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## Revista IBRACON de Estruturas e Materiais

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*On-line version* ISSN 1983-4195

### Rev. IBRACON Estrut. Mater. vol.6 no.4 São Paulo Aug. 2013

#### https://doi.org/10.1590/S1983-41952013000400008

**Analysis of the assembling phase of lattice slabs **

**A. L. Sartorti ^{I}; A. C. Fontes^{I}; L. M. Pinheiro^{II}**

^{I}Centro Universitário Adventista de São Paulo, Engenheiro Coelho, SP, Brasil 13165-000. artur.sartorti@unasp.edu.br, anacfontes89@hotmail.com

^{II}Escola de Engenharia de São Carlos, Universidade de São Paulo, São Carlos-SP, Brasil 13566-590. libanio@sc.usp.br

**ABSTRACT**

Lattice slabs are usual in Brazil. They are formed by precast joists with latticed bars on a base of concrete, and a cover of concrete placed at the jobsite. The assembly of the joists and the filling elements is simple and do not require manpower with great skill, presenting low cost-benefit ratio. However, it is precisely in assembling phase that arise questions related to the scaffold support distance. A mistake in the proper positioning can lead to two undesirable situations. In one of them, a small space between the support lines increases the cost of scaffold, and in other an excessive space can generate exaggerated displacements, and even the collapse of the slab in the stage of concreting. The objective of this work is to analyze the bearing capacity of lattice joists in assembling phase, looking for information that is useful in defining the scaffold support distance. Several joists were tested to define the failure modes and their load bearing capacities. The results allowed to determine equations for calculating the appropriate distance between the support lines of the joists.

**Keywords**: lattice slabs; lattice joists; self-portance; support lines; buckling.

**1. Introduction**

Since ancient times, the art of building has been improved in order to ensure economy, safety, and comfort. Therefore, it was necessary to develop techniques that could help to transpose the great engineering challenges, such as: support large loads, implement elevated floors, and win large spans. In this context, there was the need to create new systems for slabs in order to conciliate the desired structural characteristics with the parameters of economy and speed of construction. In this way, the slabs formed by lattice joists, also called lattice joist slabs or simply lattice slabs, acquired space and became one of the most used systems in the Brazilian civil construction currently.

A common lattice slab is composed by lattice joists, or trussed joists (TR), with a base of precast concrete and lattice reinforcement partially embedded. Among the lattice joists are placed filling elements that reduce the self-weight and complete the lower part of the slab. Over this part is placed a layer of concrete called cover which concludes the ribbed slab. Figure 1 illustrates the parts that compose an ordinary lattice slab. Besides the arrangement shown in this figure, are still found joists with flanges of concrete in bottom and top (I section), and massive ribs.

However, a problem that persists in all configurations of lattice joist slabs is the question of distance of the scaffold lines, which support the slab during the transitional phase of assembly and concreting.

The load bearing capacity of a lattice joist slab in the assembling phase is directly connected with the resistant capacity of the parts that compose the truss reinforcement, weld of the bars, and lattice joist itself. The characteristics of the lattice joist are illustrated in Figure 2. The bars of the joist are specified by the Brazilian Code ABNT NBR 7480 (1996) [2]. In Figure 3, the dimensions of the joists are illustrated according to the ABNT NBR 14862 (2002) [3].

Joists are indicated by a code TR, followed by five digits: the first two represent the height of the joist, in centimeters, and the last three represent the diameters, in millimeters, of the upper bar, the sinusoid (diagonals), and the lower bars respectively, without consideration of decimal places. Ex.: TR08 634 – trussed joist composed by steel with characteristic yielding strength of 600 MPa, 8 cm of height, upper bar with 6 mm, sinusoid with 3.4 mm, and lower bars with 4.2 mm.

Both GASPAR [4] and DROPPA JR. [5] show that the diagonals in joist reinforcement provide rigidity to the set and good conditions of shipping and handling, in addition to resist the shear stresses and ensure that the system is monolithic after placing the concrete cover. The bottom bars serve to combat the tensile stresses resulting from bending. When necessary, should be placed additional reinforcement to resist the tensile stresses.

Is still asserted by GASPAR [4] that the top bar is the main responsible for the stiffness in transportation and also by the maximum scaffold support distance.

ABNT NBR 14860-1 (2002) [6] in section 5.2 states that "the spacing between scaffold support lines should be determined in the project, considering the type of slab and the loads in the phase of assembly and concreting".

Although there is a recommendation that the design of the spacing between support lines be done in order to guarantee safety to the slab failure at the time of concreting, only some studies are found in the technical literature which effectively consider the subject, and not much give a practical orientation about the calculation of the scaffold support distance.

GASPAR [4] studied the transitory phase of assembly for joists with 8 cm and 12 cm of height requested by a positive bending moment. TERNI *et al.* [7] carried out a finite element modeling using as a base some tests made by EL DEBS and DROPPA JR. (2000)^{1}*, apud* TERNI *et al.* [7].

CARVALHO *et al.* [8] performed an extensive literature review about the state of the art of precast slabs with lattice joists. In this study the researches of GASPAR [4], EL DEBS e DROPPA JÚNIOR (1999)^{2}, and FORTE *et al.* (2000)^{3} are indicated.

Thus, in this work, an experimental study was developed, which made possible to get results that can be used in the calculation of scaffold support distance of lattice joist slabs.

The paper is about tests of positive bending moment and shear, performed in the Laboratory of Materials and Structures of the Adventist University Center of Sao Paulo (Centro Universitário Adventista de São Paulo), as described in the following items. These tests led to the collapse and allowed the analysis of the displacements and the failure modes for lattice joists, answering the aim of the study which is to describe the failure mechanisms and generate recommendations that can be used in the calculation of the scaffold support distance.

**2. Experimental analysis**

In this item will be considered the characterizations of materials and the tests.

*2.1 Characterization of the materials*

Will be characterized the lattices and the concrete bases.

**2.1.1 Joists**

The characteristics of the used lattice joists are indicate in the Table 1. The trussed joist TR06 is also considered, which was tested in spite of not being present in the current standard of lattice reinforcement (NBR 14862 (2002) [10]), because it was already included in the revision of the standard mentioned.

The cross and longitudinal sections of the lattice joists are illustrated in the Figures 4 and 5. The cover of 1.5 cm showed in the Figure 4 was guaranteed by spacers. The upper bar is also denominated upper flange, the lower ones, lower flange, and the sinusoids are also called diagonals.

**2.1.2 Concrete bases**

The concrete bases of the joists were molded in two stages due to the amount of available molds.

The first molding with mix-design in mass 1 : 2.9 : 2.84 : 0.65 was made in March 2nd, 2012 and included the joists TR16745, TR20745, TR25756, TR30856. It was made the slump test with result of 55 mm. The compression characteristic strength predicted for 28 days was 25 MPa. For its determination were molded six cylindrical specimens of 10 cm x 20 cm.

The second molding of concrete bases with mix-design in mass 1 : 2.9 : 2.84 : 0.5 was in March 15th, 2012 completing the rest of the joists: TR6634, TR08644 and TR12644. The change in the amount of water was due to the weather variation between molding days. The slump test gave a result of 50 mm. As in the first molding, the compression characteristic strength predict for 28 days was 25 MPa. Also in this stage, six specimens were molded to determine the concrete strength.

The procedure for molding the concrete bases consisted in four steps: (1) wetting the molds with demoulding oil (Figure 6); (2) filling of the molds with a fresh concrete layer (Figure 7); (3) vibration of the concrete with the aid of a rubber hammer (Figures 8); (4) insertion of the reinforcement (Figure 9).

*2.2 Characterization of tests*

The tests were made on April 19th and April 20th, 2012, respectively bending and shear tests.

**2.2.1 Equipments used in the tests**

The equipments used in the tests were:

▪ Universal testing machine servo hydraulic, capacity 1000 kN, mark Contenco;

▪ Two dial gauges to measure displacements, capacity 50 mm, precision 0,01 mm;

▪ Two magnetic supports for the dial gauges;

▪ Steel beam for support;

▪ Wood devices for load application;

▪ Neoprene plates for testing of models.

**2.2.2 Bending tests**

Figure 10 shows a bending test and Figure 11 represents a corresponding schematic drawing. The lattice joist was placed on two movable supports (avoiding the introduction of horizontal forces) which, on its turn, were supported on a steel beam with cross section in form of I, which served as a base for performing the test.

The mobile supports were placed 20 cm from the extremities of the joist, generating a theoretical span of 260 cm. The adopted dimensions were chosen according to the capacity of the laboratory and test equipments. If larger spans were used, it would be impracticable the assembly of the tests. Very small spans would present problems for the measuring of vertical displacement. Therefore, it is noted that the obtained deflection for the span in study is perfectly measurable, justifying the adopted span. In the middle of this span two deflection indicators were placed (R1 and R2) in order to measure the vertical displacements at this position.

The loading speed was 3 mm/min in the piston^{4} of the jack, and it was applied through a hydraulic cylinder fixed in the steel beam, in a way that the wood dispositive distributed the total force F in two application points distant 86.66 cm from the supports (in addition to this load it was considered the self-weight of the piece). Two specimens were tested for each lattice joist height, totalizing 14 tests.

An important observation is that the concentrated load in the middle thirds generates a stretch of positive bending moment "almost" constant. The "almost" is due to the presence of the distributed self-weight. Another aspect is that in the central portion can occur buckling of the upper bar.

**2.2.3 Shear tests**

Figure 12 illustrates a shear test, and Figure 13 represents a corresponding schematic drawing.

It was used a base composed by a steel beam with cross section in form of I, which served as movable supports that sustain the lattice joist. The left movable support was placed at 60 cm from the extremity of the joist while the right movable support was placed at 20 cm from the opposite extremity.

The loading was applied through a hydraulic cylinder on the metallic beam and the fixed wood support which transferred the force to the position at 30 cm from the left support. If the loading was applied closer to the left support, the transference to the concrete base of the joist could be through the alternative mechanisms of shear strength of concrete.

Two deflection indicators were used (R1 e R2) in the application point of the loading to measure the vertical displacements. The load speed was 3 mm/min in the piston of the press, and two joists of each height were submitted to this test, totalizing 14 tests.

**3. Test results**

Will be present the test results of the specimens and the bending and shear tests.

*3.1 Concrete specimens*

Six pairs of cylindrical specimens of 10 cm x 20 cm molded with the base of the concrete joists were tested on April 25th, 2012 and presented the results summarized in Table 2. Analyzing this table it is noted that the medium strength to compression of the specimens molded on March 2nd, 2012 is f_{cm}= 36.2 MPa, and on March 15th, 2012 is f_{cm}= 38 MPa.

When a standard deviation of 5.5 MPa is considered (FUSCO [9]), the characteristic strengths to compression are 27.15 MPa e 28.95 MPa respectively. Utilizing these characteristic values the concrete modulus of elasticity was estimated by the equations 1 and 2, according to Brazilian Code ABNT NBR 6118 (2007) [10].

E_{ci} is the initial tangent elasticity modulus of the concrete, E_{cs }is the concrete secant modulus of elasticity, and f_{ck} is the characteristic strength of concrete to compression (all in MPa).

*3.2 Bending tests results*

Each test generated a graphic for applied force versus vertical displacement as illustrated in Figure 14 from which was obtained the maximum force resisted by the joist and the corresponding force to limit deflection. The obtained results in the flexion tests with positive bending moment are synthesized in Table 3. Figures 15 to 17 illustrate buckling of the upper bar, rupture of a welded node, and buckling of the diagonals respectively.

*3.3 Shear test results *

As well as for bending test, each shear test generated a graphic of applied force versus vertical displacement, as illustrated in Figure 18, indicating the maximum force resisted by the joist. The obtained results are summarized in Table 4, and Figures 15 to 17 illustrate the indicated types of failure.

**4. Analysis of the results ^{5}**

For bending and shear tests, will be considered the results and its applications.

*4.1 Bending tests*

In the positive bending tests, most of the joists reached failure by buckling of the upper bar or compressed diagonals, with exception of the joist with 25 cm of height, which by a deficiency in the welding, broke also in the weld (Figure 16).

Another objective of the analysis of the results is determine effective lengths of buckling for parts of the lattice, since the consideration of the articulated nodes (Classic Mechanics) is not real in these structures. The actual length of buckling allows the determination of a limit loading for the structure.

Tables 5 and 6 present the values of the resisting moment and the lengths of buckling, calculated based on test results, according to the following procedure for buckling of the upper bar and buckling of the diagonals. The values shown in Tables 5 and 6 were obtained according to the sections 4.1.1 and 4.1.2.

**4.1.1 Bending test with failure by buckling of the upper bar **

The resisting moment and the length of buckling will be considered in this subsection.

*a) Resisting moment*

The test resisting moment is calculated by Equation 3.

The value 260 cm is the theoretical span of the test; 86.67 cm is the length of the middle third of the span, relative to the application of the load; *PD* is the weight of the test device; *P _{CR,test}* is the critical loading that caused the buckling; and

*h*is the height of the joist.

*b) Length of buckling*

Equations from 4 to 6 were used to calculate the length of buckling when failure was by buckling of upper bar.

*P _{CR,test }*is the critical load that causes buckling;

*I*is the moment of inertia of the cross section of upper bar;

_{BS}*E*is the modulus of elasticity of the steel, assumed with value 21000 kN/cm²;

_{s}*ℓ*is effective length of buckling;

_{f,test}*M*is the bending moment relative to the test; and

_{test}*h*is the height of the joist.

**4.1.2 Bending test with failure by buckling of the diagonals**

In this subsection will be considered the shear force, the normal force in a diagonal, and its respective length of buckling.

*a)Shear force*

Shear force of the test (*V _{test}*) is calculated by Equation 7.

*PD*is the weight of the test device; *F _{failure}* is the maximum force of the test; and

*pp*is the self-weight of the joist.

Axial force in a diagonal

The axial force of the test in a diagonal (*N _{test}*) is determined by Equation 8

^{6}.

*V _{test}* is the shear force of the test;

*h*is the height of the joist; and ℓ

*is the theoretical length of buckling of the diagonal (Equation 9).*

_{f ,theoret}*c) Length of buckling *

The length of buckling (ℓ* _{f ,test}*) is obtained by Equations 10 e 11.

*E _{s}*

_{}is the modulus of elasticity of the steel, assumed as 21000 kN/cm²;

*I*is the moment of inertia of the cross section of the diagonal bar; and

_{D}*N*is the axial force in a diagonal.

_{test}In Table 5 can be noted that, for the joists with height less than or equal 20 cm, the effective length of buckling for the upper bar is smaller than the distance between the nodes (20 cm). This is explained by the stiffness that the welded node provides to this upper bar. In theoretical predictions, this node is considered as a perfect articulation. However, when the length of the diagonals increases (joist with 25 cm of height) the stiffness given by the welded node is small, increasing the length of buckling.

Looking at Table 6, it is noted that the concrete base provides an additional stiffness to the diagonals, decreasing the length of buckling obtained with the test. The length of buckling of the diagonal of the joist 25 cm height is relatively larger than that of the joist with 30 cm. The possible explanation for this fact is that the failure of the joist of 25 cm was characterized simultaneously by buckling of upper bar, buckling of the diagonals, and eventually by rupture of the weld. These combined effects reduced in a drastic way the stiffness of the diagonals, approximating the effective length of buckling of their respective theoretical value. Possibly these value would be different if the weld execution was better.

**4.1.3 Analysis of the maximum displacement (deflection)**

On the Table 7 are presented the values of flexural rigidity (EI), calculated based on the results of the tests, according to the procedure described in this subsection.

*a) Limit deflection*

The limit deflection is calculated using the Equations 12 and 13.

**F _{lim}**

_{}is the force corresponding to the deflection of 5.2 mm;

**a**is the limit deflection obtained by the division of the span

_{lim}**ℓ**by 500, equal to 5.2 mm in this case;

**ℓ**is the span between the supports (260 cm); and

**is the product of stuffiness relating to the test.**

*(E)*_{test}*b)Theoretical value of (EI)*

The (EI)_{theoretical} value was calculated to allow determination of the ratio (EI)_{test}/(EI)_{theoretical }. It was determined by homogenization of the section in stage I (non-cracked concrete), and considering the secant elasticity modulus of the concrete given by Equation 2. The modular ratio is determined by Equation 14. The position of the gravity center of the homogenized section and its moment of inertia are obtained by Equations 15 and 16.

The variables indicated in Equations 15 and 16 are illustrated in Figure 19: **x** is the position of the gravity center of the homogenized section with reference in the base; ** I_{H}** is the moment of inertia of the homogenized section; Φ

*is the diameter of the upper bar; Φ*

_{BS}*is the diameter of the lower flange bars;*

_{BI }*is the height of the lattice;*

**h***is the concrete cover of the lower bars always equal to 1.5 cm in the tests;*

**c**_{nom}*is the lower width of the concrete base always equal to 11 cm in the tests;*

**b**_{s}*is the height of the concrete base always equal to 2,5 cm in the tests.*

**h**_{s}

The theoretical stiffness product (EI)_{theoretical} is given by Equation 17.

In Table 7 it is observed that the concrete strength influences more the effective product of stiffness of lower joists (less than 12 cm of height) than the value for higher joists.

*4.2 Shear test*

The shear test results are shown in Tables 8 and 9. Looking at these results it is clear that the upper bar buckling occurred in the lowest joists (heights of 6 cm to 12 cm). For higher heights (16 cm a 30 cm) took place buckling of the diagonals. This is due to the fact of the bucking length of the diagonals be reduced by the embedding in the concrete base, lower height of the joists, and stiffness of the welded node.

Table 8 refers to shear tests in which the failure occurred by buckling of the upper bar. It presents values of resistant moment and buckling length calculated in accordance with procedure indicated in section 4.2.1.

Table 9 regarding to buckling of the diagonals, in addition to bucking lengths, indicated values of shear force and axial force on the diagonals, obtained with information presented in section 4.2.2.

**4.2.1 Shear force test with failure by buckling of the upper bar**

**a) Resistant moment**

The resistant moment of the test (* M_{test}*) was determined by Equation 18.

*PD*is the weight of the test device; *F _{failure} *is the force that causes failure;

*pp*is the self-weight;

*P*is the critical loading that causes buckling; and

_{CR,test }*h*is the height of the joist.

**b) Length of buckling**

The effective length of buckling concerning the test (ℓ* _{f,test}* ) was calculated using the Equations 19 to 21.

*P _{CR,test}* is the critical load that causes buckling;

*E*is the modulus of elasticity of steel, with assumed the value of 21000 kN/cm²;

_{s }*I*is the moment of inertia of the cross section of the upper bar;

_{Bs }*M*is the maximum moment relative to the test; and

_{test}**is the height of the joist.**

*h***4.2.2 Shear test with failure by buckling of diagonals**

*a) Shear force*

The shear force of the test (*V _{test}*) is given by Equation 22.

*PD * is the weight of the test device; *F _{failure}*

_{}is the maximum applied force;

*pp*is the self-weight.

*b) Axial force on a diagonal*

To calculate the axial force of test on a diagonal (*N _{test}*) Equation 23

^{7}was used.

*V _{test}*

_{}is the shear force of the test; ℓ

_{f,theoret}is the buckling theoretical length of the diagonal (Equation 9); h is the height of the joist.

**c) Length of buckling**

The effective length of buckling (*ℓ _{f,test}*) is given by Equations 24 and 25.

** I_{D}** is the moment of inertia of the cross section of the diagonal bars;

**is the modulus of elasticity of steel, with the assumed value of 21000 kN/cm²;**

*E*_{s}**is the axial force of test on a diagonal.**

*N*_{test}_{}It is observed in Table 8 that the effective length of buckling obtained in the test for the upper bar is smaller than the distance of 20 cm between the nodes. This is explained by the stiffness that the welded nodes provide to the upper bar. In a theoretical calculation these nodes are considered as perfect articulations.

In Table 9 it is noted that the concrete base provides additional stiffness to diagonals, decreasing the effective length of buckling obtained through the test. Again it is noted that the welded node with a finishing deficiency in the joist of 25 cm of height generated a relative length of buckling larger than in the joist of 30 cm.

*4.3 Application of the results*

As mentioned in item 1, in the assembly of slabs with lattice joists there is a space between the scaffold support lines. As indicated, the main objective of this work is to provide information for calculating the maximum spacing that can be used.

The position of the support lines defines a static scheme of the joist, where each line can be simulated as a simple support, as illustrated in Figure 20. With this static scheme are obtained bending moments and shear forces due to self-weight of the joist, weight of fresh concrete, filling of the slab, workers and equipments used in the phases of assembly and concreting. These efforts must be resisted by the joists, as it was already commented.

The resistant efforts of the lattice joist are function of the buckling lengths of the bars which compose the lattice. These buckling lengths were determined in the tests. Therefore, the application of the test results consists in finding the resistant moment and the resistant shear force of each joist.

The failure modes observed in the tests were: buckling of the upper bar under effect of positive bending moment; buckling of the diagonals due to shear; and failure of the weld in a node, also by effect of shear. In the sequence, are determined equations for obtain resistant moments and shear forces related to buckling of the diagonals and rupture of the weld.

**4.3.1 Buckling of the upper bar due to bending moment**

Figure 21 shows the internal forces scheme of a joist solicited by a positive bending moment.

The design resistant moment (** M_{d,res}**) and the effective length of buckling (

*) are calculated using Equations 26 to 29:*

**ℓ**_{f,test}*P _{CR}*is the critical load of buckling of the upper bar;

*h*is the height of the lattice;

*E*is the modulus of elasticity of steel, with the assumed value of 21000 kN/cm²;

_{S}*I*is the moment of inertia of the cross section of the upper bar; and

_{BS }*Average*is the value indicate in the last column of Table 8.

Safety is guaranteed when respected the condition:

*M _{Sd }*is the design bending moment.

**4.3.2 Buckling of the diagonals due to shear**

Figure 22 illustrates the scheme of internal forces of a joist subjected to shear.

The value of axial force (*N*) which compresses a diagonal is given by Equation 31.

*V _{Sd }*is the shear force of design; ℓ

*is the theoretical length of buckling of the diagonal, given by Equation 9; and*

_{f ,theoretical,D}*h*is the height of the joist.

Critical axial force (*P _{CR,D}*) that causes buckling of a diagonal is given by Equations 32 and 33.

*E _{S}* is the modulus of elasticity of steel, with the assumed value of 21000 kN/cm²;

*I*is the moment of inertia of the cross section of each diagonal bar;

_{D}*ℓ*is the effective length of buckling; ℓ

_{f,test}*is the theoretical length of buckling; and*

_{f ,theoretical,D}*Average*is the value indicate in the last column of Table 9.

Safety is guaranteed when respected the condition:

**4.3.3 Failure of the weld**

The shear force (V) relating to the weld strength of the top node of the lattice must satisfy to Equation 35, adapted from NBR 14862 (2002) [3].

Φ* _{BS}* is the diameter of the bar which composes the superior flange of the lattice;

*h*is the height of the lattice;

**ℓ**is the length between the nodes of the lattice, fixed in 20 cm.

_{node}Being *V _{Sd}* the shear force of design in the transitory phase; safety is guaranteed when is respected the condition:

**4.3.4 Calculation of displacement**

In the transitory phase is recommendable that the maximum displacement of the joist is smaller than the value of the span divided by 500 (ℓ*/*500). The values of the product of stiffness (EI) shall be calculated as shown in Equation 37, using Equations 3, 15, and 16.

*E _{CS }*is the concrete secant modulus of elasticity, calculated with the characteristic strength

*f*is the moment of inertia of the homogenized section; and

_{ck};I_{H}*Average*is the value indicated in the last column of Table 7.

**4.3.5 Example of application**

This example considers the equations presented in items 4.3.1 to 4.3.4. The goal is to find the maximum span (ℓ) between two supports for the joist TR 16 745.

*a) Data of the example*

Figure 23 illustrates the static scheme of the joist. A concrete cover of 5 cm was adopted, with main inter-axis of 49 cm, and transversal inter-axis of 129 cm. The width of the rib is 9 cm and the filling is in expanded polystyrene (EPS) as Figure 24 illustrates. The concrete base of the joist was admitted with *f _{ck} = 35 MPa*.

**p** is the total load uniformly distributed; **g **is the permanent load (includes self-weight of the joist; filling, and fresh concrete placed on the slab); **q** is the variable load (includes workers and equipments for concreting).

*b) Loading*

With the indicated characteristics a permanent load of 2.23 kN/m² acts in the slab. A variable load of 1.50 kN/m² is adopted.

The loading for the verifications of ultimate limit state (ULS) is considered with the coefficients of increasing for combination of construction actions indicated in ABNT NBR 6118 [9] (Equation 38).

However, the loading for verifying excessive deformation in Serviceability Limit State is indicated in Equation 39, for almost permanent combination of actions.

*c) Efforts in ULS*

For the isostatic joist the values of the efforts are (Equation 40):

ℓ is the searched span in centimeters.

*d) Buckling of the upper bar due to bending moment (ULS)*

This verification uses the equations from item 4.3.1. By Equation 29 is determined *ℓ _{f,test}* (Equation 41), where

*Average*value is obtained in Table 5 for joist TR16745.

The resisting moment of design is calculated with Equation 28 and the result is (Equation 42):

The maximum value of *ℓ* in this verification of buckling is given by Equation 43, using the condition:

*e) Buckling of the diagonals due to shear (ULS)*

This verification is made with the equations from item 4.3.2. The value of *ℓ _{f,test}* is determined by Equation 33, with

*Average*value obtained in Table 9, for joist TR16745 (Equation 44).

The critical normal force (*P _{CR,D}*) which causes buckling of the diagonals is given by Equation 32, with the result being shown in Equation 45.

The value of the normal force of compression (*N*) in the diagonal is given by Equation 31 and the result is shown in Equation 46.

In this verification of buckling of the diagonals, the maximum value of ℓ is given by Equation 47 using the condition:

*f) Rupture of the weld (ULS)*

The maximum shear force that can be applied in the joist so there is no weld rupture of the upper node is given by Equation 35, item 4.3.3. The result is shown in Equation 48.

Safety is guaranteed when is respected the condition:

*g) Serviceability limit state of deflection*

To determine the deflection is necessary to calculate the effective stiffness product of the joist according to Equation 37 in which the *Average* value is obtained in Table 7 for joist TR16745. Equation 50 illustrates this calculation considering the concrete secant modulus of elasticity *E _{CS}*obtained with Equation 2, and the moment of inertia

*I*of homogenized section, with Equation 16.

_{H }The maximum displacement of this joist happens in middle of the span ℓ. According to the Classical Mechanic, its value is given by (Equation 51):

The serviceability limit state of deflection is verified when is respected the condition:

*h) Maximum span *

Observing the four values obtained for ℓ, it is verified under the conditions of this example that the maximum is the relative to buckling of upper bar. Therefore, the maximum span that can be used in this case is 207.14 cm.

**5. Conclusion**

As explained in section 1 the lattice slabs are composed of independent elements (lattice joists and filling elements) disposed in a way to form a panel that when it receives a concrete layer, begin to work as a single system.

During assembly of this structure must be placed scaffold support lines to ensure the positioning of these elements, even when the structure is subjected to loads such as the weight of concrete cover, movement of workers, equipments, etc.

The aim of this study was to provide useful data for calculating the economic scaffold support distance which ensures safety for the workers during the construction of the slab and results in a structure without pathologies of execution.

For this, it was necessary to carry out tests of lattice joists in laboratory in order to verify the actual behavior of these elements when subjected to loading.

It was verified that, both for the bending tests as for the shear ones, the joists with height lower than 20 cm had failure by buckling of the upper bar, while for greater heights, failure occurred by buckling of the diagonals, except the joist of 25 cm, which presented failure of the weld.

By analyzing Tables 5 and 8 it is concluded that the joists with lower height presented lengths of buckling of the upper flange smaller than the distance of 20 cm between the nodes. Therefore, these nodes contributed to increase the stiffness of the upper flange.

In a similar way, with Tables 6 and 9 it is noticed that the concrete base provides an additional stiffness to the diagonals, decreasing the effective length of buckling obtained in the test.

The lengths of buckling obtained from the tests were useful to calculate the maximum compression force which can be resisted by the respective bars of the lattice. With this maximum resistant force to compression, it was determined the resistant moment and the resistant shear force of each joist.

In lattice slabs with any scaffold support distance, bending moments and shear forces are generated. These efforts must be smaller than the resistant ones. The resistant moment is always equal for the joists of same height and the same diameters of the bars because the length of buckling is constant for them. This length of buckling was defined and calculated in the tests.

The deflection is determined by the elastic line of the joist, which depends on the static scheme and the scaffold support distance.

To calculate the deflection was necessary to determine a product of stiffness (EI) that represents what occurs actually in a lattice joist, since the theoretical value of EI can not be used because the material is not elastic and not linear and homogeneous how admit the Classical Mechanics.

Therefore, it was determined in the test the actual value of EI for the joist, which was used to calculate the deflection in the transitory phase of assembling and concreting of the slab. It should be emphasized that this deflection must be lesser than ℓ/500, threshold value for visual acceptability according to ABNT NBR 6118:2007 [10] in the verification of serviceability limit state relating to excessive deflection.

The example detailed in item 4.3.5 demonstrates the applicability of the results and equations given in this paper. It is noted that the presented calculation is simple and easy for computational programming.

This research does not close the subject and, therefore, further tests should be done in order to refine the results and analyze joists with bars of other diameters.

**6. References**

[01] BOUNASSAR, J. Elaboração de normas: projeto, fabricação e execução de lajes mistas pré-moldadas. Coletânea Habitare, Paraná, v. 3, p. 79-109, 1997. [ Links ]

[02] ASSOCIAÇÃO BRASILEIRA DE NORMAS TÉCNICAS. NBR 7480. Barras e fios de aço destinados a armaduras para concreto armado. Rio de Janeiro. ABNT: 1996. [ Links ]

[03] ASSOCIAÇÃO BRASILEIRA DE NORMAS TÉCNICAS. NBR 14862. Armaduras treliçadas eletrossoldadas – requisitos. Rio de Janeiro. ABNT: 2002. [ Links ]

[04] GASPAR, R. Análise da segurança estrutural das lajes pré-fabricadas na fase de construção. São Paulo, 1997. 103f. Dissertação (Mestrado em Engenharia de Estruturas) – Escola Politécnica da Universidade de São Paulo. [ Links ]

[05] DROPPA JÚNIOR, A. Análise estrutural das lajes formadas por elementos pré-moldados tipo vigota com armação treliçada. São Carlos, 1999. 177f. Dissertação (Mestrado em Engenharia de Estruturas) – Escola de Engenharia de São Carlos da Universidade de São Paulo. [ Links ]

[06] ASSOCIAÇÃO BRASILEIRA DE NORMAS TÉCNICAS. NBR 14860-1. Laje pré-fabricada – Pré-laje – requisitos – Parte 1: Lajes unidirecionais. Rio de Janeiro. ABNT: 2002. [ Links ]

[07] TERNI, A. W.; MELÃO, A. R.; OLIVEIRA, L. E. A utilização do método dos elementos finitos na análise comportamental da laje treliçada na fase construtiva. Congresso Brasileiro do Concreto, 50. IBRACON. Salvador, 2008. [ Links ]

[08] CARVALHO, R. C.; PARSEKIAN, G. A.; FIGUEIREDO FILHO, J. R.; MACIEL, A. M. Estado da arte do cálculo das lajes pré-fabricadas com vigotas de concreto. Encontro Nacional de Pesquisa-Projeto-Produção em Concreto Pré-moldado, 2. EESC – USP. São Carlos, 2010. [ Links ]

[09] FUSCO, P. B. Tecnologia do concreto estrutural. 1. ed. PINI. São Paulo, 2008. [ Links ]

[10] ASSOCIAÇÃO BRASILEIRA DE NORMAS TÉCNICAS. NBR 6118. Projeto de estruturas de concreto – procedimento. Rio de Janeiro. ABNT: 2007. [ Links ]

Received: 07 Sep 2012

Accepted: 03 Jan 2013

Available Online: 12 Aug 2013

1 EL DEBS, M. K.; DROPPA JÚNIOR, A. Um estudo teórico-experimental do comportamento estrutural de vigotas e painéis com armação treliçada na fase de construção. In: Congresso Brasileiro do Concreto, 42. Fortaleza – CE. Anais Eletrônicos. 2000.

2 EL DEBS, M. K., DROPPA JÚNIOR, A. (1999). Critérios para dimensionamento de vigotas com armação treliçada nas fases de construção. Relatório Técnico. EESC-USP, São Carlos, SP.

3 FORTE F. C.; FANGEL L.; ARADO F. B. G.; CARVALHO, R. C.; FURLAN JUNIOR, S.; FIGUEIREDO FILHO, J. R. Estudo experimental do espaçamento de escoras em lajes pré-moldadas com nervuras do tipo treliça. Congresso Brasileiro de Concreto, 42. IBRACON. Fortaleza, CE. Anais Eletrônicos. 2000.

4 The equivalent static load varies with the stiffness of the piece. Those with a larger stiffness have a larger equivalent static load. It was also observed that the loading curve is nonlinear, but as a reference it can be adopted a medium value of 100 kgf/min = 1.0 kN/min.

5 In this section several equations are presented. They were obtained by summation of bending moments, shear forces, and homogenization of cross section. The equation of Euler's critical loading is also used, and in section 4.3 this equation is adapted to make it suitable for use in this work.