Simplified Bukling Analysis of Skeletal Strutures B.A. Izzuddin 1 ABSRAC A simplified approah is proposed for bukling analysis of skeletal strutures, whih employs a rotational spring analogy for the formulation of the geometri stiffness matrix. he benefit of this analogy is that it offers an intuitive framework, whih is based on the ommon notions of linear strutural analysis. Assuming that the strutural defletions prior to bukling are negligible, a linear eigenvalue problem, utilising the geometri and material stiffness matries, an be easily formulated and solved for the ritial bukling loads. his an be further simplified using an assumed mode, where the rotational spring analogy is shown to provide onsiderable omputational benefits and signifiant insight into the bukling of various forms of skeletal struture. In this ontext, the use of different assumed modes an be oneived as a proess of probing the struture to establish the most likely mode for bukling and the orresponding ritial load. It is also shown that the approximation inherent in the assumed mode approah together with the disrete form of the rotational spring analogy an be signifiantly improved through modal ombinations and inreasing the number of elements, respetively, where onvergene to the exat bukling solution is demonstrated. Several illustrative examples are provided in the paper, whih highlight the simpliity of the proposed approah, its appliation using a linear strutural analysis tool, and its ability to shed signifiant light on important issues in bukling analysis of skeletal strutures. Keywords: bukling; simplified analysis; skeletal strutures; frames; trusses. 1 Reader in Computational Strutural Mehanis, Department of Civil and Environmental Engineering, Imperial College, London SW7 2AZ, United Kingdom. 1
1. INRODUCION he bukling of various strutural forms attrated signifiant researh interest over many years 1-5, sine it normally provides a pratial limit on the load arrying apaity of strutures. he lowest bukling load is of most pratial signifiane, and is normally ahieved when the tangent stiffness assoiated with a mode of deformation beomes zero, suh a mode then referred to as the bukling mode. Of ourse, numerous sophistiated proedures and omputational tools have been developed over the past few deades that deal with strutural bukling, both in terms of simplified linear eigenvalue analysis and through traing the geometrially nonlinear response, in the latter ase often dealing with the influene of material nonlinearity as well. While the approah proposed in this paper does not deal with a new lass of problem, it sheds new light on the bukling analysis of skeletal strutures, enabling better understanding of the bukling mehanisms, and it provides a simplified and pratial framework for bukling preditions, importantly, using linear analysis priniples. As mentioned above, bukling an be related to the singularity of the tangent stiffness matrix, whih in turn onsists of two parts 6,7. he first part is the material stiffness matrix whih is related to the deformational stiffness of the omponents, taking into aount the onnetivity of omponents in the urrent geometri onfiguration of the struture. For linear elasti omponents, the material stiffness is idential to the linear elasti stiffness, but updating the strutural geometry to inlude the effet of any displaements. he seond part is the geometri stiffness matrix, whih is related to the omponent fores, and in some ases to the applied loading, taking into aount the effet of a hange in geometry from the urrent onfiguration. For typial strutures, the material stiffness is positive for all deformation modes, mathematially referred to as positive-definite, whereas the geometri stiffness an admit negative values for ertain modes, depending on the omponent fores and applied loading. It is therefore the effet of a negative geometri stiffness that an lead to a singular overall tangent stiffness matrix, and hene bukling. 2
his paper presents a pratial approah to the bukling analysis of skeletal strutures, based on a rotational spring analogy for formulating the geometri stiffness ontribution. he paper proeeds with desribing the rotational spring analogy, highlighting its benefits in relation to simplified bukling analysis. After outlining the appliability of this analogy to detailed bukling preditions using iterative tehniques, its pratial benefits in simplified bukling analysis using an assumed mode are disussed in detail. In the latter ontext, two approahes are proposed, based on individual probing modes and ombined modes, respetively, both of whih maintain the pratial benefits of the rotational spring analogy. Finally, the oneptual benefits of the new approah are highlighted, espeially in relation to the bukling interation between various applied loads. Several examples are provided throughout the paper, whih demonstrate the pratiality of the proposed approah and the oneptual power it furnishes towards understanding the bukling of skeletal strutures. 2. ROAIONAL SPRING ANALOGY he geometri stiffness of skeletal strutures, subjeted to onservative nodal loads that maintain a fixed diretion, an be determined from equivalent rotational springs using linear analysis priniples, hene the term rotational spring analogy. In the present ontext, the main soure of geometri nonlinearity is onsidered to be the hange to the path of axial fores within the struture due to a hange in the deformed geometry. o illustrate this analogy, onsider an axially loaded element whih remains straight, as shown in Figure 1. he only first-order hange in fores required to maintain equilibrium under an infinitesimal rigid body rotation ( ) is represented by a ouple proportional to the axial fore (F) and the element length (L), and hene this geometri stiffness is equivalent to a rotational spring with a stiffness: k FL (1) Clearly, the equivalent rotational spring is stabilising ( k 0) for a tensile axial fore ( F 0) and is onversely destabilising ( k 0) for a ompressive axial fore ( F 0 ). 3
When the element is prone to bending due to bukling, the equivalent rotational springs beome distributed over the length, as shown in Figure 2. his effet, however, is equivalent to a single disrete rotational spring onneted to the element hord, as given by (1), in addition to loal distributed rotational springs onneting the element hord to the deformed element referene line (Figure 2). hese two sets of equivalent spring are responsible for global and loal geometri nonlinearity, or the P- and P- effets 8, respetively, and both an be onsidered in the formulation of the geometri stiffness matrix using linear analysis priniples. However, sine the loal spring deformations diminish relative to the global spring deformations as the number of elements is inreased (Figure 3), the loal distributed springs are ignored in the proposed approah for simplified bukling analysis. he above rotational spring analogy an be used for a variety of strutural forms, inluding trusses, frames, ables, and link-spring idealisations, for both 2D and 3D strutures. he most general appliation of this analogy entails the formulation of the geometri stiffness matrix from the equivalent rotational springs using the priniple of virtual work, where only simple linear kinematis are required, as disussed next for 2D and 3D analysis. However, a signifiant benefit of the rotational spring analogy is that bukling analysis, in its most simplified form, may be undertaken without the formulation of a geometri stiffness matrix, as elaborated in Setion 4. 2.1. 2D Analysis In 2D analysis, the ontribution of an equivalent rotational spring to the geometri stiffness matrix is determined by the linear kinemati relationship between the spring rotation and the four translational freedoms: s U2 U1 V2 V1 U (2.a) L L with, 1 1 2 2 U U V U V (2.b) 1 s s L (2.) 4
os( ); s sin( ) (2.d) where, L = orresponding element length, and = element angle with global X-axis, as shown in Figure 4(a). Using the priniple of virtual work, the ontribution of the equivalent spring to the geometri stiffness matrix is simply obtained as: G k K (3) with k given by (1). If the strutural defletions up to bukling are negligible, then L and orrespond to the initial undeformed onfiguration, though the above expression (3) is equally appliable for the ase involving signifiant pre-bukling defletions, in whih ase F, L and must be obtained for the deformed onfiguration. Importantly, in the ontext of simplified bukling analysis, it is often suffiient to evaluate the quadrati work assoiated with the geometri stiffness, whih is simply represented by: 2 G k k U K U U U (4) Clearly, in suh a ase, the rotational spring analogy offers onsiderable simplifiation, where the ontribution of the equivalent springs to quadrati work is obtained from (4) with the spring rotations orresponding to U determined readily from (2). 2.2. 3D Analysis In 3D analysis, eah axial omponent is assoiated with two orthogonal equivalent rotational springs, as shown in Figure 4(b), where the orresponding rotations are determined as: y z U (5.a) with, 1 1 1 2 2 2 U U V W U V W (5.b) 1 y,1 y,2 y,3 y,1 y,2 y,3 L z,1 z,2 z,3 z,1 z,2 z,3 (5.) 5
where, y, z = diretion osines of the transverse loal element axes. Again, using the priniple of virtual work, the ontribution of the equivalent spring to the geometri stiffness matrix is simply obtained as: G k K (6) and its ontribution to the quadrati work is represented by: 2 2 G k k y z U K U U U (7) where y, z are readily obtained from (5). 3. BUCKLING ANALYSIS Consider a struture subjeted to a ombination of onstant initial loads ( 0 P ) and proportional loads onsisting of nominal loads ( P ) varying aording to a ommon load fator (). Negleting pre-bukling defletions and assuming a linear elasti material response, the axial fores within the members similarly onsist of initial onstant fores ( F ) n and nominal fores ( F ), and the stiffnesses of the orresponding equivalent rotational o springs are obtained from (1) as ( k ) and ( k ), respetively. his leads to ontributions whih an be assembled into two orresponding geometri stiffness matries, ( K n G n n o G o ) and ( K ), whih ombine with the material stiffness matrix ( K E ) to determine the overall tangent stiffness matrix as follows: where and 2.2. o n E G G K K K K (8) o n o n KG, K G an be obtained from, k k, respetively, as disussed in Setions 2.1 Bukling analysis is onerned with the solution of the following linear eigenvalue problem for the ritial load fator ( ) and assoiated bukling mode (U): o n, K U K K K U 0 U 0 (9) E G G If the bukling mode (U) is known, then the ritial load fator an be obtained from premultiplying (9) by U as follows: 6
o E G n U KGU U K U U K U (10) In pratie, however, the bukling mode is unknown, although an approximate mode may be determined from linear analysis using some load pattern (P): E 1 E P K U U K P (11) Of ourse, when the mode is approximate the predition in (10) beomes also approximate, though always providing an upper bound on the lowest ritial load fator for typial strutures that are stable in the unloaded state ( i.e. K positive definite for 0). In view of (4) and (7), the rotational spring analogy simplifies the expression in (10) suh that the assembly of the geometri stiffness matries beomes unneessary: m o 2 o U P k,i i ke kg i1 n m kg k n 2,i i i1 (for 2D analysis) (12.a) m o 2 2 o U P k,i y,i z,i ke kg i1 n m kg k n 2 2,i y,i z,i i1 (for 3D analysis) (12.b) where, m = number of equivalent rotational springs. Clearly, the term k E U P represents the work done by the onsidered load pattern, whih an be easily obtained using a linear analysis tool. he remaining terms simply represent the energy stored or imparted by the equivalent rotational springs, involving the o n o n spring stiffnesses k, k as obtained for F, F using (1), and the orresponding rotations due to the onsidered mode (U) as determined from (2) or (5). he predition of the lowest ritial load fator in (12) may be improved by revising the approximate mode using, for example, the method of subspae iterations 9. With this method, the load pattern is iteratively revised after eah estimate of U and from the geometri stiffness: as the fores resulting 7
o n G G P K K U (13) leading to new values for U and using (11) and (12). One disadvantage of the method of subspae iterations is that it requires in (13) the assembly of the geometri stiffness matries, o K G and K n G, or at least the orresponding geometri fores. An alternative, yet approximate, approah is proposed in Setion 4, whih allows an improvement of the initial bukling predition using the rotational spring analogy without the need for assembly of the geometri stiffness matrix. 3.1. Example: Column with Clamped-Pinned Ends Consider an elasti olumn with one end lamped and the other pinned, as shown in Figure 5(a), where bukling analysis is undertaken using the rotational spring analogy with subspae iterations. With 4 elements, 4 equivalent rotational springs are employed to represent the geometri stiffness, where the spring properties are provided in Figure 5(b). Clearly, and as easily verified from (2), the equivalent spring rotations are dependent only on the transverse defletions of the 3 internal nodes. Similarly, the only fores arising from the geometri stiffness of the equivalent springs in (13) are transverse fores at the 3 internal nodes. herefore, the proposed bukling approah an be applied with a linear analysis tool that models the response of the elasti olumn under 3 arbitrary transverse fores at the internal nodes, though the use of subspae iterations requires also the assembly of the geometri stiffness matrix as disussed previously. In applying subspae iterations, the starting mode is obtained from a transverse load applied to the middle node, where in all iterations the load pattern and the orresponding mode are identially saled to normalise the work done (i.e. U P 1N.m ). he rotations of the equivalent springs () are easily determined from U using (2), and the estimated ritial load fator ( ) is simply obtained from (12.a). able 1 provides the iterative estimates, and Figure 6 depits the starting/final modes, where it is lear that onvergene to the lowest ritial load fator ( 22.36 ) is ahieved after only 4 iterations. his predition exat 20.19 overestimates the exat theoretial solution ( ) by around 11%, whih is due to 8
the neglet of the loal geometri stiffness. Considering Figure 6, the signifiane of the loal geometri stiffness for the predited mode is manifested in the disrepany between the element hords (dotted line) and the olumn referene line (solid line), where it is lear that improved auray is most effetively ahieved with more elements next to the lamped support and near midspan. Following on from the last point, the use of 8 elements with the same starting load pattern leads to the iterative estimates given in able 2, where the onverged value ( 20.73) overestimates the exat solution by only 2.7%. his improved auray is refleted in Figure 7 by a smaller disrepany between the element hords and the olumn referene line in omparison with the previous ase using 4 elements. Clearly, the proposed simplified bukling approah onverges to the exat solution as the number of elements is inreased, though the neglet of the loal geometri stiffness, whih is equivalent to loal distributed rotational springs that are all negative, leads to onvergene from above. 4. SIMPLIFIED APPROACH Considering a speifi mesh of elements, when the starting mode is only an approximation of the atual bukling mode that satisfies (9), subspae iterations may be used to onverge to the lowest bukling solution. However, it is often possible to selet a load pattern (P) leading to a defletion mode (U) suh that the starting bukling predition is already of suffiient auray, thus avoiding the need for subspae iterations. For example, Stevens 10 suggested that a reasonable approximation for multi-storey frames may be ahieved with a load pattern onsisting of horizontal fores that are diretly proportional to the applied vertial loading at various floor levels. In this ontext, the rotational spring analogy offers onsiderable simplifiation, where only a linear analysis model of the skeletal struture is required: o n 1. to determine the axial fores, o n loads, o n springs k, k, and F F orresponding to the initial and nominal P P, hene establishing the stiffnesses of the equivalent rotational 9
2. to determine the mode (U) assoiated with the load pattern (P), from whih the orresponding spring rotations are obtained aording to (2) or (5), leading to the estimation of the bukling load fator ( ) using (12). Clearly, sine the load pattern is assumed, there is no need with the rotational spring analogy for assembly of the geometri stiffness matrix, though the result is an approximate whih typially overestimates the lowest bukling solution of the eigenvalue problem in (9). However, the onsideration of various reasonable load patterns and the seletion of the smallest often enables a good estimate of the lowest bukling load fator. his an be visualised as a probing proess, where the struture is subjeted to various disturbanes (U) with the objetive of determining the most likely bukling mode assoiated with the smallest. For eah probing mode (U), the value of is most easily oneived as that for whih the energy stored in the linear struture, ½U KEU, is idential to the energy imparted by o n the equivalent rotational springs, ½ U K G K G U. 4.1. Improvement of Assumed Mode Approximation Despite the pratial benefits of the rotational spring analogy with the assumed mode approah, there are ases when it is diffiult to establish a suffiient number of probing modes to onverge on the lowest bukling load fator with reasonable auray. Suh ases benefit from a pratial modifiation, where two previously onsidered modes, A U and B U, are ombined to formulate a rank 2 redued eigenvalue problem 9 of the following form: with, o n 0, k u k k k u 0 u (14.a) E G G A B U U U u (14.b) where 1 2 u u u represents the two weighting parameters of the redued problem. In performing problem redution, the redued 2 2 material stiffness matrix is simply given in terms of the work done by the load patterns, A P and B P, over their respetive modes: 10
U A UAPA UAP B ke KE UA UB UB U BPA UBPB (15) On the other hand, the redued 2 2 geometri stiffness matries are easily determined using the rotational spring analogy, and are obtained in terms of the energy stored/imparted by the equivalent rotational springs over the rotations orresponding to A U and B U : o n U A o n kg KG UA U U B B (16.a) k m m o n 2 o n k,i A,i k,i A,i B,i o n i1 i1 G m m o n o n 2 k,i B,i A,i k,i B,i i1 i1 (2D analysis) (16.b) k o n G m 2 m ya,i o n o n ya,i yb,i k,i 2,i i1 k za,i i1 za,i zb,i m m 2 o n yb,i ya,i o n yb,i k,i k,i 2 i1 zb,i za,i i1 zb,i (3D analysis) (16.) Of ourse, the solution of the redued eigenvalue problem in (14.a) is straightforward, where an improved estimate of the lowest bukling load fator ( ) is easily obtained from the ondition det( k ) 0 representing a quadrati equation. Importantly, the above simplified approah allows reasonably aurate preditions of the lowest using the rotational spring analogy with a linear analysis tool, without the need for o n assembly of the overall geometri stiffness matries KG, K G. Obviously, the auray of these preditions depends on the extent to whih modes A U and B U an, in ombination, represent the lowest bukling mode. However, the use of a few representative probing modes for the seletion of A U and B U often produes very good estimates of the lowest. Finally, the most effetive mode B U whih is omplementary to a probing mode A U, leading to the losest approximation of the lowest shown to orrespond to the following load pattern: from the redued problem, an be 11
P B A o A n A G P K K G U (17.a) B 1 B E U K P (17.b) Clearly, the determination of the above mode requires the assembly of the geometri stiffness matries, or at least the fores resulting from the equivalent rotational springs, and hene suffers from the same shortoming of the method of subspae iterations. However, the use of the two modes, A U and B U, within a redued eigenvalue problem leads to muh faster onvergene than with standard subspae iterations as disussed in Setion 3. ypially with the redued problem, onsisting of a probing mode A U and a related mode B U using (17), the value of obtained as the solution of (14.a) provides a very good approximation of the lowest bukling load fator. However, it is possible to refine the predition further with the redued problem, where the mode U obtained from (14.b) may be used as A U, with B U determined again from (17), for a subsequent iteration. 4.2. Example: Frame Subjet to Vertial Loading Consider an elasti plane frame subjet to proportional vertial loading applied diretly to the olumns, as shown in Figure 8(a), where two elements are used to represent eah olumn. With negligible axial fores in the beams, and negleting pre-bukling defletions, the geometri stiffness is represented by 10 equivalent rotational springs in the olumns, as illustrated in Figure 8(b), where the nominal spring stiffnesses are provided in the same figure. Clearly, the rotations of these springs depend only on the horizontal displaements of 10 nodes, and therefore only these omponents, denoted by U, need to be extrated from linear analysis under probing load patterns. For omparison purposes, the exat lowest bukling mode for the frame is determined using a exat 0.9765 suffiient number of subspae iterations, where ( ) and the orresponding mode is shown in Figure 9(a). Now, simplified bukling analysis is first undertaken by onsidering three probing modes, (I, II, III), orresponding to horizontal loads applied at three different nodes, as illustrated in Figures 9(b-). able 3 provides the load patterns, nodal displaements and rotations of the 12
equivalent springs for the three probing modes, normalised suh that ( U P 1N.m ). he bukling load fator ( ) is estimated for eah of the modes diretly from (12.a), where as expeted all preditions overestimate approximation to within 1.1%. exat, though probing mode (I) offers an exellent Next, improved bukling predition is onsidered by ombining any two of the three previous probing modes in a redued problem, where the redued material and geometri stiffness matries are obtained diretly from (15) and (16), as given in able 4. he solution of the redued problem, expressed by (14), leads to improved estimates of the lowest and the orresponding weighting fators of the two ombined modes (u). Considering the results in able 4, it is lear that amongst the three possible ombinations, ombining modes (I / II) leads to the greatest improvement in simplified bukling predition, now to within 0.4%. On the other hand, ombinations involving mode (III) with modes (I) or (II) lead only to a marginal improvement in the single-mode preditions. However, if a related mode obtained aording to (17) from mode (III), onsidered hene as mode ( III A B (III ) is ), a near perfet predition is ahieved to within 0.02%. Of ourse, the only shortoming of the last ombination A (III B / III ) is that it requires the assembly of fores resulting from the equivalent rotational springs due to mode ( III suh requirement. A ), whereas the other ombinations have no Finally, it is noted that the relatively oarse disretisation with 10 olumn elements leads to further inauray in the above preditions due to the neglet of loal geometri nonlinearity, the extent of whih is manifested in the small disrepany between the element hords and referene lines for the lowest bukling mode, as shown in Figure 9(a). his, however, results in only a 3.4% disrepany in the lowest bukling load fators between the adopted oarse oarse 0.9765 fine 0.944 disretisation ( ) and a very fine disretisation ( ), thus highlighting the pratiality of the proposed simplified bukling approah. 5. CONCEPUAL BENEFIS he bukling of onservative strutural systems under stati loading is defined by the singularity of the tangent stiffness matrix ( K ), whih onsists of the material stiffness ( K ) E 13
and the geometri stiffness ( K ), where the latter is diretly proportional to the applied G loading and the equilibrating internal fores. For typial skeletal strutures, bukling ours at a load level that produes a suffiiently negative in a speifi deformation mode, leading to a singular adjaent equilibrium states. he relative influene of K E and K G that overomes the positive K E K and the presene of infinitesimally K G for a speifi mode an be established in terms of the quadrati energy whih they store and impart, respetively, where the sum indiates the nature of the overall strutural response to an external disturbane. Prior to bukling, the negative influene of mode, hene K G is not suffiient to overome the positive resistane of K E for any K is positive definite and strutural equilibrium is thoroughly stable. At bukling, the negative influene of speifi bukling mode, hene ritial. Finally, when the negative influene of for some modes, K G beomes equal to the positive resistane of K E for a K beomes positive semi-definite and equilibrium beomes K G exeeds the positive resistane of K admits negative values and equilibrium beomes thoroughly unstable. For elasti struture with negligible pre-bukling defletions, bukling along the fundamental equilibrium path an be expressed as a linear eigenvalue problem, where onstant and K E K E remains K G is linearly dependent on the loading. Of ourse, bukling preditions based on an assumed mode, as disussed in Setion 4, annot by definition orrespond to a positive definite K. herefore, for the aforementioned strutures, assumed mode preditions lead to K whih admits negative values, and hene the lowest bukling load is overestimated, as previously indiated. he only exeption is when the assumed mode is idential to the bukling mode, in whih ase the exat bukling load is obtained, and the resulting positive semi-definite. K is Now, the use of the rotational spring analogy brings signifiant oneptual benefits, whether it is employed for formulating K G in detailed bukling analysis or for establishing the orresponding quadrati energy in simplified bukling analysis. Sine the stiffness of an equivalent rotational spring is proportional to the axial fore and length of the represented element, the greater the ompressive axial fore, element length and the spring rotation 14
orresponding to a given mode, the greater its ontribution to the negative influene of for that mode, as refleted by the quadrati energy ontribution. herefore, bukling an be oneived as a phenomenon in whih the negative equivalent springs an just about apply enough moment ouples in a bukling mode to overome the material resistane, and possibly that of other positive equivalent springs, suh that equilibrium an be maintained to a seond-order. Perhaps the greatest oneptual benefit of the rotational spring analogy, however, arises in onsidering the bukling interation between different loads or, alternatively, in establishing the relationship between bukling preditions for related loads. o elaborate, onsider a struture for whih the lowest bukling load fator ( ) and orresponding mode have already been established under a speifi loading. he influene of a small variation in the initial/nominal load distribution an be assessed, using the original mode, in terms of the effet of the orresponding load inrement on the quadrati energy. his in turn depends on the orresponding inrement in the geometri stiffness, and an be simply evaluated in terms of the additional equivalent rotational springs due to the hange in element axial fores. Given that, with the neglet of pre-bukling defletions, any predition original mode is an upper bound on the new lowest bukling load fator (i.e. following outomes an be asertained: O K G P assuming the N P ), the 1. If all axial fores due to the inremental initial/nominal loads are ompressive, the orresponding additional equivalent springs will all be negative. herefore, the inrement in quadrati energy will either be zero resulting in ( P O ), if none of the additional equivalent springs are rotated due to the original mode, or more typially negative leading to ( P O ). herefore, the exat and predited values an in N P O general be related for this ase aording to ( ). 2. If all axial fores due to the inremental initial/nominal loads are tensile, the orresponding additional equivalent springs will all be positive. herefore, the inrement in quadrati energy will either be zero resulting in ( P O ), if none of the additional equivalent springs are rotated due to the original mode, or more typially 15
positive leading to ( P O ). It an be shown that the exat and predited values are O N P generally related for this ase aording to ( ). 3. If the inremental axial fores are of different signs, the same will apply to the additional equivalent springs. herefore, the inrement in quadrati energy may be evaluated as negative, zero or positive, depending on the relative ontributions of the equivalent springs over the original mode, leading to ( ), ( ( P O P O P O ) or ), respetively. Clearly, for the two former senarios, the exat and predited N P O values are related aording to ( ), while for the latter senario no suh N P P O ordering in possible, only that (, ). ypially, however, good estimate of P offers a N, espeially for relatively small hanges to the load distribution for whih the original bukling mode remains reasonably aurate. Finally, another oneptual benefit of the rotational spring analogy relates to the signifiane of negleting loal geometri nonlinearity. Along similar lines of argument as above, if loal geometri nonlinearity is ignored in elements whih are all ompressive, this orresponds to the neglet of distributed equivalent rotational springs whih are all negative. his in turn leads to an underestimation of the negative quadrati energy and, onsequently, to an overestimation of the lowest bukling load. Of ourse, if the negleted loal geometri nonlinearity is only for elements whih are tensile, then the lowest bukling load is underestimated. In any ase, it is emphasised again that the influene of loal geometri nonlinearity redues as the number of elements is inreased, due to the redution in the loal rotations relative to the element hord for a given mode, thus guaranteeing onvergene to the exat solution using the proposed simplified approah. 5.1. Example: Spae russ Subjet to Vertial Loading Consider the elasti spae truss, shown in Figure 10(a), whih is symmetri about its midspan. he bukling of the truss is to be investigated under three related load ases, as depited in Figures 10(a-), where the exat lateral-torsional bukling mode and the orresponding load fator are determined for load ase (I) as follows (Figure 11): 16
6.230, 53.70, 6.267, 11.05, 54.64, 10.52, 1.929, 16.27, 1.537, U 0.0, 274.6, 94.98, 0.0, 277.1, 99.30, 0.0, 15.63, 1.706, (18.a) ( m) 6.230, 53.70, 6.267, 11.05, 54.64, 10.52, 1.929, 16.27, 1.537 I k 1.3645 (18.b) k In the above, E n G k E and n k G represent the quadrati energy stored and imparted by the material and nominal geometri stiffness, respetively. Alternatively, they orrespond to the redued material and nominal geometri stiffness, and are obtained with referene to (12.b) as: E k (N.m) U P 1 (19.a) n m n 2 2 G,i y,i z,i i1 k (N.m) k 0.73287 (19.b) Now with load ase (II), onsider the influene of initial loads applied diretly to the four orner truss members, as shown in Figure 10(b). hese lead to initial axial fores ( F ) in the various truss members, the ontribution of whih to the geometri stiffness is determined in terms of initial equivalent rotational springs ( k ). he orresponding quadrati energy assoiated with the original mode in (18.a) is obtained as: o m o 2 2 G,i y,i z,i i1 k (N.m) k 0.010984 (20) and hene the lowest bukling load fator is estimated from (12.b) as: o o o II ke kg n kg 1.3795 (21.a) ompared to the exat value: II 1.3755 (22.b) Interestingly for load ase (II), the appliation of initial vertial loading inreases the bukling load fator (i.e. II I ), whih is due to the positive additional quadrati energy in (21). Although, the initial vertial loading auses ompression in the vertial members, it also auses signifiant tension in the top horizontal hords, thus leading to positive equivalent 17
rotational springs that restrain the lateral bukling of these hords. Clearly, the predition of II in (22.a) using the original mode offers a very good assessment of the effet of initial loading to within 0.3%, refleting that the new mode is similar to the original mode. Finally, load ase (III) onsiders the effet of applying part of the nominal vertial load on the lower hord, whih an be seen in terms of an additional self-equilibrating set of nominal loads, as demonstrated in Figure 10(). his additional set of nominal loads leads to axial fores in only three members, highlighted in Figure 10() with a thiker line, where the orresponding equivalent springs, spring rotations due to the original mode and quadrati energy are obtained as follows: F n (MN) 0.83333 2.6352 2.6352 (22.a) k n (MN.m) 1.25 6.25 6.25 (23.b) y (mrad) 0.12952 0.13010 0.12898, (mrad) 0 (23.) n m n 2 2 G,i y,i z,i i1 k (N.m) k 0.18880 (23.d) herefore, in view of (12.b), the lowest bukling load fator is estimated as: z k 1.8380 III E n n kg kg (23.a) ompared to the exat value: III 1.7664 (24.b) he above inrease in the bukling load fator (i.e. III I ) is learly due to the relatively large inremental tensile fores in elements 2 and 3 in omparison with the ompressive fore of element 1, as obtained in (23.a). Considering the stiffness of the additional equivalent springs in (23.b), and with the orresponding rotations due to the original mode being almost idential, the inremental quadrati energy in (23.d) is positive, leading to inreased resistane to the original bukling mode. Importantly, the predition of III in (24.a) using 18
the original mode offers a good assessment to within 4%, again refleting that the new mode is similar to the original mode. 6. CONCLUSION his paper presents a simplified approah for bukling analysis of skeletal strutures, whih adopts a rotational spring analogy to assess the influene of the geometri stiffness, enabling the use of ommon notions from linear strutural analysis. he proposed approah ombines simpliity with auray, thus providing a pratial framework for bukling analysis. It also offers signifiant oneptual benefits, shedding onsiderable light that an aid in the teahing and understanding of strutural bukling. he rotational spring analogy is first desribed, and the formulation of the geometri stiffness matrix from equivalent disrete springs is disussed for 2D and 3D analysis. It is noted, however, that the main pratial benefit of the rotational spring analogy arises in the ontext of simplified bukling analysis. In this ontext, only the quadrati energy stored/imparted by the equivalent springs due to a deformation mode is required, while the assembly of the geometri stiffness matrix beomes unneessary. Detailed bukling analysis is then outlined, where the appliability of the rotational spring analogy to bukling preditions using a linear strutural analysis tool is highlighted. It is shown that iterative methods, suh as subspae iterations, may be used to establish the exat lowest bukling mode, though this requires the assembly of the geometri stiffness matrix, or at least the resulting fores due to an arbitrary deformation mode. In this ontext, the rotational spring analogy does not offer major pratial benefits, apart from the ease of formulation of the geometri stiffness. Simplified bukling analysis using an assumed mode is subsequently disussed, where it is noted that the approximate preditions represent an upper bound on the exat lowest bukling load. For this type of analysis, the rotational spring analogy offers a very simple framework, where only the rotations of the equivalent springs orresponding to the assumed mode are 19
required. hese are diretly used to establish the quadrati energy due to the geometri stiffness, enabling the approximation of the lowest bukling load fator with relative ease. wo approahes are then proposed for simplified bukling analysis. he first approah involves the onsideration of individual andidate modes, orresponding to individual probing load patterns, and the seletion of the mode whih ahieves the lowest bukling load. he seond approah is a refinement whih ombines two probing modes in a redued eigenvalue problem, enabling a more aurate predition of the lowest bukling load. Both approahes maintain the pratial benefits of the rotational spring analogy in that the assembly of the geometri stiffness matrix is not required. Notwithstanding, an even more aurate version of the latter approah is proposed, where the seond probing mode is diretly related to the first, though this modifiation requires the assembly of the geometri stiffness ontributions. Finally, the oneptual benefits of the rotational spring analogy are highlighted. It is suggested that bukling an be oneived as a phenomenon driven by negative equivalent rotational springs apable of overoming the positive stiffness of the material and other equivalent springs. When seen in these terms, the influene of parameters suh as the magnitude of ompressive/tensile fores and the member lengths on the bukling load and assoiated mode an be naturally assessed using ommon priniples of linear analysis. he most signifiant oneptual benefits, however, arise in onsidering the bukling interation between different loads, where it is shown that the rotational spring analogy offers a powerful means to predit and explain the hange in the lowest bukling load fator due to additional initial/nominal loading. Several illustrative examples are provided throughout the paper to demonstrate the pratial framework and oneptual power offered by the rotational spring analogy for bukling analysis of skeletal strutures. It is believed that this new view of strutural bukling an play an important role in providing simplified design-oriented tools and explaining the results of more detailed bukling analysis. 20
7. REFERENCES [1] imoshenko, S.P., and Gere, J.M. (1961), heory of Elasti Stability, MGraw Hill. [2] Netherot, D.A., and Rokey, K.C. (1971), A Unified Approah to the Elasti Lateral Bukling of Beams, he Strutural Engineer, Vol. 49, No. 7, pp. 321-330. [3] Horne, M.R. (1975), An Approximate Method for Calulating the Elasti Critial Loads of Multi-Storey Plane Frames, he Strutural Engineer, Vol. 53, No. 6, pp. 242-248. [4] Allen, H.G., and Bulson, P.S. (1980), Bakground to Bukling, MGraw Hill. [5] Bradford, M.A. (1992), Lateral-Distortional Bukling of Steel I-Setion Members, Journal of Construtional Steel Researh, Vol. 23, No. 1-3, pp. 97-116. [6] Crisfield, M.A. (1991), Nonlinear Analysis of Solids and Strutures, Vol. 1, Wiley. [7] Belytshko,., Liu, W.K., and Moran, B. (2000), Nonlinear Finite Elements for Continua and Strutures, Wiley. [8] Chan, S.L., and Zhou, Z.H. (2000), Non-linear Integrated Design and Analysis of Skeletal Strutures by 1 Element per Member, Engineering Strutures, Vol. 22, No. 3, pp. 246-257. [9] Hughes,.J.R. (1987), he Finite Element Method, Prentie-Hall. [10] Stevens, L.K. (1967), Elasti Stability of Pratial Multi-Storey Frames, Proeedings of the Institution of Civil Engineers, Vol. 36, pp. 99-117. 21
8. ABLES Iteration P (kn) U (mm) (mrad) 1 0 3.492 0 0.1279 0.2864 0.2199 0.1705 0.2114 0.0887 0.2932 23.87 2 0.3181 2.333 1.591 0.1163 0.2799 0.2414 0.1551 0.2181 0.0513 0.3219 22.49 3 0.4718 2.016 2.024 0.1115 0.2756 0.2456 0.1486 0.2188 0.0400 0.3274 22.37 4 0.5237 1.930 2.143 0.1099 0.2741 0.2466 0.1466 0.2190 0.0368 0.3288 22.36 able 1. Critial load fator for elasti olumn using 4 elements Iteration 1 2 3 4 22.14 20.86 20.74 20.73 able 2. Critial load fator for elasti olumn using 8 elements 22
Mode P (kn) U (mm) (mrad) (I) P 0.8451 4 P 0 (i 4) i 0.7280 0.7787 0.7413 1.1832 1.1814 1.1805 1.2737 1.3058 1.3392 1.3393 0.4853 0.5191 0.4942 0.3035 0.2685 0.2928 0.0615 0.0835 0.0436 0.0223 0.9874 (II) P 1.1826 2 P 0 (i 2) i 0.6618 0.8456 0.6815 1.0896 1.0902 1.0893 1.1321 1.1906 1.1938 1.1940 0.4412 0.5637 0.4544 0.2852 0.1631 0.2718 0.0279 0.0675 0.0411 0.0023 1.0656 (III) P 0.6369 9 P 0 (i 9) i 0.6274 0.6429 0.6026 1.0092 1.0099 1.0101 1.2872 1.3231 1.5701 1.5693 0.4183 0.4286 0.4017 0.2545 0.2446 0.2717 0.1849 0.2087 0.1886 0.1641 1.2650 able 3. Probing modes for elasti frame Modes (A / B) k n E (N.m) k G (N.m) u (I / II) 1.0 0.9209 0.9209 1.0 1.0127 0.9637 0.9637 0.9384 0.7809 0.2334 0.9804 (I / III) 1.0 0.8529 0.8529 1.0 1.0127 0.8708 0.8708 0.7905 0.9742 0.0301 0.9872 (II / III) 1.0 0.7603 0.7603 1.0 0..9384 0.8209 0.8209 0.7905 0.7330 0.3221 1.0269 A B (III / III ) 1.0 0.0 0.0 1.0 0.7905 0.3836 0.3836 0.3932 0.8453 0.5197 0.9767 able 4. Combined modes for elasti frame 23
Figure 1. Equivalent geometri stiffness for straight element Izzuddin: Simplified Bukling Analysis of Skeletal Strutures Using a Rotational Spring Analogy
Figure 2. Equivalent geometri stiffness for bending element Izzuddin: Simplified Bukling Analysis of Skeletal Strutures Using a Rotational Spring Analogy
Figure 3. Redution of loal rotations with refinement Izzuddin: Simplified Bukling Analysis of Skeletal Strutures Using a Rotational Spring Analogy
Figure 4. Linear kinematis of equivalent rotational springs Izzuddin: Simplified Bukling Analysis of Skeletal Strutures Using a Rotational Spring Analogy
Figure 5. Elasti olumn with lamped-pinned ends Izzuddin: Simplified Bukling Analysis of Skeletal Strutures Using a Rotational Spring Analogy
Figure 6. Lowest bukling mode for elasti olumn using 4 elements Izzuddin: Simplified Bukling Analysis of Skeletal Strutures Using a Rotational Spring Analogy
Figure 7. Lowest bukling mode for elasti olumn using 8 elements Izzuddin: Simplified Bukling Analysis of Skeletal Strutures Using a Rotational Spring Analogy
Figure 8. Elasti frame subjet to vertial loading Izzuddin: Simplified Bukling Analysis of Skeletal Strutures Using a Rotational Spring Analogy
Figure 9. Bukling and probing modes for elasti frame Izzuddin: Simplified Bukling Analysis of Skeletal Strutures Using a Rotational Spring Analogy
Figure 10. Elasti spae truss subjet to vertial loading Izzuddin: Simplified Bukling Analysis of Skeletal Strutures Using a Rotational Spring Analogy
Figure 11. Bukling mode of truss under load ase (I) Izzuddin: Simplified Bukling Analysis of Skeletal Strutures Using a Rotational Spring Analogy