# Analysis of Two Way Slabs by FINITE ELEMENT METHOD using MATLAB

DOI : 10.17577/IJERTV3IS110446

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#### Analysis of Two Way Slabs by FINITE ELEMENT METHOD using MATLAB

Mr. Kallakury .Srinivas

Department of Civil Engineering V R Siddhartha Engineering College

Sri V.Ramesh

Associate Professor Department of Civil Engineering

V R Siddhartha Engineering College Vijayawada, India

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Abstract – The Slabs can be two way or oneway based on Long span to short span ratio. From the years the slabs are being analyzed based on (I)RankineGrashoff Theory and IS 456 procedure for slabs simply supported on four sides with corners not held down (II) PigeaudMethod,IS 456 and Marcus Method for Slabs with edges fixed or continuous and carrying uniformly distributed load.

In the present study of slabs, it is aimed at study of two way slabs with different edge conditions using plate bending models using Finite Element Analysis assuming plates to be isotropic and for linear analysis. The results are compared with Timoshenko Tables based on exact analysis for uniformly distributed loads.Later , the concept is extended and for study on slabs subjected to concentrated loads or patch loads at two different locations i.e. one concentric patch load and patch load eccentric to both the axes , as in the case of bridges and these results are compared with Pigeaud analysis.

The thin plates are analyzed based on Kirchhoffs theory using 4noded rectangular elements with 3dof. The moderately thick plates are studied as per Mindlin Reissner Theory using 4 Noded Elements for all-round simply supported and all-round clamped plates and 8Noded Elements for all round clamped plates with 3dof at each node. These results are obtained for thickness span ratio of 0.05. The Poissonsratio of

0.2 has been used in analysis. The software used for the analysis is MATLAB.

The Results are tabulated and the following conclusions are derived from the thesis.

ii)Negative Moments at edges i.e. Mx is varying from 33.53% to 40.16%,My is varying from 40.16% to 59.72% from exact analysis It can be readily understood that negative moments i.e. at edges are not converging in the case of 4N elements but Positive moments are converging towards exact analysis. But it is contrary in the case of 8N Elements where in negative moments are converging but positive moments are away from 25%.

When thin plate theory has been used for the above case, the findings are like this

1. Positive Moments at centrei.e. Mx is varying from 0.0% to 6.05%, My is varying from 0.00% to 4.29% from exact analysis

2. Negative Moments at edges i.e. Mx is varying from 0.00% to 2.14%, My is varying from 2.14% to 14.36% from exact analysis. It is observed that the behaviour of positive moments in case of 4Noded Mindlin Plates is consistent than that of thin plate theory. The My Negative is decreasing regularly from span ratios of 1.3 in case of 4N and thin, and from 1.4 in case of 8N elements. But in exact analysis it is increasing up to 1.6 and from then onwards it is constant.

mxx(+ve) vs spanratio (all sides clamped)

Exact Thin

Mindlin 4n

Mindlin 8n

0.05

0.045

*p*b2- – – >

0.04

0.035

When subjected to udl with all-round clamped plates a 8N isoparametric element with 4*4 subdivision have exhibited the following results for span ratios varying from 1.00 to 2.00

1. Positive Moments at centrei.e. Mx is varying from 15.05 to 17.34%, My is varying from 15.96% to 24.58% from exact analysis

2. Negative Moments at edges i.e. Mx is varying from 5.00% to 7.6%, My is varying from 7.6% to 24.04% from exact analysis. The same properties in case of 4N elements using Mindlin theory with 8*8 subdivisions are like this

1. Positive Moments at centrei.e. Mx is varying from 0.00% to 2.7%, My is varying from 0% to 3.35% from exact analysis

0.03

0.025

0.02

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

longspan(a)/shortspan(b)- – ->

0.03

0.028

0.026

0.024

*p*b2- – – >

0.022

0.02

0.018

0.016

0.014

0.012

myy(+ve) vs spanratio (all sides clamped)

Exact Thin

Mindlin 4n

Mindlin 8n

1. Positive Moments at centrei.e. Mx is varying from 0.0 to 0.94%, My is varying from 0.00% to 0.99% from exact analysis When CPTwith 8*8 subdivisions is used for the above case, the findings are like this

i) Positive Moments at centrei.e. Mx is varying from 0.23% to 0.89%, My is varying from 0.00% to 0.42% from exact analysis

It is observed that the behaviour of the elements is roughly similar. Both the thin Plates and Mindlin Plates are behaving roughly in the same way. Thus, it can be stated that positive moments are alike in the above two cases even the plates to be analysed fall within thin plate classification.

mxx vs spanratio (all sides simply supported)

0.11

0.01

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

longspan(a)/shortspan(b)- – ->

Exact Thin

Mindlin 4n

Mindlin 8n

mxx(-ve) vs spanratio (all sides clamped)

Exact Thin

0.1 Mindlin

0.08

0.07

*p*b2- – – >

0.06

Comparision between exact vs

thin an

0.09

*p*b2- – – >

0.08

0.07

0.05

0.06

Comparision between exact vs thin and

0.04

0.05

0.03

0.02

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

longspan(a)/shortspan(b)- – ->

myy(-ve) vs spanratio (all sides clamped)

Exact Thin

0.04

0.05

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

longspan(a)/shortspan(b)- – ->

myy vs spanratio (all sides simply supported)

0.08 Mindlin 4n

Mindlin 8n

0.07

0.045

Exact Thin

Mindlin

*p*b2- – – >

0.06

Comparision between exact vs thin and

0.04

*p*b2- – – >

0.05 0.035

0.04

0.03

0.03

0.025

0.02

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

longspan(a)/shortspan(b)- – ->

0.02

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

longspan(a)/shortspan(b)- – ->

When Subjected to udl with all-round simply supported condition, a 4Noded Mindlin isoparametric element with 8*8 subdivision have exhibited the following results for span ratios varying from

1.00 to 2.00

For all other cases, only 8*8 subdivision Kirchhoff elements are considered and the observations are like this.

In the case of 3 sides discontinuous and one short side continuous, the observations are like this.

1. Positive Moments at centre i.e. Mx is varying from 1.3% to 1.52%,My is varying from 0.26% to 1.98% from exact analysis

2. My at middle of fixed edge is ranging from 2.6% to 9.05%

mxx(+ve)at center of plate vs spanratio (3 sides discont and 1short side conti)

mxx(+ve)at center of plate vs spanratio (2 adj sides conti)

0.08

Exact Thin

Exact Thin

0.09 0.07

0.08

0.06

*p*b2- – – >

0.07

0.05

*p*b2- – – >

0.06

0.05

Comparision between exact vs thin by

0.04

Comparision between exact vs thin by

0.04

0.03

0.03

0.02

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

longspan(a)/shortspan(b)- – ->

0.02

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

longspan(a)/shortspan(b)- – ->

myy(+ve)at center of plate vs spanratio (2 adj sides conti)

0.04

myy(+ve)at center of plate vs spanratio (3 sides discont and 1short side conti)

Exact

0.035

Exact Thin

0.044

0.043

0.042

Thin

0.03

*p*b2- – – >

0.025

Comparision between exact vs thin by

*p*b2- – – >

0.041

0.02

0.04

0.039

0.038

0.037

0.015

0.01

0.005

0.036

0.035

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

0

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

longspan(a)/shortspan(b)- – ->

longspan(a)/shortspan(b)- – ->

myy(-ve)at middle of fixed edge vs spanratio (3 sides discont and 1short side con

mxx(+ve)at center of plate vs spanratio (2 adj sides conti)

0.08

Exact

Thin

Exact Thin

0.125

0.12

0.115

0.07

0.06

0.11

*p*b2- – – >

0.105

0.1

0.095

0.09

0.085

0.08

Comparision between exact vs thin by

0.05

*p*b2- – – >

0.04

0.03

0.02

Comparision between exact vs thin by

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

longspan(a)/shortspan(b)- – ->

0.075

mx(-ve)at middle of fixed edge vs spanratio (2 adj sides conti)

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

longspan(a)/shortspan(b)- – ->

In the case of 2 adjacent sides continuous, the observations are like this.

1. Positive Moments at centre of plate i.e. Mx is varying from 1.6% to 2.32%,My is varying from 0% to 1.85 % from exact analysis

2. Negative Moments at middle of fixed edges i.e. Mx is varying from 0.26% to 1.77%, and My is varying from 1.77% to 7.88% iii)Mmax at x=0.4a and y=0.4b from discontinuous edges, it is varying from 0.66 to 3.02%.

0.12

0.11

*p*b2- – – >

0.1

0.09

0.08

0.07

0.06

0.05

Exact Thin

Comparision between exact vs thin by

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

longspan(a)/shortspan(b)- – ->

myy(-ve)at middle of fixed edge vs spanratio (2 short sides conti)

Exact Thin

0.12

I conclude by paying my tributes to Clough and Zienckewicz for this wonderful concept. Can we call them Finiteers.

0.11

0.1

*p*b2- – – >

0.09

0.08

0.07

0.06

0.05

Comparision between exact vs

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

longspan(a)/shortspan(b)- – ->

thin by

REFERENCES:

1. Theory of Plates and Shells , S.Timoshenko and S.W.Kreiger,2nd edition

2. Finite Element Analysis,S.SBhavikatti,2nd edition

3. Finite Element Analysis ,Theory and Programming, C.S.Krishna Moorthy2nd edition

4. Energy and Finite Element Methods in Structural Mechanics I.H.Shames and C.L.Dym 1st Edition

5. Concepts and Applications of Finite Element AnalysisR.D.Cook, D.S.Malkus eta l3rd

6. The Finite Element Method (Volume 1 the basis)

mmax(+ve)at 0.4a and 0.4b from dis edges vs spanratio (2 adj sides conti

0.07

Exact

Thin

0.065

0.06

0.055

*p*b2- – – >

0.05

0.045

n by FEM

Comparision between exact vs thi

0.04

0.035

0.03

0.025

O.C.Zienkiewicz and R.L.Taylor 5th

7. The Finite Element Method using MATLABYoung

.W.Kwon and Hyochoong Bang2nd

8. The Finite Element Method (Volume 2 solid mechanics) O.C.Zienkiewicz and R.L. Taylor 5th

9. Essentials of Bridge Engineering D. Johnson Victor6th

10. IS 456 -2000 Plain and Reinforced Concrete – Code of Practice 4th Revision

11. MATLAB an Introduction with Application Amos GilatReprint 2011

0.02

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

longspan(a)/shortspan(b)- – ->

Likewise, the Stress resultants at all important locations are studied for different edge conditions for span ratios from 1.0 to

2.0. It has been observed that the Positive Moments about both the spans and Negative Moments about short span are within a range of 3% from exact analysis. But Negative Moments about long span are oscillating up to 10%.

Pigeaud curve for all sides simply supported condition is also prepared for the span ratios from 0.0 to 3.0 and it is observed that the error is about 3.0% compared to Pigeaud curve.

Moment coeffecients for slab completely loaded with udl using FEM (Pigaud meth

0.06

MOMENT COFFEEIENT m1 or m2 – – – >

0.05

0.04

0.03

0.02

0.01

0

0 0.5 1 1.5 2 2.5 3

values of K or 1/K/ – – ->

From the above observations, it can be concluded that FEM results are in line with exact analysis with any type of loading and boundary conditions. But judicious use of Modelling and software are essential.