite beam Subjected to Loads at Its End

Consider the semi-infinite beam subjected to a point load P and a positive bending moment at its end

can be obtained by applying boundary conditions

the deflection of the beam is ]By rearranging,

A steel I-beam (E = 200GPa ) has a depth of 102mm , a width of 68mm , a moment of inertia of and a length of 4m. It is attached to a rubber foundation for which. A concentrated load p = 30.0 KN is applied at one end of the beam. Determine the maximum deflection, the maximum bending stress in the beam, and their locations.

Solution: The spring coefficient, k = b

value of

The maximum deflection occurs at the end where load P is applied (z =0 ), since is maximum we have and

FINITE BEAMS:• Finite beams are defined as beams for which • Finite beams can be analysed using the analysis results

of infinite beams• Finite beam can be split into loading case namely

symmetric and anti-symmetric load acting on infinite beam.

PRACTICAL SIGNIFICANCE • Historically the first application of this theory was to

rail road track • Another application of this soon after the rail road

track is grid works of beams • Mat foundation under certain structures, such as silos

water storage tanks, coal storage tanks, etc., and footing foundation supporting group of columns, are frequently designed and constructed in the form of beams resting on soil.

• No doubt it satisfies the actual conditions of real elastic theory of soil.

• Arthur P. Boresi, Richard J.Schindt, “Advanced Mechanics of materials”, Sixth edition John Wiley &Sons. Inc., New Delhi, 2005

• Thimoshenko & J N Goodier, “Mechanics Of Solids”, Tata McGraw-Hill publishing Co.Ltd, New Delhi, 1997

• Seely Fred B. and Smith James O., “ Advanced Mechanics of Materials”, 2nd edition, John Wiley & Sons Inc, New York, 1952, pp.112-136

• Srinath L.S., “Advanced Mechanics of Solids”, Tata McGraw-Hill publishing Co.Ltd, New Delhi, 1980, pp180-191

• Thimoshenko S., “Strength of Materials”, Part-1, Elementary Theory and Problems, 3rd Edition, D. Van Nostrand company Inc., New York, 1955, pp.227-244

• Boresi A.P and Chong K.P(2000), “Elasticity In Engineering Mechanics” 2nd edition New York ; Wiley – Interscience.

• N Krishna Raju & D R Gururaja, “Advance Mechanics Of Solids & Structures”, 1997

• B C Punmia & A K Jain. “Strength of Materials and Theory of Structures”, Vol.2 Lakshmi publications (P) Ltd.

REFERENCES