EARTHQUAKE-RESISTANT BRIDGE COMPETITION STUDENT GUIDELINES A PROJECT DEVELOPED FOR THE UNIVERSITY CONSORTIUM ON INSTRUCTIONAL SHAKE TABLES Developed by: Kurt McMullin Assistant Professor Department of Civil and Environmental Engineering College of Engineering San Jose State University This project is supported in part by the National Science Foundation Grant Nos. DUE-9950340 and CMS-9733272. Additional support is provided by the College of Engineering at San Jose State University Goal and Objective The goal of this project is to use basic engineering concepts to design and build a bridge that can resist the forces induced during an earthquake. The objective of this project is to build a small bridge from balsa wood that can withstand severe ground shaking recorded during actual earthquakes. The winning bridge will be selected as the entry that supports the highest mass while being shaken by the most severe earthquake motion. 7/10/01 Page 1 of 11
Design Constraints The bridge must meet the following constraints. Violation of any constraint without the approval of the course instructor will result in disqualification of the bridge. 1. The only materials to be used for construction are square balsa wood and glue. 2. The maximum length of material that can be used for the bridge is limited to 4000 mm of balsa. The maximum cross-section of the balsa wood is 4 mm by 4 mm (1/8 inch square balsa). 3. Bridges must meet the limitations on dimensions as shown in Figures 1 through 3. In addition, an aluminum tube 40 mm wide and 60 mm tall must be able to be passed completely through the interior of the bridge. 4. An area designated for the restraining blocks in Figures 1 and 2 must be free of obstruction to allow for connection of the bridge to the abutment. Likewise, a similar region in the middle of the bridge must exist to allow for the loading blocks. 5. All parallel members must be separated by at least 6 mm face-to-face. 6. The shortest member of balsa wood in the bridge is to be 10 mm. 7. When gluing one segment of balsa wood to another, glue should be confined to within 10 mm of the center of the connection. 8. All wood members must be securely connected. Sliding, slipping, or hinging of portions of the bridge are not allowed. 9. The maximum weight of the bridge is limited to 25 grams. Testing Bridge entries will be tested for three criteria; ability to support vertical weight within a maximum vertical deflection, ability to support transverse weight within a maximum transverse deflection, and ability to resist transverse acceleration developed using an earthquake simulator (Shaker Table). Gravity Test The specimen will be mounted on abutments that are 400 mm apart (center-to-center) as shown in Figure 1. A vertical weight of 2000 grams will be applied to the loading blocks. The deflection of the bridge due to the weight will be measured in the center of both chord members. If the average deflection is more than 2 mm, the entry will be disqualified from the competition. Figure 4 shows a gravity test for a similar bridge competition. 7/10/01 Page 2 of 11
Transverse Test The specimen will be mounted on abutments that are 400 mm apart similar to those shown for the gravity load test. A weight of 1000 grams will be applied to the mounts. The deflection of the bridge due to the weight will be measured in the center of one lower chord member. If the chord member deflects more than 2 mm, the entry will be disqualified from the competition. Earthquake Test The earthquake test will be conducted using the Quanser Shaker Table. The bridge will be installed on the shake table so that acceleration is applied transversely to the axis of the bridge. The protocol has been designed to test entries at lower force levels and slowly progress to higher levels. Testing Protocol 1. A mass of 2000 gms will be connected to the interior of the bridge. The bridge will then be installed on the shake table abutments. The bridge will be secured to the abutment by the restraining blocks. 2. A ground motion recorded during the 1940 El Centro, CA earthquake will be applied to the base of the abutments using the shaker table. 3. A ground motion recorded at the Kobe station of the 1995 Kobe, Japan earthquake will be applied to the abutments using the shaker table. 4. The mass will be increased to 3000 gms and steps 2 and 3 will be repeated. 5. The mass will be increased to 4000 gms and steps 2 and 3 will be repeated. Each bridge will be tested according to the protocol until the bridge is unable to support the loaded mass. The step of the protocol where the bridge collapses will be recorded. The winning bridge will be the one that survives the most steps of the protocol. In the event of a tie, the bridge with the lightest weight will be chosen as the winner. 7/10/01 Page 3 of 11
Figure 1: Plan View of Bridge Figure 2: Elevation View of Bridge 7/10/01 Page 4 of 11
Figure 3: Transverse View of Bridge 7/10/01 Page 5 of 11
Figure 4: Photo of Gravity Test Figure 5: Bridge Installed on Shake Table 7/10/01 Page 6 of 11
Definition of Trusses Design Guidelines for Truss Bridges Trusses are a common way to carry the weight of a structure over a certain span. Truss bridges have been used for centuries to allow people to travel across rivers and streams. Trusses can also be used in other applications, for example in roofs of basketball arenas and in the construction of aircraft and satellites. Trusses may be made of any material, but wood, steel and aluminum are the most common. The efficiency of a truss is usually evaluated as the load supported divided by the weight of the truss. Trusses are defined as assemblies that resist all forces by having the members in tension or compression. The connections between the members are assumed to be hinges and although often not a true assumption, it is accurate enough for engineering purposes. A truss will meet this requirement when it is made from a series of triangles. Truss members are referred to as chords or web members. The chord members run parallel to the ground in a bridge, and the web members connect the two chords to each other. A truss must have both chords and web members or else it will not work the way that we intend. For some trusses (an example being the trusses that support the roof of a house) the top chord member is sloped at an angle. This usually is more efficient, but requires more engineering analysis to accurately design. Trusses are very common for bridges, but have limitations on the length of the span. It is seldom that we find trusses that span over 100 meters. Bridges are designed with trusses to span over rivers and highways. The truss is supported by concrete abutments at the two ends and may have interior supports in the form of piers. Online examples of truss bridges can be found in many applications: Broadway Bridge in Portland, Oregon http://www.bizave.com/cgibin/photoalbum.cgi?photoalbum=pdxbridges&slidenum=1 Quebec Bridge (check out the two that collapsed!) http://www.civeng.carleton.ca/exhibits/quebec_bridge/intro.html Encyclopaedia Britannica http://www.britannica.com/bcom/eb/article/4/0,5716,127644+7,00.html A Fink Truss Bridge (check out the Northeast Elevation view) http://www.tuscazoar.org/zsb.htm An early design of a steel truss bridge http://www.ce.ufl.edu/~historic/esb/esbframe.htm Bridges actually require trusses in two directions. The first is in the vertical longitudinal plane. This truss resists the gravity load that is applied by the weight of the bridge and the traffic that crosses it. Usually two trusses are used, so there is a truss on each side of 7/10/01 Page 7 of 11
the traffic lanes. When this occurs, each truss is expected to carry half of the traffic weight. These gravity trusses are the ones most commonly seen in a bridge. In addition, a horizontal truss is located in the bridge to carry transverse loads, such as the force developed during an earthquake. These trusses are seen in the plan view of the bridge and are often covered by the deck that supports the cars and trucks. Analysis of the Member Forces The engineering of a truss usually begins with the engineer sketching a rough layout of the members. When doing this, it is important to keep aware of the required span, the location of applied loads, and any geometric limitations on the layout. Although not necessary, it is highly recommended that the members be straight. The location of the connections for the members should be identified in this preliminary drawing. It is also recommended that a connection be placed near the location of applied loads and also near the abutments. Add a coordinate system, where the x-axis passes through the two supports. The next step is to draw the force that the truss is expected to resist. Usually this force will be in the y-direction, and is often in the negative direction. This force will be resisted by the two supports. If the truss is symmetric about the centerline and the load is applied at the center of the truss, then each support will resist half of the applied load. The force in each truss member will be parallel to the direction of the member. In addition, the relative size of the force in each member will have a magnitude. A vector can easily be used to represent the magnitude and direction of the force. The analysis of the truss can then be determined using the fact that the sum of all vectors at a connection must equal zero. For those unfamiliar with vector addition, another means is by requiring that the sum of the forces along the x-axis and the forces along the y-axis must both add to zero. One way to simplify the math of this analysis is to have most of the members parallel to either the x or y-axis. At each connection in the truss, two equations can be written (the sum of the forces along the x-axis and the sum of the forces along the y-axis). The unknowns are the magnitudes of the forces in the members. Starting at either the point of applied load or at the support, equations of equilibrium can be written and solved. Depending upon the complexity of the truss, these equations may be easy or difficult to solve (in actual engineering problems we usually use computer programs to solve complicated trusses). If the truss is symmetric and the loads are symmetric, then the analysis can be done for only one half of the truss and the member forces will be symmetrically distributed. (For this and many other reasons it is highly recommended that you design your truss to be symmetric. Antisymmetry may have architectural benefits, but few engineering ones!) If you find a connection where you cannot balance the forces in both the x-direction and the y-direction, then your truss is unstable (and will almost certainly collapse with little 7/10/01 Page 8 of 11
load). To relieve this problem, add more members to the connections that are problematic. It is possible that your truss cannot be solved using the process stated above. If so, this means that the truss is indeterminant. Indeterminancy has some benefits for civil engineers, but causes more work than we want to take on in this project. If your truss is indeterminant, then it can safely be built with fewer members. Try removing members from your drawing. (Be careful though, removing the wrong members will create an unstable situation! See the previous paragraph.) Having solved for the force in each member, we can compare it to the expected strength of the balsa wood to predict if the truss member will be strong enough. In addition, we can think about alternatives to our truss design to come up with alternative bridge geometries that will result in lower forces in the main members. The truss analysis can also help us in determining the care that needs to be used in gluing our connections. Connections that resist high forces should be cut and glued with more accuracy. Example Calculation of a Simple Truss We will analyze a very simple truss as shown: y-axis Since the truss has a single load at the center and is symmetric, then each support will receive half the x-axis load or 5 N. If we begin by solving for the force at the middle joint of the bottom chord, we will 3m high receive the following two equations: x-direction => F 1 = F 2 4 m 4m y-direction => F 3 = 10 N Applied Load at Middle = 10 N F 3 Now if we move to the top center joint, F 1 F 2 we will receive another set of equations: 10 N x-axis: 4/5F 4 = 4/5F 5 y-axis: 3/5F 4 + 3/5F 5 + F 3 = 0 So now we know that F 1 must be equal to F 2, and F 4 F 5 that F 4 must be equal to F 5. Since we know that F 3 is 10 N, then F 4 and F 5 must be -8.33 N. We have F 3 = 10 N originally drawn the arrows assuming that the truss members all act in tension (arrow points away from connection). Since the number came out negative, this indicates that the arrow is backwards and that the truss member is actually in compression. 7/10/01 Page 9 of 11
Finally, we can use the joint at the lower left corner, and finish the problem. Now we set up two more equations: x-direction: F 1 = 4/5F 4 F 4 y-direction: 3/5F 4 = F A F A F 1 Since we know that F 4 is 8.33 (note that we drew the arrow as compression), then F1 must be 6.67 N and F A must be 5N (which confirms our analysis because we already knew that the reaction at the abutment should be half the applied load). Two items to note about the example, first, that the calculations are much easier since the truss is laid out in the shape of a 3-4-5 triangle. Also the symmetry makes the calculations very easy. Deflection due to Applied Load The expected deflection that a truss will experience due to an applied load can be calculated quite accurately using basic linear mechanics. These courses are generally taken in the sophomore or junior year of an undergraduate engineering program. Unfortunately they are a bit too involved for simple guidance for students in this competition. Instead, a simpler formula can be used to estimate the deflection, and can likely be within 100% of the actual deflection (not really very accurate for engineering but at least it is a good starting point). The deflection of a truss can be estimated as: = (PL 3 ) / (48EI) Where: = the estimated deflection P = the applied load at the middle of the truss E = the modulus of elasticity of the material, might be assumed to be 1,000,000 psi for balsa wood L = the distance between the supports I = second moment of area of the cross-section of the truss For a truss, the value of I can be estimated as: I = Ad 2 Where: A = d = the cross-sectional area of the chord the distance between the centers of the chords 7/10/01 Page 10 of 11
Estimation of Peak Force Developed during Earthquake Ground Motion The truss analysis discussed above will give us very good understanding about the behavior of the bridge for the static tests (gravity and transverse tests). Unfortunately the dynamic behavior of the bridge during an earthquake may be significantly more complex. Examples of truss bridges that performed poorly in earthquakes are bountiful. One means that engineers use to predict the effect of earthquakes is to consider that the acceleration of the mass supported times the mass can be considered as a force and the truss is analyzed statically, as shown in the previous example. In addition, the most critical time during the earthquake is probably coincident with the time of the peak acceleration of the ground. If we look at the time-history plot of the acceleration (see Figure 6) and identify the peak value, we can use this to help design the bridge. Figure 6: Time-history Plot of El Centro and Kobe Earthquake Ground Motions GROUND ACCELERATION, cm/s/s 1000 500 0-500 -1000 EL CENTRO EARTHQUAKE MAY 18, 1940 - MAGNITUDE 7.1 PEAK GROUND ACCELERATION = 343 cm/s/s 0 5 10 15 20 TIME, seconds KOBE EARTHQUAKE JAN. 17, 1995 - MAGNITUDE 7.2 PEAK GROUND ACCELERATION = 818 cm/s/s If a bridge is very stiff transversely, it will move exactly the same as the abutments, which are rigidly attached to the ground. In this case, the peak acceleration of the mass should be equal to the peak acceleration of the ground. This usually only occurs on very stiff bridges (and your balsa wood probably will not be this rigid). For flexible bridges the maximum acceleration of the mass on the bridge will be some value times the peak acceleration of the ground. The actual value may be greater than or less than 1.0. Engineers use complex analytical models to determine this value, many of which you may be introduced to in senior or graduate level courses. For our introduction to the problem, we should use an estimate of this value. Most likely the value for your bridge will be between 1.2 and 2.0. 7/10/01 Page 11 of 11