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The influence of the dimension and configuration of geometric imperfections on the static strength of a typical façade scaffolding
archives of civil and mechanical engineering 16 (2016) 269–281
Available online at www.sciencedirect.com
ScienceDirect journal homepage: http://www.elsevier.com/locate/acme
Original Research Article
The influence of the dimension and configuration of geometric imperfections on the static strength of a typical façade scaffolding E. Błazik-Borowa *, J. Gontarz 1 Lublin University of Technology, Faculty of Civil Engineering and Architecture, 40 Nadbystrzycka St., 20-618 Lublin, Poland
article info
abstract
Article history:
This paper describes the results of static non-linear calculations in reference to a typical
Received 23 July 2015
facade scaffolding. The influence of the dimension and localisation of imperfections was
Accepted 23 November 2015
analysed in the researches. It was found that the geometrical imperfections cause the
Available online
increase of internal forces, and the highest increase occurs in the lowest elements. Higher normal stresses were also obtained when imperfections were modelled as regular horizontal
Keywords:
displacements of decks than in the case, when they were arranged according to the form of
Frame scaffolding
buckling. The magnitudes of imperfections primarily influence the increase of the internal
Imperfections
forces in the standards of frames and bracings. Internal forces in bracings are responsive
Static calculations
mainly to imperfections parallel to the scaffolding, regardless of the type of load. In contrast,
Critical buckling analysis
internal forces in standards are sensitive to imperfections in any direction, but mainly when
Finite element method (FEM)
the scaffolding is subjected to a vertical load. # 2015 Politechnika Wrocławska. Published by Elsevier Sp. z o.o. All rights reserved.
1.
Introduction
This article looks at an analysis of the impact of the size and configuration of geometric imperfections on the static strength of scaffolding. A structure is perfect only as an idea, e.g. in the phase of computer calculations. In reality the shapes of designed elements are no longer immaculate, which is especially true for scaffolding. All the elements are to some degree faulty, either in the area of geometry or materials, which results from the production process (thermal and
mechanical treatment), transport, storing, assembly and usage. Moreover, due to the specificity of scaffolding i.e. multiple usage in various configurations, there exists a possibility of permanent plastic deformations. Finally, the last reason of the high importance of the question of geometric imperfections of scaffolding – the play of joints, which may result in tilting of adjacent elements. Such inaccuracies cannot be foreseen or prevented as they tend to be random quantities. Hence the necessity of taking the imperfections into consideration in the design phase as their impact on the static strength of a construction is usually negative.
* Corresponding author. Tel.: +48 81 5384433. E-mail addresses: [email protected] (E. Błazik-Borowa), [email protected] (J. Gontarz). 1 Tel.: +48 81 5384437. http://dx.doi.org/10.1016/j.acme.2015.11.003 1644-9665/# 2015 Politechnika Wrocławska. Published by Elsevier Sp. z o.o. All rights reserved.
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The strength analyses of scaffoldings and the problem of the safety use of scaffoldings are the issues of many papers. Beale described 260 papers in his paper [1]. In this paper authors pay attention only for exemplary papers. Hence the problems of accidents, related to the use of scaffolding on building sites, are described, for example, in such articles as [2] and [3]. In the article [2] authors described the causes of accidents in scaffoldings. They concluded that accidents are caused mainly using of defective components, modification of the structure with no authorisation, and neglecting the barriers. The paper [3] is also about the reasons of accidents, with statistics concerning the accidents and also the probability of its occurrence. The scaffolding static strength analysis and the critical buckling analysis constitute subjects of such papers as [4] and [5]. The article [4] presents a test for a real scaffolding in full scale. The test concerns the changes of behaviour of the structure, due to the increase of the applied load. The article [5] describes the numerical analysis of a singular node from modular scaffolding, where different maximum loads have been designated. Among others papers one can find articles on the static analysis of scaffolding construction, stability analysis, joint rigidity analysis, load capacity of joint testing, as well as three papers [6], [7] and [8] on imperfections in real structures and their influence on the scaffolding behaviour. The article [6] presents results of the tests of local arc imperfections and global tilt imperfections of scaffolding used on various construction sites in Australia. Testing involved 302 measurements of local imperfections (the relation of the standard deviation in the middle to its height) and 80 measurements of global imperfections (the relation of the top standard end deviation to the bottom standard to the total height of the scaffolding). The tests were performed on scaffolding with 200 mm mandrel connected stakes. Observations showed that there is no correlation between local and global imperfections, and, moreover, that their directions were random. An average joint deviation i.e. global imperfections noted amounted to 1.6 mm/m, and the maximal one – 2.8 mm/m. These sizes were calculated as the quotient of deviation to the height of the structure, e.g. for the scaffolding of the height of 24.2 m which, according to the European standards [9], is regarded as the maximal height of the typical average deviation from the surface, it would amount to 38.4 mm, and the maximal deviation – even 67.2 mm. According to the article [6], these values are smaller than the ones approved in the Australian standards. According to the European standard [9] for the pin length at least of 150 mm, the maximal deviation approved equals 5.0 mm/m, but only with regard to the distance equal to the height of one standard, which is generally 10 mm for a 2.0 m standard. Therefore this is a much smaller value than imperfections quoted in paper [6], but they have to be applied on every level of the scaffolding. The sequels to article [6] are articles [7] and [8], where the influence of eccentricities of the clearance point of vertical forces on the scaffolding load bearing capacity with random geometric imperfections has been examined, and the reliability of the design has been assessed. Unfortunately, during the designing stage of scaffolding there is usually no possibility of performing the complete analysis of the influence of the random imperfections arrangement and eccentricities of the
clearance points of forces, therefore in standards [10] and [11] the following solutions have been proposed: applying the horizontal replacement load [11], taking into account the displacement of nodes on platform levels, where displacement of the successive platforms have to be directed in the opposite direction and the direction of the displacement should be consistent with the horizontal load, it usually being a wind [10], applying imperfections according to a buckling form [10] and [11]. If the choice were to be made between the two first methods, then it could be immediately concluded that the execution of geometry changes by the displacements of nodes would be less labour-intense than applying the horizontal replacement load and, additionally, it does not bias the results of the calculations. Moreover, both of these methods should yield similar results. The application of the third method, however, looks utterly different. On the one hand, it is a labour-intense method on account of it requiring the application of complicated geometry describing a buckling. On the other hand, it should be assumed that the arrangement of imperfections in a structure in a way similar to the form of buckling is the most unfavourable and most dangerous situation in the case of scaffolding. For this reason this paper compares the influence of imperfections applied by the second and third methods on the example of façade scaffolding of the height of 24.2 m, it being the typical scaffolding according to standard [9].
2.
Description of numerical analyses
2.1.
Description of scaffolding and its static scheme
The research consists in the comparison of the results of the static calculation analyses with geometric nonlinearity, whose results are obtained on the basis of two methods of arrangement and different magnitudes of imperfections. Material nonlinearity has not been included because the yield strength has not been exceeded in the analysis. The calculations were performed by means of a programme called Autodesk Simulation Mechanical system (earlier Autodesk Algor Simulation Professional). The subject of research was the façade scaffolding. The height of the scaffolding module (the height of the standard) equalled 2.0 m, the length (the length of the ledger) being 3.07 m, and the dimension perpendicular to the wall (the length of the transom) – 0.73 m. The scaffolding consisted of 9 modules, thus the total length of the scaffolding along the wall was 27.63 m. The scaffolding height of 24.2 m (12 modules) was analysed. Three verticals of the scaffolding (the middle one and the two outer ones) were stiffened with the vertical bracings of the length of 3.66 m. On every floor of the scaffolding there were two railing elements applied at the height of 0.5 m and 1.0 m respectively. The scaffolding was based on base jacks which are modelled on joint supports with blocked movement in three directions in numeric models and attached to the walls with anchorage elements of the length of 0.2 m with the blocked movement on the horizontal direction
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Fig. 1 – The schemes of scaffolding: (a) the technical drawing, (b) the calculation model, & the location of anchorage.
perpendicular and parallel to the wall. Anchorages were fixed in standard and transom connections, where every second node (checkerboard pattern) and every node on the outer verticals were anchored. A technical drawing of the analysed structure is shown in Fig. 1a. All the elements were modelled as three-dimensional frame elements, except for truss elements modelling the platform. Exact data about number and type of elements is shown in Table 1. The static scheme of the scaffolding is shown in Fig. 1b. The FEM model consists of 3656 nodes. In Fig. 2 individual model elements are shown and their geometric characteristics are summarised in Table 2. The elements connections were set as follows: a base jack and a frame standard – rigid connection, a frame standard and a frame standard – rigid connection,
a u-transom standard and a frame standard – rigid connection, a ledger and the frame standard – the perfect hinge at the end of the ledger, angle brace (diagonal in the horizontal plane) and a frame standard – the perfect hinge at the end of the brace, a wall tie and the frame standard – rigid connection, a railing horizontal element and the frame standard – the perfect hinge at the end of the railing horizontal element. Each of the frame elements is divided into four FEA elements with the exception of the bracing which are divided into five FEA elements, which means that the elements have a length of about 0.5 m. The static scheme does not include elements irrelevant to the construction stiffness, e.g., toe boards, ladders.
Table 1 – Finite element characteristics. No.
1 2 3 4 5 6 7 8 9
Element
Number of finite elements
Type of finite elements
Base jack f38 4 Standard f48.3 2.7 U-transom Bracing in vertical plane f42.0 2.7 Railing f38.0 2.7 Anchorage f48.3 3.2 Horizontal brace f42.0 2.7 Platform replacement element Platform element used for applying the pressure
80 984 260 180
Beam Beam Beam Beam
912 78 72 756 36
Beam Beam Beam Truss Shell
Fig. 2 – Scaffolding elements: 1 – base jack, 2 – standard, 3 – transom, 4 – bracing in vertical plane, 5 – railing, 6 – anchorage, 7 – horizontal brace, 8 – platform-side, 9 – platform-truss, 10 – surface for distributed load.
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Table 2 – Geometrical characteristics of scaffolding elements. No.
Element
A
J1 2
1 2 3 4 5 6 7
Base jack f38 4 Standard f48.3 2.7 U-transom Bracing in vertical plane f42.0 2.7 Railing f38.0 2.7 Anchorage f48.3 3.2 Horizontal brace f42.0 2.7
J2 4
J3 4
W2 4
3
W3
[cm ]
[cm ]
[cm ]
[cm ]
[cm ]
[cm3]
4.273 3.868 4.175 3.333 2.994 4.534 3.333
12.519 20.178 0.14573 12.932 9.382 23.171 12.932
6.259 10.089 12.751 6.466 4.691 11.586 6.466
6.259 10.089 17.027 6.466 4.691 11.586 6.466
3.294 4.178 4.558 3.079 2.469 4.797 3.079
3.294 4.178 7.095 3.079 2.469 4.797 3.079
A – area of cross section, J1 – torsional resistance, J2 and J3 – moments of inertia, W2 and W3.
The platforms were substituted by truss elements with the horizontal stiffness in both directions as in the real steel deck on the basis of the paper [12], i.e. truss elements arranged as trusses with replacement cross-section A = 3.1 mm2 and sides of the platform with replacement cross-section A = 5.65 mm2 and the modulus of elasticity E = 2108 kPa. In the model there were plate elements with very small stiffness which are used to apply a service load. The structure is made of steel S335J2 with the following properties: mass densityr = 7.87 t/m3, modulus of elasticity E = 2.03108 kPa, Poisson's ratio v = 0.29, strain hardening modulus Eh = 2.03106 kPa, yield strength fy = 2.9105 kPa.
2.2.
Description of calculation variants and methods
Since the object of calculations was primarily to analyse the influence of imperfections, then it was decided not to apply the standard loads, but only their equivalents. Moreover the numerical research is performed separately for each of load type. It is come from the fact that the use of the scaffolding is possible only if the wind velocity is small (EN 12811-1 [10]). It means that the scaffolding should be loaded by one type of an action and it can be either an service vertical load, or horizontal forces along the scaffolding caused by wind or horizontal forces in the perpendicular direction to the scaffolding caused by wind. Therefore, in the numerical analysis the influence of imperfections on scaffolding at the following variants of load configurations was tested (Fig. 3): Case No. 1 – dead load and horizontal load of concentrated forces Fy in the direction perpendicular to the surface of scaffolding, equivalent to the wind pressure of the value of qy;
Case No. 2 – dead load and horizontal load of concentrated forces Fx in the direction parallel to the surface of scaffolding, equivalent to the wind pressure of the value of qx; Case No. 3 – dead load and a uniformly distributed vertical load of the value p, equivalent to the service load. Concentrated forces Fy and Fx from cases no. 1 and no. 2 were determined in the same way as concentrated forces in the case of wind loads interacting on an element surface. In other words, forces were collected from a half of every element which connected to a given node, which are all common nodes of standards and platforms. As stated in the introduction, this work presents the analyses of two methods of applying the imperfections: Method No. 1 – imperfections of the value d applied in a way shown in Fig. 4 i.e. platforms are shifted with respect to each other so that the distance between the standard nodes from neighbouring levels equals d, Method No. 2 – imperfections applied according to the first form of the buckling of the scaffolding of perfect geometry, and the maximal displacement should equal d. In the first method the imperfections are implemented by regular shifts of individual decks with respect to each other by the value d (Fig. 4). In method no. 2, however, the nodes are moved in order to obtain the form of buckling. This procedure is preceded by finding a maximally moved node and then the geometry of the whole arrangement is changed so that the scaffolding receives the shape equivalent to the form of buckling while the displacement of the determined node, the value d changing from 0 mm to 40 mm. The calculations from these two methods of modelling the imperfections were compared in the analysis of results at the same maximum
Fig. 3 – The scheme of load configurations: (a) case No. 1, (b) case No. 2, (c) case No. 3.
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the next stages and as such constitutes an exception. The analysis of the results of numerical researches is shown in the following section.
Fig. 4 – The schema of modelling the geometric imperfections of value d.
values d because during the assembly of scaffolding (there appears no need to check) the frequency of the occurrence of imperfections is not checked, but how large the maximal deviation of the scaffolding is. For every variant of load and a method of taking imperfections into account the models of the following levels of imperfections d: 0, 10, 20, 30, 40 mm were created. For each of the three cases of a load configuration with perfect construction geometry the static analysis was conducted and the critical load multipliers were designated from the linear critical buckling analysis. Whereas the nonlinear static analyses were conducted for every possible parameter configuration described above. The load in nonlinear calculations was increasing evenly in the following steps. This graph also shows that the self weight applied integrally during the first stage of calculations has the constant value in
3.
The analysis of calculation results
3.1.
The results of the critical buckling analysis
The critical buckling analysis has been conducted in order to determine the forms of buckling with regard to the three cases of loads qx = qy = p = 1.0 kN/m2. The first forms of buckling for each case of load configurations are shown in Fig. 5. Also, in this picture the nodes with a maximum resultant displacement are indicated. As it is shown in Fig. 5, for the first buckling form, both the load on the horizontal surface perpendicular to the scaffolding and the vertical one give similar forms with the global stability loss and with the similar value of the critical load multiplier acr. In both cases the standards are seen to deflect in the plane parallel to the plane of the scaffolding. It means that in the cases no. 1 and 3 it is the vertical load that primarily determines the loss of stability. In the first case it is the weight of the structure, while in the case no. 3 it is the sum of the dead load and the vertical load p. However, the horizontal load arranged along the scaffolding causes a completely different buckling form, the buckling of only the lowest central vertical bracing occurring in this case.
3.2.
The results of the nonlinear static analysis
The execution of the planned range of research has required conducting nonlinear calculations for 31 tasks. Such a large number of data requires a preliminary selection of results. For this purpose, Figs. 6 and 7 which include some changes of the maximum stresses in individual elements depending on the changes of the value d were made. As it can be seen in Figs. 6 and 7 the normal stresses in railings are low comparing to the ones in other elements and do not show any sensitivity to the size of the imperfections. In anchorages there exist stresses with significant values for the horizontal perpendicular load qx. This load causes the bending of anchorages and hence the values of stresses increase even in comparison with the situation, when the horizontal load is perpendicular to the scaffolding and anchorages are simply
Fig. 5 – The first forms of buckling of the scaffolding: (a) case no. 1, acr = 2.64, (b) case no. 2, acr = 3.77, (c) case no. 3, acr = 2.80.
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Fig. 6 – The maximal normal stresses in individual groups of elements depending on the value d, using method no. 1 of modelling the imperfections: (a) case no. 1 with imperfections perpendicular to the surface of scaffolding at qy = 2.0 kPa, (b) case no. 2 with imperfections parallel to the surface of scaffolding at qy = 2.0 kPa, (c) case no. 3 with imperfections perpendicular to the scaffolding at p = 8.0 kPa, (d) case no. 3 with imperfections parallel to the scaffolding for p = 8.0 kPa.
pulled out from the wall. However, the values of normal stresses in the cross-section of anchorages are low. The influence of imperfections on stresses in anchorages is visible only in the case of the vertical load p, where, as before, the values in normal stresses in anchorages are low. The main elements ensuring the correct structure behaviour are scaffolding frames consisting of standards and
transoms. Stresses in the transoms of frames show no sensitivity to imperfections. Normal stresses which occur in these elements are caused by the weight of platforms and the vertical load p. If there is no vertical load, then the stress values fall. Here, it is worth noting that at the horizontal parallel load qx stresses are almost twice lower than the ones at the horizontal perpendicular load qy. It results from the fact that
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Fig. 7 – The maximal normal stresses in individual groups of elements depending on the value d, using method no. 2 of modelling the imperfections: (a) case no. 1 at qy = 2.0 kPa, (b) case no. 2 at qx = 2.0 kPa, (c) case no. 3 at p = 8.0 kPa.
transoms in every other non-anchored node are significantly bent, but it has no relation to the values of the imperfections. Normal stresses in standards of frames adopt the highest values. In the case of horizontal loads standards are bent, and in the case of vertical loads most of them compressed. So this happens in the situations when element compressing decides about the value of stress that the clear influence of imperfections on stress values can be seen. In Figs. 8–10 changes of internal forces in standards depending on values of loads and imperfections are shown. These drawings refer to the situation when the structure is subjected to the vertical load p, and the results of calculations relate to the most strained element i.e. the node located in middle standard on the lowest level (point A in Fig. 1b). At the same time this is a point of the biggest displacement in the buckling form caused by the vertical load p. In these drawings one can clearly see that the axial force does not depend on imperfections, while imperfections exert a
significant influence on the bending moments. The values of bending moments M2 (bending moments M2 – bending on the plane of the scaffolding and bending moments M3 – bending on the plane perpendicular to the scaffolding) at p = 4.5 kPa (the sum of design loads from the highest platform and the security platform for the third class of loads according to the standard [9]) amounts to 0.3 kNm (Fig. 9b), which constitutes 25% of the value at which the yield strength is achieved (Mzg = fy W2 = 290,000 kPa 4.17810 6 m3 = 1.21 kNm). This significant increase in bending results of course from the increase in the eccentrics in a structure subjected to vertical loads. Figs. 9 and 10 refer to two methods of modelling imperfections, however, as it can be seen, the results of the calculations shown in these drawings are close to each other. This is due to the fact that Fig. 9 shows the results of the calculations conducted with regard to imperfections modelled by method no. 1 and in the surface perpendicular to the
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Fig. 8 – The internal forces in the standard at point A resulting from the load p at imperfections perpendicular to the scaffolding and applied by method no 1: (a) axial forces N, (b) bending moments M3, the values of bending moments M2 are very small.
Fig. 9 – The internal forces in the standard in point A from the load p with imperfections parallel to the scaffolding and applied by method No 1: (a) axial forces N, (b) bending moments M2, the values of bending moments M3 are very small.
Fig. 10 – The internal forces in the standard at point A resulting from the service load p at imperfections introduced by method no. 2: (a) axial forces N, (b) bending moments M2, the values of bending moments M3 are very small.
scaffolding, while Fig. 10 shows imperfections modelled by method no. 2, where the imperfections make the buckling form that mainly consisted of shifts in the surface parallel to the scaffolding. Of course, in method No. 2 there are imperfections to be seen with the component perpendicular to the surface of the scaffolding because in Fig. 10c the values of bending moments are significantly higher than in Fig. 9c where they can be considered negligibly small. This results in higher stresses in standards (compared to Figs. 6d and 7c)
obtained by means of calculations in which the imperfections are modelled by method no. 2. Normal stresses in bracings in all situations are sensitive to the changes of imperfections. Both changes of axial forces and bending moments have an influence on them. Figs. 11 and 12 show the changes of axial forces in bracings arranged in the scheme of the scaffolding. The bar graphs are placed on the scaffolding scheme at bracings for which they present results. As shown, some changes of axial forces in bracings occur over
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Fig. 11 – The distribution of axial forces in bracings depending on d when using method no. 1 of modelling the imperfections: (a) case no. 1 with imperfections directed perpendicularly to the surface of the scaffolding at qy = 2.0 kPa, (b) case no. 2 with imperfections directed in parallel to the surface of the scaffolding at qy = 2.0 kPa, (c) case no. 3 with imperfections perpendicular to the surface of scaffolding at p = 8.0 kPa, (d) case no. 3 with imperfections parallel to the surface of scaffolding at p = 8.0 kPa. d = 0.0 cm, d = 2.0 cm, d = 4.0 cm.
the whole structure. Larger differences between the values of axial forces, obtained for various values of imperfections, are visible when imperfections are modelled by method no. 2. The remaining results of calculations for bracings i.e. (which means) changes of internal forces depending on the load and imperfections are shown in Figs. 13–17. The results relate to the most strained bracing in a node located in point B (Fig. 1b) and simultaneously the one in which the node with the biggest displacement) in the buckling form obtained for the horizontal parallel load (Fig. 5b) was found. This convergence results from bracings working mainly at the horizontal load qx. In the case of the horizontal load perpendicular to the surface of the scaffolding qy, the values of all internal forces are much lower (Figs. 13 and 16), which results in far fewer stresses
occurring in bracings in this case, whatever the method of modelling imperfections is applied. In the case of the horizontal load qx, it is not only the stresses with significant values that are achieved but also distinct sensitivity to the changes of imperfections is noticeable. Like in the case of standards, it is a result of the formation of an eccentric subjected to the axial force, which in turn results in a significant increase in the bending moment M2 associated with the bending in the surface of the scaffolding (Figs. 14 and 17). In the analysed range of loads and imperfections the maximum value M2 = 0.6 kNm has been reached, which constitutes 33% of the value at which the yield strength is achieved (Mzg = fyW2 = 290,000 kPa 3.08010 6 m3 = 0.89 kNm). The values of axial forces alter slightly (Figs. 11b, 12b, 14a and 17a) following an increase of
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Fig. 12 – The distribution of axial forces in bracings depending on d when using method no. 2 of modelling the imperfections: (a) case no. 1 at qy = 2.0 kPa, (b) case no. 2 at qx = 2.0 kPa, (c) case no 3 at p = 8.0 kPa. d = 0.0 cm, d = 2.0 cm, d = 4.0 cm.
Fig. 13 – Changes in the internal forces in the bracing at point B resulting from the load qy at imperfections perpendicular to the scaffolding and introduced by method no. 1: (a) axial forces N, (b) bending moments M2, the values of bending moments M3 are very small.
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Fig. 14 – Changes in the internal forces in the bracing at point B resulting from the load qx at (with) imperfections parallel to the scaffolding and introduced by method no. 1: (a) axial forces N, (b) bending moments M2, the values of bending moments M3 are very small.
Fig. 15 – Changes in the internal forces in the bracing at point B resulting from the load p at imperfections parallel to the scaffolding and introduced by method no. 1: (a) axial forces N, (b) bending moments M2, the values of bending moments M3 are very small.
Fig. 16 – Changes in the internal forces in the bracing at point B resulting from the load qy at imperfections applied by method no. 2: (a) axial forces N, (b) bending moments M2, the values of bending moments M3 are very small.
imperfections even though they have fairly large values. In the case of the analysed loads, axial forces even reached as high a value as that of 8.0 kN at the critical force for this element amounting to 9.50 kN.
In the case of the vertical load p (case no. 3), the values of each of the internal forces in the bracings are higher than those in the case of the horizontal perpendicular load qy (case No. 1), however significantly lower than in the case
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Fig. 17 – Changes in the internal forces in the bracing at point B resulting from the load qx at imperfections introduced by method no. 2: (a) axial forces N, (b) bending moments M2, the values of bending moments M3 are very small.
of the horizontal load parallel to the scaffolding qx (case No. 2). In conclusion, the standards are subject to an effort increase due to the presence of imperfections in the case of the vertical load p (case No. 3), and the bracings in the case of the horizontal parallel load qx (case No 2). The paper does not discuss in detail the behaviour of the other elements because either normal stresses in the elements adopt negligible values compared to a yield strength or show no sensitivity to imperfections.
4.
Conclusions
The article confirms that the imperfections exert an adverse effect on the internal forces in the elements of the scaffolding. The magnitudes of imperfections primarily influence the increase of the internal forces in the standards of frames and bracings. Internal forces in bracings are responsive mainly to imperfections parallel to the scaffolding, regardless of the type of load. In contrast, internal forces in standards are sensitive to imperfections in any direction, but mainly when the scaffolding is subjected to a vertical load. Moreover, the influence of imperfections on the internal forces is the strongest in the lower elements of scaffolding, there appearing the greatest axial forces. Imperfections cause that the axial forces in elements affect eccentricities, which leads to the increase in bending moments. With the increase in loads the sensitivity of internal forces to imperfections increases. In some cases, also a significant increase in the sensitivity of bending moments to the increase of imperfections can also be seen. Internal forces and stresses in the structure with imperfections modelled by means of various methods adopt different values. The differences are the most remarkable in the upper part of the structure, where in the case of method no. 1 there appear large imperfections, while in the case of method no. 2 the deflections of nodes are negligible. In the lower part of the structure the differences between internal forces decrease on account of the decrease in the differences in magnitudes of
imperfections. Finally, as it can be seen in Figs. 6 and 7, if the imperfections have the same direction, the maximum stresses in the elements are on a similar level. In other words, the values of maximum stresses are not determined so much by the number of imperfections, but rather by what values they adopt. Finally, some practical conclusions can be drawn from the work, which are as follows: actually, both methods of modelling imperfections in scaffolding of the height of up to 24 m give similar values of stresses, so it is recommended that at the design stage of scaffolding the method of modelling imperfections that is easier to use in a given computer programme should be adopted in engineering practice, since the largest increase in internal forces occurs in the lower layer of scaffolding, in the case of high scaffolding, the construction of stronger elements of lower layers of the structure should be considered. The study analysed the typical scaffolding, where the range of generated stresses does not exceed the yield strength. In the future, higher scaffolding should be examined, while taking into account the non-linearity of materials and also the problem, not mentioned here, concerning anchorages. In some cases, some anchorages ceased to work, which resulted in overloading the other ones. This paper does not discuss this problem in detail because in the situations when the sign of an axial force changes, in the anchorages their small values were found. In the case of higher wind loads, e.g. higher scaffolding, taking into account the load changes together with the height of the structure or with the net, the problem may be significant. Therefore, a separate paper is expected to examine this problem.
Acknowledgement This work was financially supported by the statutory budget funds of Faculty of Civil Engineering and Architecture in Lublin University of Technology.
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[6] T. Chandrangsu, K.J.R. Rasmussen, Investigation of geometric imperfections and joint stiffness of support scaffold system, Journal of Constructional Steel Research 67 (2011) 576–579. [7] T. Chandrangsu, K.J.R. Rasmussen, Structural modeling of support scaffold system, Journal of Constructional Steel Research 67 (2011) 866–875. [8] H. Zhang, T. Chandrangsu, K.J.R. Rasmussen, Probabilistic study of the strength of steel scaffold system, Structural Safety 32 (2010) 393–401. [9] EN 12810-1: Façade scaffolds made of prefabricated components – Part 1: Products specifications. [10] EN 12811-1. Temporary works equipment – Part 1: Scaffolds – Performance requirements and general design. [11] EN 1993-1: Design of steel structures – Part 1: Generals rules and rules for buildings. [12] A. Robak, Capacity analysis of steel scaffolding decks, Civil Engineering and Architecture 13 (2) (2014) 357–365.
Available online at www.sciencedirect.com
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Original Research Article
The influence of the dimension and configuration of geometric imperfections on the static strength of a typical façade scaffolding E. Błazik-Borowa *, J. Gontarz 1 Lublin University of Technology, Faculty of Civil Engineering and Architecture, 40 Nadbystrzycka St., 20-618 Lublin, Poland
article info
abstract
Article history:
This paper describes the results of static non-linear calculations in reference to a typical
Received 23 July 2015
facade scaffolding. The influence of the dimension and localisation of imperfections was
Accepted 23 November 2015
analysed in the researches. It was found that the geometrical imperfections cause the
Available online
increase of internal forces, and the highest increase occurs in the lowest elements. Higher normal stresses were also obtained when imperfections were modelled as regular horizontal
Keywords:
displacements of decks than in the case, when they were arranged according to the form of
Frame scaffolding
buckling. The magnitudes of imperfections primarily influence the increase of the internal
Imperfections
forces in the standards of frames and bracings. Internal forces in bracings are responsive
Static calculations
mainly to imperfections parallel to the scaffolding, regardless of the type of load. In contrast,
Critical buckling analysis
internal forces in standards are sensitive to imperfections in any direction, but mainly when
Finite element method (FEM)
the scaffolding is subjected to a vertical load. # 2015 Politechnika Wrocławska. Published by Elsevier Sp. z o.o. All rights reserved.
1.
Introduction
This article looks at an analysis of the impact of the size and configuration of geometric imperfections on the static strength of scaffolding. A structure is perfect only as an idea, e.g. in the phase of computer calculations. In reality the shapes of designed elements are no longer immaculate, which is especially true for scaffolding. All the elements are to some degree faulty, either in the area of geometry or materials, which results from the production process (thermal and
mechanical treatment), transport, storing, assembly and usage. Moreover, due to the specificity of scaffolding i.e. multiple usage in various configurations, there exists a possibility of permanent plastic deformations. Finally, the last reason of the high importance of the question of geometric imperfections of scaffolding – the play of joints, which may result in tilting of adjacent elements. Such inaccuracies cannot be foreseen or prevented as they tend to be random quantities. Hence the necessity of taking the imperfections into consideration in the design phase as their impact on the static strength of a construction is usually negative.
* Corresponding author. Tel.: +48 81 5384433. E-mail addresses: [email protected] (E. Błazik-Borowa), [email protected] (J. Gontarz). 1 Tel.: +48 81 5384437. http://dx.doi.org/10.1016/j.acme.2015.11.003 1644-9665/# 2015 Politechnika Wrocławska. Published by Elsevier Sp. z o.o. All rights reserved.
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The strength analyses of scaffoldings and the problem of the safety use of scaffoldings are the issues of many papers. Beale described 260 papers in his paper [1]. In this paper authors pay attention only for exemplary papers. Hence the problems of accidents, related to the use of scaffolding on building sites, are described, for example, in such articles as [2] and [3]. In the article [2] authors described the causes of accidents in scaffoldings. They concluded that accidents are caused mainly using of defective components, modification of the structure with no authorisation, and neglecting the barriers. The paper [3] is also about the reasons of accidents, with statistics concerning the accidents and also the probability of its occurrence. The scaffolding static strength analysis and the critical buckling analysis constitute subjects of such papers as [4] and [5]. The article [4] presents a test for a real scaffolding in full scale. The test concerns the changes of behaviour of the structure, due to the increase of the applied load. The article [5] describes the numerical analysis of a singular node from modular scaffolding, where different maximum loads have been designated. Among others papers one can find articles on the static analysis of scaffolding construction, stability analysis, joint rigidity analysis, load capacity of joint testing, as well as three papers [6], [7] and [8] on imperfections in real structures and their influence on the scaffolding behaviour. The article [6] presents results of the tests of local arc imperfections and global tilt imperfections of scaffolding used on various construction sites in Australia. Testing involved 302 measurements of local imperfections (the relation of the standard deviation in the middle to its height) and 80 measurements of global imperfections (the relation of the top standard end deviation to the bottom standard to the total height of the scaffolding). The tests were performed on scaffolding with 200 mm mandrel connected stakes. Observations showed that there is no correlation between local and global imperfections, and, moreover, that their directions were random. An average joint deviation i.e. global imperfections noted amounted to 1.6 mm/m, and the maximal one – 2.8 mm/m. These sizes were calculated as the quotient of deviation to the height of the structure, e.g. for the scaffolding of the height of 24.2 m which, according to the European standards [9], is regarded as the maximal height of the typical average deviation from the surface, it would amount to 38.4 mm, and the maximal deviation – even 67.2 mm. According to the article [6], these values are smaller than the ones approved in the Australian standards. According to the European standard [9] for the pin length at least of 150 mm, the maximal deviation approved equals 5.0 mm/m, but only with regard to the distance equal to the height of one standard, which is generally 10 mm for a 2.0 m standard. Therefore this is a much smaller value than imperfections quoted in paper [6], but they have to be applied on every level of the scaffolding. The sequels to article [6] are articles [7] and [8], where the influence of eccentricities of the clearance point of vertical forces on the scaffolding load bearing capacity with random geometric imperfections has been examined, and the reliability of the design has been assessed. Unfortunately, during the designing stage of scaffolding there is usually no possibility of performing the complete analysis of the influence of the random imperfections arrangement and eccentricities of the
clearance points of forces, therefore in standards [10] and [11] the following solutions have been proposed: applying the horizontal replacement load [11], taking into account the displacement of nodes on platform levels, where displacement of the successive platforms have to be directed in the opposite direction and the direction of the displacement should be consistent with the horizontal load, it usually being a wind [10], applying imperfections according to a buckling form [10] and [11]. If the choice were to be made between the two first methods, then it could be immediately concluded that the execution of geometry changes by the displacements of nodes would be less labour-intense than applying the horizontal replacement load and, additionally, it does not bias the results of the calculations. Moreover, both of these methods should yield similar results. The application of the third method, however, looks utterly different. On the one hand, it is a labour-intense method on account of it requiring the application of complicated geometry describing a buckling. On the other hand, it should be assumed that the arrangement of imperfections in a structure in a way similar to the form of buckling is the most unfavourable and most dangerous situation in the case of scaffolding. For this reason this paper compares the influence of imperfections applied by the second and third methods on the example of façade scaffolding of the height of 24.2 m, it being the typical scaffolding according to standard [9].
2.
Description of numerical analyses
2.1.
Description of scaffolding and its static scheme
The research consists in the comparison of the results of the static calculation analyses with geometric nonlinearity, whose results are obtained on the basis of two methods of arrangement and different magnitudes of imperfections. Material nonlinearity has not been included because the yield strength has not been exceeded in the analysis. The calculations were performed by means of a programme called Autodesk Simulation Mechanical system (earlier Autodesk Algor Simulation Professional). The subject of research was the façade scaffolding. The height of the scaffolding module (the height of the standard) equalled 2.0 m, the length (the length of the ledger) being 3.07 m, and the dimension perpendicular to the wall (the length of the transom) – 0.73 m. The scaffolding consisted of 9 modules, thus the total length of the scaffolding along the wall was 27.63 m. The scaffolding height of 24.2 m (12 modules) was analysed. Three verticals of the scaffolding (the middle one and the two outer ones) were stiffened with the vertical bracings of the length of 3.66 m. On every floor of the scaffolding there were two railing elements applied at the height of 0.5 m and 1.0 m respectively. The scaffolding was based on base jacks which are modelled on joint supports with blocked movement in three directions in numeric models and attached to the walls with anchorage elements of the length of 0.2 m with the blocked movement on the horizontal direction
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Fig. 1 – The schemes of scaffolding: (a) the technical drawing, (b) the calculation model, & the location of anchorage.
perpendicular and parallel to the wall. Anchorages were fixed in standard and transom connections, where every second node (checkerboard pattern) and every node on the outer verticals were anchored. A technical drawing of the analysed structure is shown in Fig. 1a. All the elements were modelled as three-dimensional frame elements, except for truss elements modelling the platform. Exact data about number and type of elements is shown in Table 1. The static scheme of the scaffolding is shown in Fig. 1b. The FEM model consists of 3656 nodes. In Fig. 2 individual model elements are shown and their geometric characteristics are summarised in Table 2. The elements connections were set as follows: a base jack and a frame standard – rigid connection, a frame standard and a frame standard – rigid connection,
a u-transom standard and a frame standard – rigid connection, a ledger and the frame standard – the perfect hinge at the end of the ledger, angle brace (diagonal in the horizontal plane) and a frame standard – the perfect hinge at the end of the brace, a wall tie and the frame standard – rigid connection, a railing horizontal element and the frame standard – the perfect hinge at the end of the railing horizontal element. Each of the frame elements is divided into four FEA elements with the exception of the bracing which are divided into five FEA elements, which means that the elements have a length of about 0.5 m. The static scheme does not include elements irrelevant to the construction stiffness, e.g., toe boards, ladders.
Table 1 – Finite element characteristics. No.
1 2 3 4 5 6 7 8 9
Element
Number of finite elements
Type of finite elements
Base jack f38 4 Standard f48.3 2.7 U-transom Bracing in vertical plane f42.0 2.7 Railing f38.0 2.7 Anchorage f48.3 3.2 Horizontal brace f42.0 2.7 Platform replacement element Platform element used for applying the pressure
80 984 260 180
Beam Beam Beam Beam
912 78 72 756 36
Beam Beam Beam Truss Shell
Fig. 2 – Scaffolding elements: 1 – base jack, 2 – standard, 3 – transom, 4 – bracing in vertical plane, 5 – railing, 6 – anchorage, 7 – horizontal brace, 8 – platform-side, 9 – platform-truss, 10 – surface for distributed load.
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Table 2 – Geometrical characteristics of scaffolding elements. No.
Element
A
J1 2
1 2 3 4 5 6 7
Base jack f38 4 Standard f48.3 2.7 U-transom Bracing in vertical plane f42.0 2.7 Railing f38.0 2.7 Anchorage f48.3 3.2 Horizontal brace f42.0 2.7
J2 4
J3 4
W2 4
3
W3
[cm ]
[cm ]
[cm ]
[cm ]
[cm ]
[cm3]
4.273 3.868 4.175 3.333 2.994 4.534 3.333
12.519 20.178 0.14573 12.932 9.382 23.171 12.932
6.259 10.089 12.751 6.466 4.691 11.586 6.466
6.259 10.089 17.027 6.466 4.691 11.586 6.466
3.294 4.178 4.558 3.079 2.469 4.797 3.079
3.294 4.178 7.095 3.079 2.469 4.797 3.079
A – area of cross section, J1 – torsional resistance, J2 and J3 – moments of inertia, W2 and W3.
The platforms were substituted by truss elements with the horizontal stiffness in both directions as in the real steel deck on the basis of the paper [12], i.e. truss elements arranged as trusses with replacement cross-section A = 3.1 mm2 and sides of the platform with replacement cross-section A = 5.65 mm2 and the modulus of elasticity E = 2108 kPa. In the model there were plate elements with very small stiffness which are used to apply a service load. The structure is made of steel S335J2 with the following properties: mass densityr = 7.87 t/m3, modulus of elasticity E = 2.03108 kPa, Poisson's ratio v = 0.29, strain hardening modulus Eh = 2.03106 kPa, yield strength fy = 2.9105 kPa.
2.2.
Description of calculation variants and methods
Since the object of calculations was primarily to analyse the influence of imperfections, then it was decided not to apply the standard loads, but only their equivalents. Moreover the numerical research is performed separately for each of load type. It is come from the fact that the use of the scaffolding is possible only if the wind velocity is small (EN 12811-1 [10]). It means that the scaffolding should be loaded by one type of an action and it can be either an service vertical load, or horizontal forces along the scaffolding caused by wind or horizontal forces in the perpendicular direction to the scaffolding caused by wind. Therefore, in the numerical analysis the influence of imperfections on scaffolding at the following variants of load configurations was tested (Fig. 3): Case No. 1 – dead load and horizontal load of concentrated forces Fy in the direction perpendicular to the surface of scaffolding, equivalent to the wind pressure of the value of qy;
Case No. 2 – dead load and horizontal load of concentrated forces Fx in the direction parallel to the surface of scaffolding, equivalent to the wind pressure of the value of qx; Case No. 3 – dead load and a uniformly distributed vertical load of the value p, equivalent to the service load. Concentrated forces Fy and Fx from cases no. 1 and no. 2 were determined in the same way as concentrated forces in the case of wind loads interacting on an element surface. In other words, forces were collected from a half of every element which connected to a given node, which are all common nodes of standards and platforms. As stated in the introduction, this work presents the analyses of two methods of applying the imperfections: Method No. 1 – imperfections of the value d applied in a way shown in Fig. 4 i.e. platforms are shifted with respect to each other so that the distance between the standard nodes from neighbouring levels equals d, Method No. 2 – imperfections applied according to the first form of the buckling of the scaffolding of perfect geometry, and the maximal displacement should equal d. In the first method the imperfections are implemented by regular shifts of individual decks with respect to each other by the value d (Fig. 4). In method no. 2, however, the nodes are moved in order to obtain the form of buckling. This procedure is preceded by finding a maximally moved node and then the geometry of the whole arrangement is changed so that the scaffolding receives the shape equivalent to the form of buckling while the displacement of the determined node, the value d changing from 0 mm to 40 mm. The calculations from these two methods of modelling the imperfections were compared in the analysis of results at the same maximum
Fig. 3 – The scheme of load configurations: (a) case No. 1, (b) case No. 2, (c) case No. 3.
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the next stages and as such constitutes an exception. The analysis of the results of numerical researches is shown in the following section.
Fig. 4 – The schema of modelling the geometric imperfections of value d.
values d because during the assembly of scaffolding (there appears no need to check) the frequency of the occurrence of imperfections is not checked, but how large the maximal deviation of the scaffolding is. For every variant of load and a method of taking imperfections into account the models of the following levels of imperfections d: 0, 10, 20, 30, 40 mm were created. For each of the three cases of a load configuration with perfect construction geometry the static analysis was conducted and the critical load multipliers were designated from the linear critical buckling analysis. Whereas the nonlinear static analyses were conducted for every possible parameter configuration described above. The load in nonlinear calculations was increasing evenly in the following steps. This graph also shows that the self weight applied integrally during the first stage of calculations has the constant value in
3.
The analysis of calculation results
3.1.
The results of the critical buckling analysis
The critical buckling analysis has been conducted in order to determine the forms of buckling with regard to the three cases of loads qx = qy = p = 1.0 kN/m2. The first forms of buckling for each case of load configurations are shown in Fig. 5. Also, in this picture the nodes with a maximum resultant displacement are indicated. As it is shown in Fig. 5, for the first buckling form, both the load on the horizontal surface perpendicular to the scaffolding and the vertical one give similar forms with the global stability loss and with the similar value of the critical load multiplier acr. In both cases the standards are seen to deflect in the plane parallel to the plane of the scaffolding. It means that in the cases no. 1 and 3 it is the vertical load that primarily determines the loss of stability. In the first case it is the weight of the structure, while in the case no. 3 it is the sum of the dead load and the vertical load p. However, the horizontal load arranged along the scaffolding causes a completely different buckling form, the buckling of only the lowest central vertical bracing occurring in this case.
3.2.
The results of the nonlinear static analysis
The execution of the planned range of research has required conducting nonlinear calculations for 31 tasks. Such a large number of data requires a preliminary selection of results. For this purpose, Figs. 6 and 7 which include some changes of the maximum stresses in individual elements depending on the changes of the value d were made. As it can be seen in Figs. 6 and 7 the normal stresses in railings are low comparing to the ones in other elements and do not show any sensitivity to the size of the imperfections. In anchorages there exist stresses with significant values for the horizontal perpendicular load qx. This load causes the bending of anchorages and hence the values of stresses increase even in comparison with the situation, when the horizontal load is perpendicular to the scaffolding and anchorages are simply
Fig. 5 – The first forms of buckling of the scaffolding: (a) case no. 1, acr = 2.64, (b) case no. 2, acr = 3.77, (c) case no. 3, acr = 2.80.
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Fig. 6 – The maximal normal stresses in individual groups of elements depending on the value d, using method no. 1 of modelling the imperfections: (a) case no. 1 with imperfections perpendicular to the surface of scaffolding at qy = 2.0 kPa, (b) case no. 2 with imperfections parallel to the surface of scaffolding at qy = 2.0 kPa, (c) case no. 3 with imperfections perpendicular to the scaffolding at p = 8.0 kPa, (d) case no. 3 with imperfections parallel to the scaffolding for p = 8.0 kPa.
pulled out from the wall. However, the values of normal stresses in the cross-section of anchorages are low. The influence of imperfections on stresses in anchorages is visible only in the case of the vertical load p, where, as before, the values in normal stresses in anchorages are low. The main elements ensuring the correct structure behaviour are scaffolding frames consisting of standards and
transoms. Stresses in the transoms of frames show no sensitivity to imperfections. Normal stresses which occur in these elements are caused by the weight of platforms and the vertical load p. If there is no vertical load, then the stress values fall. Here, it is worth noting that at the horizontal parallel load qx stresses are almost twice lower than the ones at the horizontal perpendicular load qy. It results from the fact that
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Fig. 7 – The maximal normal stresses in individual groups of elements depending on the value d, using method no. 2 of modelling the imperfections: (a) case no. 1 at qy = 2.0 kPa, (b) case no. 2 at qx = 2.0 kPa, (c) case no. 3 at p = 8.0 kPa.
transoms in every other non-anchored node are significantly bent, but it has no relation to the values of the imperfections. Normal stresses in standards of frames adopt the highest values. In the case of horizontal loads standards are bent, and in the case of vertical loads most of them compressed. So this happens in the situations when element compressing decides about the value of stress that the clear influence of imperfections on stress values can be seen. In Figs. 8–10 changes of internal forces in standards depending on values of loads and imperfections are shown. These drawings refer to the situation when the structure is subjected to the vertical load p, and the results of calculations relate to the most strained element i.e. the node located in middle standard on the lowest level (point A in Fig. 1b). At the same time this is a point of the biggest displacement in the buckling form caused by the vertical load p. In these drawings one can clearly see that the axial force does not depend on imperfections, while imperfections exert a
significant influence on the bending moments. The values of bending moments M2 (bending moments M2 – bending on the plane of the scaffolding and bending moments M3 – bending on the plane perpendicular to the scaffolding) at p = 4.5 kPa (the sum of design loads from the highest platform and the security platform for the third class of loads according to the standard [9]) amounts to 0.3 kNm (Fig. 9b), which constitutes 25% of the value at which the yield strength is achieved (Mzg = fy W2 = 290,000 kPa 4.17810 6 m3 = 1.21 kNm). This significant increase in bending results of course from the increase in the eccentrics in a structure subjected to vertical loads. Figs. 9 and 10 refer to two methods of modelling imperfections, however, as it can be seen, the results of the calculations shown in these drawings are close to each other. This is due to the fact that Fig. 9 shows the results of the calculations conducted with regard to imperfections modelled by method no. 1 and in the surface perpendicular to the
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Fig. 8 – The internal forces in the standard at point A resulting from the load p at imperfections perpendicular to the scaffolding and applied by method no 1: (a) axial forces N, (b) bending moments M3, the values of bending moments M2 are very small.
Fig. 9 – The internal forces in the standard in point A from the load p with imperfections parallel to the scaffolding and applied by method No 1: (a) axial forces N, (b) bending moments M2, the values of bending moments M3 are very small.
Fig. 10 – The internal forces in the standard at point A resulting from the service load p at imperfections introduced by method no. 2: (a) axial forces N, (b) bending moments M2, the values of bending moments M3 are very small.
scaffolding, while Fig. 10 shows imperfections modelled by method no. 2, where the imperfections make the buckling form that mainly consisted of shifts in the surface parallel to the scaffolding. Of course, in method No. 2 there are imperfections to be seen with the component perpendicular to the surface of the scaffolding because in Fig. 10c the values of bending moments are significantly higher than in Fig. 9c where they can be considered negligibly small. This results in higher stresses in standards (compared to Figs. 6d and 7c)
obtained by means of calculations in which the imperfections are modelled by method no. 2. Normal stresses in bracings in all situations are sensitive to the changes of imperfections. Both changes of axial forces and bending moments have an influence on them. Figs. 11 and 12 show the changes of axial forces in bracings arranged in the scheme of the scaffolding. The bar graphs are placed on the scaffolding scheme at bracings for which they present results. As shown, some changes of axial forces in bracings occur over
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Fig. 11 – The distribution of axial forces in bracings depending on d when using method no. 1 of modelling the imperfections: (a) case no. 1 with imperfections directed perpendicularly to the surface of the scaffolding at qy = 2.0 kPa, (b) case no. 2 with imperfections directed in parallel to the surface of the scaffolding at qy = 2.0 kPa, (c) case no. 3 with imperfections perpendicular to the surface of scaffolding at p = 8.0 kPa, (d) case no. 3 with imperfections parallel to the surface of scaffolding at p = 8.0 kPa. d = 0.0 cm, d = 2.0 cm, d = 4.0 cm.
the whole structure. Larger differences between the values of axial forces, obtained for various values of imperfections, are visible when imperfections are modelled by method no. 2. The remaining results of calculations for bracings i.e. (which means) changes of internal forces depending on the load and imperfections are shown in Figs. 13–17. The results relate to the most strained bracing in a node located in point B (Fig. 1b) and simultaneously the one in which the node with the biggest displacement) in the buckling form obtained for the horizontal parallel load (Fig. 5b) was found. This convergence results from bracings working mainly at the horizontal load qx. In the case of the horizontal load perpendicular to the surface of the scaffolding qy, the values of all internal forces are much lower (Figs. 13 and 16), which results in far fewer stresses
occurring in bracings in this case, whatever the method of modelling imperfections is applied. In the case of the horizontal load qx, it is not only the stresses with significant values that are achieved but also distinct sensitivity to the changes of imperfections is noticeable. Like in the case of standards, it is a result of the formation of an eccentric subjected to the axial force, which in turn results in a significant increase in the bending moment M2 associated with the bending in the surface of the scaffolding (Figs. 14 and 17). In the analysed range of loads and imperfections the maximum value M2 = 0.6 kNm has been reached, which constitutes 33% of the value at which the yield strength is achieved (Mzg = fyW2 = 290,000 kPa 3.08010 6 m3 = 0.89 kNm). The values of axial forces alter slightly (Figs. 11b, 12b, 14a and 17a) following an increase of
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Fig. 12 – The distribution of axial forces in bracings depending on d when using method no. 2 of modelling the imperfections: (a) case no. 1 at qy = 2.0 kPa, (b) case no. 2 at qx = 2.0 kPa, (c) case no 3 at p = 8.0 kPa. d = 0.0 cm, d = 2.0 cm, d = 4.0 cm.
Fig. 13 – Changes in the internal forces in the bracing at point B resulting from the load qy at imperfections perpendicular to the scaffolding and introduced by method no. 1: (a) axial forces N, (b) bending moments M2, the values of bending moments M3 are very small.
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Fig. 14 – Changes in the internal forces in the bracing at point B resulting from the load qx at (with) imperfections parallel to the scaffolding and introduced by method no. 1: (a) axial forces N, (b) bending moments M2, the values of bending moments M3 are very small.
Fig. 15 – Changes in the internal forces in the bracing at point B resulting from the load p at imperfections parallel to the scaffolding and introduced by method no. 1: (a) axial forces N, (b) bending moments M2, the values of bending moments M3 are very small.
Fig. 16 – Changes in the internal forces in the bracing at point B resulting from the load qy at imperfections applied by method no. 2: (a) axial forces N, (b) bending moments M2, the values of bending moments M3 are very small.
imperfections even though they have fairly large values. In the case of the analysed loads, axial forces even reached as high a value as that of 8.0 kN at the critical force for this element amounting to 9.50 kN.
In the case of the vertical load p (case no. 3), the values of each of the internal forces in the bracings are higher than those in the case of the horizontal perpendicular load qy (case No. 1), however significantly lower than in the case
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Fig. 17 – Changes in the internal forces in the bracing at point B resulting from the load qx at imperfections introduced by method no. 2: (a) axial forces N, (b) bending moments M2, the values of bending moments M3 are very small.
of the horizontal load parallel to the scaffolding qx (case No. 2). In conclusion, the standards are subject to an effort increase due to the presence of imperfections in the case of the vertical load p (case No. 3), and the bracings in the case of the horizontal parallel load qx (case No 2). The paper does not discuss in detail the behaviour of the other elements because either normal stresses in the elements adopt negligible values compared to a yield strength or show no sensitivity to imperfections.
4.
Conclusions
The article confirms that the imperfections exert an adverse effect on the internal forces in the elements of the scaffolding. The magnitudes of imperfections primarily influence the increase of the internal forces in the standards of frames and bracings. Internal forces in bracings are responsive mainly to imperfections parallel to the scaffolding, regardless of the type of load. In contrast, internal forces in standards are sensitive to imperfections in any direction, but mainly when the scaffolding is subjected to a vertical load. Moreover, the influence of imperfections on the internal forces is the strongest in the lower elements of scaffolding, there appearing the greatest axial forces. Imperfections cause that the axial forces in elements affect eccentricities, which leads to the increase in bending moments. With the increase in loads the sensitivity of internal forces to imperfections increases. In some cases, also a significant increase in the sensitivity of bending moments to the increase of imperfections can also be seen. Internal forces and stresses in the structure with imperfections modelled by means of various methods adopt different values. The differences are the most remarkable in the upper part of the structure, where in the case of method no. 1 there appear large imperfections, while in the case of method no. 2 the deflections of nodes are negligible. In the lower part of the structure the differences between internal forces decrease on account of the decrease in the differences in magnitudes of
imperfections. Finally, as it can be seen in Figs. 6 and 7, if the imperfections have the same direction, the maximum stresses in the elements are on a similar level. In other words, the values of maximum stresses are not determined so much by the number of imperfections, but rather by what values they adopt. Finally, some practical conclusions can be drawn from the work, which are as follows: actually, both methods of modelling imperfections in scaffolding of the height of up to 24 m give similar values of stresses, so it is recommended that at the design stage of scaffolding the method of modelling imperfections that is easier to use in a given computer programme should be adopted in engineering practice, since the largest increase in internal forces occurs in the lower layer of scaffolding, in the case of high scaffolding, the construction of stronger elements of lower layers of the structure should be considered. The study analysed the typical scaffolding, where the range of generated stresses does not exceed the yield strength. In the future, higher scaffolding should be examined, while taking into account the non-linearity of materials and also the problem, not mentioned here, concerning anchorages. In some cases, some anchorages ceased to work, which resulted in overloading the other ones. This paper does not discuss this problem in detail because in the situations when the sign of an axial force changes, in the anchorages their small values were found. In the case of higher wind loads, e.g. higher scaffolding, taking into account the load changes together with the height of the structure or with the net, the problem may be significant. Therefore, a separate paper is expected to examine this problem.
Acknowledgement This work was financially supported by the statutory budget funds of Faculty of Civil Engineering and Architecture in Lublin University of Technology.
archives of civil and mechanical engineering 16 (2016) 269–281
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