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1.Objective 2.Theory 3.Experimental Set-up 4.Data Analysis 5.Sources of error 6.Discussion questions 7.Conclusion

Summer 2013
CIVE 302
T. Johnson
Dr. Dowell
Lab 0. Introduction to Laboratory Concepts and Presentation
While engineers rely heavily on equations and mathematics to perform design and analysis, it is
important to remember that the basis for these techniques comes from physics and applied
science. Thus, to properly use an equation and trust its results as accurate, an understanding of
the physical basis for each input is essential. This lab will explore many concepts relevant to this
process, including measurement, data processing, data/calculation presentation, comparison of
theory versus measured, and error analysis.
To start, let us cover some basic definitions that will be encountered throughout this lab. Data
refers to any quantity physically measured by equipment (the diameter of a circular bar); a
calculation, however, is any number that is derived from data (the area of a circular bar).
Distinguishing and labeling these two items not only good practice for double checking your
own work, but a clear way to present concepts and results to others not immediately familiar
with the project or experiment. In either case, any number presented should always be clearly
labeled with appropriate units, such as a diameter with units of inches or an area with units of
inches squared. Numbers by themselves have no meaning – only by giving them units and
labels do they have any value or usefulness. Table 1-1 provides a list of common units and
Table 1-1: Common Parameters with Associated Units
Inches (in)
Feet (ft)
Meter (m)
Millimeter (mm)
Pound (lb)
Kip (k)
Newton (N)
In practice, data can come in many different forms. Some measurements are discrete, single
values – such as the length of beam – but others may have hundreds of thousands of
measurements. An example of this is shown below Figure 1-1, which shows 3000 measured
acceleration values taken over 30 seconds during the Northridge earthquake in 1994. While the
actual data comes in a text file, note that it is presented in graphical format. Remember,
whenever presenting numerical values in calculations, reports, or other mediums, the goal is to
convey information and not to simply list numbers. In this case, we can quickly identify the
approximate maximum accelerations, the distribution of acceleration over time, and the
duration of the earthquake all at a glance; this is certainly more useful than listing several pages
of raw spreadsheet data as presented results.
Ground Acceleration (g)
Time (sec)
Figure 1-1: Example of Continuous Data. Northridge Earthquake, Beverly Hills Station
Additionally, let us look carefully at the formatting of the plot and identify several key features:

The plot has a clear title and unit-labeled x- and y- axes.

The numerical values on the axes are listed at easy-to-read, yet still useful, intervals with an
appropriate number of significant figures.

The plot is enclosed with a clearly defined border and does not have gridlines. This is prevent
visual clutter; unless you have a specific need or purpose to use items like gridlines, they are

The line representing the data is thin and clean. For multiple lines on the same plot varying,
easily-readable colors and line types should be chosen.
An important concept to keep in mind when working with measured numbers, too, is the concept of
significant figures. Using the appropriate number of significant figures is not only beneficial in making
presented values and calculations easily interpreted, but it also represents the level of confidence or
accuracy in the values being presented. For example, many engineering values of relevance are taken
from visual inspection off graphs; if asked to find the maximum acceleration and its associated time
from Figure 1-1, how would this be approached? One could print the graph, take a ruler, and estimate
about 0.4g at 8 seconds. If asked to find this directly from the data, however, the actual values are
0.4158g at 8.05 seconds. While carrying appropriate numbers of decimal places is good practice for
minimizing error, be mindful of not confusing decimal place accuracy with the actual accuracy of your
data or calculations.
Another important concept is measurement. When reading outputs from laboratory equipment,
understanding both the physics and tolerance behind how the values are being generated is vital to their
proper application. Most common are either mechanical or electrical measuring devices. Mechanical
devices operate using precisely-engineered intervals, such as the length intervals on a tape measure or
highly sensitive springs to measurement displacements. Electrical devices, the theory behind which will
be discussed later in this lab, use changes in an object’s resistance or input voltage to measure values.
Using physical relationships, these electrical changes can be correlated to other quantities such as
displacements, forces, or accelerations. Figure 1-2 shows a mechanical and electrical potentiometer – a
displacement measuring device – placed side by side.
Figure 1-2: Electrical potentiometer (left). Mechanical potentiometer (right).
Images and tables are often valuable tools for presentation as well. Whenever including images, be
professional – posting no image is sometimes preferable to posting low resolution or blurry photos. If
using images from an outside source, always cite that source. On that same note, posting random
images from a search engine is in very poor taste. Also, whenever including tables or figures always
provide a title/caption and discuss them in the text; if the figure is not being discussed, it should not be
included in the report.
When writing a laboratory report, keep in mind that it is intended to provide a technical description and
presentation of an experiment or procedure. Aim to be concise and to the point, but do not sacrifice
detail to so. Examples of a sentence in the experimental setup section, for example, may be:
Poor Statement: “The professor set up everything in the lab.”
Good Statement: “A steel coupon with threaded ends was loaded into the grips of a tension testing
Note the difference in these statements beyond the obvious absurdity. The second statement notes
specifics about the setup that would be required to reciprocate it in another setting: the coupon has
threaded ends which, logically, must fit into grips on machine loading heads. Additionally, the coupon is
fixed into a machine capable of applying tensile forces to the coupon. A detailed explanation of exactly
what laboratory reports for this course should contain is provided in the syllabus, but do not simply
write to the template: think about the exact details of what happened in the experiment and what is
needed to verbally, graphically, and logically explain this information.
Lastly, think rationally about results obtained in the laboratory experiment when analyzing results. Error
analysis should be far more specific than “the equipment was old” or “humans operating the equipment
make mistakes.” The purpose for doing error analysis is not to simply state that numbers may or may
differ from what was expected, but to examine why they differed and determine, using proper
judgment, whether they are acceptable or not. Examples of questions to consider while doing error
analysis are:

How were numerical values measured in the experiment? Were they approximated visually or
read precisely?

What is the significant figure accuracy of the measuring equipment being used? Furthermore,
what is the sensitivity of the equipment being used?

Were the calculations performed correctly?
The last question is not one that should only be asked at the end of performing calculations, but one
that should be asked throughout. Calculation mistakes are common and normal, but can often be
caught by critically examining results. For instance, if the expected yield stress for a material is 50 ksi
and you calculate 6500 ksi, this is a clear indication of a user-end mistake. Try to thoroughly analyze
your own work for significant errors before attributing faulty results to other sources.
Tour the structural engineering laboratory and examine several different pieces of measurement and
testing equipment. Carefully record the order in which the tour was performed and use this as an
experimental procedure. This lab will have no experimental setup or theory sections.
Using the earthquake data provided on Blackboard, develop a scatter plot representing this information.
Follow the general rules stated above to generate a properly formatted graph, complete with labeled
axes, units, and distinctly labeled lines. Present this in a clearly labeled “data” section of your lab report.
You will have no calculations for this lab report.
Provide a brief description of at least three pieces of equipment and at least three measuring devices
discussed in the lab. Include its proper name and how it measures or performs its function. An example
of each would be:
Tinius-Olsen Tension-Compression Testing Machine. Specimens are placed between two rigid loading
heads. Hydraulic pressure is used to move the lower head up or down to apply loads.
Electrical Potentiometer. A displacement-measuring device which tracks the elongation or contraction
of a rod. Correlates changes in length to an electrical current that can be read by a computer.
CIVE 302 – Lab #1
Tensile Behavior of Steel
T. Johnson
Spring 2013
How do forces influence objects?
Say we have a block of material sitting on the ground and apply a load P to it:
Just by intuition, we expect that the force required to crush the small block will not
crush the large block. This is because objects do not feel forces, they feel stresses.
Forces are important to system-level performance, but stresses are important to
material performance.
So what is a stress?
A stress is a forced normalized by the area over which it is applied. There are two
definitions of stress which are commonly used: true stress and engineering stress. True
stress considers the area at any given point in time after loading begins, while
engineering stress only considers the initial area. For axial loads:
 true (t ) 
P (t )
A(t )
 engineerin g (t ) 
P(t )
where : P(t) = axial force at a given time t
A(t) = cross-sectional area at a given time t
Ai = original cross-sectional area
While similar, these two definitions state an important difference: true stress takes into
the consideration the cross-section at a given time during loading, while engineering
stress only considers the original cross-section. The latter is by far the most common in
engineering practice and will be the only type discussed in this course.
True vs. Engineering Stress
True Stress
Engineering Stress
Comparing the two, it is important to note the deception that engineering stress
indicates that the material is getting weaker as is it more heavily. In actuality, the
opposite is true: materials gain strength as they are heavily deformed up until failure.
Normalizing Displacement
While stresses provide us with a normalized way in which to evaluate how an object
will react to a load, it is important to recognize that they only take into consideration an
object’s geometry. Realistically speaking, however, we know that material plays an
important role as well.
To examine this relationship, let’s introduce the concept of strain. We define
engineering strain as the change in length normalized by the original length:
 engineerin g (t ) 
where :
L(t )
ΔL(t) = net change in length at a given time t
Li = original length of the member
Defining Gage Lengths
This definition of strain seems very straightforward, but applying it to real situations is
tricky. By taking cuts in coupon we can use statics to calculate stresses, but what about
strains? Let’s look at the threaded-end steel coupon we will be testing today:
(a) Coupon
(b) Stress:
Take cut and use internal
force & cross section of
(c) Strain:
Define using different
criteria (discussed
later). Select specific
Stress vs. Strain Behavior
Important Locations:
σy = Yield Stress
σu = Ultimate Stress
σr = Rupture Stress
εy = Yield Strain
εh = Hardening Strain
εu = Ultimate Strain
εr = Rupture Strain
Stress (psi)
Linear-Elastic Region:
0 < ε < εy 20000 Plastic Region: εy < ε < εr 10000 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Strain (in./in.) εy εh εu εr Important Observations When a material is linear elastic it will return to its original shape after being unloaded. This is the linear region we see on the previous chart. In this region, we can relate stresses to strains via Young’s modulus (also known as the modulus of elasticity), E: σ  Eε E σy εy Keep in mind E is just the slope of a line representing a stiffness of a material. E should always be plotted on top of your data to make sure the slope you calculated is accurate. - Once a material exceeds its yield stress, it becomes plastic and permanently deformed. - Some materials have a yield plateau in which the material strains without developing additional stresses. - Once a material reaches its ultimate strain, significant local strains begin to develop. In steel, this is called necking. Ductility Another extremely important behavior to pull from the stress-strain curve of a material is its ductility. Ductility is a measure of how much plastic deformation an object can sustain prior to reaching its ultimate strain. Relevant to our lab will be the concept of strain ductility, taken as the ratio of the ultimate strain to the yield strain: ε ductility  εu εy This value measures how much deformation an object can sustain after it yields. Materials with low strain ductility are considered brittle – that is, they reach their ultimate strength shortly after leaving the linear-elastic region. Ductile materials, however, have high strain ductility. Materials with high ductility have larger areas under the stress-strain curve which ensures two things: a material will visually show deformation as it is accumulates damage, and a larger amount of energy will be required to deform it to this state. Poorly Defined Yield Some materials lack yield plateaus or have otherwise hard-to-define yield points. A common technique is to visually fit a slope E to the linear region and then offset this line to a strain of 0.002 (0.2%). The intersection of this line with the actual plot is taken to be the yield point. This technique is known as the 0.2% offset. σ E 1 σy E 1 0.002 εy ε 0.2% Offset We can calculate the yield stress and yield strain once we solve for the modulus of elasticity, E. Do this as you would normally for a material with a well-defined yield. A line with slope E may then be plotted, using a starting strain of 0.002 instead of 0, and the point where this new line intersects the original curve is the yield stress. σ For yield strain, the following can be derived from simple geometry: E 1 σy ε y  0.002  E 1 0.002 εy ε σy E CIVE302 Fall 2013 Dr. Dowell T. Johnson Lab 1. Tensile Test of Steel A computer-controlled Tinius-Olsen testing machine will be used to perform experiments on steel coupons by loading them to failure in tension. This lab project will demonstrate typical stress-strain responses of hot-rolled (Figure 1-1) and cold-rolled (Figure 1-2) steels, starting with the initial linear-elastic behavior and going through yield, the yield plateau, strain-hardening, and finally rupture. Failure occurs when the single axial member fractures into two pieces following necking – a strain localization signified by a visible reduction of the cross-sectional area. Strain is defined as the change in length L of a member, for a given load, divided by the member length. While there are several methods for defining strain, engineering strain is the most common and is very useful for structural engineering, and herein will be referred to simply as strain. Engineering strain  considers the original member length L prior to being loaded and may thus be written as  L L (1-1) For an axially loaded member, stress is defined as the force divided by the cross-sectional area. Engineering stress considers the member force F and the original, un-stressed, cross-sectional area A when determining axial stress. As with strain, there are varying definitions of stress that allow for the changing cross-sectional area as the member is being loaded. However, engineering stress  is the most useful definition for practical applications and from here on will be referred to simply as stress. Thus stress is given as  F A (1-2) For the axially loaded member discussed here the stresses and strains are the same at all locations within a cross-section; however, it is important to realize that the concepts of stress and strain are not limited to axial loading as in this lab project, but can be used for 2-D and 3-D objects. In 3-D analysis these values can vary from point to point, with stress and strain quantities defined at a single point rather than over a member length as in Eq. (1-1). In the finite element approach, for example, solid objects of arbitrary shape can be modeled by many interconnected elements, with stress and strain contours developed that provide an understanding of locations of high stress and strain. Stress Useful range of stress-strain curve Necking su Fracture sy E ey eh eu Figure 1-1. Typical stress-strain diagram for hot-rolled steel 2 Strain In Eq. (1-1) the member length can be replaced by a gage length (shorter than the member length), providing a different measured displacement while retaining the same strain. For example, if the elongation were measured over half of the member length (gage length = 1/2 of the member length) the measured elongation would also be 1/2 of the full member length value, resulting in the same strain. This is an important concept that regardless of the chosen gage length the measured strain will be the same, with one exception as discussed below. As the member begins to experience significant deformation, local imperfections in material structure cause a large increase in strain over a short length of the member and typically only over a portion of a chosen gage length. When this localization takes place the measured strain will directly depend on the gage length and, therefore, the very concept of strain will lose meaning and any engineering significance. This is why the useful range of the stress-strain curve for steel ends at peak stress (as indicated in Figures 1-1 and 1-2). Note that peak stress is equal to the ultimate stress. Thus, regardless of the gage length the stress-strain curve will be consistent until the peak stress and associated strain are reached. The shape of the stressstrain curve at strains larger than the ultimate strain (necking) will depend on the chosen gage length and is, therefore, not unique – that is, the measured strain value will vary for a given stress dependent upon a selected gage length. Figure 1-3 shows a coupon fixed at the base and loaded at the top. The dark lines on the left figure indicate the initial geometry of the coupon with length L. With load F applied, an equal and opposite reaction develops at the fixed base and the member elongates L as shown with the light gray outline on the same figure on the left. In addition to elongating, the cross-section reduces in size as shown in the figure. It is assumed here that the fixed base allows the crosssection to reduce size (fixed axially only). It is clear that the figure on the left is below the ultimate stress because there is no localization with the same strains throughout the member length. The figure in the middle shows necking (localization) leading to the failure and fracture 3 shown on the right figure. Ductile materials develop extensive necking while brittle materials fracture abruptly. Stress Useful range of stress-strain curve Necking su Fracture sy E E 0.002 ey eu Strain Figure 1-2. Typical stress-strain diagram for cold-rolled steel 4 F F F DL L R=F R R Figure 1-3. Constant strain, necking and fracture of steel coupon Experiment Tension tests will be conducted for hot-rolled and cold-rolled steels using circular test coupons that have two threaded ends. The threaded end regions are larger diameter than the center portion of interest, resulting in lower stresses due to the larger cross-sectional area (stress = force/area) and preventing yield in the attachment regions. Proper attachment of the specimen to the machine grips will ensure most of the deformations will occur over the member length that has constant cross-section. Displacements will be recorded by computer, measuring the elongation of a machined length using a potentiometer. Since most of the 5 specimen deformations occur between the threaded ends, and because the machine is stiff, use this distance as the gage length (member length with constant cross-section). Axial force will also be recorded throughout the experiment and saved to computer. Using a digital caliper, the diameter of the specimen at the center of the gage length should be measured before testing. Force and Displacement data will be posted on Blackboard for each test, with one column providing measured axial displacement and a second column giving measured force. Students will be responsible for changing the force and displacement data to stress and strain and plotting the axial stress-strain curve for each experiment. From the recorded data the following is required. REQUIRED CALCULATIONS: Hot-Rolled Steel Plot of stress versus strain Plot of force versus displacement (this should be presented in data section) Member-end-stiffness K (kips/inch) at yield stress and at ultimate stress* Modulus of elasticity E (ksi) Yield stress (ksi) Yield strain Strain at first hardening Ultimate stress (ksi) Ultimate strain (strain associated with ultimate stress) 6 Strain ductility Modulus of toughness (ksi) Cold-Rolled Steel Plot of stress versus strain Plot of force versus displacement (this should be presented in data section) Initial member-end-stiffness K (kips/inch) Modulus of elasticity E (ksi) Yield stress (ksi) – from 0.2% offset Yield strain (strain associated with yield stress) – from 0.2% offset Ultimate stress (ksi) Ultimate strain (strain associated with ultimate stress) Strain ductility Modulus of toughness (ksi) REQUIRED DISCUSSION: From the measured results, discuss any differences and similarities between hot-rolled steel and cold-rolled steel (numerically and/or conceptually). Comment on the overall behavior of the materials in the force-displacement and stress-strain curves (be specific). Provide example applications in the real world for both types of steel. 7 Definitions: Modulus of elasticity E (ksi). This is the initial, linear slope of the stress-strain diagram. In this range the material (and any structure that is made of this material) will be linear-elastic, meaning that if the load is doubled the deformations will double (linear), and upon removal of the load it will return to its original position (elastic) with no permanent deformation. Yield stress (ksi). For hot-rolled steel the yield stress is the stress at the end of the proportional limit, which is followed by the yield plateau. For cold-rolled steel there is no well-defined yield stress due to the nonlinear nature of the stress-strain curve. Therefore the yield stress is found by drawing a line on the stress-strain diagram with the same slope as the initial E, but offset by a strain of 0.002 (0.2%), and finding where it intersects the measured stress-strain diagram. Yield strain. Yield strain is the strain associated with the yield stress (at the same point on the stress-strain diagram). Strain at first hardening. For hot-rolled steel, strain hardening occurs at the end of the yield plateau. Beyond yield strain, the stress remains approximately constant (equal to the yield stress) for quite a while until strain hardening begins. This does not apply to cold-rolled steel due to the shape of the stress-strain curve. Ultimate stress (ksi). This is the maximum stress achieved. At strains beyond this point the stresses decrease and necking develops, followed soon after by fracture and failure of the specimen. 8 Ultimate strain (ksi). This is the strain associated with ultimate stress. It is not the strain at rupture as this strain is not unique, with different values for different arbitrary gage lengths. Strain ductility. This is the ratio of ultimate strain to yield strain. Modulus of toughness (ksi). This is the area under the stress-strain curve to ultimate stress and strain (not to rupture). Member-end-stiffness (kips/inch). Axial force divided by displacement, which is the slope of the force-displacement curve. In the plastic region stiffness is defined as the tangent slope to the stress-strain curve. 9 Purchase answer to see full attachment

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