Building Frequency-Teacher s Manual

Transcription

1 Objective The goal of this experiment is to give students insight into the behavior of two common earthquake-resisting systems for buildings, the moment frame and the brace frame, through introductory concepts of mass, stiffness and frequency. Table of Contents Objective... 1 Background... 2 Moment Frame... 2 Brace Frame... 2 Frequency... 3 Mass... 3 Stiffness... 4 Moment Frame... 4 Brace Frame... 5 Single Degree of Freedom Model... 6 Multiple Degree of Freedom... 7 Model Construction Braced-Frame Construction Moment Frame Brace Frame/Moment Frame Combination Model Model Testing Load-Displacement Test(Grades 6~Undergrad) Materials Procedure Natural Frequency with a Shake Table (Grades 9~Undergrad) Materials Natural Frequency by Calculation Natural Frequency by Observation Procedure Natural Frequency by Accelerometer Procedure Natural Frequency by Pull-Back Test (Grades 6~Undergrad) Materials Procedure Page

2 Building Behavior Test(Grades 6-12) Materials Procedure Background Moment Frame A moment frame is an earthquake force resisting system comprised of two columns and a connecting beam, as illustrated in Figure 1. The beam-column connection is a moment connection, which means that it does not allow the beam or column to rotate relative to one another. Unlike a pinned connection, which allows its members to rotate, a moment connection restrains the members to a right angle and transfers the load from the beam to the columns and ultimately down to the ground. Notice the right angle is maintained when the frame is pushed. Figure 1. Moment Frame Brace Frame A brace frame is an earthquake force resisting system comprised of two columns, a connecting beam, and an angled member (known as a brace). Unlike the moment frame, the connections between these members are all pin connected. A pin connection allows the members to rotate relative to each other. The brace transfers the load from the beam to the ground. Figure 2 depicts a typical brace-frame configuration. 2 Page

3 Notice that the members are free to rotate when frame is pushed Figure 2. Brace Frame Frequency The natural frequency of a building is an important earthquake engineering concept because it is one of the factors that defines the behavior of a specific building during a seismic event. Natural frequency is the number of oscillations per time period. The unit of measure used for frequency is Hertz (Hz), which is cycles per second. Much like a pendulum s frequency is dependent on length and mass, each building has its own frequency that is dependent on stiffness and mass. Natural frequency, mass, and stiffness are related according to the following equation. Natural Frequency: f = k M 2π units: Hz Mass According to Newton s Second Law: F = m a units: pounds (lbs) Where: m=mass (slugs) a=acceleration (ft/s 2 ) F=force (lbs) 3 Page

4 Weight and mass are not the same thing but rather are related according to following form of Newton s Second Law: W = m g units: pounds (lbs) Where: g=gravitational acceleration=32.2 ft/s 2 W=weight (lbs) Clearly, every building has weight, which means that it has an inherent mass as well. With all other factors held constant, the greater the mass of a building, the lower the natural frequency. Stiffness Every building has an inherent stiffness. When a force is applied to a building, it will move in the direction of the force. Stiffness is defined as the ratio of the force applied to the displacement experienced. Therefore, the higher the stiffness, the less a building will deform under a given load. Stiffness: k = Force Displacment Moment Frame There are several ways to model the moment frame, each of which results in a unique stiffness. If the base connection is modeled as fixed (like the column-beam moment connection), the stiffness of the frame is defined by following equation: k moment frame = 2 12EI h3 units: lbs/in Where: I=Moment of inertia of the column (in 4 ) [See Mechanics of Materials by Beer, Johnston, DeWolf & Maurek] If the base connection is modeled as a pinned connection, as shown in Figure 3a, the stiffness of the frame is as follows: k moment frame = 2 3EI h 3 4 Page

5 Figure 3a. Moment Frame For the K NEX model, the base connections are somewhere between fixed and pinned. Therefore, its stiffness cannot simply be determined by the theoretical equations above. It is recommended that the actual stiffness be determined experimentally by a load displacement test. Instructions for such a test are included later in this document. Brace Frame The stiffness of a brace frame can be determined in the same manner. However, there is also a specific equation that defines its stiffness based on the cross section properties of the brace. The equation is shown below: k brace = EA L units: lbs/in where: E=modulus of elasticity of the brace (lbs/in 2 ) A=Area of the brace (in 2 ) L=the length of the brace (in) The modulus of elasticity is a material property, and varies for each material [steel, concrete, wood, etc.]. Additionally, the lateral (horizontal) stiffness that it provides to a frame is defined below. k brace frame = EA L cos2 θ units: lbs/in 5 Page

6 Figure 3b. Brace Frame Stiffness Factors Single Degree of Freedom Model Once the concepts of mass, stiffness, and frequency are understood, one can begin to discuss different ways of analyzing the building. The first and most simple approach is to model the structure as a single degree of freedom model. A degree of freedom is an independent direction in which an object can move. Since the primary concern of earthquake engineering is lateral (or sideways) movement, we assign one horizontal degree of freedom at an effective height along the building. This single degree of freedom simplifies the building s behavior to that of a lollipop structure, which has a single mass and stiffness. All vertical and rotational motions are ignored when using this method. This idea is displayed in Figure 4. Figure 4. Equivalent Single Degree of Freedom Structure 6 Page

7 To use this simplified method, the stiffnesses and masses of each floor cannot simply be added together. Rather, an effective mass and stiffness must be used. To do this, a complex procedure known as static condensation must be utilized. However, this process is beyond the scope of these experiments. Instead, a total mass and stiffness will be used to find a frequency, which will then be adjusted by a factor to achieve an accurate prediction. This factor takes into account the effective height, stiffness, and mass, each of which will not be computed directly. To find the total mass (not the effective mass), simply sum the mass of each floor. The total stiffness is slightly more difficult to calculate. The equation for combining the stiffnesses of adjacent floors is as follows. k tot = k 1 k 2 k 3 Where: k 1, k 2, and k 3 are the stiffnesses of the first, second, and third floor, respectively. These are calculated using the equations mentioned above. Use the total mass and stiffness in the equation below, which includes the adjustment factor. f = 1.28 k tot M tot 2π Multiple Degree of Freedom Another, more accurate approach to analyze a structure is to use multiple degrees of freedom. There are numerous ways to assign degrees of freedom, but one common method is to place one horizontal degree of freedom at each level, as shown in Figure 5. 7 Page

8 Figure 5. Equivalent Multiple Degree of Freedom Structure To determine the frequency of the structure, a procedure called Eigen Analysis must be utilized (see Dynamics of Structures by Chopra, Chapters 9-10). This approach makes use of matrices which are mathematical arrays of numbers arranged in rows and columns. For Eigen Analysis, they are used to store values of various modes and degrees of freedom. In this case the mass and stiffness matrices would take on the following forms: m Mass: M = 0 m m 3 Where: m 1 = the mass of floor one m 2 = the mass of floor two m 3 =the mass of floor three k 1 =the stiffness of floor one k 2 =the stiffness of floor two k 3 =the stiffness of floor three k 1 + k 2 k 2 0 Stiffness: k = k 2 k 2 + k 3 k 3 0 k 3 k 3 8 Page

9 For a three degree of freedom system, there are three natural mode shapes. These modes describe the various displacement behaviors of the building due to the relative motion of the degrees of freedom. Examples of the three modes are shown diagrammatically below. Figure 6. Mode Shapes From here, an Eigen analysis is performed to determine the angular frequency. In a TI-89 calculator, the following process may be used to perform Eigen Analysis. 1. Set A=(M -1 )k 2. Select function eigvl and input A. This will give an output of three values, which are the squared angular frequencies (ω 2 ) of the three modes. The smallest value is the squared angular frequency of mode one, which is the fundamental mode. The second largest value is associated with the second mode and the largest value is related to the third mode. The natural frequency of each mode is related to the angular frequency by the following equation. Where: n=a particular mode f n = ω n 2π 9 Page

10 Model Construction The class will be split into three different groups. Each group shall construct a three story building model out of K NEX with a specific structural system. Team one will build a moment frame model, team two will build a braced-frame model, and team three will construct a combination of both braced and moment frame (this will be referred to as the mixed model). Each group will construct their model as follows. Braced-Frame Construction 1. Construct the base as shown in Figure 7. Notice the alternating orientation of the connection pieces to allow for brace connection points. Figure 7. Base 2. Construct 6 of the frames shown in Figure 8. Notice the alternating orientation of the connection points Figure 8. Brace Frame 10 P age

11 3. Place two of the frames on opposite sides of the base as shown in Figure 9. Figure 9. Partially Constructed Brace Frame Story 4. Insert the colored members, as indicated in Figure 10. Figure 10. Brace Frame Story 11 P age

12 5. Insert floor stiffeners, which are shown in color in Figure 11. Figure 11. Floor Stiffeners 6. Repeat steps 3-5 twice, attaching the frames to the previously created story below instead of the base. 7. Next, create the floor masses. Each mass consists of properly sized pieces of paper stapled together and tied securely to the frame as shown in Figure 12. Figure 12. Floor Mass 12 P age

13 8. Lastly, fasten the building to a base as shown in Figure 13. Figure 13. Base Connection Moment Frame Construction 1. Construct the base as shown in Figure 14. Notice the matching orientation of the connection pieces. Figure 14. Base 13 P age

14 2. Construct 6 of the frames shown in Figure 15. Notice the matching orientation of the connection pieces. Figure 15. Moment Frame 3. Place two of the frames on opposite sides of the base as shown in Figure 16. Figure 16. Partial Moment Frame Story 4. Insert the colored members, as indicated in Figure P age

15 Figure 17. Complete Moment Frame Story 5. Repeat steps 3-5 twice, attaching the frames to the previously created story below instead of the base. 6. Next, create the floor masses. Each mass consist of properly sized pieces of paper stapled together and tied securely to the frame as shown in Figure Lastly, fasten the building to a base as shown in Figure 13. Brace Frame/Moment Frame Mixed Model Construction 1. Construct the base as shown in Figure 18. Notice the matching orientation of the connection pieces. Figure 18. Base 15 P age

16 2. Construct 3 of the frames shown in Figure 19. Figure 19. Brace Frame 3. Construct 3 of the frames shown in Figure 20. Figure 20. Moment Frame 16 P age

17 4. Place one of each of the frames on opposite sides of the base as shown in Figure 21. Figure 21. Partial Story 5. Insert the colored members, as indicated in Figure 22. Figure 22. Complete Story 6. Repeat steps 3-5 twice, attaching the frames to the previously created story below instead of the base. 7. Next, create the floor masses. Each mass consist of properly sized pieces of paper stapled together and tied securely to the frame as shown in Figure Lastly, fasten the building to a base as shown in Figure P age

18 Model Testing Building Frequency-Teacher s Manual Load-Displacement Test This procedure allows students to experimentally find the stiffness of a one-story moment frame by means of forced displacement. Materials Moment Frame Model w/ base 5 weights (each weighing around 0.5 lbs.) Measuring tape or ruler Procedure 1. Construct a one-story K NEX moment frame as described in steps 1-4 in the moment frame construction procedure. Secure the frame to a base, as shown in Figure Measure and record the masses of four to five weights individually. 3. Convert the recorded masses to weights by the following equation. weight = m g Where: g=gravitational acceleration=32.2 ft/s 2 4. Place the moment frame along with its base securely in the horizontal position. (To do this, one student can simply hold the base against the wall. See Figure 23.) 5. Place the first weight on the end of the moment frame and record the displacement. (You will want some sort of consistent point from which to measure this displacement) 6. With the first weight still on the frame, add the next weight to the free end of the moment frame and record the displacement. Continue this process until all the weights are suspended from the end of the moment frame. 7. Next, plot the displacement vs. the total weight suspended from the moment frame. Estimate a straight line to fit the data (If using excel, use a linear trendline. Make sure to select show trend-line equation. ). The slope of this line is the stiffness of the 1 story moment frame. 18 P age

19 Weights Figure 23. Moment Frame Displacement Test Natural Frequency with a Shake Table In this experiment, natural frequency will be determined by calculation, observation and means of an accelerometer. These results will then be compared. Materials Moment Frame Model Shake Table or Board on Rollers with Metronome Accelerometer that records maximum acceleration Natural Frequency by Calculation 1. Depending on the age of your students, have them calculate the building stiffness and natural frequency by either Eigen Analysis or a single degree of freedom approach as in the Background. If you have not performed the Load Displacement Test to find the stiffness of a one story moment frame use: k MF =0.767 lbs/in [one story only] Where: n=number of stories k tot = 1 1 n k MF 19 P age

20 Natural Frequency by Observation Procedure 1. Secure the accelerometer to the top of the structure and fasten the model to the shake table, as shown in Figure 24 Figure 24. Model Secured to Shake Table 2. Begin the shake table frequency at 1.2 Hz and an amplitude of 0.2 inches. 3. Continue to the step up the frequency by 0.1 Hz until you reach the natural frequency, which will occur when the displacements are largest. As you raise the table frequency beyond the natural frequency, displacements will decrease. Natural Frequency by Accelerometer Procedure Note: If you do not have an accelerometer but do have a smart phone, look for an acceleration measurement app. You can use this in place of the accelerometer. 1. Begin the shake table at a frequency of 1.2 Hz and amplitude of 0.2 inches. Record the maximum acceleration the building experiences. 2. Step up the frequency by 0.1 Hz (if you are using the board/metronome shaker, you will have to use a larger frequency step than 0.1 Hz) and record the maximum acceleration. 3. Continue to step up the frequency, recording the maximum acceleration for each frequency. The maximum acceleration should continue to increase until a certain frequency, after which, it will start to decrease. Go several frequency steps past this maximum point. 20 P age

21 4. Plot a graph of maximum acceleration vs. frequency. The highest point on this graph is the natural frequency. Natural Frequency by Pull-Back Test Materials Moment Frame Model Accelerometer that measures acceleration vs. time Procedure 1. Depending on the age of your students, have them calculate the building stiffness and natural frequency by either Eigen Analysis or a single degree of freedom approach as described in the Background. Don t forget to take into account the mass of the accelerometer at the roof. If you have not performed the Load Displacement Test to find the stiffness of a one story moment frame use: k MF =0.767 lbs/in [one story only] Where: n=number of stories k tot = 1 1 n k MF 2. Secure the accelerometer to the top of the moment frame structure. 3. Pull the top of the structure back, begin recording acceleration, and then release the structure. 4. You should have an acceleration vs. time graph similar to that shown in Figure 25. One cycle Figure 25. Example Acceleration vs. Time Graph 21 P age

22 5. Using the graph, measure the time between two adjacent peaks or valleys. This is the time it took the building to complete one oscillation, and is called the period of the building. 6. Take 1 divided by the period. This is the natural frequency of the building. Building Behavior Test Materials Brace Frame Model Moment Frame Model Mixed Model Shake Table or Board on Rollers with Metronome Procedure 1. Place the moment frame model on the shaker, and shake at a set frequency of 1.5 Hz and amplitude of 0.4 inches. Have the students record observations of how the building behaves. 2. Shake the brace frame model at the same frequency and amplitude as before, having the students record their observations. 3. Shake the mixed model as before, again having the students record their observations. Note: This building model has extreme torsional effects, which is an adverse building characteristic. 4. Take the brace frame model and remove the braces just at the first floor. 5. Shake the model as before and record the behavior. Note: This is called a soft story and can occur, most often, when large storefronts occur in buildings. It creates an undesirable building behavior. 6. Let the students modify the existing model or create their own. Have them predict how the building will behave and then shake it as before. 22 P age