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Three-dimensional geometrically nonlinear analysis of slack cable structures
Computers and Structures 128 (2013) 160–169
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Three-dimensional geometrically nonlinear analysis of slack cable structures M. Ahmadizadeh ⇑ Department of Civil Engineering, Sharif University of Technology, P.O. Box 11155-9313, Tehran, Iran
a r t i c l e
i n f o
Article history: Received 13 August 2012 Accepted 11 June 2013 Available online 13 September 2013 Keywords: Slack cable Large displacement Instability Numerical analysis Collapse analysis
a b s t r a c t An effective iterative method is proposed for three-dimensional large-displacement analysis of structures consisting of slack cables as the primary load-bearing system. This approach takes advantage of substructuring technique that is specially tailored to address the analysis and form-finding problem of slack cables subjected to arbitrary loads, and to reduce the overall analysis costs. At the overall structural level, a robust procedure is employed for detecting the instabilities and limiting the corresponding incremental displacements. The effectiveness of the proposed method for the analysis of highly unstable or variableform structures is demonstrated by a number of analysis examples of slack cable systems. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction Tensile cable structures have received close attention for several decades as a means of construction with minimum use of material. The recent developments of new and improved construction materials further support the use of such structures to cover relatively large spans. On the other hand, the analysis and construction of cable-stayed systems is more challenging than most ordinary structures. Numerical analysis is more problematic particularly when the cables are not fully pretensioned, leading to considerable cable sag [1]. Such design may be chosen to reduce the internal member forces or to simplify the construction by eliminating the pre-tensioning procedure, particularly in large structures. In these structures, large displacements and rotations occur in response to the applied loads until reaching an equilibrium state. As a result, the structural form is highly dependent on the applied loads, and unlike other common types of structures, the structural geometry is not known before the analysis. That is, following any alteration of the applied loads, the structure is essentially unstable and ordinary numerical analysis methods (e.g. stiffness approach – commonly with tension-only link elements to model the cables) usually break down due to singularities. These structures often have irregular shapes and low self-weights which makes them more sensitive to the applied loads. Hence, the first step in the analysis of such structures is finding the initial equilibrium configuration, which in turn affects the load-bearing capacity of the structure. After determination of the equilibrium configura⇑ Tel.: +98 21 6616 4241; fax: +98 21 66014828. E-mail address: [email protected] 0045-7949/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compstruc.2013.06.005
tion, the stiffness matrices of the structural members can be used to describe the structural stiffness to resist the applied loads. In this paper, cable suspended roofs are considered, where the loads are directly carried by the cables [2]. These systems can be divided into the categories of (i) simply suspended cables, (ii) pretensioned cable trusses, and (iii) pretensioned cable nets [3]. This study focuses on the analysis of simply suspended cables, where the pretension is negligible and the cables show considerable amounts of sag. Furthermore, the cladding is assumed not to significantly increase the stiffness of the system. These roof systems have a single or double curvature and show little or no stiffness. Most of the literature on the analysis of slack cables is essentially limited to two-dimensional or planar loading and deformations [4–7]. These planar elements cannot be used to accurately describe the shape of three-dimensional cables. As a result, these elements are widely used to analyze two dimensional cable net structures, or those with essentially straight or pre-tensioned cables [8–15]. On the other hand, the works on three-dimensional analysis of slack cable structures involve complex formulations or simultaneous solutions of several nonlinear Equations [16–17], and thus may be restricted to the analysis of relatively small structures. In this study, an effective and simple iterative method is developed for three-dimensional large-displacement analysis of structures consisting of slack cables as the primary load bearing system. The effectiveness of the proposed analysis method stems from (i) the reduction of the memory and processing costs by treatment of each slack cable element as a substructure, whose point coordinates and end reactions are updated in each iteration, and (ii) the detection of the overall structural instabilities (that may arise before reaching the equilibrium state) in the stiffness matrix
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measured from the lowest point to an arbitrary point p on the cable can then be obtained from:
by recognizing degrees of freedom (DF’s) with near zero stiffness, and allowing only limited displacement increments in them. The slack cable substructures are treated as tension-only members subjected to arbitrary concentrated loads along their length, and are allowed to deform in three dimensions. The form of each slack cable and its internal forces are found in an iterative procedure within the overall iterations by satisfying the compatibility and equilibrium requirements for that cable. After sufficient number of iterations, this procedure naturally leads to a reconfiguration of the structural elements in a way that the internal forces of the structure will be in equilibrium with the applied loads. The proposed analysis method is applied to a number of example structures using a computer program written in MATLAB [18] environment. A procedure is also developed for the initial formfinding of these structures when only subjected to dead gravity loads. The example structures consist of pin-ended bar elements and both taut and slack cables. Finally, this method is successfully utilized to analyze a large structure of this type – a design alternative for birds’ park of Tehran, Iran, with a plan area of more than 50,000 square meters. It is shown that this procedure is an effective approach for the analysis of such structures with high degrees of instability and large displacements. Due to its effectiveness and modest memory and processing costs, this approach also paves the way for dynamic analysis of slack cable-bar structures subjected to time varying lateral loads such as earthquake and wind.
x0p
z0 ¼ ac cosh
x0 ac 1
Lðac ; x0p Þ
¼
Z 0
x0p
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 2 x0p dz 0 1þ dx ¼ ac sinh 0 ac dx
2.1. Slack cable as a substructure In order to avoid the numerical instabilities resulting from small flexural stiffness of the cable and reduce the computational costs, each cable is treated separately as a substructure that its internal nodes do not add to the overall DF’s of the system. The substructure is formed by dividing the cable into short straight segments with concentrated loads at their joints. These segments can be uniformly spaced, or they can be selected based on the locations of the concentrated loads applied to the cable. Distributed loads can also be replaced with equivalent concentrated loads at these points. The initial form of the cable substructure is shown in Fig. 1(b). Similar to other finite element analysis problems, the number of cable divisions can be selected based on the required precision and considering the analysis costs. One can use a response parameter, such as cable sag or maximum internal force, to study the effects
where x0 and z0 are horizontal and vertical cable coordinates measured from the lowest point of the parabola as shown in Fig. 1, and ap is the parameter of the parabola. The algebraic cable length
B zB
(b)
z’ z
y Z Y
y’
A
x
x’
ð4Þ
Knowing the total cable length (or more probably, the desired cable sag) and the relative locations of its end supports, the initial shape of the cable is governed using Eqs. (1) and (2) or Eqs. (3) and (4). This shape constitutes the initial coordinates of the points along the cable and is used to determine the initial values of the end reactions assuming a uniform loading. As shown in Fig. 1, the initial point coordinates in the local transverse y-direction are all zero, that conform to zero lateral reactions at ends. It is also shown that the cable local axes are defined such that the horizontal x- and vertical z-axes lie in a vertical plane including support points.
ð1Þ
xB
ð3Þ
where ac is the parameter of the catenary. The cable length measured from the lowest point to an arbitrary point p is then:
A cable analysis method is introduced herein that determines the cable shape, internal forces and deformations, and end reactions. The three-dimensional cable form highly depends on the applied loads and is unknown in the beginning of the analysis. To create a starting point for the iterative procedure that finds this form, an initial trial form needs to be chosen for the cable. This selection is essentially arbitrary, since the final cable shape is independent of this initial form. On the other hand, selection of a reasonable initial shape will help determine the final shape in fewer iterations. In this paper, a two-dimensional initial shape is assumed for simplicity, that can be used to initialize the locations of points along the cable and the support reactions. If this shape is chosen to be parabolic (occurs when cables are subjected to gravitational loads that are uniform along the horizontal), it is given by Beer et al. [19]:
(a)
ð2Þ
If p is selected to be the cable support at one end, the above equation yields the cable length at one side of its lowest point. Special considerations need to be taken into account for the cases where the lowest point of the parabola governing the cable shape falls outside of the span. Alternatively, if an initial catenary shape is assumed for the cable (uniform load along cable length) Eq. (1) is replaced by:
2. Analysis of slack cables
z0 ¼ ap x02
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 2 dz 0 1þ dx ¼ 0 dx 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 2ap x0p 1 þ 4a2p x02 1 þ 4a2p x02 p þ ln 2ap xp þ p ¼ 4ap Z
Lðap ; x0p Þ
z
y A
x 0 1 2 3 4
X Fig. 1. (a) Global and local coordinates, (b) cable substructure in its initial form.
N N-1 N-2
B
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where p is the segment number and Lpd is the deformed length of that segment. Note that the negative sign in the above equation shows that the cable internal tension is in the opposite direction of RFp. The segment deformed length can be obtained from its undeformed length Lp using the cable cross-sectional area A and modulus of elasticity E:
of the number of divisions in the analysis of a representative cable example. A suitable value for the number of divisions is the one beyond which no appreciable variation is observed in the considered response parameter. Following the determination of the initial positions of the cable points in its local coordinates and the applied nodal loads, the cable shape and forces are updated iteratively until the associated compatibility and equilibrium equations are satisfied. Selection of straight elements between nodal points facilitates the use of equilibrium to determine the trial cable forms and internal forces as described in the next section.
kRFp k Lpd ¼ Lp 1 þ AE
The key point here is that the new location of point N will not necessarily coincide with the location of the support at far end, and needs to be adjusted using the method described in Step 5. (4) Using the updated point positions, the deviation of cable end (point N) from the far support (point B) is determined using:
2.2. Cable form and internal forces The initial parabolic or catenary forms assumed for cables do not satisfy the equilibrium and compatibility requirements, unless the cable is only subjected to a uniform load conforming to the assumed shape. The following iterative procedure is proposed herein that updates the cable form, internal forces and end reactions to satisfy these requirements.
ðDx; Dy; DzÞ ¼ ðxN ; yN ; zN Þ ðxB ; yB ; zB Þ
ðRAx ; RAy ; RAz Þiþ1 ¼ ðRAx ; RAy ; RAz Þi þ ðcx ; cy ; cz Þi
cx ¼ c
z y
RAz
Dx RAx ; xB
cy ¼ c
Dy jRAx j; xB
Dx < e; xB
Dz jRAx j xB
ð10Þ
Dy < e; xB
Dz
ð11Þ
where e is a predetermined convergence tolerance. In addition, the iterative changes in the cable point coordinates should be small enough to consider the iterations as converged:
ð6Þ
(a)
z
x
(b)
x
y
RAy 0
0 ∑F y
1
RAx
cz ¼ c
In the above equations, c is a proportional gain that governs the magnitude of corrections made on the reactions, and hence controls the convergence. This value can be conservatively selected as 0.5 to limit the iterative variations of the form and ensure analysis convergence. Alternatively, one can omit using this gain and instead limit the maximum magnitudes of the adjustments, say, to a value of 0.1|RAx|. Several numerical analyses have shown that in most cases, more relaxed limits can be chosen for the above corrections to speed up the convergence. However, it should be noted that a large gain may occasionally destabilize the analysis procedure, particularly when the load pattern or support locations are significantly different than what corresponds to the last converged cable shape. The logic behind the development of Eq. (10) is explained later in this section. (6) The iterations are stopped and the reaction at far end is calculated using force equilibrium if convergence is achieved; otherwise, the next iteration begins starting from Step 1 with recalculation of point locations. The criteria used for convergence are described below. The relative error in the prediction of cable point N should be sufficiently small; that is:
ð5Þ
Lpd ðRF x ; RF y ; RF z Þp kRFp k
ð9Þ
where the subscript i denotes the iteration number and the adjustments are given by:
which must be in equilibrium with the internal force of this segment. Then, by calculating the deformed length of segment 12 and using the new orientation of the resultant force RF2, the updated location of point 2 is determined, as shown in Fig. 2(b). Similarly for the next segment, the force sum is increased by the force components applied at point 2. This procedure is repeated until the locations of all subsequent points of the cable are updated. In general, the updated point coordinates can be obtained from:
ðxp ; yp ; zp Þ ¼ ðxp1 ; yp1 ; zp1 Þ
ð8Þ
(5) Next, the reactions (RAx, RAy, RAz) applied to the cable at end 0 are modified according to the following rules:
(1) The forces applied to the cable are transformed from global coordinates to the cable local coordinates. (2) Starting from end 0 (support A) shown in Fig. 1, the initial value of the sum of the forces applied to the cable RF1 = (RFx, RFy, RFz)1 consists only of the reactions applied from the support at this point, namely (RAx, RAy, RAz). These reactions are calculated either in the previous iteration, or with the initial assumption of cable shape. These forces must be in equilibrium with the internal force in the first segment of the cable, denoted 01. Hence, the deformed length of this segment can be calculated using its internal force and axial stiffness. Then, knowing the deformed length of segment 01, the new location of point 1 (x1, y1, z1) is determined such that segment 01 lies along the direction of the resultant force RF1 as shown in Fig. 2(a). (3) Moving to the next segment 12, the force sum is increased by the load components (Fx, Fy, Fz)1 applied at point 1:
RF2 ¼ RF1 þ ðF x ; F y ; F z Þ1
ð7Þ
∑F z 1
2
2
1 ∑F x
1 2
Fig. 2. Updating the locations of (a) point 1, and (b) point 2 of the cable.
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M. Ahmadizadeh / Computers and Structures 128 (2013) 160–169 2
N dp 1X
ð12Þ
2
in which dp is the square of distance of point p to the local origin and d2p is the square of distance of point p to its position in the previous iteration. In order to understand how the above procedure updates the cable shape and internal forces towards achieving equilibrium and compatibility, the update rules for point positions and reactions should be considered. First, the cable point coordinates are updated such that the internal force in each cable segment is compatible with its deformation and in equilibrium with the applied forces. This serves to satisfy the equilibrium and compliance requirements for the cable. Next, to satisfy the compatibility requirements, the reactions at end 0 are modified to move the end point N closer to the far end support B. For example, Dx in the first one of Eq. (10) is indicative of error in the prediction of point N along the local x-axis. A positive value of Dx shows that the cable is stretched too much, and it should simply be released to some extent to increase its slack, as illustrated in Fig. 3(a). For this reason, the first one of Eq. (10) tends to reduce the x-axis reaction in this case, and increase it otherwise. The equations governing the corrections along y- and z-axes also modify the reactions based on errors Dy and Dz, respectively. A positive error shows that the far end of the cable needs to move in the negative direction; that is, the cable should rotate clockwise in xy or xz plane as illustrated in Fig. 3(b). To apply a clockwise moment, the reaction at A should algebraically increase, and the reaction at B should in turn decrease. This can be achieved by making a positive correction in the corresponding reaction component at A using Eq. (10). Instead of updating the cable form through modifying the reactions, it may seem intuitive to simply make proportional corrections in point locations to address the above compatibility error, and then recalculate the reactions based on the new cable form. However, such modification has the adverse effect of changing the cable length from what satisfies equilibrium and compliance requirements, which is observed to significantly delay the achievement of convergence. For this reason, the above procedure does not directly modify the point coordinates in the corrections made on the cable form. Finally, the convergence is controlled at the end of each iteration, and the cable end reactions are transformed to the global coordinates for use in the overall structural analysis. The performance of the proposed procedure is demonstrated through the analysis of a cable in Section 4 of this paper.
z
The block diagram of the iterative analysis procedure described in the preceding section is illustrated in Fig. 4. As shown, this procedure is analogous to a feedback control system. Here, in addition to the applied loads and basic cable properties, the ‘‘variable driving signal’’ that governs the position of point N is the reaction at point A. Using a given set of reaction values at point A, the equilibrium and compliance equations are used to determine the cable form and internal forces. The resulting position of point N can be considered as the output signal of the ‘‘plant’’ representing the cable. The difference between the position of point N and that of support B is then considered as the error signal, which is used to adjust the reactions at point A. In this analogy, the ‘‘plant’’ and its governing equations are highly nonlinear. For this reason, the stability and convergence of the feedback system cannot be easily determined in closed form, using the classical control theory. On the other hand, very frequently in control of highly nonlinear mechanical systems, controllers with linear or simple structures (such as PID) have been successfully implemented, that do not directly compensate the plant dynamics. Similarly, in the proposed cable analysis method, the reactions at point A are determined using equations that are essentially in the simplest possible form. Considering Eq. (10), one can observe that they simply change the dimension of the closing errors from length to force, and then adjust their magnitude using a proportional gain, c. This gain is the primary parameter that affects the stability and convergence of the analysis scheme. To this end, the best value of c to achieve convergence with the least possible number of iterations should be found based on the considered problem. In other words, similar to simple control structures, a general value for c that is considered to be the best for convergence and stability may not be easily prescribed for all problems. Since this is not a desirable property for an analysis method, the above-mentioned recommended value for c is rather conservative, such that it can be applied to the majority of the problems. As a result, often larger values of the gain can be chosen to accelerate the convergence without destabilizing the analysis scheme. Further studies regarding selection of more suitable gain values and its stability limits are currently underway.
2.4. Special considerations Considering the cable form correction scheme outlined in the preceding section, one can observe that the internal forces and reactions have a central role in the success of the analysis procedure. For this reason, it is recommended that end 0 of the cable is selected as the end with larger reaction forces. This can be done
Δx
(a)
Δx
z RBx
B N
A RAx
2.3. Stability and convergence
B
Reduce reactions
RAx
0
RAy/z A 0
(b)
RBy/z
y/z
N
Δy/z B x
x
0
RBy/z y/z
RBx
N
A
x
Increase reaction at A, reduce at B
RAy/z
A
Δy/z N
B
0
Fig. 3. Correction of reaction forces to adjust the position of point N (a) along x-direction, (b) along y- or z- directions.
x
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M. Ahmadizadeh / Computers and Structures 128 (2013) 160–169
Position of Support A Initial Form Cable Length and Stiffness Applied Loads
End Reactions at Support A
Cable Analysis (Steps 1-3)
Position of Support B
Position of Point N
– +
Step 4
Cable Form and Internal Forces Start
Calculate Reaction Corrections (Step 5)
No
Conv. Check
Record Results End Yes Step 6
Fig. 4. Block diagram showing the iterative procedure to determine the cable form and internal forces.
by comparing, say, the x-component of reactions at ends, which is widely used in corrections given by Eq. (10). A rare special case may arise when a cable segment has a nearzero amount of internal force. In this case, the coordinates of the next point cannot be accurately determined using the procedure described in Steps 2 and 3. Consequently, the cable can in fact be separated into two or more self-equilibrating statically-determinate sections between supports and zero-force segments. The cable sections directly connected to the supports can be used to determine the support reactions by summing up the forces applied to those sections. The shape of each section can then be immediately determined using the above-mentioned procedure starting from the supported ends. In the extremely rare case of a cable having more than one zeroforce segment, only the internal forces and shapes of cable sections between these segments can be determined using the above-mentioned procedure; that is, the exact positions of the intermediate sections relative to cable ends would not be clear, since the zeroforce segments are not fully straight and the relative coordinates of their end points are unknown. One can only check the distance between the segment ends, which should not exceed the length of the segment for compatibility. This is usually the case for these segments, as otherwise, their internal force would not be zero. In any case, this situation will not cause any issues in the overall structural analysis, since only the end reactions are required for this purpose, and the loads applied to the intermediate sections sum up to zero.
3. Overall analysis procedure – static relaxation method A large-displacement analysis procedure called static relaxation is presented herein that is tailored to handle the numerical instabilities resulting from small flexural stiffness of cables and unknown structural geometry. The existing methods for analysis of such structures may introduce equivalent stiffness matrices for cables and artificially supplement the overall stiffness matrix to avoid instabilities [4], or introduce structural impedance by including artificial mass and damping (dynamic relaxation method [13–15]). Instead, the procedure adopted herein limits the amounts of displacement increments only in DF’s with near zero stiffness and allows for gradual changes in the structural form (and hence, the locations of cable end supports and loads) until the external loads are balanced. Therefore, there is no need to determine and update the equivalent stiffness of the cables, and by only modifying the stiffness of the unstable DF’s, the internal forces in stable DF’s will be closer to their final values, which help reduce the number of iterations required for convergence. Furthermore, it should be noted that with a singular stiffness matrix, the fundamental frequency of the structure will be zero, which in
dynamic relaxation method, makes the selection of suitable values for artificial mass and damping difficult. 3.1. Initial form of the slack cable system The initial form of the structure is usually obtained when the structure is subjected to its self weight. In order to start the analysis, the overall structural DFs are temporarily restrained. In other words, the locations of end supports for primary cables (cables that are directly connected to supports or column top nodes) are initially fixed. After assuming initial shapes for all slack cables, their shapes and internal forces are updated according to the procedure described in Section 2.2. Then, if the primary cables support any other cables (named herein the secondary cables), the updated forms of primary cables are used to update the support locations of secondary cables. Next, the secondary cables are analyzed with updated support locations. This in turn alters their end reactions applied to the supporting primary cables. The forces applied to the primary cables are then updated accordingly, and the above procedure is repeated until the displacement increments become sufficiently small. Following the above-mentioned procedure, the equilibrium and compatibility is achieved within the slack cable system, but the primary cable support nodes have not yet moved to their actual positions. It may be possible to leave the achievement of this state to the next step, where the overall DF’s are released and the structural geometry is allowed to change; however, several simulations have shown that this may result in a considerable increase in the number of iterations necessary to achieve convergence. Furthermore, if desired, one may choose to adjust cable lengths at this stage to achieve a uniform cable sag through the system as required by design. However, this usually intensifies the lateral cable deformations that may be undesired from an architectural point of view. 3.2. Determination of the equilibrium state After achieving equilibrium within the cable system, the columns are released to allow them to move to their equilibrium locations. In these structures, the columns are usually pin-ended, as otherwise the horizontal cable reactions will lead to extremely large moments at their bases. For this reason, the columns that do not have any means of physical restraint (such as pre-tensioned anchorage cables) will have significant rigid body movements. A two-dimensional illustrative example of such movement is shown schematically in Fig. 5. It is illustrated that since the cable to the left of the column is longer (and hence has a larger tension) the initially-vertical column tilts to the left, so the cable sags in the left and right cables are increased and reduced, respectively. This reduces the horizontal reaction of cable AB and increases that of
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M. Ahmadizadeh / Computers and Structures 128 (2013) 160–169
A
B
C
B’
RABy
RBCy
RABy
RBCy
RBCx RABx
B’ Rcol
(a)
RBCx RABx
B Rcol
(b) D
Fig. 5. (a) Analysis of a two-dimensional slack cable system in the cases of cable supports fixed and released, (b) equilibrium at point B.
BC, and continues until the sum of cable end reactions and column compressive force cancel out as shown in Fig. 5(b). It can be seen that since the pin ended columns that ‘‘lean’’ on slack cables do not have an appreciable lateral stiffness, the external forces can be balanced only by a reconfiguration of the structure. For this reason, a stiffness matrix consisting of DF’s at column top nodes is essentially singular, because it only consists of the axial stiffness of the columns and fully taut cables. To get around this issue, the following procedure is adopted: (1) The displacement vector uc consisting of column top displacements is initialized to zero, or the values obtained from the last converged analysis step. (2) The unbalanced load vector fu is calculated as the sum of forces fc applied to the column top nodes (from primary cables or directly-applied concentrated loads) and the column internal forces applied to their top nodes. (3) A stiffness matrix K is formed for the DF’s of column top nodes based on the current orientation of the columns and the current state of tensioned or straight cables. Here, the slack cables are assumed not to provide any stiffness for the system. (4) The eigenvalues ki and eigenvectors ui of the stiffness matrix are obtained. (5) The structural instabilities are then recognized by a transformation of the stiffness matrix K and unbalanced force vector fu to modal coordinates, where the stiffness matrix is diagonal:
Km ¼ UT KU ¼ diagðki Þ
ð13Þ
f um ¼ UT f u
ð14Þ
in which U is the modal matrix consisting of normalized eigenvectors of K [20]. (6) Then, the iterative modal displacement increments duum are determined by dividing each modal unbalanced force by the corresponding nonzero modal stiffness. If the modal stiffness is obtained to be near zero, the modal displacement in that DF is allowed to increase in the direction of its modal unbalanced force by a small amount, say a displacement increment coefficient m times the unbalanced force. As shown in Fig. 5, this movement should gradually reduce the unbalanced forces. It is recommended to take m as the inverse of the average modal stiffness of the structural DF’s (or more conservatively, that of the stable DF’s), to avoid abrupt changes in structural geometry. In the modal coordinate system, this value in fact replaces the impedance provided by mass and damping in the dynamic relaxation method [10], or the additional stiffness provided by supplementing the equivalent stiffness matrix by a variable diagonal matrix [4]. A small value of m would increase the number of required iterations, particularly when large displacements are expected to achieve equilibrium, but is recommended to avoid divergence.
(7) The calculated modal displacement increments are transformed to the global coordinates system using:
duu ¼ Uduum
ð15Þ
(8) The system geometry is updated based on the current values P of column top displacements uc = duu, and the internal forces of columns and fully tensioned cables are recalculated. (9) Based on the new locations of cable supports, the cable forms are updated following the procedure described in Section 3.1, and the cable support reactions are recalculated. (10) The above procedure is repeated from Step 2, unless convergence is achieved, in which case the load increment is completed. If after a large number of iterations, no further reduction in the unbalanced forces is observed while displacement increments remain significant, this may be a sign of an unstable structure, where an equilibrium configuration cannot be found. It can be seen that similar to the cable analysis procedure presented in Section 2.2, the static relaxation procedure outlined above is also based on the reduction of an error quantity, namely the unbalanced force vector. 4. Analysis examples Several numerical analyses have been carried out to verify the accuracy and efficiency of the proposed analysis procedure. A few illustrative examples are presented below. 4.1. Analysis of a single cable in three dimensions In order to demonstrate the behavior of the proposed procedure for the analysis of cables, the progressive determination of the form of a cable is illustrated in Fig. 6. The span length of the 36-meter-long cable is taken to be 35 m, support B being 2.5 m higher than support A. This cable with EA = 105 kN is treated as a substructure with 20 segments. The loads applied to the cable at its internal nodes are also shown in Fig. 6. As illustrated, the cable form is initially taken to be parabolic (shown by the dashed curve). In the first iteration, a relatively large change in the cable form is seen, resulting from the large differences of reactions at A from what are needed for equilibrium and compatibility. Then, the proposed procedure attempts to gradually modify the form (dotted curves) towards its final configuration (solid curve). The cable elongation at the end of analysis is obtained to be 20.7 cm. With a convergence tolerance of 0.001 and a proportional gain of 1.0, the number of required iterations to find the final form is 58. The number of required iterations for the analysis of the abovementioned cable is plotted against the proportional gain c in Fig. 7. It can be seen that a large gain (more than 2 in this case) results in analysis instability, while arbitrarily small values only increase the number of required iterations. Hence, a suitable value of gain appears to be the lower limit of the nearly flat part of the curve,
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M. Ahmadizadeh / Computers and Structures 128 (2013) 160–169
Fy , kN
20 0 2
0
5
10
15
20
25
30
35
x, m
z, m
-20
0 -2
Fz , kN
0 -4
-20
4 2 0
-40 0
5
10
15
20
25
30
-2
35
-4
y, m
3
4
2
3
1
2
0
1
y, m
z, m
x, m
-1
30
40
x, m
0
-2
-1
-3
-2
-4
-3
-5
10
0
20
-4 0
5
10
15
20
25
30
35
40
0
x, m
5
10
15
20
25
30
35
40
x, m
Fig. 6. Cable loads along local x-axis and progressive determination of cable form.
position shown by the dash-dot line in Fig. 8. It can be seen that the final result obtained by the proposed method and shown by the solid curves is in agreement with other published results, giving the displacement of the central node as (26.6, 43.1, 2.6). The convergence behavior of the proposed analysis procedure within an analysis step is compared with that of dynamic relaxation method through considering the iterative values of the displacements of the center node in Fig. 9. In order to ensure the stability of dynamic relaxation method, a mass M is assumed to exist in any DF, which should satisfy [10]:
Number of Iterations
1000 800 600 400 200 0 0
0.5
1 γ
1.5
2
M¼
Dt 2 K 2
ð16Þ
Fig. 7. Number of required iterations versus the proportional gain c.
which results in a near-minimum number of iterations and is adequately far from the stability limit. As shown, a larger proportional gain could be selected in this analysis, to safely reduce the number of the required iterations to less than 40. It should be noted that the number of iterations required to determine the cable shape in subsequent analysis steps will be considerably smaller, since they usually constitute smaller changes in cable form than those of the first step. 4.2. Analysis of a benchmark cable-spring assembly The 3DF structure shown in Fig. 8 – considered in reference [4] and some other studies – consists of three interconnected cables that are attached to a vertical spring at the center node. This node can move freely along the horizontal X- and Y-directions. Consistent units are used throughout this section. The structure is subjected to the cable self weights, a temperature change, and a concentrated horizontal load of 1000 units applied to the center node in the positive Y-direction. The system is released from the
where Dt is an arbitrary time step and K is the stiffness in that DF. Then, to achieve a suitable convergence speed, the damping coefficient C should be selected such that a critically-damped system results. However, since the stiffness of the center node is zero in the horizontal plane, such dynamic properties cannot be determined in this manner. For this reason, with a unit time step, the mass is taken to be 500 in all three DF’s (to observe the above stability limit in the vertical direction), while different values of damping coefficient are examined. Here, the residual forces are obtained using the same cable analysis procedure described in Section 2.2. The results show that while the proposed static relaxation method with m = 3/1000 satisfies the tolerance of 0.001 within 56 iterations, none of the selected damping values lead to convergence within the considered 100 iterations. For the proposed static relaxation method, the number of required iterations is plotted against the displacement increment coefficient in Fig. 10, which shows that the above-mentioned value for m is somewhat conservative. For dynamic relaxation method, one can reduce the number of required iterations by rigorously determining the oscillation frequencies in the three DF’s of the center node using direct observation of the response, and then use them to find the critical damping
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300 Original Release Position Concentrated Load at Center in Y -Direction
200
Vertical ( Z ) Spring Constant = 1000
400 350
0
-100
-200
-300 -400
300
w=1 EA = 290000 L = 580
250
z
y
100
Distributed Loads Along Cable Lengths in Y -direction
w=2 EA = 290000 L = 510
-6
Thermal Constant: 6.5x10 Temperature Change: 100 (All Consistent Unites) -300
-200
-100
200 150
w=2 EA = 290000 L = 510
100 50
0
100
200
300
0 -400
400
-300
-200
-100
0
x
100
200
300
400
x
Fig. 8. Three-degree-of-freedom structure subjected to concentrated and distributed loads.
Displacement X
Static R 50
DR C=10000
DR C=1000
DR C=100
DR C=10
DR Kin Damp
40 30 20 10 0
Displacement Y
-10 0
10
20
30
40
0
10
20
30
40
50
60
70
80
90
100
50
60
70
80
90
100
60 40 20 0
Iterations Fig. 9. Comparison of the iterative values of displacements of the center node.
artificial stiffness of 10 units in the horizontal DF’s helps the solution to converge after 82 iterations supports the above statement. In a second analysis of the 3DF structure using the proposed method, the concentrated load is replaced by equivalent
200 Number of Iterations
coefficients in each DF. Still, it should be noted that the previously introduced substructuring technique is used in both analysis methods compared herein to limit the number of DF’s of the problem to three; otherwise, the problem of determination of dynamic relaxation parameters along the cables will be much more difficult. Dynamic relaxation method can also be used with kinetic damping, in which method the kinetic energy of the system is reset to zero whenever it reaches a peak. Again, the mass cannot be determined using the previously-mentioned stability criterion, since two of the three DF’s of the structure have zero stiffness. Using the same mass of 500 units in the top node, it is observed that convergence cannot be achieved due to small oscillations of center node as shown in Fig. 9. This problem persists with other choices of mass, which do not meaningfully affect the number of required iterations. It appears that resetting the maximum kinetic energy to zero fails to reduce and keep the displacement variations below the tolerance. This can be attributed to the fact that the underlying assumption of kinetic damping approach (that the maximum kinetic energy corresponds to a position near equilibrium) cannot be extended to structures with zero or highly nonlinear stiffness in a number of DF’s. The fact that only adding a small
150
100
50
0 0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
ν Fig. 10. Number of required iterations versus the displacement increment coefficient t.
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M. Ahmadizadeh / Computers and Structures 128 (2013) 160–169
Fig. 11. Progressive analysis of a slack cable roof system (from left to right: initial cable forms, updated cable forms with supports fixed, and final form with column tops released).
Fig. 12. Extreme loading scenario: gravitational dead load with removal of one of the anchorage cables.
uniformly-distributed loads of 0.625 units acting along the cable lengths in the same positive Y-direction. This is carried out to demonstrate the effectiveness of the proposed method for analysis of cables subjected to lateral loads. The result is shown by dashed curves in Fig. 8. As expected, a reduced displacement of the central point from the original release position is obtained, being (23.4, 23.9, 2.4). 4.3. Analysis of a slack-cable-suspended roof The irregular hexagonal roof system shown in Fig. 11 consists of seven compression-only pin-ended columns located on a slope with lengths ranging from 24 to 30 m. The plan dimensions of this structure are 85 and 70 m along global X and Y directions, respectively. Each of the six columns on the perimeter of the structure are restrained with a pair of fully taut cables that anchor the column tops to the ground. These columns are tilted with a slope of 10% to the sides to increase the stability and load-bearing capacity of the structure. The center column is initially assumed to be vertical. The column tops are interconnected with primary slack cables, which in turn support several uniformly spaced secondary cables that span between primary ones and make a spider web form. This
structure is in fact a typical module of a design candidate for the birds’ park in Tehran (having 15 interconnected hexagonal units covering an area of more than 50,000 m2 – only one unit is presented herein for brevity and clarity of the results). The roof system shown herein has a total of 3297 unrestrained DF’s. By using the approach presented in this paper, the analysis problem reduces to a number of smaller problems, being the analysis of individual cables with at most 60 DF’s, and the overall analysis of columns having 21 DF’s. Fig. 11 shows the analysis results of the slack cable roof system subjected to a uniform gravitational dead load of 250 N/m2. Note that each analysis step is plotted against a dashed outline of the previous step to highlight the corresponding displacements. In the first step, all cables are assumed to have parabolic shapes in their vertical planes. The cable lengths are selected to achieve a near-uniform sag throughout the structure, being 9% of the span length for primary cables and 6% for secondary cables. Next, the cable forms are updated while preventing the translation of their supports; as a result, the primary cables deform in three dimensions in response to the unsymmetrical loads applied from secondary cables. In this step, the cable lengths are revised again to alleviate the uneven sags of the cables as a result of lateral
M. Ahmadizadeh / Computers and Structures 128 (2013) 160–169
movements of primary cables. Finally, the columns are released to gradually move towards achieving equilibrium and balancing the external loads. This usually leads to a lower energy state, through which the lateral deformations of cables are generally reduced. The amount of final displacement at the top of the central column is calculated to be 95.3 cm, mostly towards east. In order to demonstrate the versatility of the proposed procedure to handle very large deformations and resolve numerical instabilities, an extreme loading scenario is assumed for the considered structure. After reaching the equilibrium state for the dead load (Fig. 11), it is assumed that one of the anchorage cables of the southwest column is removed and the structure is reanalyzed. The results shown in Fig. 12 demonstrate a large displacement at the top of the column with a failed cable: 291.1 cm towards east and 43.5 cm in the south direction. The displacement of the central column towards east also increases to 119.2 cm. It should be noted that reaching the equilibrium state in these conditions that constitute sudden changes in the internal force system may take several hundred iterations using the proposed procedure. 5. Conclusions An effective iterative approach is developed for the analysis of variable-form structures consisting of slack cables and bar elements. A specialized substructuring technique is used for the analysis of slack cables to reduce the processing costs and memory requirements, and to alleviate the cable instability issues before reaching the equilibrium state. The procedure to determine the cable form and internal forces uses sensible relations that progressively adapt the cable form to satisfy the equilibrium and compatibility requirements. Furthermore, the singularities at the overall structural level are addressed by recognizing the unstable DF’s and restricting their displacement increments. Using these procedures, the structural forms corresponding to load combinations consisting of gravitational and lateral loads can be accurately determined. The proposed method is successfully applied to the analysis of several slack-cable structures that exhibit considerable degrees of instability. It is shown that this method can be used in
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the analysis of structures with very large rigid-body displacements and rotations, such as the cases wherein collapse scenarios are examined. References [1] Friere AMS, Negrao JHO, Lopes AV. Geometrical nonlinearities on the static analysis of highly flexible steel cable-stayed bridges. Comput Struct 2006;84:2128–40. [2] Krishna P. Cable-suspended roofs. New York: McGraw-Hill; 1978. [3] Buchholdt HA. An introduction to cable roof structures. Cambridge: Cambridge University Press; 1985. [4] Peyrot AH, Goulois AM. Analysis of cable structures. Comput Struct 1979;10:805–13. [5] Bouaanani N, Ighouba M. A novel scheme for large deflection analysis of suspended cables made of linear or nonlinear elastic materials. Adv Eng Softw 2011;42:1009–19. [6] Z. Wang, T. McCarthy, M.N. Sheikh. Taut-slack algorithm for analyzing the geometric nonlinearity of cable structures. In: 21st International offshore and polar engineering conference, United States, 2011. [7] Irvine HM. Cable structures. Dover Publications; 1992. [8] Ozdemir H. A finite element approach for cable problems. Int J Solids Struct 1979;15:427–37. [9] Atai AA, Mioduchowski A. Equilibrium analysis of elasto-plastic cable nets. Comput Struct 1998;66:163–71. [10] Lewis WJ. Tension structures form and behavior. London: Thomas Telford; 2003. [11] West HH. Discretized initial-value analysis of cable nets. Int J Solids Struct 1973;9:1403–20. [12] Jayaraman HB, Knudson WC. A curved element of the analysis of cable structures. Comput Struct 1981;14:325–33. [13] Ivanyi P, Topping BHV. Parallel dynamic relaxation form finding. In: Papadrakakis M, Topping B, editors. Innovative computational methods for structural mechanics. Stirlingshire, UK: Saxe-Coburg Publications; 1999. p. 127–47. [14] Barnes MR. Form finding and analysis of tension structures by dynamic relaxation. Int J Space Struct 1999;14:89–104. [15] Day AS. An introduction to dynamic relaxation. The Engineer 1965;219:218–21. [16] Gosling PD, Korban EA. A bendable finite element for the analysis of flexible cable structures. Finite Elem Anal Des 2001;38:45–63. [17] Impollonia N, Ricciardi G, Saitta F. Statics of elastic cables under 3D point forces. Int J Solids Struct 2011;48:1268–76. [18] The MathWorksÒ Inc, MATLAB user’s guide. Natick, MA; 1994–2011. [19] Beer FP, Johnston ER, Eisenberg ER, Staab GH. Vector mechanics for engineers, statics. 7th ed. McGraw-Hill Science; 2003. [20] Greenberg MD. Advanced engineering mathematics. 2nd ed. Prentice Hall; 1998.
Contents lists available at ScienceDirect
Computers and Structures journal homepage: www.elsevier.com/locate/compstruc
Three-dimensional geometrically nonlinear analysis of slack cable structures M. Ahmadizadeh ⇑ Department of Civil Engineering, Sharif University of Technology, P.O. Box 11155-9313, Tehran, Iran
a r t i c l e
i n f o
Article history: Received 13 August 2012 Accepted 11 June 2013 Available online 13 September 2013 Keywords: Slack cable Large displacement Instability Numerical analysis Collapse analysis
a b s t r a c t An effective iterative method is proposed for three-dimensional large-displacement analysis of structures consisting of slack cables as the primary load-bearing system. This approach takes advantage of substructuring technique that is specially tailored to address the analysis and form-finding problem of slack cables subjected to arbitrary loads, and to reduce the overall analysis costs. At the overall structural level, a robust procedure is employed for detecting the instabilities and limiting the corresponding incremental displacements. The effectiveness of the proposed method for the analysis of highly unstable or variableform structures is demonstrated by a number of analysis examples of slack cable systems. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction Tensile cable structures have received close attention for several decades as a means of construction with minimum use of material. The recent developments of new and improved construction materials further support the use of such structures to cover relatively large spans. On the other hand, the analysis and construction of cable-stayed systems is more challenging than most ordinary structures. Numerical analysis is more problematic particularly when the cables are not fully pretensioned, leading to considerable cable sag [1]. Such design may be chosen to reduce the internal member forces or to simplify the construction by eliminating the pre-tensioning procedure, particularly in large structures. In these structures, large displacements and rotations occur in response to the applied loads until reaching an equilibrium state. As a result, the structural form is highly dependent on the applied loads, and unlike other common types of structures, the structural geometry is not known before the analysis. That is, following any alteration of the applied loads, the structure is essentially unstable and ordinary numerical analysis methods (e.g. stiffness approach – commonly with tension-only link elements to model the cables) usually break down due to singularities. These structures often have irregular shapes and low self-weights which makes them more sensitive to the applied loads. Hence, the first step in the analysis of such structures is finding the initial equilibrium configuration, which in turn affects the load-bearing capacity of the structure. After determination of the equilibrium configura⇑ Tel.: +98 21 6616 4241; fax: +98 21 66014828. E-mail address: [email protected] 0045-7949/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compstruc.2013.06.005
tion, the stiffness matrices of the structural members can be used to describe the structural stiffness to resist the applied loads. In this paper, cable suspended roofs are considered, where the loads are directly carried by the cables [2]. These systems can be divided into the categories of (i) simply suspended cables, (ii) pretensioned cable trusses, and (iii) pretensioned cable nets [3]. This study focuses on the analysis of simply suspended cables, where the pretension is negligible and the cables show considerable amounts of sag. Furthermore, the cladding is assumed not to significantly increase the stiffness of the system. These roof systems have a single or double curvature and show little or no stiffness. Most of the literature on the analysis of slack cables is essentially limited to two-dimensional or planar loading and deformations [4–7]. These planar elements cannot be used to accurately describe the shape of three-dimensional cables. As a result, these elements are widely used to analyze two dimensional cable net structures, or those with essentially straight or pre-tensioned cables [8–15]. On the other hand, the works on three-dimensional analysis of slack cable structures involve complex formulations or simultaneous solutions of several nonlinear Equations [16–17], and thus may be restricted to the analysis of relatively small structures. In this study, an effective and simple iterative method is developed for three-dimensional large-displacement analysis of structures consisting of slack cables as the primary load bearing system. The effectiveness of the proposed analysis method stems from (i) the reduction of the memory and processing costs by treatment of each slack cable element as a substructure, whose point coordinates and end reactions are updated in each iteration, and (ii) the detection of the overall structural instabilities (that may arise before reaching the equilibrium state) in the stiffness matrix
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measured from the lowest point to an arbitrary point p on the cable can then be obtained from:
by recognizing degrees of freedom (DF’s) with near zero stiffness, and allowing only limited displacement increments in them. The slack cable substructures are treated as tension-only members subjected to arbitrary concentrated loads along their length, and are allowed to deform in three dimensions. The form of each slack cable and its internal forces are found in an iterative procedure within the overall iterations by satisfying the compatibility and equilibrium requirements for that cable. After sufficient number of iterations, this procedure naturally leads to a reconfiguration of the structural elements in a way that the internal forces of the structure will be in equilibrium with the applied loads. The proposed analysis method is applied to a number of example structures using a computer program written in MATLAB [18] environment. A procedure is also developed for the initial formfinding of these structures when only subjected to dead gravity loads. The example structures consist of pin-ended bar elements and both taut and slack cables. Finally, this method is successfully utilized to analyze a large structure of this type – a design alternative for birds’ park of Tehran, Iran, with a plan area of more than 50,000 square meters. It is shown that this procedure is an effective approach for the analysis of such structures with high degrees of instability and large displacements. Due to its effectiveness and modest memory and processing costs, this approach also paves the way for dynamic analysis of slack cable-bar structures subjected to time varying lateral loads such as earthquake and wind.
x0p
z0 ¼ ac cosh
x0 ac 1
Lðac ; x0p Þ
¼
Z 0
x0p
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 2 x0p dz 0 1þ dx ¼ ac sinh 0 ac dx
2.1. Slack cable as a substructure In order to avoid the numerical instabilities resulting from small flexural stiffness of the cable and reduce the computational costs, each cable is treated separately as a substructure that its internal nodes do not add to the overall DF’s of the system. The substructure is formed by dividing the cable into short straight segments with concentrated loads at their joints. These segments can be uniformly spaced, or they can be selected based on the locations of the concentrated loads applied to the cable. Distributed loads can also be replaced with equivalent concentrated loads at these points. The initial form of the cable substructure is shown in Fig. 1(b). Similar to other finite element analysis problems, the number of cable divisions can be selected based on the required precision and considering the analysis costs. One can use a response parameter, such as cable sag or maximum internal force, to study the effects
where x0 and z0 are horizontal and vertical cable coordinates measured from the lowest point of the parabola as shown in Fig. 1, and ap is the parameter of the parabola. The algebraic cable length
B zB
(b)
z’ z
y Z Y
y’
A
x
x’
ð4Þ
Knowing the total cable length (or more probably, the desired cable sag) and the relative locations of its end supports, the initial shape of the cable is governed using Eqs. (1) and (2) or Eqs. (3) and (4). This shape constitutes the initial coordinates of the points along the cable and is used to determine the initial values of the end reactions assuming a uniform loading. As shown in Fig. 1, the initial point coordinates in the local transverse y-direction are all zero, that conform to zero lateral reactions at ends. It is also shown that the cable local axes are defined such that the horizontal x- and vertical z-axes lie in a vertical plane including support points.
ð1Þ
xB
ð3Þ
where ac is the parameter of the catenary. The cable length measured from the lowest point to an arbitrary point p is then:
A cable analysis method is introduced herein that determines the cable shape, internal forces and deformations, and end reactions. The three-dimensional cable form highly depends on the applied loads and is unknown in the beginning of the analysis. To create a starting point for the iterative procedure that finds this form, an initial trial form needs to be chosen for the cable. This selection is essentially arbitrary, since the final cable shape is independent of this initial form. On the other hand, selection of a reasonable initial shape will help determine the final shape in fewer iterations. In this paper, a two-dimensional initial shape is assumed for simplicity, that can be used to initialize the locations of points along the cable and the support reactions. If this shape is chosen to be parabolic (occurs when cables are subjected to gravitational loads that are uniform along the horizontal), it is given by Beer et al. [19]:
(a)
ð2Þ
If p is selected to be the cable support at one end, the above equation yields the cable length at one side of its lowest point. Special considerations need to be taken into account for the cases where the lowest point of the parabola governing the cable shape falls outside of the span. Alternatively, if an initial catenary shape is assumed for the cable (uniform load along cable length) Eq. (1) is replaced by:
2. Analysis of slack cables
z0 ¼ ap x02
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 2 dz 0 1þ dx ¼ 0 dx 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 2ap x0p 1 þ 4a2p x02 1 þ 4a2p x02 p þ ln 2ap xp þ p ¼ 4ap Z
Lðap ; x0p Þ
z
y A
x 0 1 2 3 4
X Fig. 1. (a) Global and local coordinates, (b) cable substructure in its initial form.
N N-1 N-2
B
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where p is the segment number and Lpd is the deformed length of that segment. Note that the negative sign in the above equation shows that the cable internal tension is in the opposite direction of RFp. The segment deformed length can be obtained from its undeformed length Lp using the cable cross-sectional area A and modulus of elasticity E:
of the number of divisions in the analysis of a representative cable example. A suitable value for the number of divisions is the one beyond which no appreciable variation is observed in the considered response parameter. Following the determination of the initial positions of the cable points in its local coordinates and the applied nodal loads, the cable shape and forces are updated iteratively until the associated compatibility and equilibrium equations are satisfied. Selection of straight elements between nodal points facilitates the use of equilibrium to determine the trial cable forms and internal forces as described in the next section.
kRFp k Lpd ¼ Lp 1 þ AE
The key point here is that the new location of point N will not necessarily coincide with the location of the support at far end, and needs to be adjusted using the method described in Step 5. (4) Using the updated point positions, the deviation of cable end (point N) from the far support (point B) is determined using:
2.2. Cable form and internal forces The initial parabolic or catenary forms assumed for cables do not satisfy the equilibrium and compatibility requirements, unless the cable is only subjected to a uniform load conforming to the assumed shape. The following iterative procedure is proposed herein that updates the cable form, internal forces and end reactions to satisfy these requirements.
ðDx; Dy; DzÞ ¼ ðxN ; yN ; zN Þ ðxB ; yB ; zB Þ
ðRAx ; RAy ; RAz Þiþ1 ¼ ðRAx ; RAy ; RAz Þi þ ðcx ; cy ; cz Þi
cx ¼ c
z y
RAz
Dx RAx ; xB
cy ¼ c
Dy jRAx j; xB
Dx < e; xB
Dz jRAx j xB
ð10Þ
Dy < e; xB
Dz
ð11Þ
where e is a predetermined convergence tolerance. In addition, the iterative changes in the cable point coordinates should be small enough to consider the iterations as converged:
ð6Þ
(a)
z
x
(b)
x
y
RAy 0
0 ∑F y
1
RAx
cz ¼ c
In the above equations, c is a proportional gain that governs the magnitude of corrections made on the reactions, and hence controls the convergence. This value can be conservatively selected as 0.5 to limit the iterative variations of the form and ensure analysis convergence. Alternatively, one can omit using this gain and instead limit the maximum magnitudes of the adjustments, say, to a value of 0.1|RAx|. Several numerical analyses have shown that in most cases, more relaxed limits can be chosen for the above corrections to speed up the convergence. However, it should be noted that a large gain may occasionally destabilize the analysis procedure, particularly when the load pattern or support locations are significantly different than what corresponds to the last converged cable shape. The logic behind the development of Eq. (10) is explained later in this section. (6) The iterations are stopped and the reaction at far end is calculated using force equilibrium if convergence is achieved; otherwise, the next iteration begins starting from Step 1 with recalculation of point locations. The criteria used for convergence are described below. The relative error in the prediction of cable point N should be sufficiently small; that is:
ð5Þ
Lpd ðRF x ; RF y ; RF z Þp kRFp k
ð9Þ
where the subscript i denotes the iteration number and the adjustments are given by:
which must be in equilibrium with the internal force of this segment. Then, by calculating the deformed length of segment 12 and using the new orientation of the resultant force RF2, the updated location of point 2 is determined, as shown in Fig. 2(b). Similarly for the next segment, the force sum is increased by the force components applied at point 2. This procedure is repeated until the locations of all subsequent points of the cable are updated. In general, the updated point coordinates can be obtained from:
ðxp ; yp ; zp Þ ¼ ðxp1 ; yp1 ; zp1 Þ
ð8Þ
(5) Next, the reactions (RAx, RAy, RAz) applied to the cable at end 0 are modified according to the following rules:
(1) The forces applied to the cable are transformed from global coordinates to the cable local coordinates. (2) Starting from end 0 (support A) shown in Fig. 1, the initial value of the sum of the forces applied to the cable RF1 = (RFx, RFy, RFz)1 consists only of the reactions applied from the support at this point, namely (RAx, RAy, RAz). These reactions are calculated either in the previous iteration, or with the initial assumption of cable shape. These forces must be in equilibrium with the internal force in the first segment of the cable, denoted 01. Hence, the deformed length of this segment can be calculated using its internal force and axial stiffness. Then, knowing the deformed length of segment 01, the new location of point 1 (x1, y1, z1) is determined such that segment 01 lies along the direction of the resultant force RF1 as shown in Fig. 2(a). (3) Moving to the next segment 12, the force sum is increased by the load components (Fx, Fy, Fz)1 applied at point 1:
RF2 ¼ RF1 þ ðF x ; F y ; F z Þ1
ð7Þ
∑F z 1
2
2
1 ∑F x
1 2
Fig. 2. Updating the locations of (a) point 1, and (b) point 2 of the cable.
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M. Ahmadizadeh / Computers and Structures 128 (2013) 160–169 2
N dp 1X
ð12Þ
2
in which dp is the square of distance of point p to the local origin and d2p is the square of distance of point p to its position in the previous iteration. In order to understand how the above procedure updates the cable shape and internal forces towards achieving equilibrium and compatibility, the update rules for point positions and reactions should be considered. First, the cable point coordinates are updated such that the internal force in each cable segment is compatible with its deformation and in equilibrium with the applied forces. This serves to satisfy the equilibrium and compliance requirements for the cable. Next, to satisfy the compatibility requirements, the reactions at end 0 are modified to move the end point N closer to the far end support B. For example, Dx in the first one of Eq. (10) is indicative of error in the prediction of point N along the local x-axis. A positive value of Dx shows that the cable is stretched too much, and it should simply be released to some extent to increase its slack, as illustrated in Fig. 3(a). For this reason, the first one of Eq. (10) tends to reduce the x-axis reaction in this case, and increase it otherwise. The equations governing the corrections along y- and z-axes also modify the reactions based on errors Dy and Dz, respectively. A positive error shows that the far end of the cable needs to move in the negative direction; that is, the cable should rotate clockwise in xy or xz plane as illustrated in Fig. 3(b). To apply a clockwise moment, the reaction at A should algebraically increase, and the reaction at B should in turn decrease. This can be achieved by making a positive correction in the corresponding reaction component at A using Eq. (10). Instead of updating the cable form through modifying the reactions, it may seem intuitive to simply make proportional corrections in point locations to address the above compatibility error, and then recalculate the reactions based on the new cable form. However, such modification has the adverse effect of changing the cable length from what satisfies equilibrium and compliance requirements, which is observed to significantly delay the achievement of convergence. For this reason, the above procedure does not directly modify the point coordinates in the corrections made on the cable form. Finally, the convergence is controlled at the end of each iteration, and the cable end reactions are transformed to the global coordinates for use in the overall structural analysis. The performance of the proposed procedure is demonstrated through the analysis of a cable in Section 4 of this paper.
z
The block diagram of the iterative analysis procedure described in the preceding section is illustrated in Fig. 4. As shown, this procedure is analogous to a feedback control system. Here, in addition to the applied loads and basic cable properties, the ‘‘variable driving signal’’ that governs the position of point N is the reaction at point A. Using a given set of reaction values at point A, the equilibrium and compliance equations are used to determine the cable form and internal forces. The resulting position of point N can be considered as the output signal of the ‘‘plant’’ representing the cable. The difference between the position of point N and that of support B is then considered as the error signal, which is used to adjust the reactions at point A. In this analogy, the ‘‘plant’’ and its governing equations are highly nonlinear. For this reason, the stability and convergence of the feedback system cannot be easily determined in closed form, using the classical control theory. On the other hand, very frequently in control of highly nonlinear mechanical systems, controllers with linear or simple structures (such as PID) have been successfully implemented, that do not directly compensate the plant dynamics. Similarly, in the proposed cable analysis method, the reactions at point A are determined using equations that are essentially in the simplest possible form. Considering Eq. (10), one can observe that they simply change the dimension of the closing errors from length to force, and then adjust their magnitude using a proportional gain, c. This gain is the primary parameter that affects the stability and convergence of the analysis scheme. To this end, the best value of c to achieve convergence with the least possible number of iterations should be found based on the considered problem. In other words, similar to simple control structures, a general value for c that is considered to be the best for convergence and stability may not be easily prescribed for all problems. Since this is not a desirable property for an analysis method, the above-mentioned recommended value for c is rather conservative, such that it can be applied to the majority of the problems. As a result, often larger values of the gain can be chosen to accelerate the convergence without destabilizing the analysis scheme. Further studies regarding selection of more suitable gain values and its stability limits are currently underway.
2.4. Special considerations Considering the cable form correction scheme outlined in the preceding section, one can observe that the internal forces and reactions have a central role in the success of the analysis procedure. For this reason, it is recommended that end 0 of the cable is selected as the end with larger reaction forces. This can be done
Δx
(a)
Δx
z RBx
B N
A RAx
2.3. Stability and convergence
B
Reduce reactions
RAx
0
RAy/z A 0
(b)
RBy/z
y/z
N
Δy/z B x
x
0
RBy/z y/z
RBx
N
A
x
Increase reaction at A, reduce at B
RAy/z
A
Δy/z N
B
0
Fig. 3. Correction of reaction forces to adjust the position of point N (a) along x-direction, (b) along y- or z- directions.
x
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M. Ahmadizadeh / Computers and Structures 128 (2013) 160–169
Position of Support A Initial Form Cable Length and Stiffness Applied Loads
End Reactions at Support A
Cable Analysis (Steps 1-3)
Position of Support B
Position of Point N
– +
Step 4
Cable Form and Internal Forces Start
Calculate Reaction Corrections (Step 5)
No
Conv. Check
Record Results End Yes Step 6
Fig. 4. Block diagram showing the iterative procedure to determine the cable form and internal forces.
by comparing, say, the x-component of reactions at ends, which is widely used in corrections given by Eq. (10). A rare special case may arise when a cable segment has a nearzero amount of internal force. In this case, the coordinates of the next point cannot be accurately determined using the procedure described in Steps 2 and 3. Consequently, the cable can in fact be separated into two or more self-equilibrating statically-determinate sections between supports and zero-force segments. The cable sections directly connected to the supports can be used to determine the support reactions by summing up the forces applied to those sections. The shape of each section can then be immediately determined using the above-mentioned procedure starting from the supported ends. In the extremely rare case of a cable having more than one zeroforce segment, only the internal forces and shapes of cable sections between these segments can be determined using the above-mentioned procedure; that is, the exact positions of the intermediate sections relative to cable ends would not be clear, since the zeroforce segments are not fully straight and the relative coordinates of their end points are unknown. One can only check the distance between the segment ends, which should not exceed the length of the segment for compatibility. This is usually the case for these segments, as otherwise, their internal force would not be zero. In any case, this situation will not cause any issues in the overall structural analysis, since only the end reactions are required for this purpose, and the loads applied to the intermediate sections sum up to zero.
3. Overall analysis procedure – static relaxation method A large-displacement analysis procedure called static relaxation is presented herein that is tailored to handle the numerical instabilities resulting from small flexural stiffness of cables and unknown structural geometry. The existing methods for analysis of such structures may introduce equivalent stiffness matrices for cables and artificially supplement the overall stiffness matrix to avoid instabilities [4], or introduce structural impedance by including artificial mass and damping (dynamic relaxation method [13–15]). Instead, the procedure adopted herein limits the amounts of displacement increments only in DF’s with near zero stiffness and allows for gradual changes in the structural form (and hence, the locations of cable end supports and loads) until the external loads are balanced. Therefore, there is no need to determine and update the equivalent stiffness of the cables, and by only modifying the stiffness of the unstable DF’s, the internal forces in stable DF’s will be closer to their final values, which help reduce the number of iterations required for convergence. Furthermore, it should be noted that with a singular stiffness matrix, the fundamental frequency of the structure will be zero, which in
dynamic relaxation method, makes the selection of suitable values for artificial mass and damping difficult. 3.1. Initial form of the slack cable system The initial form of the structure is usually obtained when the structure is subjected to its self weight. In order to start the analysis, the overall structural DFs are temporarily restrained. In other words, the locations of end supports for primary cables (cables that are directly connected to supports or column top nodes) are initially fixed. After assuming initial shapes for all slack cables, their shapes and internal forces are updated according to the procedure described in Section 2.2. Then, if the primary cables support any other cables (named herein the secondary cables), the updated forms of primary cables are used to update the support locations of secondary cables. Next, the secondary cables are analyzed with updated support locations. This in turn alters their end reactions applied to the supporting primary cables. The forces applied to the primary cables are then updated accordingly, and the above procedure is repeated until the displacement increments become sufficiently small. Following the above-mentioned procedure, the equilibrium and compatibility is achieved within the slack cable system, but the primary cable support nodes have not yet moved to their actual positions. It may be possible to leave the achievement of this state to the next step, where the overall DF’s are released and the structural geometry is allowed to change; however, several simulations have shown that this may result in a considerable increase in the number of iterations necessary to achieve convergence. Furthermore, if desired, one may choose to adjust cable lengths at this stage to achieve a uniform cable sag through the system as required by design. However, this usually intensifies the lateral cable deformations that may be undesired from an architectural point of view. 3.2. Determination of the equilibrium state After achieving equilibrium within the cable system, the columns are released to allow them to move to their equilibrium locations. In these structures, the columns are usually pin-ended, as otherwise the horizontal cable reactions will lead to extremely large moments at their bases. For this reason, the columns that do not have any means of physical restraint (such as pre-tensioned anchorage cables) will have significant rigid body movements. A two-dimensional illustrative example of such movement is shown schematically in Fig. 5. It is illustrated that since the cable to the left of the column is longer (and hence has a larger tension) the initially-vertical column tilts to the left, so the cable sags in the left and right cables are increased and reduced, respectively. This reduces the horizontal reaction of cable AB and increases that of
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A
B
C
B’
RABy
RBCy
RABy
RBCy
RBCx RABx
B’ Rcol
(a)
RBCx RABx
B Rcol
(b) D
Fig. 5. (a) Analysis of a two-dimensional slack cable system in the cases of cable supports fixed and released, (b) equilibrium at point B.
BC, and continues until the sum of cable end reactions and column compressive force cancel out as shown in Fig. 5(b). It can be seen that since the pin ended columns that ‘‘lean’’ on slack cables do not have an appreciable lateral stiffness, the external forces can be balanced only by a reconfiguration of the structure. For this reason, a stiffness matrix consisting of DF’s at column top nodes is essentially singular, because it only consists of the axial stiffness of the columns and fully taut cables. To get around this issue, the following procedure is adopted: (1) The displacement vector uc consisting of column top displacements is initialized to zero, or the values obtained from the last converged analysis step. (2) The unbalanced load vector fu is calculated as the sum of forces fc applied to the column top nodes (from primary cables or directly-applied concentrated loads) and the column internal forces applied to their top nodes. (3) A stiffness matrix K is formed for the DF’s of column top nodes based on the current orientation of the columns and the current state of tensioned or straight cables. Here, the slack cables are assumed not to provide any stiffness for the system. (4) The eigenvalues ki and eigenvectors ui of the stiffness matrix are obtained. (5) The structural instabilities are then recognized by a transformation of the stiffness matrix K and unbalanced force vector fu to modal coordinates, where the stiffness matrix is diagonal:
Km ¼ UT KU ¼ diagðki Þ
ð13Þ
f um ¼ UT f u
ð14Þ
in which U is the modal matrix consisting of normalized eigenvectors of K [20]. (6) Then, the iterative modal displacement increments duum are determined by dividing each modal unbalanced force by the corresponding nonzero modal stiffness. If the modal stiffness is obtained to be near zero, the modal displacement in that DF is allowed to increase in the direction of its modal unbalanced force by a small amount, say a displacement increment coefficient m times the unbalanced force. As shown in Fig. 5, this movement should gradually reduce the unbalanced forces. It is recommended to take m as the inverse of the average modal stiffness of the structural DF’s (or more conservatively, that of the stable DF’s), to avoid abrupt changes in structural geometry. In the modal coordinate system, this value in fact replaces the impedance provided by mass and damping in the dynamic relaxation method [10], or the additional stiffness provided by supplementing the equivalent stiffness matrix by a variable diagonal matrix [4]. A small value of m would increase the number of required iterations, particularly when large displacements are expected to achieve equilibrium, but is recommended to avoid divergence.
(7) The calculated modal displacement increments are transformed to the global coordinates system using:
duu ¼ Uduum
ð15Þ
(8) The system geometry is updated based on the current values P of column top displacements uc = duu, and the internal forces of columns and fully tensioned cables are recalculated. (9) Based on the new locations of cable supports, the cable forms are updated following the procedure described in Section 3.1, and the cable support reactions are recalculated. (10) The above procedure is repeated from Step 2, unless convergence is achieved, in which case the load increment is completed. If after a large number of iterations, no further reduction in the unbalanced forces is observed while displacement increments remain significant, this may be a sign of an unstable structure, where an equilibrium configuration cannot be found. It can be seen that similar to the cable analysis procedure presented in Section 2.2, the static relaxation procedure outlined above is also based on the reduction of an error quantity, namely the unbalanced force vector. 4. Analysis examples Several numerical analyses have been carried out to verify the accuracy and efficiency of the proposed analysis procedure. A few illustrative examples are presented below. 4.1. Analysis of a single cable in three dimensions In order to demonstrate the behavior of the proposed procedure for the analysis of cables, the progressive determination of the form of a cable is illustrated in Fig. 6. The span length of the 36-meter-long cable is taken to be 35 m, support B being 2.5 m higher than support A. This cable with EA = 105 kN is treated as a substructure with 20 segments. The loads applied to the cable at its internal nodes are also shown in Fig. 6. As illustrated, the cable form is initially taken to be parabolic (shown by the dashed curve). In the first iteration, a relatively large change in the cable form is seen, resulting from the large differences of reactions at A from what are needed for equilibrium and compatibility. Then, the proposed procedure attempts to gradually modify the form (dotted curves) towards its final configuration (solid curve). The cable elongation at the end of analysis is obtained to be 20.7 cm. With a convergence tolerance of 0.001 and a proportional gain of 1.0, the number of required iterations to find the final form is 58. The number of required iterations for the analysis of the abovementioned cable is plotted against the proportional gain c in Fig. 7. It can be seen that a large gain (more than 2 in this case) results in analysis instability, while arbitrarily small values only increase the number of required iterations. Hence, a suitable value of gain appears to be the lower limit of the nearly flat part of the curve,
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Fy , kN
20 0 2
0
5
10
15
20
25
30
35
x, m
z, m
-20
0 -2
Fz , kN
0 -4
-20
4 2 0
-40 0
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-2
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-4
y, m
3
4
2
3
1
2
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1
y, m
z, m
x, m
-1
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40
x, m
0
-2
-1
-3
-2
-4
-3
-5
10
0
20
-4 0
5
10
15
20
25
30
35
40
0
x, m
5
10
15
20
25
30
35
40
x, m
Fig. 6. Cable loads along local x-axis and progressive determination of cable form.
position shown by the dash-dot line in Fig. 8. It can be seen that the final result obtained by the proposed method and shown by the solid curves is in agreement with other published results, giving the displacement of the central node as (26.6, 43.1, 2.6). The convergence behavior of the proposed analysis procedure within an analysis step is compared with that of dynamic relaxation method through considering the iterative values of the displacements of the center node in Fig. 9. In order to ensure the stability of dynamic relaxation method, a mass M is assumed to exist in any DF, which should satisfy [10]:
Number of Iterations
1000 800 600 400 200 0 0
0.5
1 γ
1.5
2
M¼
Dt 2 K 2
ð16Þ
Fig. 7. Number of required iterations versus the proportional gain c.
which results in a near-minimum number of iterations and is adequately far from the stability limit. As shown, a larger proportional gain could be selected in this analysis, to safely reduce the number of the required iterations to less than 40. It should be noted that the number of iterations required to determine the cable shape in subsequent analysis steps will be considerably smaller, since they usually constitute smaller changes in cable form than those of the first step. 4.2. Analysis of a benchmark cable-spring assembly The 3DF structure shown in Fig. 8 – considered in reference [4] and some other studies – consists of three interconnected cables that are attached to a vertical spring at the center node. This node can move freely along the horizontal X- and Y-directions. Consistent units are used throughout this section. The structure is subjected to the cable self weights, a temperature change, and a concentrated horizontal load of 1000 units applied to the center node in the positive Y-direction. The system is released from the
where Dt is an arbitrary time step and K is the stiffness in that DF. Then, to achieve a suitable convergence speed, the damping coefficient C should be selected such that a critically-damped system results. However, since the stiffness of the center node is zero in the horizontal plane, such dynamic properties cannot be determined in this manner. For this reason, with a unit time step, the mass is taken to be 500 in all three DF’s (to observe the above stability limit in the vertical direction), while different values of damping coefficient are examined. Here, the residual forces are obtained using the same cable analysis procedure described in Section 2.2. The results show that while the proposed static relaxation method with m = 3/1000 satisfies the tolerance of 0.001 within 56 iterations, none of the selected damping values lead to convergence within the considered 100 iterations. For the proposed static relaxation method, the number of required iterations is plotted against the displacement increment coefficient in Fig. 10, which shows that the above-mentioned value for m is somewhat conservative. For dynamic relaxation method, one can reduce the number of required iterations by rigorously determining the oscillation frequencies in the three DF’s of the center node using direct observation of the response, and then use them to find the critical damping
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300 Original Release Position Concentrated Load at Center in Y -Direction
200
Vertical ( Z ) Spring Constant = 1000
400 350
0
-100
-200
-300 -400
300
w=1 EA = 290000 L = 580
250
z
y
100
Distributed Loads Along Cable Lengths in Y -direction
w=2 EA = 290000 L = 510
-6
Thermal Constant: 6.5x10 Temperature Change: 100 (All Consistent Unites) -300
-200
-100
200 150
w=2 EA = 290000 L = 510
100 50
0
100
200
300
0 -400
400
-300
-200
-100
0
x
100
200
300
400
x
Fig. 8. Three-degree-of-freedom structure subjected to concentrated and distributed loads.
Displacement X
Static R 50
DR C=10000
DR C=1000
DR C=100
DR C=10
DR Kin Damp
40 30 20 10 0
Displacement Y
-10 0
10
20
30
40
0
10
20
30
40
50
60
70
80
90
100
50
60
70
80
90
100
60 40 20 0
Iterations Fig. 9. Comparison of the iterative values of displacements of the center node.
artificial stiffness of 10 units in the horizontal DF’s helps the solution to converge after 82 iterations supports the above statement. In a second analysis of the 3DF structure using the proposed method, the concentrated load is replaced by equivalent
200 Number of Iterations
coefficients in each DF. Still, it should be noted that the previously introduced substructuring technique is used in both analysis methods compared herein to limit the number of DF’s of the problem to three; otherwise, the problem of determination of dynamic relaxation parameters along the cables will be much more difficult. Dynamic relaxation method can also be used with kinetic damping, in which method the kinetic energy of the system is reset to zero whenever it reaches a peak. Again, the mass cannot be determined using the previously-mentioned stability criterion, since two of the three DF’s of the structure have zero stiffness. Using the same mass of 500 units in the top node, it is observed that convergence cannot be achieved due to small oscillations of center node as shown in Fig. 9. This problem persists with other choices of mass, which do not meaningfully affect the number of required iterations. It appears that resetting the maximum kinetic energy to zero fails to reduce and keep the displacement variations below the tolerance. This can be attributed to the fact that the underlying assumption of kinetic damping approach (that the maximum kinetic energy corresponds to a position near equilibrium) cannot be extended to structures with zero or highly nonlinear stiffness in a number of DF’s. The fact that only adding a small
150
100
50
0 0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
ν Fig. 10. Number of required iterations versus the displacement increment coefficient t.
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Fig. 11. Progressive analysis of a slack cable roof system (from left to right: initial cable forms, updated cable forms with supports fixed, and final form with column tops released).
Fig. 12. Extreme loading scenario: gravitational dead load with removal of one of the anchorage cables.
uniformly-distributed loads of 0.625 units acting along the cable lengths in the same positive Y-direction. This is carried out to demonstrate the effectiveness of the proposed method for analysis of cables subjected to lateral loads. The result is shown by dashed curves in Fig. 8. As expected, a reduced displacement of the central point from the original release position is obtained, being (23.4, 23.9, 2.4). 4.3. Analysis of a slack-cable-suspended roof The irregular hexagonal roof system shown in Fig. 11 consists of seven compression-only pin-ended columns located on a slope with lengths ranging from 24 to 30 m. The plan dimensions of this structure are 85 and 70 m along global X and Y directions, respectively. Each of the six columns on the perimeter of the structure are restrained with a pair of fully taut cables that anchor the column tops to the ground. These columns are tilted with a slope of 10% to the sides to increase the stability and load-bearing capacity of the structure. The center column is initially assumed to be vertical. The column tops are interconnected with primary slack cables, which in turn support several uniformly spaced secondary cables that span between primary ones and make a spider web form. This
structure is in fact a typical module of a design candidate for the birds’ park in Tehran (having 15 interconnected hexagonal units covering an area of more than 50,000 m2 – only one unit is presented herein for brevity and clarity of the results). The roof system shown herein has a total of 3297 unrestrained DF’s. By using the approach presented in this paper, the analysis problem reduces to a number of smaller problems, being the analysis of individual cables with at most 60 DF’s, and the overall analysis of columns having 21 DF’s. Fig. 11 shows the analysis results of the slack cable roof system subjected to a uniform gravitational dead load of 250 N/m2. Note that each analysis step is plotted against a dashed outline of the previous step to highlight the corresponding displacements. In the first step, all cables are assumed to have parabolic shapes in their vertical planes. The cable lengths are selected to achieve a near-uniform sag throughout the structure, being 9% of the span length for primary cables and 6% for secondary cables. Next, the cable forms are updated while preventing the translation of their supports; as a result, the primary cables deform in three dimensions in response to the unsymmetrical loads applied from secondary cables. In this step, the cable lengths are revised again to alleviate the uneven sags of the cables as a result of lateral
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movements of primary cables. Finally, the columns are released to gradually move towards achieving equilibrium and balancing the external loads. This usually leads to a lower energy state, through which the lateral deformations of cables are generally reduced. The amount of final displacement at the top of the central column is calculated to be 95.3 cm, mostly towards east. In order to demonstrate the versatility of the proposed procedure to handle very large deformations and resolve numerical instabilities, an extreme loading scenario is assumed for the considered structure. After reaching the equilibrium state for the dead load (Fig. 11), it is assumed that one of the anchorage cables of the southwest column is removed and the structure is reanalyzed. The results shown in Fig. 12 demonstrate a large displacement at the top of the column with a failed cable: 291.1 cm towards east and 43.5 cm in the south direction. The displacement of the central column towards east also increases to 119.2 cm. It should be noted that reaching the equilibrium state in these conditions that constitute sudden changes in the internal force system may take several hundred iterations using the proposed procedure. 5. Conclusions An effective iterative approach is developed for the analysis of variable-form structures consisting of slack cables and bar elements. A specialized substructuring technique is used for the analysis of slack cables to reduce the processing costs and memory requirements, and to alleviate the cable instability issues before reaching the equilibrium state. The procedure to determine the cable form and internal forces uses sensible relations that progressively adapt the cable form to satisfy the equilibrium and compatibility requirements. Furthermore, the singularities at the overall structural level are addressed by recognizing the unstable DF’s and restricting their displacement increments. Using these procedures, the structural forms corresponding to load combinations consisting of gravitational and lateral loads can be accurately determined. The proposed method is successfully applied to the analysis of several slack-cable structures that exhibit considerable degrees of instability. It is shown that this method can be used in
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the analysis of structures with very large rigid-body displacements and rotations, such as the cases wherein collapse scenarios are examined. References [1] Friere AMS, Negrao JHO, Lopes AV. Geometrical nonlinearities on the static analysis of highly flexible steel cable-stayed bridges. Comput Struct 2006;84:2128–40. [2] Krishna P. Cable-suspended roofs. New York: McGraw-Hill; 1978. [3] Buchholdt HA. An introduction to cable roof structures. Cambridge: Cambridge University Press; 1985. [4] Peyrot AH, Goulois AM. Analysis of cable structures. Comput Struct 1979;10:805–13. [5] Bouaanani N, Ighouba M. A novel scheme for large deflection analysis of suspended cables made of linear or nonlinear elastic materials. Adv Eng Softw 2011;42:1009–19. [6] Z. Wang, T. McCarthy, M.N. Sheikh. Taut-slack algorithm for analyzing the geometric nonlinearity of cable structures. In: 21st International offshore and polar engineering conference, United States, 2011. [7] Irvine HM. Cable structures. Dover Publications; 1992. [8] Ozdemir H. A finite element approach for cable problems. Int J Solids Struct 1979;15:427–37. [9] Atai AA, Mioduchowski A. Equilibrium analysis of elasto-plastic cable nets. Comput Struct 1998;66:163–71. [10] Lewis WJ. Tension structures form and behavior. London: Thomas Telford; 2003. [11] West HH. Discretized initial-value analysis of cable nets. Int J Solids Struct 1973;9:1403–20. [12] Jayaraman HB, Knudson WC. A curved element of the analysis of cable structures. Comput Struct 1981;14:325–33. [13] Ivanyi P, Topping BHV. Parallel dynamic relaxation form finding. In: Papadrakakis M, Topping B, editors. Innovative computational methods for structural mechanics. Stirlingshire, UK: Saxe-Coburg Publications; 1999. p. 127–47. [14] Barnes MR. Form finding and analysis of tension structures by dynamic relaxation. Int J Space Struct 1999;14:89–104. [15] Day AS. An introduction to dynamic relaxation. The Engineer 1965;219:218–21. [16] Gosling PD, Korban EA. A bendable finite element for the analysis of flexible cable structures. Finite Elem Anal Des 2001;38:45–63. [17] Impollonia N, Ricciardi G, Saitta F. Statics of elastic cables under 3D point forces. Int J Solids Struct 2011;48:1268–76. [18] The MathWorksÒ Inc, MATLAB user’s guide. Natick, MA; 1994–2011. [19] Beer FP, Johnston ER, Eisenberg ER, Staab GH. Vector mechanics for engineers, statics. 7th ed. McGraw-Hill Science; 2003. [20] Greenberg MD. Advanced engineering mathematics. 2nd ed. Prentice Hall; 1998.