CSCE 155N Fall 2012 Homework Assignment 2: Stress-Strain Curve Assigned: September 11, 2012 Due: October 02, 2012 Note: This assignment is to be completed individually - collaboration is strictly prohibited. Points: 100 points Objectives The objectives of this homework assignment: 1. Master the use of standard I/O in Matlab. 2. Master the use of plots in Matlab. 3. Master the use of selection statements in Matlab. 4. Master the use of loops in Matlab. 5. Familiarize with the use of File Input/Output (I/O) in Matlab. 6. Familiarize with the user of functions in Matlab. 7. Familiarize with the use of matrices in Matlab. 8. Familiarize with the concept of control statements (selection and loop) in solution design 9. Be exposed to the concept of problem decomposition and the design-implementationtest cycle. 10. Appreciate and understand the application of computational thinking in solving engineering problems Memo To: The programmer Memo From: Your boss Well, another job has come in! And you are my only hope! This time, the customer is from a mechanical engineering company. They want a program that takes in the parameters of a material specimen, loading force and elongation corresponding to the load, and returns the area of the specimen and the stress- strain curve. I don t know much about the setup all I have is this description of the tensile testing and the stress- strain curve. Please take care of this task! Problem Description Have a nice day! Your boss Studying the properties of different materials is a crucial field in mechanical engineering, because it helps engineers select appropriate materials for different engineering applications. Generally, materials undergo a set of tests; different tests give information about a specific property of the studied material. One of the well-known tests is the tension testing. In tension testing, a uniaxial tension force is applied to a sample of the material (specimen) we are interested to study. This sample is called tensile specimen. A typical tensile specimen (see Figure 1) has enlarged ends or shoulders used for gripping, and a gage section. The two most commonly used cross section shapes of the tensile specimen are the circular and square cross sections. In both of them, the gage section has a smaller cross sectional area than the shoulders. 1
The tension testing involves mounting the specimen in a machine, from the two shoulders, and then tension force is applied until it fractures. (Note from boss: This is cool!) As the load increases, the test sample elongates. This elongation in the gage section is recorded against the applied load (force). F F L o Figure 1: Typical tensile specimen An example of data recorded during a real tension testing is shown in the following table: Applied Force (N) Elongation (mm) 0 50.8000 7248.7792 50.8510 14497.5584 50.9020 21604.2047 50.9520 28852.9839 51.0030 36101.7631 51.0540 40103.6987 51.3080 43005.4237 51.8160 46508.0824 52.8320 47908.1165 53.8480 49009.6138 54.8640 49202.6332 55.8800 47802.5992 56.8960 46501.6484 57.6580 44301.2273 58.4200 38097.5838 59.1820 Two important values are calculated from the force applied and the elongation experienced by the specimen, namely: engineering stress and engineering strain, respectively. Engineering stress (σ) is a value that describes how the body having the force exerted upon it counteracts the pressure that results from this force, and it is defined as: σ =!!! N/m! (1) where F is the applied force, A! is the cross section area of the gage section assuming a circular specimen with diameter D!, or a specimen with a square cross section of side length D!. Engineering strain (ε) describes the deformation and change of the specimen's length. By definition, it is the ratio between the change in the length of the specimen and the initial length before the test., and it is defined as: 2
ε =!!!! =!!!!!! (2) where ΔL, L!, and L are the change in gage length, initial gage length, and the final gage length, respectively. Strain is a unit-less value since it represents a ratio of parameters of the same units. Hence, you have to make sure of unit consistency, i.e., that all of the lengths used are of the same unit (either meters or millimeters). A typical stress-strain curve is shown in Figure 2. IMPORTANT: For each force-elongation pair, we have to compute the corresponding engineering stress and strain. However, usually a large set of data points are recorded in this kind of material tests. This makes the process of calculating the engineering stress and strain at each point manually, a time-consuming process. (Note from boss: Consider using loops!) As you may have noticed, a stress-strain curve for a ductile material always starts with a linear region. This linear region is called the elastic range. In the elastic range if the load is removed, the material goes back to its original length, i.e., the deformation exists as long as the load is applied but the deformation disappears if the load is removed. However, after the elastic range ends, the plastic range starts, in which the material suffers permanent deformation. In other words, the material cannot get back to its original state even if the load is removed. Figure 2: Stress-Strain Curve. Figure 3: Stress-Strain curve (Colored Elastic Region) Using visuals are helpful in making data interpretation an easy process. One way to make it easy for the engineers to recognize different phases of a material's stress-strain curve is to use colors to differentiate these different phases. For example, we can use the green color to represent the elastic range (the straight line in the curve), as shown in Fig. 3. To identify this elastic region, we need to identify the points that form it. And, in order to identify these points, we have to characterize them first. In other words, we need to find the common characteristics that the points in the elastic range have in common, and make them different from other points in the curve. Let's have a closer look at the given data, so we can identify these characteristics. In Table 1 below, we can see, the segments that define the elastic region are all of the same slope value, which is equal to 56,332. This makes them different from 3
all other points, because any other consecutive points that form a segment that has this exact slope, should be part of this straight line. Point number (i) Applied Force (N) Elongation (mm) Stress (MPa) Strain 1 0.0000 50.8000 0.0000 0.0000 2 7248.7792 50.8510 56.3320 0.0010 3 14497.5584 50.9020 112.6640 0.0010 4 21604.2047 50.9520 167.8915 0.0030 5 28852.9839 51.0030 224.2235 0.0040 6 36101.7631 51.0540 280.5555 0.0050 7 40103.6987 51.3080 311.6555 0.0100 8 43005.4237 51.8160 334.2055 0.0200 9 46508.0824 52.8320 361.4255 0.0400 10 47908.1165 53.8480 372.3055 0.0600 11 49009.6138 54.8640 380.8655 0.0800 12 49202.6332 55.8800 382.3655 0.1000 13 47802.5992 56.8960 371.4855 0.1200 14 46501.6484 57.6580 361.3755 0.1350 15 44301.2273 58.4200 344.2755 0.1500 16 38097.5838 59.1820 296.0655 0.1650 Slope of the segment (i, i+1) 6220.0001 2255.0000 1361.0000 544.0000 428.0000 75.0000-544.0000-674.0000-1140.0000-3214.0000 Table 1. Points and interactions between applied force and elongation, and the slope of each successive pair of points (segment). You tasks are as follows: 1. Design and implement a program that takes in an input file with the following format. a. The type of the specimen's cross section, either circular or square. (You need to give the user the option of choosing either. You must use either an "if" or a "switch" statement in your solution.) b. The diameter (or the side length) of the specimen D! in millimeters. If the user selects 1 as an answer for the previous step, then the question should ask him/her for the diameter, and if s/he selects 2, the question should ask for the side length. c. The initial length of the specimen L! in millimeters. 2. Load the two vectors of the loads and resultant elongations from two input files, ForceVector.dat and ElongationVector.dat, respectively. Note: We will provide two sample input files on the course website at: http://www.cse.unl.edu/~lksoh/classes/csce155n_fall12/homeworks.html 4
3. Calculate the engineering stress σ a. Calculate the cross section area A! (based on the inputs of the user in 1-b and 1-c). b. Design a "for" loop that uses the cross-sectional area A! (found in 2-a) and the vector of forces applied to find the engineering stress at each load applied using Equation (1). 4. Calculate the engineering strain (ε) a. Design a "for" loop that uses both; the initial length of the specimen L! and the vector of the resultant elongations to calculate the engineering strain using Equation (2). 5. Calculate the slopes of the segments. In the following an illustration of how to find the slope. For any two points on the straight line A = (x!, y! ) and B = (x!, y! ), the slope is defined as: slope = y! y! x! x! 6. Define a vector to store the indices of the set of points that form the elastic region. Of course you do not know the length of this vector ahead of time. 7. Use a "while" loop to fill in this vector of indices. You should scan the vector of slopes you formed in Step 4, and check if the slope satisfies the condition that characterizes the points in the elastic range or not. If yes, then the indices of the two points that form this segment should be placed in the vector of indices. Generally, we take the first slope, which corresponds to the first segment, as the condition. Then any other segment has a slope that matches this condition, is considered an extension of the first segment For example, in the given problem, the condition of an elastic region is "while the slope is equal to 56,332, place the point in the vector of the elastic range". Scanning the vector of slopes, we find that the first slope is 56,332, and the indices 1 and 2 should be placed in the vector of Step 5, because these two points from the segment (1, 2) which has the slope 56,332. Then, we take the slope of the next segment (2, 3), and compare it with the condition 56,332. As we can see, the slope of the segment (2, 3) satisfies the condition, and hence, we should place the index 3 in the vector (because 2 has already been placed there). The while loop should stop once the slope value is no longer 56,111, in this example. (Note: In our test case, we will use different set of values and thus the condition will not be always 56,111. But it will always be based on the slope value of the first segment derived from the input points.) 8. The output of the program should show the following: a. The cross-sectional area A! b. A matrix of size (M x 5). Columns from 1 to 5 represent the load, length, stress and strain and slopes, respectively. (Note: In the table shown in the previous page, each cell in the vector of slopes is centered offset between two of the data points, this is done just to illustrate that this slope is obtained using these two data points. However, in Matlab, we will not be able to generate table with such an offset; instead, a segment s slope is aligned with the row of the first point of the segment. Please see the sample output in the next page.) c. The vector of the indices of the points that form the elastic region. 9. Generate plots 5
a. The stress-strain curve must be plotted. This curve shows the relation between the stress and the strain you calculated previously in Steps 2 and 3. This graph should show the elastic range plotted in green color, while the rest of the curve is plotted in the black color See Figure 3 for an example. A sample run of the required program is shown below. (Note: Bold texts are user input.) Sample Output (SSCalc) Welcome to SSCalc: Please enter the following: Specimen's Type (1-Circular 2-Square): 1 Diameter of the specimen (Do) in millimeter: 12.8 Length of the specimen (Lo) in millimeter: 50.8 Filename for the load s vector in N: ForceVector Filename for the elongation s vector in millimeter: ElongationVector Output: Cross-sectional area (Ao) in mm^2= 128.6796 The values of the stress and strain are shown in the following table Load(N)-Length(mm)-Stress(MPa)-Strain - Slope SSvalues_output= 1.0e+04 * 0.0000 0.0051 0 0 5.6111 0.7249 0.0051 0.0056 0.0000 5.6111 1.4498 0.0051 0.0113 0.0000 5.6111 2.1604 0.0051 0.0168 0.0000 5.6111 2.8853 0.0051 0.0224 0.0000 5.6111 3.6102 0.0051 0.0281 0.0000 0.6220 4.0104 0.0051 0.0312 0.0000 0.2255 4.3005 0.0052 0.0334 0.0000 0.1361 4.6508 0.0053 0.0361 0.0000 0.0544 4.7908 0.0054 0.0372 0.0000 0.0428 4.9010 0.0055 0.0381 0.0000 0.0075 4.9203 0.0056 0.0382 0.0000-0.0544 4.7803 0.0057 0.0371 0.0000-0.0674 4.6502 0.0058 0.0361 0.0000-0.1140 4.4301 0.0058 0.0344 0.0000-0.3214 3.8098 0.0059 0.0296 0.0000 0 6
Elastic_Rregion_Indices: 1 2 3 4 5 6 Thank you for using SSCalc. Note: The "..." is used to split a long vector of input into two lines as shown above. IMPORTANT: Note that this program will be more complicated than the one you did to solve the chemical engineering problem last week. Thus, you should consider using functions to increase the modularity of your solution. That means, you should have multiple m-files in your solution package. Your boss will be looking at the quality of modularity of your work. (Hint: Think about how to break this task into multiple, smaller steps, and how each step is to process an input and then return an output to be used by the subsequent step.) Challenge - Extra Credit (10 points) As mentioned above, after the elastic range, the plastic range begins. This transition in the properties of the material concerns the designers, because it has a great impact on the design calculations. For example, while designing certain mechanical application, the engineers should know the maximum load expected to be applied to this application; and subsequently, based on this expected maximum load, the designers should use a material that will remain in its elastic range even under this maximum load. That is why it is very important to know the stress at which the elastic range ends and the plastic range starts, which is called the yield strength σ!. Sometimes this yield strength is clear and can be easily detected. But sometimes the material s properties make it hard to be detected. For such materials, a method called the offset yield method is used to determine the yield strength. In the offset yield method, a straight line that is parallel to the elastic range is plotted. This line crosses the x-axis at the point corresponding to ε = 2%. The intersection point of this straight line with the stress-strain curve represents the yield strength point. Your task is to add to your program such that it plots the straight line of the offset yield method. The benefit of this addition is twofold: (1) it will save engineers time spent on drawing this line manually, and (2) since the engineers plot this line manually, there is a high probability of error while drawing, and consequently, the value of the yield strength observed might be erroneous. On the other hand, plotting this straight line using the computer will reduce the probability of this error. An example is shown in Figure 4. As we can see, the intersection between the straight line and the stress-strain curve is almost at 280 MPa. Hence, σ! = 280 MPa. Since this straight line is parallel to the line of the elastic range, then they have the same slope. So what you have to do is to use the slope of the elastic range, you have already computed in Step 6, along with the information that this straight line passes through the point (0.002,0), to get the equation of this straight line. You need to do these calculations using paper and pencil (you should document these calculations in a document file named ANALYSIS_Challenge.doc), and then use Matlab to plot it. Your code also should print the statement "The 2% straight line of the offset yield method" beside the plotted straight line, as shown in Figure 4. You can't write this 7
statement on the curve using the plot editor, you have to do it using the Matlab commands. (Hint: Consider using the functions "str" and "text".) IMPORTANT: Note that you have to adjust the x-axis scale to make sure that (1) the plot is readable and (2) the straight line of the offset yield method is clear, as shown in Figure 4. As a check for the correctness of your calculations and code, make sure that the point of intersection is located inside the range between 250 MPa and 300 MPa. You can do this step visually, and just write down your observation that it is located within this range. In other words, you do not need to verify this using Matlab. Figure 4: The 2% straight line in the offset yield method. Submission Procedure This assignment is due at the start of class (2:00 PM). Assignments five minutes late will NOT be accepted. It is highly recommended that you read the grading policy and grading guidelines on the course website for a complete explanation of how the assignments will be graded. Remember, your program should follow a good programming style, include plenty of comments both inline documentation and Matlabdoc documentation, and perform all of the functionality outlined above. Also, in the welcome message of your program, state whether you are implementing the extra credit functionality. There are two submission steps: (1) You must handin the following files on-line: Source files: SSCalc.m and other m-files and m-file functions Readme file: README Testing file: TEST (Also, if you complete the Challenge, please submit your input text file, TEST_Challenge, and a document that shows the work and calculations done by you, ANALYSIS_Challenge.) (2) You must submit a stapled paper copy of a coversheet, your source files, the README file, and the TEST file. Both of these steps must be done by the start of the class on the day the assignment is due. Please download this coversheet from the instructor s course website, under the Homework Assignments link. This coversheet allows the grader to give comments and categorize the points for your homework. 8