study of Plastic Hinge Formation in Steel Beams

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Institute of Technology,

Nirma University.

M.Tech CASAD Semester I

CL1205 Structures Lab 2014-2015

Lab Report

Study of Plastic Hinge Formation in Steel

Beams

Neeraj Khatri (14MCLC12)Pragnesh Patel (14MCLC17)

Ravi Patel (14MCLC18)Sachin Patel (14MCLC19)Tejas Patil (14MCLC22)

M. Tech. 1st Year

April 30, 2015



Contents

1 Introduction 5

2 Theoretical Background 62.1 Plastic Analysis: . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Example: Failure of Simply Supported Beam . . . . . . . . . . . 62.3 Plastic Modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4 Principles of Plastic Analysis . . . . . . . . . . . . . . . . . . . . 82.5 Shape Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.6 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Design Calculations, Experimental Program and Test Set up 103.1 Hollow Section : Calculations . . . . . . . . . . . . . . . . . . . . 103.2 I-Section : Calculations . . . . . . . . . . . . . . . . . . . . . . . 113.3 Castellated Beam: Calculations . . . . . . . . . . . . . . . . . . . 12

3.3.1 Calculation of Property of Section . . . . . . . . . . . . . 12

3.3.2 Section Classification . . . . . . . . . . . . . . . . . . . . . 123.3.3 Check for Shear . . . . . . . . . . . . . . . . . . . . . . . . 123.3.4 Check for Deflection . . . . . . . . . . . . . . . . . . . . . 13

3.4 Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.5 Practical Setup and Procedure: Hollow Section . . . . . . . . . . 133.6 Practical Setup and Procedure: I-Section . . . . . . . . . . . . . 153.7 Practical Setup and Procedure: Castellated Beam . . . . . . . . 16

4 Results and Discussions 204.1 For Rectangular hollow section . . . . . . . . . . . . . . . . . . . 204.2 For I-section and castellated section: . . . . . . . . . . . . . . . . 214.3 Load Deflection curves . . . . . . . . . . . . . . . . . . . . . . . . 21

5 Conclusions 26

2



List of Figures

2.1 A typical stress-strain curve for mild steel. . . . . . . . . . . . . . 62.2 Simply Supported Beam . . . . . . . . . . . . . . . . . . . . . . . 72.3 Elastic Deflection . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.4 Kink in the beam : Plastic hinge . . . . . . . . . . . . . . . . . . 72.5 Diagram of Structure Featuring Plastic Hinges . . . . . . . . . . 72.6 Common Shape Function Values . . . . . . . . . . . . . . . . . . 8

3.1 Setup for Hollow Section . . . . . . . . . . . . . . . . . . . . . . . 133.2 Deformed Shape of Hollow Section . . . . . . . . . . . . . . . . . 143.3 Setup for I- Section . . . . . . . . . . . . . . . . . . . . . . . . . . 153.4 Deformed Shape of I- Section . . . . . . . . . . . . . . . . . . . . 163.5 Setup for Castellated Beam . . . . . . . . . . . . . . . . . . . . . 173.6 Deformed Shape of Castellated Beam . . . . . . . . . . . . . . . . 18

4.1 Castellated Beam : Load vs Deflection . . . . . . . . . . . . . . . 234.2 I-section : Load vs Deflection . . . . . . . . . . . . . . . . . . . . 244.3 Comparison : Load vs Deflection . . . . . . . . . . . . . . . . . . 25

3



List of Tables

4.1 Load and displacement for Rectangular section . . . . . . . . . . 214.2 Load and displacement for castellated and I-section . . . . . . . . 22

4



Chapter 1

Introduction

In plastic analysis and design of a structure, the ultimate load of the structureas a whole is regarded as the design criterion. The term plastic has occurreddue to the fact that the ultimate load is found from the strength of steel inthe plastic range. This method is rapid and provides a rational approach forthe analysis of the structure. It also provides striking economy as regards theweight of steel since the sections required by this method are smaller in size thanthose required by the method of elastic analysis.This Report include testing ondifferent steel elements like hollow rectangular beam,Castellated beam and findout their load carrying capacity,plastic hinge formation point.

5



Chapter 2

Theoretical Background

2.1 Plastic Analysis:

Fully plastic condition is defined as one at which a sufficient number of plastichinges are formed to transform the structure into a mechanism, i.e., the struc-ture is geometrically unstable. Additional loading applied to the fully plasticstructure would lead to collapse. Design of structures based on the plastic orlimit state approach is increasingly used to find out. It was accepted by variouscodes of practice, particularly for steel construction. Figure 2.1 shows a typi-cal stress-strain curve for mild steel and the idealized stress-strain response forperforming plastic analysis.

Figure 2.1: A typical stress-strain curve for mild steel.

2.2 Example: Failure of Simply Supported Beam

The experiment shows that when the load is increased ,collpase occurs by theformation of kink in the beam, known as plastic hinge.

Initially the behaviour is elastic as shown in Figure 2.3.

Eventually further curvature becomes concentrated under the load, at theplastic hinge as shown in Figure 2.4.

To understand this phenomenon completely the testing on the steel shouldbe done and plastic hinge phenomenon should be studied.

6



Figure 2.2: Simply Supported Beam

Figure 2.3: Elastic Deflection

Figure 2.4: Kink in the beam : Plastic hinge

Figure 2.5: Diagram of Structure Featuring Plastic Hinges

7



2.3 Plastic Modulus

Definition:”The plastic modulus Z is defined as the ratio of the plastic moment M to theyield stress Fy”.It can also be defined as ”the first moment of area about theneutral axis when the areas above and below the neutral axis are equal”.

2.4 Principles of Plastic Analysis

Fundamental conditions for plastic analysis,-

• Mechanism condition:The ultimate or collapse load is reached when amechanism is formed. The number of plastic hinges developed should be just sufficient to form mechanism.

• Equilibrium condition: Σ Fx= 0, Σ Fy= 0, Σ Mxy= 0.

• Plastic moment condition: The bending moment at any section of thestructure should not be more than the fully plastic moment of the section.

2.5 Shape Function

Ratio of Maximum elastic moment, My=Zeσ y where Z=I/Ymax(the elasticsection modulus) to the Ultimate (fully plastic) moment,Mp=Zpσy.The ratioof the fully plastic moment to the yield moment depends on the shape of thecross-section and is known as the shape factor, f (Megson’s notation, but also

called S and sometimes, v). f is a measure of the ’reserve strength’ in a beamthat has reached its maximum elastic moment, My.

f= MpMy

= ZpZy

Some Shape function Values are illustrated in Figure 2.6:

Figure 2.6: Common Shape Function Values

2.6 Objective

• To understand the plastic hinge formation phenomenon by testing on steelsections.

• To Study the load carrying capacity of the hollow rectangular and castel-lated beam.

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• Behaviour of the castellated beam under point load and it’s comparison

with I-section.

2.7 Scope

The purpose of the study is to see the formation plastic hinge in various steelsection. Compare the theoretical result with testing result. Plot load vs dis-placement curve.Three different sections selected for testing which are:

• Rectangular hollow section.

• I-Section.

• Castellated beam section.

Work to de done:

1. To design the section as per plastic theory and find the plastic momentand finally the ultimate load is to be calculate.

2. Find the failure load by testing the section in lab.

3. Plot load vs displacement curve. Study the characteristics of the curve.Find the load at which plastic hinge is to be formed.

9



Chapter 3

Design Calculations,

Experimental Program andTest Set up

In this experiment, we studied plastic hinge for hollow sections, an I-section,and Castellated section. The design calculations for them are explained further.

3.1 Hollow Section : Calculations

Dimensions:Length(l)= 750mmWidth(b)= 40mmDepth(d)= 80mmThickness(t)= 3mm

Step-1: Calculate section modulus of the section (Z p) :

Z p= bh2

4 − (v − 2t) × [

h

2 − t]2

= 40 × 802

4 − (40 − 2 × 3) × [80

2 − 3]2

= 64000 − 34 × 1369

= 17454 mm3

Step-2: Calculate plastic moment of the section (M p) :

M p = Z P × f y = 17454 × 415 = 7.24 KNm

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Step-3: Calculate ultimate load of the section (Wu) :

W u = 1.5 × 4 pl

= 1.5 × 4 × 7.24

0.75

= 38.61 KN

Step-4: Calculate permissible deflection of the section (δ ) :

Moment of inertia:I = Z p × y = 17454 × 40 = 698160 mm4

Deflection(δ ) = W × l3

48EI = 38.61 × 1000 × 750

3

48 × 2 × 105 × 698160 = 2.43 mm

3.2 I-Section : Calculations

Dimensions:Length(l)= 1000 mmWidth of flange= 70 mmDepth of section= 150 mmThickness of web = 4 mmThickness of flange = 6 mm

Step-1: Calculate section modulus of the section (Z p) :

Moment of inertia:

I = BD3

12 −

bd3

12

= 70 × 1503

12 −

66 × 1383

12

= 5233104 mm4

Section modulus:

Z p = I

y =

5233104

75 = 69774.72 mm3

Step-2: Calculate plastic moment of the section (M p) :

M P = Z P × f y = 69774.72 × 415 × 10−6 = 28.9565 KN m

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Step-3: Calculate ultimate load of the section (Wu) :

W u = 1.5 × 4 × M p

l

= 1.5 × 4 × 28.9565

1

= 173.739 KN

Step-4: Calculate permissible deflection of the section (δ ) :

Deflection (δ ) = W × l3

48EI =

173.739 × 1000 × 10003

48 × 2 × 105 × 5233104 = 3.458 mm

3.3 Castellated Beam: Calculations

Take factored load = 140 KNMaximum moment = Wl/4 =35 KN mMaximum shear force= 70 KNThe capacity of castellated beam will be reduce due to secondary effect of stresses. Hence Z required should be increased to find suitable trial section.Zp,req = 1.05 M/(fy/1.1) = 82409mm3

Choose ISMB 100 to make castellated with Z available=83729mm3

3.3.1 Calculation of Property of Section

Area of T chord =556mm2

Position of centroid of castellated T section= 7.9 mm from topIz=6279675 mm4

It = 35461 mm4

3.3.2 Section Classification

Flange outstand= btf = 35/6 = 5.83 ≤ 9.4e

Web slenderness= d

t= 34.5 ≤ 84e

Hence section is plastic

3.3.3 Check for Shear

The elastic shear stress= V Q

ItQ=Ay=40848 mm3

Shear stress= 56.92 ≤ 0.7fy...OK.

12



3.3.4 Check for Deflection

Bending deflection = P l3

48EI

= 46.66 × 1012

48 × 2 × 105 × 6279675=7.68 mmShear deflection =0.05*7.68=.038 mmTotal deflection =0.81 mmPermissible deflection= 1000/240=4.16 mm

3.4 Apparatus

UTM, Supports, Mechanical strain gauge, Roller.

3.5 Practical Setup and Procedure: Hollow Sec-

tion

1. As shown picture 3.1, Test set up was minimal, consisted of test specimenplaced on two I-sections to act as simply supported beam.

Figure 3.1: Setup for Hollow Section

2. For measurement of deflection at the centre point of beam, mechanicalstrain gauge instrument was used.

3. To achieve concentrated load acting at the middle of test specimen, roundbar was placed between UTM machine and the specimen, ensuring pointload acting at the centre.

4. First, the specimen was placed on the supports so that the centre of thespan of specimen coincides with point load acting from UTM.

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5. Both the supports were properly adjusted to meet the ends of specimen

at the centres of the supports.6. Contact point of the Mechanical strain gauge was placed right underneath

centre of the span of the specimen for measuring the highest deflection.

7. Load was gradually applied by the UTM from the top at the proper inter-vals until the formation of plastic hinge typically accompanied with ob-servations of large deflections occurred at the same steady load as shownin Picture .

Figure 3.2: Deformed Shape of Hollow Section

8. Measurements from mechanical strain gauge were taken at each interval.

9. Graph of Load vs Deflection was plotted from the gathered data.

3.6 Practical Setup and Procedure: I-Section





14



Figure 3.3: Setup for I- Section

5. Both the supports were properly adjusted to meet the ends of specimenat the centres of the supports.

6. Contact point of the Mechanical strain gauge was placed right underneathcentre of the span of the specimen for measuring the highest deflection.

7. Load was gradually applied by the UTM from the top at the proper inter-vals until the formation of plastic hinge typically accompanied with ob-

servations of large deflections occurred at the same steady load as shownin Picture 3.4.

Figure 3.4: Deformed Shape of I- Section



15



3.7 Practical Setup and Procedure: Castellated

Beam


Figure 3.5: Setup for Castellated Beam




5. Both the supports were properly adjusted to meet the ends of specimen

at the centres of the supports.6. Contact point of the Mechanical strain gauge was placed right underneath

centre of the span of the specimen for measuring the highest deflection.

7. Load was gradually applied by the UTM from the top at the proper inter-vals until the formation of plastic hinge typically accompanied with ob-servations of large deflections occurred at the same steady load as shownin Picture 3.6.



16



Figure 3.6: Deformed Shape of Castellated Beam

17



Chapter 4

Results and Discussions

Test result for various type of steel sections are shown below:

4.1 For Rectangular hollow section

Load(KN) δ 1(mm)0 01.0 0.0052.0 0.008

3.0 0.0064.0 0.0085.0 0.00956.0 0.017.0 0.0118.0 0.019.0 0.0110.0 0.0111.0 0.011512.0 0.1513.0 0.2714.0 0.37

15.0 0.4516.0 0.5217.0 0.618.0 0.6519.0 0.7020.0 0.7521.0 0.8222.0 0.9023.0 0.9124.0 0.9425.0 0.95

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Load(KN) δ 1(mm)

26.0 1.0027.0 1.0828.0 1.229.0 1.2830.0 1.4031.0 1.5432.0 1.6033.0 1.6834.0 1.7035.0 1.8036.0 1.8837.0 2.00

38.0 2.0839.0 2.1940.0 2.3041.0 2.4742.0 2.8043.0 3.0044.0 3.1045.0 3.4846.0 3.80

Table 4.1: Load and displacement for Rectangular section

4.2 For I-section and castellated section:

Load(KN) δ C (mm) δ I (mm)0 0 05 0.005 0.0210 0.006 0.02215 0.008 0.02320 0.01 0.02425 0.01 0.0630 0.02 0.1535 0.22 0.440 0.54 0.5745 0.84 0.74

4.3 Load Deflection curves

The load Deflection graph was obtained from readings for castellated section, Isection and Comparison of between the both.

19



Load(KN) δ C (mm) δ I (mm)

50 1.13 0.955 1.44 0.9960 1.73 1.1265 2.05 1.2570 2.34 1.3675 2.55 1.4780 2.79 1.5785 3.00 1.6790 3.21 1.7795 3.44 1.88100 3.69 1.99105 3.96 2.11

110 4.35 2.20115 4.77 2.33120 5.35 2.41125 6.05 2.55130 6.97 2.69135 8.50 2.78140 10.72 2.95145 - 3.10150 - 3.27155 - 3.47160 - 3.75165 - 4.28

170 - 4.65

Table 4.2: Load and displacement for castellated and I-section

Figure 4.1: Castellated Beam : Load vs Deflection

20



Figure 4.2: I-section : Load vs Deflection

Figure 4.3: Comparison : Load vs Deflection

21



Chapter 5

Conclusions

Castellated beams has holes in its web, as holes incorporated various local effectsin beams, increase in load causes beams to be failed in different failure mode,which resist them to take load up to their actual carrying capacity. So we cannotcompare beams with different modes of failure directly for strength criteria. Dueto the presence of holes in the web, the structural behavior of castellated steelbeam will be different from that of the solid web beams. It make structurehighly indeterminate, which may not analyzed by simple methods of analysis.So we have to design beam to avoid local effects, for improved performance of castellated beam. Following points are worth noting:

Upto servicibility limit the deflection in castellated beam of the depth sameas the solid section is higher but the unit weight of castellated beam is much

smaller thus proving to be more cost effective.After serviceability limit when the load is increased continuously, due topresence of holes in the web opening it starts introducing some local effect inthe castellated beam, due to which its deflection increases rapidly and momentcarrying capacity decreases. Thus for cases when structure is expected to en-counter heavy loads non-castellated section shall be preferred.

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study of Plastic Hinge Formation in Steel Beams