FLORIDA A&M UNIVERSITY — FLORIDA STATE UNIVERSITY

COLLEGE OF ENGINEERING

ELASTIC BEHAVIOR OF STEEL

(Not Standardized)

By

MELISSA PENNINGTON

Group A3 Black Dragon

A Laboratory Report submitted to theDepartment of Civil and Environmental Engineering

in partial fulfillment of therequirements for the Civil Materials Laboratory

Submitted:April 21st, 2015

TABLE OF CONTENTS

List of Tables 4

List of Figures 5

1 Introduction 6

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Research Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4 Research Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.5 Chapter Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Background 7

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Stress Created in Beams in Third Point Loading . . . . . . . . . . . 7

2.3 Modulus of Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Experimental Program 9

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.2 Experimental Methodology . . . . . . . . . . . . . . . . . . . . . . . 9

3.3 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.4 Test Devices and Equipment . . . . . . . . . . . . . . . . . . . . . . 9

3.4.1 Equipment for Attaching Strain Gauge to Steel Beam . . . . 9

3.4.2 Equipment Used for Testing Steel Beam . . . . . . . . . . . . 11

3.5 Specimen Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.6 Test Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.6.1 Attaching Leads to Strain Gauge . . . . . . . . . . . . . . . . 12

3.6.2 Testing of Steel Beam . . . . . . . . . . . . . . . . . . . . . . 13

4 Experimental Results 14

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.1.1 Load vs. Displacement . . . . . . . . . . . . . . . . . . . . . . 14

4.1.2 Load vs. Strain . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5 Analysis 15

2

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5.2 Stress and Strain Curves . . . . . . . . . . . . . . . . . . . . . . . . . 15

5.3 Modulus of Elasticity Analysis . . . . . . . . . . . . . . . . . . . . . 16

6 Discussion 17

7 Concluding Remarks 18

7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Bibliography 19

Appendix A Laboratory Data Sheets 21

A.1 Experimental Data Sheets . . . . . . . . . . . . . . . . . . . . . . . . 21

3

LIST OF TABLES

3.1 Steel Beam Quantitative Data . . . . . . . . . . . . . . . . . . . . . . 9

5.1 Moduli of Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5.2 Statistical Analysis for Modulus of Elasticity . . . . . . . . . . . . . 17

4

LIST OF FIGURES

2.1 Stress created in Pure Bending . . . . . . . . . . . . . . . . . . . . . 8

2.2 Modulus of Elasticity from Hooke’s Law . . . . . . . . . . . . . . . . 8

4.1 Load vs. Displacement . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.2 Load vs. Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5.1 Stress vs. Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

5

ABSTRACT

This report presents the findings of the experiment based study on the elasticbehavior of steel. The elastic section region of stress strain curves has a linearproperty, where the stress increased so did the strain. This term elastic region meansthat within this range when load is applied to the material there is deformationoccurring but it is not permanent, when the load is removed the material will returnto the original state. Beyond this region, all deformation is permanent and is oftenvisible by signs of necking and fracture.

In this third point loading system, where the load was applied at the center ofthe beam, an equal distance from each support, stress and strain was recorded foreach specific load. The beams were all generally of the same geometric properties,with few variance. These geometric properties were considered when calculating thestress formula listed in chapter 2. These stresses were then compared to the strainreported from the strain gauge output and these stress strain curves are shown inthe analysis in chapter 5.

The modulus of elasticity, a main component of this report, is valued as the slopeof the elastic region of the stress strain curve. The experimentally determined mod-ulus was calculated to be 29.96*106 psi which was very similar to the actual valueof 29*1106 psi (Gere and Timoshenko, 1991) . The congruence of the experimentaland actual values encourages the understanding that steel is a stable, predictablematerial when analyzed in the elastic region.

CHAPTER 1: INTRODUCTION

1.1 Introduction

The understanding of how materials perform under stress is of utmost impor-tance when building any type of structure. The knowledge of how timber, concreteand various metals reacted when a load was applied to them was necessary to con-duct structural analysis. Presented in this report was an experiment completed tounderstand the Modulus of Elasticity of a steel beam. With the following researchobjectives in mind, the experiment was performed to gain insight into the elasticproperties of steel.

1.2 Problem Statement

Since steel is an necessary material used in majority of building projects it isimportant to understand how it reacts when loads and stress is applied to thespecimen. This research provides insight into these properties of steel when thismaterial is within it’s elastic state.

6

1.3 Research Objective

The goal of this research was to discover the elastic modulus of steel whenapplied with third point loading by analyzing the strain and displacement given bythe digital readout.

1.4 Research Scope

This research was performed to analyze the elastic behavior of steel when appliedwith third point loading. Since the experiment was performed to study the elasticbehavior of the steel, the load applied did not increase 8000 lbs as to not push thesteel past the point at which it begins to permanently deform. This lack of full loadanalysis separates this research from others performed.

1.5 Chapter Overview

Chapter 1 defined the purpose of the experiment. Chapter 2 summarized in-formation and findings from previous research performed on this topic, includingequations used later in chapters 4 and 5. Chapter 3 introduced and explainedthe testing method and materials used during the research. Chapter 4 displayedgraphical data obtained during the research. Chapter 5 provided tables describingcalculations and information about the results. Chapter 6 related results and anal-ysis while explaining what the findings of this report meant. Chapter 7 addressedthe problem from Chapter 1 to briefly summarize the report.

CHAPTER 2: BACKGROUND

2.1 Introduction

Similar to any other material used in construction, steel undergoes physicalchanges in response to applied loads. There is many stages of deformation that thematerials undergo when this load is applied, and the region that this report focusedon was the elastic region. The elastic region of a stress strain curve, which is createdwhen the stress that the material, in this case a beam of steel, is found to undergo,is compared to the strain measured in the material. In this experiment the strainwas measured using an applied strain gauge. In the elastic region, the steel beamdoes not undertake permanent deformation, but the modulus of elasticity is enoughto allow the beam to return back to its’ original condition.

2.2 Stress Created in Beams in Third Point Loading

Testing of the steel beams during this research was in the form of center pointloading, where the load was applied to the specimen at the midpoint between the

7

two supports. This type of loading created a stress in the beam that was related tothe specimen qualities including beam length and moment of inertia. Although allof the specimens tested during this experiment were similar in shape, the formulaused to calculate the pure bending stress slightly changes in response to any slightchange in the specimen geometry. The stress in each of the beams due to the loadapplied at the center of the beam as defined as the formula below:

σ =My

I(2.1)

Where the variables:

σ = Stress

M = Moment

y = Distance from the neutral axis to the point of loading

I = Moment of Inertia for the given beam

Figure 2.1: Stress created in Pure Bending

The value calculated from the above formula is then used in the following equa-tion to find Young’s Modulus (Gere and Timoshenko, 1991).

2.3 Modulus of Elasticity

The modulus of elasticity is a property of a given material that characterizesthe ability for the material to return to it’s original state after a load was applied.The modulus of elasticity is derived as the slope of the stress strain curve of a givenmaterial when the curve is in the linear region. Thus, the modulus of elasticity isdefined as simply a calculation of stress divided by strain. (Beardmore, 2013)

E =σ

ε(2.2)

Where the variables:

E= Young’s Modulus

σ = Stress

ε = Strain

Figure 2.2: Modulus of Elasticity from Hooke’s Law

8

The value of the Modulus of Elasticity for steel is 29*106 psi (Gere and Timo-shenko, 1991) . This empirically proven value will be used throughout this reportto compare experimentally gained results to proven results.

CHAPTER 3: EXPERIMENTAL PROGRAM

3.1 Introduction

This chapter describes the experimental process of testing and analyzing theelastic behavior of steel including the materials used, specimens prepared and pro-cess of testing. The information provided in this section was largely referencing Dr.Kampmanns’ Elastic Behavior of Steel presentation (Kampmann, 2015).

3.2 Experimental Methodology

A strain gauge was attached to each steel beam. Each steel specimen was testedunder third point loading.

3.3 Materials

Each steel beam’s qualities were recorded and are listed on following figure 3.1 :

SpecimenBeamWeight(lbs)

BeamLength(in)

WebThickness

(in)

FlangeThickness

(in)

CrossSectional

Area(inˆ2)

BeamDepth(in)

WebHeight(in)

FlangeWidth(in)

Moment ofInertia(inˆ4)

A1 42.8 40.875 0.25 0.25 3.06 6 5.5 3.375 17.42A3 43.2 41.0625 0.25 0.375 3.84 6 5.25 3.375 23.07A4 43.6 41 0.25 0.25 3 6 5.5 3.25 16.92A5 43 40.9375 0.75 0.1875 5.4375 5.98375 5.5625 3.38 22.22B1 42.8 41 0.25 0.46 4.6 5.88 5.5 3.5 14.24B2 43 41 0.25 0.3125 3.453 6 5.375 3.375 20.31B3 42.9 40.94 0.25 0.365 3.89 6.23 5.5 3.44 17.69B4 42.9 40.875 0.25 0.375 3.781 6 5 3.375 28.2

Table 3.1: Steel Beam Quantitative Data

3.4 Test Devices and Equipment

This section is split into two subsections: equipment used when attaching thestrain gauge to the steel beam, and the equipment used while testing of the steelbeam.

3.4.1 Equipment for Attaching Strain Gauge to Steel Beam

� Scissors

9

� Wire Strippers

� Needle Nose Pliers

� Square

� Utility Knife

� Tweezers

� Dental Probe

� Pencil (4H)

� Ballpoint Pen

� Ultra Fine Marker

� Fine Marker

� Drafting Tape

� Adhesive Tape

� Degreaser

� Conditioner

� Neutralizer

� Catalyst

� Adhesive

� Soldering Flux

� Rosin Solvent

� Solder Wire

� 320 Grit Silicon Carbide Paper

� 400 Grit Silicon Carbide Paper

� Cotton Tipped Applicators

� Gauze Sponges

10

3.4.2 Equipment Used for Testing Steel Beam

� Load Frame

� Hydraulic Actuator

� Load Cell

� LVDT

� Digital Readout

� Strain Gauge Readout

3.5 Specimen Preparation

The steel beam was degreased thoroughly with solvent and wiped in an unidirec-tional way with clean and fresh gauze. The oxidation on the steel beam was removedby first dry abrasion using 320 grit silicon carbide paper until beam was smooth.Conditioner A was then applied to wet the surface of the steel and 320-grit siliconcarbide paper was rubbed on this wetted surface to abrade further. After abrasionfrom the conditioner and 320 grit silicon carbide paper the surface was wiped cleanin a unidirectional swiping method with a gauze sponge. This process was repeatedmultiple times with the 320-grit paper and again with the 400-grit silicon carbidepaper until black oxidation is largely removed from the attachment surface of thebeam. Mark the alignment spots on the specimen wherever needed using a ballpointpen.After the beam was abraded and burnished properly, conditioner A was then repeat-edly applied and scrubbed with cotton-tipped applicators until a clean cotton-tipwas no longer discolored after rubbing on the surface. This scrubbing process withconditioner A and cotton-tipped applicators was repeated greatly to completely clearthe steel surface of oxidation. At this point the steel beam was silver at the pointof application, with all of the black rubbed off. The surface was then wiped cleanwith an unidirectional method with a gauze sponge. Care was taken to not allowany solution to dry on the steel surface, as it would have caused a contaminatingfilm to be created and would thus reduce bonding performed in the further steps.A liberal amount of neutralizer was then applied to the surface of the beam andscrubbed with a cotton-tipped applicator. A gauze sponge was used to wipe thesurface clean in a unidirectional motion, to prevent contaminants from being rede-posited.To prepare the glass plate for the strain gauge, it was chemically cleaned with neu-tralizer. The strain gauge was then placed on the surface of the glass plate bondingside down taking care not to use tweezers and not touch the gauge. The solder ter-minal was positioned adjacent to the strain gauge, with a 1.6mm space between thegauge and terminal. Adhesive tape was placed over the gauge and terminal systemmaking sure to properly center the gauge and terminal on the tape. This tape wasthen removed from the glass plate by pulling the tape up slowly at an angle less

11

than 45 degrees.The tape was then positioned on the steel specimen matching the alignment markson the gauge with the specimen layout lines drawn earlier. The tape was then an-chored to the beam on one side. After this end of the tape was securely connectedto the steel, the opposite end of the tape was lifted at a shallow angle until thegauge and terminal were free from the specimen surface. The tape was lifted ap-proximately 13 mm from beyond the soldering terminal. The lifted loose end of thetape was tucked under itself and attached to the specimen, making sure that theentire bonding surface was exposed and the gauge and terminal lay flat.The next step was applying the catalyst to aid in the hardening of the adhesive toadhere the gauge and terminal to the steel surface. A very thin and uniform coatof the catalyst was applied during this step. The catalyst was brushed out of thecatalyst bottle after wiping the brush for ten strokes against the inside of the bottleto ensure that only a small amount of catalyst was on the brush before application.The catalyst brush was placed across full gauge and terminal surface and was leftto dry for at least one minute.The bonding adhesive process was a time sensitive process and the following proce-dure for this step was completed within 3 to 5 seconds.The tucked under tape end was lifted off of the beam and was held steady while acouple (one or two) drops of the adhesive was applied at the fold between the tapeand specimen surface- approximately 13 mm outside of the actual installation area.Quickly after, the tape was rotated to 30 degrees placing the gauge bridges over theinstallation area. The tape was held steady and taut as the gauge was applied to thespecimen surface by bringing the gauge and terminal back down over the alignmentmarks burnished on the steel and a gauze sponge was used to slowly and firmlywith a single outward movement. An extremely thin, uniform layer of adhesive wasconstructed. Pressure was immediately applied by a thumb being placed and heldon the gauge and terminal for several minutes to ensure attachment to surface. Thetape was removed after approximately two minutes by slowing pulling it over itself.

3.6 Test Procedure

This section is split into two subsections: attaching the leads to the strain gaugeand testing of the steel beam.

3.6.1 Attaching Leads to Strain Gauge

The gauge grid was prepared for leadwire attachment by masking off the areawith drafting tape leaving only the leads tabs exposed. This tape was to protectthe gauge from solder during the leadwire attachment process.The soldering iron was turned on and set to the appropriate temperature range andwas left to heat up. After the iron reached the correct temperature the tip wascleaned with a gauze sponge, wetted if necessary. The iron tip was tined with freshsolder to prepare for soldering.A little amount of solder was melted onto the tip of the solder iron and the iron tip

12

was positioned on the terminal and the rosin-core solder wire was fed into the sol-dering iron tip. The solder and iron was removed from the terminal simultaneouslyafter a short time as to not cause heat damage. When this tinning of the terminalwas performed correctly, there was a even and shiny bead of solder on the tab. Alltabs on the terminal were tinned following this process.Leadwires were prepared for attachment by selecting white, red and black strandsand cutting this strands of wire to appropriate lengths. For each of the three sep-arate leads, one white, one red, one black, approximately 13 mm of insulation wasremoved on both ends using the wire strippers. A single wire strand of each theblack and red wire bundles were separated from the others. The remaining strainsof the red and black wires were twisted together, creating a bundle of wires and onesingle wire with each insulated strand. The twisted bundle of wire strands from theblack leadwire was twisted together will all of the strands of white conductors, withstill the single strand of black lead isolated. The black and white strand bundle wastinned with solder, as was the red twisted wire strand. Both bundles were trimmedto approximately 3 mm from the insulation but the two single strands were leftwithout trimming.The insulated portion of the three-lead wire was bent at a radius of 13 mm toprepare for attachment. The leadwire was anchored to the specimen with draftingtape so that the tinned end of the wire was in contact with the solder bead. Thesolder and iron tip was held to the leadwire and solder bead for one second and thenremoved, completing the connection of the leadwire to the terminal. The draftingtape was removed by loosening the mastic with rosin solvent, which then all solventwas removed with a gauze sponge using a dabbing action. The three wire conductorwas then taped to the specimen in a stress relief loop to prevent the wire from beingpulled off of the tabs.All three stripped conductors were isolated from each other and all of the bundleson each strand were twisted together and tinned. The conductors were put intotheir appropriate terminals in the quarter bridge connection of the strain indicator.The red wire bundle was connected to terminal P+, the white to terminal S- andblack to terminal D.The beam was then ready for testing.

3.6.2 Testing of Steel Beam

The steel beam was positioned on the supports with the center of the actuatorthrust aligned with the load point. An initial seating load was placed on the beam.The LVDT was placed on the specimen in the appropriate location in order toprotect the measuring device. The LVDT reading was tared on channel two. Thehydraulic hand pump was used to slowly apply force, with a maximum force of 8000lbs. For each load, the value of the load, displacement was recorded off of the digital

13

readout and the strain gauge reading was recorded off of the strain indicator.

CHAPTER 4: EXPERIMENTAL RESULTS

4.1 Introduction

This chapter contains the results gathered during the Elastic Behavior of Steelexperiment. The chapter is split into two subsections, one for the load applied tothe beam versus displacement of the steel beam data and the other containing thedata concerning the load applied versus the strain measured in the beam.

4.1.1 Load vs. Displacement

Figure 4.1 presents a comparison of the load applied to the steel beam and thedisplacement measured by the digital readout. The figure shows a linear relationshipbetween applied load and the displacement of the LDVT. As the graph illustratesas the load increases, the displacement of the beam increases as well.

Figure 4.1: Load vs. Displacement

0 1 2 3 4 5 6 7

·10−2

1,000

3,000

5,000

7,000

9,000

Deflection (in)

Loa

d(l

bs)

Beam A-1 Test 1Beam A-1 Test 2Beam A-3 Test 1Beam A-3 Test 2Beam A-4 Test 1Beam A-4 Test 2Beam A-5 Test 1Beam A-5 Test 2Beam B-1 Test 1Beam B-1 Test 2Beam B-2 Test 1Beam B-1 Test 2Beam B-3 Test 1Beam B-3 Test 2Beam B-4 Test 1Beam B-4 Test 2

14

4.1.2 Load vs. Strain

Figure 4.2 below illustrates the direct linear relationship between the load appliedto the steel beam by the hydraulic pump and the strain within the beam that wasreported by the strain gauge reading on the strain indicator.

Figure 4.2: Load vs. Strain

−150−100 −50 0 50 100 150 200 250 300 350 400 450 500

1,000

3,000

5,000

7,000

9,000

Micro-Strain

Loa

d(l

bs)

Beam A-1 Test 1Beam A-1 Test 2Beam A-3 Test 1Beam A-3 Test 2Beam A-4 Test 1Beam A-4 Test 2Beam A-5 Test 1Beam A-5 Test 2Beam B-1 Test 1Beam B-1 Test 2Beam B-2 Test 1Beam B-1 Test 2Beam B-3 Test 1Beam B-3 Test 2Beam B-4 Test 1Beam B-4 Test 2

CHAPTER 5: ANALYSIS

5.1 Introduction

This chapter presents and analysis of the results posted in chapter 4 that wereconducting using the formulas presented in chapter 2.

5.2 Stress and Strain Curves

Figure 5.1 shows the stress strain curves of the steel beams from this research.The stress was calculated using the formula 2.1 using the geometric values from

15

table 3.1 for each specific corresponding beam. Since the steel beam was tested onlyup to a applied load of 8000 lbs, the stress strain curves showed only the elasticregion of the curve. Because the curve was only shown in this elastic region, thecurve takes a linear slope. This slope of the stress strain curve, when analyzed,corresponds to the modulus of elasticity as defined by the formula 2.2. These valuesof Elastic Moduli are presented in the next section in table 5.1.

Figure 5.1: Stress vs. Strain

−150−100 −50 0 50 100 150 200 250 300 350 400 450 500

0.1

0.3

0.5

0.7

0.9

·104

Micro-Strain

Str

ess

(lbs

in2)

Beam A-1 Test 1Beam A-1 Test 2Beam A-3 Test 1Beam A-3 Test 2Beam A-4 Test 1Beam A-4 Test 2Beam A-5 Test 1Beam A-5 Test 2Beam B-1 Test 1Beam B-1 Test 2Beam B-2 Test 1Beam B-1 Test 2Beam B-3 Test 1Beam B-3 Test 2Beam B-4 Test 1Beam B-4 Test 2

5.3 Modulus of Elasticity Analysis

The moduli of elasticity for each beam, averaged from both tests performed oneach beam, are presented in table 5.1. These values were calculated by using theformula 2.2, where the appropriate stress that was fit into that equation for eachbeam as presented above in figure 5.1. The overall average modulus of elasticity wasfound to be 29.96*106 psi .

Table 5.2 presents a statistical analysis of the resultant moduli from the researchincluding two measures of variability: standard deviation and coefficent of variance.

16

Group Test 1 E Modulus (psi in 106) Test 2 E Modulus (psi in 106) Average E Modulus (psi in 106)

A1 35.444 35.786 35.61A3 28.96 28.882 28.921A4 25.48 26.56 26.02A5 32.28 35.02 33.65B1 39.967 9.250 24.56B2 35.04 31.43 33.235B3 37.11 39.44 38.28B4 22.88 22.26 22.57

AVERAGE 29.96

Table 5.1: Moduli of Elasticity

Table 5.2 presents a statistical analysis of the resultant moduli from the researchincluding two measures of variability: standard deviation and coefficent of variance.

Average E-Modulus ( lbsin2 ) 29.96*106

Standard Deviation 7.778Coefficient of Variance 0.2596E-Modulus for Steel as referenced in chapter 2 (psi) 29*106

Table 5.2: Statistical Analysis for Modulus of Elasticity

CHAPTER 6: DISCUSSION

In general all of the calculated Moduli of Elasticity were close to the proven valueof 29 10∗6 psi (Gere and Timoshenko, 1991). The specimens that were calculated tohave a higher elasticity modulus were beam A-1 and beam A-5. The two beams thatwere significantly lower than the required value were B-1 and B-4. The variance inall of the results in show in table 5.2. The errors that caused the variance shown inthis table could have been due to incorrect recording of the values while performingthe experiment, miscalculations throughout the analysis or alterations caused bythe strain gauge being incorrectly bonded to the beams.

Due to all of the values of E modulus being so similar it was safe to assumethat all of the beams were made of the same material, structural steel. Our analysispresenting the stress strain curves in figure 5.1 illustrates that all specimen beamsfollowed the same linear path where the increase in the stress correlated to the strainin the beam. The consistency of these plots describe a stability and consistency

17

throughout steel itself, that it is very predictable within the elastic region.

CHAPTER 7: CONCLUDING REMARKS

7.1 Conclusions

This chapter brought together the problem statement in chapter 1 with thediscussion in chapter 6.

� Calculated E Modulus of Steel in the research was found to be 29.96*106 psi

� This E Modulus was significantly similar to the defined value of 29*106 psi asstated in chapter 2

� Steel was concluded to be a very stable, predictable material in relation toreaction to applied loads within the elastic region

18

BIBLIOGRAPHY

Beardmore, R. (2013). (Elastic Bending Theory). Roy Beardmore.

Gere, J. and Timoshenko, S. (1991). (Mechanics of Materials). Chapman and Hall.

Kampmann, R. (2015). (Elastic Behavior of Steel). FSU/FAMU College of Engi-neering.

19

Appendices

20

APPENDIX A: LABORATORY DATA SHEETS

A.1 Experimental Data Sheets

21