Simple Analysis for Complex Structures Trusses and SDO

J. Civil Eng. Architect. Res Vol. 1, No. 3, 2014, pp. 190-202 Received: June 30, 2014; Published: September 25, 2014 Journal of Civil Engineering and Architecture Research Simple Analysis for Complex Structures Trusses and SDO Ren Song Wuyi University, Guangdong 529020, China Corresponding author: Ren Song (song_ren@163.com) Abstract: In order to further improve the complex statically determinate truss analysis, some new methods being different from the ones emerged in the teaching material of structural mechanics in recent decades are introduced in this paper, including the use of up system symmetry and the SDO method. All the methods are explained through examples selected from various teaching materials in both English and Chinese mainly. The introduction of the SDO method includes how to ascertain the SDO forms for a component or a table part of the trusses and its application in practice, especially in IL analysis, showing the prominent advantages, would be described in detail. Key words: Single-force equation, zero bar, SDO, substitution method, F-E method, dual body, lacked object. 1. Introduction The main difficulty in complex statically determinate truss analysis is have to deal with simultaneous equations as there is no single-force equation (means there is only one unknown in the equation [1]) available. To avoid simultaneous equations, some methods emerged in recent decades are the Henneberg s Method [2]. or called the method of substitute members [3-6], the access method [3, 5, 6] and the method of virtual displacements [2], as well as the constraint substitution method [7] and the application of central symmetry [8] and so on. The examples below would show how to make a breakthrough in the analysis including finding the zero bar (s) in some specific situation and applying the symmetry of up stable truss system supported unequally on two supports in two situations: with or no horizontal bar (s) connecting the two supports being in the same elevation and the SDO method. To make the figures clearer, the following notations are in this paper: Arrows with a diagonal indicate the forces being unknown before appearing in the figure to distinguish them from the knowns. The axial forces of bars in trusses are plotted: in their axes in case it is the zero bare or their axes are in vertical or horizontal, else marked the 2 components and when the 2 components are the same, only one is marked. Some numerical values of the examples selected from the references may have been made a little modification to simplify the calculation and for the exact results to be expressed in rational number. In case the load is symmetric or antisymmetric acting on a symmetric truss, the axial forces of the bares are marked on only half of the structure sometimes. For joints of type K [1], only one of the axial forces in the 2 related bars is marked sometimes, such as joint H in Fig. 1a and joint F and B in b and c in Fig. 1. 2. Examples In order to save space, in case of the symmetry is applying, the decomposed symmetric and

Simple Analysis for Complex Structures Trusses and SDO 191 antisymmetric loads are plotted directly omitting the original one. First of all, an example below will show you how the zero bare finding makes a breakthrough in analyzing a complex truss, although it has nothing to do with the symmetry. Example 1 Make a complete analysis for the complex truss supported and loaded as shown in a, b, c in Fig. 1 [2, 5, 6]. For Fig. 1a: First of all, there are 2 zero bars jointed at L and another one N BC = 0 are found at a glance and N AF = 0 would be obtained by analyzing the up part of section k-k. After the reactions are calculated, analyze the joints in order of A-J-K-E, the internal forces in all the bars would be calculated easily and plotted in Fig. 1a. For Fig. 1b and Fig. 1c: Follow the analysis order in that in Fig. 1a, all the results would be got as plotted in b and c in Fig. 1. Example 2 Make a complete analysis for the complex truss supported and loaded as shown in a and b in Fig. 2 [6]. For Fig. 2a: For the loads are balanced themself, then all the reactions would be zero; the zero bars are found and marked considering the symmetry; the analysis would be completed by taking joint method easily starting form joint F or joint G. For Fig. 2b: The zero bars are also found considering the symmetry, then the axial forces in any other bars would be calculated easily following the reaction analysis. Fig. 1 Making a breakthrough in complex trusses analysis by finding zero bare as earlier as possible. Fig. 2 The first example for making a breakthrough in complex trusses analysis by finding zero bare in utilizing the symmetry of up truss system.

192 Simple Analysis for Complex Structures Trusses and SDO Example 3 The structure is loaded by symmetric and antisymmetric loads respectively as a and b in Fig. 3 [2, 5, 6]. Make a completely analysis. For Fig. 3a: It is clear at a glance that the reactions are all zero, then all the axial forces of members are zero except CD is -6 as marked in the Fig. 3a. For Fig. 3b: For the load is antisymmetric, after determine CD is a zero bar considering the symmetry, the joint method should be taken to analyze the rest easily starting from joint C or D, the results has been also plotted in Fig. 3b. Example 4 Determine the internal forces in all the bars of the trusses shown in a and b in Fig. 4 [2]. It is easy to take the joint method to analyze the axial forces of the bares in the trusses after determining all the zero bars according to the symmetry, the result has been plotted in a and b of Fig. 4. Example 5 A symmetric truss loaded and supported as shown in a and b in Fig. 5, make a complete analysis [2]. For Fig. 5a: The joint method should be taken to analyze it in order of A-F-G-C-I-H-B considering the symmetry. For Fig. 5b: The zero bares should be determined first. Then the joint method should be taken for the rest to complete the analysis with no difficulty. Example 6 Make a complete analysis of the structures drawn in a and b in Fig. 6 [2, 5, 6]. Determined all the zero bar in a and b in Fig. 6 first, then take the stable part to be analyzed as shown in c and d in the Fig. 6.. The results have been plotted in a and b in Fig. 6. Fig. 3 The 2nd example for making a breakthrough in complex trusses analysis by finding zero bare by using up bar system symmetry. Fig. 4 The 3rd example for making a breakthrough in complex trusses analysis by finding zero bare in a symmetric up system. Fig. 5 The 4th example for making a breakthrough in complex trusses analysis by finding zero bare in a symmetric up system.

Simple Analysis for Complex Structures Trusses and SDO 193 Fig. 6 The 5th example for making a breakthrough in complex trusses analysis by finding zero bare in a symmetric up system. Example 7 Make a complete analysis for a and b in Fig. 7 [3, 4, 6] For Fig. 7a: As the load is symmetric, take the access method setting N CF = 8x; the axial force expressions for all the members would be analyzed by joint method in the order of C-F-E. Consider the horizontal balance of joint D: 4 + 4x = 8x, then x 1 and all the axial forces of the members would be found and have been plotted in Fig. 7a. For Fig. 7b: Broken down into b1 and b2. For Fig. 7b1: As the force system is an antisymmetric, all the zero bares are easily to be found and the rest of the analysis would be a cup of tea for you I believe as plotted in it. For Fig. 7b2: As the force system is symmetric, taking the access method setting N CF = 8x as in Fig. 7b2. The following analysis could follow the one in Fig. 7a. The internal forces in Fig. 7b would be got by summing up those in b1 and b2. Example 8 Make a complete analysis for the structures shown in a and b in Fig. 8 [2]. The structure here is a little complex than that in Fig. 7, but it is the same that a symmetrical up system supported simply with no horizontal bar (s) between the 2 supports. So the same analysis order could be taken as that in example 7. Supplemental instruction In order to make use of the symmetry of the structure, add a horizontal force on the support B to form a symmetric and antisymmetric force system as b1and b2 in Fig. 8, it also makes the load system to be the same of that in Fig. 8b. The analysis would be much similar to that in example 7. Example 9 Make a complete analysis for the truss shown in Fig. 9a [2-6].

194 Simple Analysis for Complex Structures Trusses and SDO The stable part D-B is separated from the structure as b in Fig. 9. As the 2 lines of the actions on B and F are clearly, the third one on D must also point to O, the intersection of the 2 lines mentioned. Then the direction of action on D of another stable part is shown as c in the Fig. 9. As it is an SDO (the abbreviation of statically determinate object), the analysis must be a piece of cake for you I believe. Brief summary From the above examples, you may find that complex trusses can be analyzed by simple methods sometimes, which can be summarized as follows: (1) By finding the zero bar(s), a breakthrough for analyzing would be made, such as example 1. If failed in (1), in case the up system is symmetry and supported simply, (2) Being with horizontal bar(s) between the two supports, the symmetry should also be made full use to find zero bar(s), such as example 2-6. (3) Being with no horizontal bar(s) between the two supports, the symmetry should also be used to find zero bar(s) after a certain force is added on a appropriate location such as the ones in example 7-8. (4) If failed in (1)-(3), the SDO method should be adopted to go another way to analyze the structure(s), such as example 9. If failed in (1)-(4), the traditional methods guided in the above section of introduction should be adopted for analyzing: In case the structure is symmetric, the author recommends the access method, otherwise suggest the method of substitute members or constraint substitution method and the application of central symmetry etc. Fig. 7 The first example for making a symmetry situation by adding force on a support.

Simple Analysis for Complex Structures Trusses and SDO 195 Fig. 8 The 2nd example for making a symmetry situation by adding force on a support. Fig. 9 The first example for analyzing a complex trusses by SDO method. 3. The Methods for Ascertaining the SDO in Complex Statically Determinate Trusses The above examples show many methods to simplify the analysis for complex trusses and the method of SDO applied in example 9 is a new one having certain application scope. In order to make the method to be practical, some method to ascertain the SDO would be introduced such as the Substitution Method and the F-E method etc. 3.1 The Substitution Method The substitution method is to take an equivalent constraint system to replace the direct constraint

196 Simple Analysis for Complex Structures Trusses and SDO force system making an SDO for a member or a stable part in the structures such as the method applied in example 9. See the example below. Example 10 Ascertain the SDO for CD and make a complete analysis for the structure shown in Fig. 10a [2, 5, 6]. Ascertain As the truss has many specific joints (type K and K [1]), an access could be found of B-G-E in Fig. 10a as plotted there and it meanes that N CB = -N DE. The direct constraint force system of component CD would be that in Fig. 10b. After reasonable equivalent conversion, replace force system with a constraint system as in Fig. 10c, this is what we were ascertaining, the SDO for component CD. Now put the load on the SDO as Fig. 10c, all the axial forces in the bars would be calculated and plotted in it easily. Restore to original one, the result is plotted as in Fig. 10d. Verification Put another load 12 on the SDO and analyzed as Fig. 10e. Restore to original one, the result is plotted as in Fig. 10f verifying the SDO. Brief summary As the SDO ascertainment is not simple sometimes and the situation of setback is quite common, some other methods should be sought, see the following. 3.2 Simple Introduction of the F-E Method The principle of this method is to make use of 1 or 2 analyzed results of the structure, called the fore example(s) to find the corresponding constraint(s) of the selected component or a stable part of the structure to form the SDO. Of course, the fore example(s) could be analyzed by any methods including the ones adopted in the above examples and guided in the introduction section above. So the method would be called the Fore Example method, simply the F-E method. See examples below. Example 11 Ascertain the SDO for GF in the structure as in Fig. 11a [5]. Ascertain Obviously, component GF has a roller constraint through a dual body of GCF, there are other 2 have to be ascertained still. Of course, the 2 constrains must be equivalent to the 3 other constrains of the structure on D, A and B. Because they have only the directions without specific quantities when there is no load acting, they could not be found immediately. So the F-E method should be had a try. Now put a vertical and a horizontal force P and Q on GF and analyzed respectively as a and b in Fig. 11. The 2 fore examples would be obtained when the loaded structures have been analyzed as potted in a and b in Fig. 11. Now, replace system of the direct restraint forces (the internal forces of the 4 bars exposed by cutting with section k-k as a and b in Fig. 11), by equivalent constraint systems as c and d in it, the lacked object (an object could bear loads in a certain direction only) and the SDO would be obtained as c and d respectively in the Fig. 11. Verification Put a load (marked in italics, and all the forces coursed by it would be marked in italics also) on the SDO as d in the Fig., the reaction would be calculated easily and marked in it. Now make them return to the original structure as in Fig. 11b. All the forces in italics would form a balance system as shown in it. Brief summary (1) Failed with the substitution method, the F-E method may be a road to success in the ascertaining. (2) The SDO is a powerful weapon in analyzing complex structures for any kinds of various loads acting on the object directly. Once the SDO has been ascertained, the analysis would be a peace of cake for every body. As it is more useful in analyzing the composite structures and frames, the introduction in detail would be offered in the companion volume of this paper some day.

Simple Analysis for Complex Structures Trusses and SDO 197 Fig. 10 The example for ascertaining the SDO for a member in a complex truss by substitution method. Fig. 11 The example for ascertaining the SDO for a member in a complex truss by F-E Method. 4. The Application of SDO to IL Analysis Mentioned above, the SDO is a powerful weapon in analyzing various loads in complex trusses; in the other words, its advantage can be revealed in analyzing difference loads acting on the SDO directly, that is the member or stable part selected for the ascertaining. Obviously, its application in IL, the abbreviation of influence line, is the typical representative, see the following examples. Example 12 Draw the IL for the reactions of all the 3 supports of the structure in Fig. 12 a (the mobile unit load moving along the string [2-6]).

198 Simple Analysis for Complex Structures Trusses and SDO Preparation The kinematic method shows that all the IL of the reactions would be composed of 2 straight line segments intersected at section C and the values on A and B are either 1 or 0 when the unit load is acting up A or B. Visible, only work out the value on C (means the mobile unit load is acting on D), the IL would be completed by connecting with the sizes of the reactions. Now, the key step is to calculate the reaction on C when the unite load is acting on D. As the analysis for the original structure is quite ado, taking the substitution method to ascertain the SDO for DB and AD according to b and c in Fig. 12 would make the analysis much simpler; no mater where the mobile unit load moved to, the reaction on any support would be calculated easily with the SDO (see the analysis in example 9 above). First of all, calculate the detail size around d as Fig. 12d according to b and c in Fig. 12, then put the mobile unit load 100 (in 1 over 100) on D as Fig 12c, the reaction would be 120 acting on d in Fig. 12c. Now, make it break up into 2 directions components of concurrent on d as Fig. 11c (for the 2 horizontal component must be contour reverse and have no influence on the analysis, the relevant digit has been omitted in the Fig. and the same bellow). Then we Y d Y 2Y 140( ) have C d3 ( 3 indicates the vertical component of the force from point d to point 3, the same bellow) and Y Y Y Y 20( ) A B dd d3 and put all the 3 reactions on Fig. 11a. The IL would be drawn as Fig. 11e. Example 13 Draw the influence lines for the reactions of all the supports of the structure in Fig. 13a (the mobile unit load moving along the string [6]). According to the analysis in example 12, the SDO for S-W in Fig. 13a would be a simply supported beam with isometric cantilever on both ends as that in Fig. 13b. Then, the detail round b and a would be symmetric as plotted in c and d in Fig. 13 and Fig 13d would be the same as Fig. 12d. Now put the unit load 816 (in 1 over 816) on the SDO as in Fig. 13b, the 2 reactions on b and a would be calculated and marked in it. All the 4 reactions would be calculated easily as: Y A 574 410 164 4 41( ) ; Y E 2 574 4 287( ) Y 2 98 4 49( ) ; J ; 98 70 4 7( ) Y N The IL for all the reactions would be drawn as Fig. 13e. Example 14 Draw the influence lines for the reactions on all the supports of the structure in Fig. 14a (the mobile unit load moving along the string [6]). Explain In order to provide enough data for drawing the IL, put the unite load on G and H respectively as in Fig. 14a. All the related data for IL would be calculated before the reaction analysis. According to the analysis in the above example, the SDO for G-H is as Fig. 14e, a simply supported beam with cantilever on both ends with different length. All the digit would be calculated according to Fig. 14b. For all the reactions being vertical, their values is nothing to do with the vertical coordinates of the supports, they won t appear in the following equations. The horizontal coordinate would be calculated first: that of a is show on Fig. 14c being the same as that in Fig 12d and Fig. 13c, but that of b has to be calculated as in Fig. 14d. Then we have (the following sizes are all in horizontal): 350 41 350 41 58 235 360 2520 Kd 235 58 58 29 203 => 72 1044 Kb 5. 1429 => 2520 1044 1476 bd. 14 203 203 203 203 Then: 1044 36540 1044 1015 34481 ba 180 5 ; 203 203 203

Simple Analysis for Complex Structures Trusses and SDO 199 7105 a H 30 5 35 ; and 203 1044 6090 1044 7134 Gb 30 ; 203 203 203 Ga 210 5 205; 34481 34481 203 35 bh ba ah 35 203 203 41586. 203 Now, put loads P = 1448202 on G and H (in 1 over 1448202) respectively as e and f in Fig. 14; then calculate all the reactions respectively and marked on Fig. 14a (The upper line is for the load on G in Traditional while the lower is for the load on H in italics). The IL for all the reactions would be drawn as that in Fig. 14g. Brief summary The examples in this section show that the key step in the IL analysis for such type of bridge is to ascertain the SDO for all the spans of the structure. As they are all simply supported beams with cantilever on one or both ends, then we can say that the key step is to determine the horizontal coordinates of the 2 supports of the beams. If the number of the spans is changing, coordinates of the 2 supports changes slightly according to Fig. 14b. Fig. 12 An example for IL analysis by SDO method for a 2-span complex truss bridge.

200 Simple Analysis for Complex Structures Trusses and SDO Fig. 13 An example for IL analysis by SDO method for a 3-span complex truss bridge. 5. Epilogue Some improvements in complex trusses analysis are put forward in this paper, especially calling for discovery the only force equation as early as possible, such as example 1, or zero bar in case the up system is symmetric, such as example 2-8, to make a breakthrough in the analysis. In addition, the concept of SDO is proposed and successfully applied in IL analysis for some type of bridge structures, showing the effect of simplifying the calculation greatly. However, as Jack C. McCormac said: Generally speaking there is little need for complex trusses because it is possible to select simple or compound trusses that will serve that desired purpose equally well [3]. This article looks not necessarily have great practical significance, is likely to be a kind of intelligence training just. However, the current text books of structural mechanics did not forget the content in recent decades still, why? I feel hesitate to conclude, if this article may provide some references only? Yet, I d like to say still that don t despise the improvement in theory for not finding its practical application today, it is hard to say whether it will shine one day, for example that the type of bridge in Fig. 12-14 have its advantage of the special of both the static determinate and indeterminate: adapting to the unknown pedestal subsidence and stressed evenly; but the analysis of short board (complex) is being

Simple Analysis for Complex Structures Trusses and SDO 201 Fig. 14 An example for IL analysis by SDO method for a 4-span complex truss bridge. eased by the SDO method now. I would like to make a bold prediction: this form of bridge would be accepted by the market again with the spreading of SDO method. What is more, the companion volume

202 Simple Analysis for Complex Structures Trusses and SDO of this paper (Simple analysis for complex structure -composite structure and frames) will be seen soon I believe. There must be much of questionable issues in this paper still, different opinions are welcome. References [1] R. Song, Tricks and Procedure in Mechanics Analysis, Beijing: Publishing House of Building Industry, 2006. (in Chinese) [2] S. Timoshenko, D.H. Young, Theory of Structures, the McGraw-Hill. Book Company, Inc., 1945 [3] J.C. McCormac, Structural Mechanics, 1st ed., Harper & Row, Yublishers, New York, 1984. [4] A. Darkov, Structural Mechanics, 3rd ed., Mir Publishers, Moscow, 1979. [5] Y.Q. Long, S.H. Bao, et al., Structural Mechanics, 2nd ed., Higher Education Press, Beijing, 1996. (in Chinese) [6] Y.Q.Yang, C.R. Tang, et al., Structural Mechanics, 3rd ed., Higher Education Press, Beijing, 1984. (in Chinese) [7] R. Song, Constraint substitution method, Science Paper Online 8 (2009) 24. (in Chinese) [8] R. Song, The application of central symmetry, Science Paper Online 9 (2009) 4. (in Chinese)