Self-supporting tests in lattice joists subject to negative bending

Resumo During the construction of bridges, cantilever roofs and eaves, assembling formworks and scaffold that will support the slabs is a point of difficulty in the construction phase. Therefore, it is relevant the study of the lattice joists which serve as self-supporting formwork, supporting its weight, the weight of the fresh concrete, the weight of workers and the weight of concreting equipment. The analysis of the bearing capacity of lattice joists subject to negative bending with base concrete opening enables checking the maximum span that each lattice truss model bears, either cantilevered or between continuous spans with reduced or no scaffold. The concrete opening enables the monolithism between the slab and its support. This paper presents the results of tests on lattice joist with concrete opening. By the results analysis, formulations for designing the spacing between prop lines were found. The results are promising and indicate great possibilities of using lattice joists with concrete opening over the supports (beams), in order to optimize the slab shuttering.


Introduction
The manufacturing of lattice slabs started in Brazil after the implementation of the first electroplating machine, using steels grade 600.According to the Brazilian code ABNT NBR 14862 [1], lattice reinforcement is a precast element with a three-dimensional prismatic mold, made of two steel wires in the bottom and one steel wire in the top which form its lower and upper flanges, respectively.These elements are connected by electrofusion to two steel wires, called sinusoid (diagonal bars), following a regular 20 cm spacing, known as step and worldwide standardized.The lattice girders are identified for a TR code followed by two digits that represent its height in centimeters.The last three digits represent respectively the upper flange, sinusoid and lower flange diameters, in millimeters.The code ABNT NBR 14859-1 [2] regulates the precast lattice joists fabrication (VT) with a base of concrete.The lattice girder partially embedded in the concrete base provides a light element easy to handle, requiring fewer shuttering components according to its selfsupporting capacity (SARTORI [3]).The joists can be reinforced or not, depending on its structural demand.These constructive elements are normalized according to ABNT NBR 14859-1 [2]; ABNT NBR 14859-2 [4]; ABNT NBR 14860-1 [5]; ABNT NBR 14860-2 [6]; ABNT NBR 14862 [1] and ABNT NBR 15696 [7], reporting to ABNT NBR 6118 [8]. Figure 1A illustrates a lattice joist cross section.The combination of two or more lattice girders comprises mini panels (Figure 1B) and lattice girder panels (Figura 1C).Gaspar [10] concluded that the lattice joist bearing capacity during the construction phase, considering sagging moments, is governed by the upper flange buckling.The upper flange is characterized by its diameter and diagonal stiffness.As the vertical forces increase, the upper flange is progressively compressed in bending (sagging moment) possibly causing instability or buckling.It is also argued that the shuttering assembly is necessary so that the structure doesn't achieve its ultimate capacity, minimizing the elements stresses.Terni et al [11] developed researches using computer programs in order to analyze this behavior in the upper flanges.Sartorti et al [3] developed experimental studies with positive bending and shear in lattice joists subject to sagging moments, aiming to provide data about the calculation of spacing between prop lines.It also aimed to obtain a more economic construction process, eliminating issues during the construction phase and ensuring security.The later authors concluded that the element ruin can also be characterized by its diagonal bars buckling -for joists with a height of 25 cm or more -besides the upper flange buckling, which is more frequent in joists with 20 cm height or less.In another occasion, Sartorti et al [9] affirmed that the following situations shall be considered for a self-supporting structure calculation in the ultimate limit state: the upper flange buckling due to sagging moments, the lower flange buckling due to hogging moments, the sinusoid shear buckling, failure for excessive plastic deformation of the tension bars and failure of the welded node due to shear.For the serviceability limit states, the calculation of this elements shall consider the analysis of its vertical displacements.The assemblage of formwork and shuttering elements is a challenge during the construction of bridges and overpasses.Regarding to this issue, the use of lattice elements is an interesting solution whereas its concrete base works as a self-supporting formwork for the slabs.

Figure 1
Lattice girder joist and lattice girder panels (cross section scheme) Source: Adapted from SARTORTI et al [9] Self-supporting tests in lattice joists subject to negative bending equipment, making it easy to handle.This combination of factors results in sustainability and cost saving.Self-supporting lattice slabs can be used usually in two different manners: simply supported, according to Figure 2A, or with discontinuous concrete joists -particularly the joist located over the supports -making it possible a monolithic joining between the bridge girder and its deck, as shown in Figure 2B.This research aimed to expand the knowledge concerning self-supporting lattice joists, analyzing its behavior when subject to negative bending and concrete opening (discontinuous concreting) over the supports (Figure 2B).Therefore, the behavior of the steel bars from the lower flange and sinusoids were analyzed, measuring the maximum load bearing capacity of a lattice joist until the serviceability limit state of excessive deformation and the ultimate limit state of instability of any of the latticed components.It was determined the real effective buckling length necessary to the calculation of the maximum span either in cantilever or between supports with no shuttering.

Characteristics of the negative bending tests
This item describes the main characteristics of the experimental program.

Lattice joists
The models of lattice trusses used for the lattice joists are described in Table 1 and its longitudinal and cross section are illustrated in Figures 1A and 3, respectively.For each truss height (6,8,10,12,16,20, 25 and 30 cm) there are nine models -three of them with 20 cm concreting interruption, three of them with 30 cm concreting interruption and other three with 40 cm concreting interruption, in a total of 72 lattice joists.The concrete bases of the joists were cast using the cement type CP V-ARI, with a self-compacting concrete ratio of  Three concrete cylinder tests were conducted for each mix of concrete, expect for the last one, where six concrete cylinders were taken for the determination of the dynamic modulus of elasticity and its characteristic bearing capacity under compression loads.

Set-up of the bending tests
The bending tests were carried out at a concrete age of 50 days.
The following equipment were used: servo-hydraulic universal testing machine (1000 kN capacity); dial indicators with a stroke of 50 mm and a precision of 0.01 mm; magnetic supports for the dial indicators; steel beam used as struts and wood elements used for loading.
The joists were positioned with its upper flange downwards and put above wooden supports which served as pinned supports located

Tests results
This item presents the results of concrete cylinder tests and negative bending tests.

Concrete specimens
The concrete cylinders for each concrete mixture were tested by the age of 50 days, measuring its Young's modulus and compressive strength.The Young's modulus was measured by an acoustic emission non-destructive test.The cylinder is exposed to an impulse which measures the dynamic modulus of elasticity.The Sonelastic ® equipment was used for this purpose.Its functioning is quite simple and the tests can be repeated many times as they are not destructive.For better understanding this equipment functioning, the following steps can be idealized: a) The weight and geometry of the specimens are measured and registered in the Sonelastic ® computer program; b) The specimen is positioned under the wires in the nodal points of flexional resonance -0.224L from the edge of the specimen, where L is its length; c) Using a pre-determined mass impactor, the specimen suffers an impact providing a sound; d) The impact sound is recorded by an acoustic conventional receptor (microphone).Two natural frequencies (flexional and longitudinal) of the specimen are contained in the sound waves; e) The computer program performs a Fast Fourier Transform (FFT) in order to identify the natural frequencies of the specimen; f) Equations from ASTM E1876-1 [12] are used for calculating the modulus of elasticity having the values of the natural frequencies.
Emphasis is given for the fact that the modulus of elasticity is a unique property of the material.The difference between flexional and longitudinal frequencies exists only because of the way they are obtained.Figure 5 illustrates the set-up of the described test.For more information about Sonelastic ® it is recommended reading Sartorti [13].
The values obtained by this method are 20% to 40% higher than the ones obtained with static tests, according to Mehta and Monteiro [14].A great advantage observed from the dynamic tests is the small variability of results, in sharp contrast with the static tests.Figure 6 shows the results of dynamic elasticity modulus for each concrete mixture.The axial compression test results are presented in Figure 7.Each specimen has a strength value ci f .The mean value for all the specimens is cm f .Fusco [15] indicates an expression for calculat- ing the characteristic strength of a tested concrete (Equation 1): (1) Where ck f is the characteristic compressive concrete cylinder strength (by the age of 28 days), with 5% probability of being (2) A variation of f ck results between different concrete mixtures was observed, even when the same ratio was used for the mixes.A small number of cylinder specimens were available for each mixture (equipment limitation), the high room temperature and the low air humidity are some of the reasons attributed by the authors to this variation.Therefore, the different range of time for molding the specimens shall have occasioned the loss of kneading water for the atmosphere, increasing the f ck results variation.

Figure 7
Results of f ck and standard deviation for each concrete mixture

Negative bending test results
Each of the bending tests resulted in a load versus vertical displacement curve, as shown in Figure 8.There were obtained two important parameters: the maximum joist load bearing capacity and the maximum load that corresponds to the maximum displacement.

Analysis of results
The analysis of the lattice joists tests aims to define a real effective buckling length for the elements which failure during the test set-up (see Table 2).After defining the real effective length of joists, mini panels and trussed panels it is possible to use this results in designing situations, in cases when the structural element has a concrete opening and is subject to hogging moments in the opening region.Aiming   to define the real effective length, the subsequent scheme for the tested lattice joists will be considered (Figure 10).
For the scheme, a is a fixed dimension of 20 cm for all the present  Self-supporting tests in lattice joists subject to negative bending weight action divided by 240 cm (Table 2); and P is the applied load (F failure ) added to the weight of the test equipment (PD) (Table 2) divided by 2. The maximum bending moment and shear force acting on each joist can be calculated for the scheme illustrated in Figure 10.The maximum values of the bending moment máx M (middle joist span) and shear force máx V (internally to any of the supports) can be calculated by Equations 3 and 4.
(3) (4) The results analysis is divided into three groups.The first one comprises the joists that failure for buckling in the lower flange, in the concrete opening region.The second group comprises the joists in which the failure occurred by the diagonal buckling.Finally, the third one discusses the results regarding the deflections.

Failure due to buckling of the lower flange
The maximum bending moment and internal forces in the truss are shown in Figure 11.
Where h is the height of the truss; c R is the compression force resulting in the lower flange; and t R is the tension force resulting in the upper flange.
The value of c R is determined by Equation 5. (

5)
As the lower flange is composed by 2 steel bars, the resulting compression force acting in one of the bars ( c F ) in given by Equation 6. (6) Resulting internal forces and bending moment in the truss

Source: Authors
The Euler critical buckling load ( cr P ) for compressed elements is determined by Equation 7.
where is the modulus of the truss taken as 21000 kN/cm²; e l is the effective length of the bar; and v , φ inf I is the gross of inertia the flange (Equation 8).(8) where φ inf is the diameter of one lower flange steel bar.
The monolithism provided by the welding in the truss nodes and the fixity of the bars in the concrete base shall interfere in the compressed elements effective buckling length.The lower flanges have theoretical effective lengths equal to 20, 30 and 40 cmwhich refer to the concrete opening widths.If equal to cr P , it is possible to calculate the real effective length ( , e real l ) of the truss element (Equation 9).

(9)
The results of , e real l for the lattice joists that failure for lower flange buckling are summarized Table 3, where 'Avrg' is the abbreviation for Average.The ratio between the real effective length and the theoretical effective length is smaller than 1.It indicates that there is some stiffening in the lower truss bars possibly due to two reasons.One reason is related to the truss nodes, regarding the fact that an electro welded truss doesn't have fully pinned nodes, as it is considered in the classical mechanics.with a height of 25 and 30 cm did not achieve lower flange failure, when the concrete opening was only of 20 cm.For these cases, the failure occurs in the diagonals.

Failure due to diagonal bars buckling
The maximum shear force in a lattice joist introduces compression stresses in its diagonals.The compression force ( Q ) in a truss diagonal is given by Equation 10. (10) where h is the height of the truss in centimeters; z in the spacing between the two bars of the lower flanges, in centimeters, and is always the value of 9 cm for all the tested lattice joists.
When the compression force Q is equal to the Euler critical

Figure 12
Joist cross section Source: Authors between real and theoretical effective length.Results for the samples that failure of the lattice joist was due to buckling of the diagonals are presented in Table 4.The results are regarding to trusses with heights of 25 and 30 cm, as far as the failure mode described in this item only occurred for these geometry.For other values of height, the buckling of diagonal bars is not the predominant failure mode for the proposed test set-up.
The fixity of the welded nodes and the embedment of the diagonal bars in the joist concrete strongly reduce the effective buckling length of the diagonal bars.

Deflection analysis
The deflection calculation is particularly complex in the step of analysis of results due to the composite cross section of the lattice joist -one of the section regions is composed by concrete and steel, while the other one is only composed by the steel truss bars.The moments of inertia are calculated separately for each region of the cross section mentioned above (Equations 16 to 22).Properties of the transformed section ( where (20) Properties of the section composed only for the steel truss bars According to the Figure 13, the parameters used in Equations 21 and 22 are: 1 x is the distance between the center of gravity of the section and its bottom base; S I is the moment of inertia of the steel section; φ BS is the diameter of the upper flange bar; φ BI is the di- ameter of the lower flange bars; h is the height of the lattice truss.
Considering Figure 10, it is observed that for a e b segments the moment of inertia is equal to H I , whereas the moment of inertia is equal to s I for segment c.The values for the experimental de- flection are taken in the application load points (P).Therefore, the theoretical deflection value, analyzing the same point, is given by Equation 23. ( Rewriting the expression, Equation 24 is obtained. ( (25) where ( ) . (26)

cs H real E I
is the mean stiffness of the testes joist that will be used for the deflection calculation.This stiffness is different from the theoretical stiffness due to the concrete cracking.  .

cs H real E I
is much smaller than the theoretical one.The discontinuity occasioned by the concrete opening and the concrete cracking explain this reduction.In the theoretical calculation, the value for the neutral axis position results in some point close to the concrete footing, indicating that concrete cracking is occurring.

Results applicability
As mentioned in item 1, during the assembly of a cantilever slab with concrete opening in the region right over the supports, there is a need of knowing the lattice joist strength during the construction phase.In this phase, the lattice joist will support itself the weight of the fresh concrete, of the workers and of the concreting equipment.The failure modes visually observed in the tests were: lower flange buckling due to hogging moments and diagonal bars buckling due to shear forces.Besides this modes of failure, it can still occur the failure of the welded node due to shear forces.When a joist, mini panel or panel is loaded and positioned over the supports, bending moments and shear loads will develop.For cases of sagging moments acting on the structure, the adequate equation can be found in Sartorti et al [3].For cases of hogging moments acting on the structure in regions where a concrete opening exists, the present article defines equations for calculating the resisting hogging moment and shear forces.

Displacements calculation
In the transitory phase, it is recommended a limit value for the maximum displacement of the lattice joist, given by the span value divided by 500 ( / 500 l ).The stiffness values shall be calculated as indicated in Equation 38, using Equations 16 to 22.

(38)
where CS E is the secant modulus of elasticity of the concrete; H I is the moment of inertia of the transformed section; Avrg is the value indicated in the last column of Table 5.

Conclusions
In order to ease the construction of bridge decks, cantilever roofs and eaves, self-supporting lattice joists can be used working as formwork, capable of supporting its own weight, the fresh concrete weight and the weight workers and equipments in the construction phase.This paper aimed to study the lattice joists behavior under negative bending moments, in cases where there is a concrete opening in the region of joist supports, by carrying out laboratory tests.For lattice joists with a height of 20 cm or less, the predominant failure mode was the buckling of the lower flanges of the truss.In the cases of joists 25 or 30 cm height, with opening of 20 cm of the concrete, the buckling of diagonal bars governed the failure of the lattice joist.For the same truss height, considering the opening of 30 and 40 cm of the concrete, the lattice joist failure was governed by the lower flanges buckling.The two mentioned modes of failure were analyzed separately for the calculation of the real effective buckling length.It was concluded that the diagonal and lower flange bars are stiffened by the electro welded truss nodes and by the fixity of the truss embedded in the concrete joist base.This results in a significant reduction of the effective buckling length.Disposing of the real effective buckling length, it is possible to calculate the maximum bending moments and shear forces resisted by the lattice joist.This values are quite important for the adequate design of the maximum cantilever span with no shuttering, or the maximum span between supports.
For the transition phase calculation, the stiffness value ( ) EI needs to be calibrated.It was verified that ( . )

CS H real E I
. is smaller than the theoretical value, due to discontinuity of the concrete in the base of the joist, and concrete cracking.
The reinforcement composing the lattice trusses helps bearing the forces acting in the structure in the serviceability state.The use of these latticed elements dispense the need of many transportation I. S. STORCH | J. G. S. DOBELIN | L. C. BATALHA | A. L. SARTORTI

Figure 2
Figure 2Usual kinds of mini panels used for bridge decks Source: SARTORTI et al[9]

Figure 3 4
Figure 3 Longitudinal section of the lattice joists models with total length of 240 cm.The concrete opening in the central region in indicated Source: Authors

Figure 8
Figure 8Load versus vertical displacement for the VT 20-30-2 joist (lattice joist 10 cm height; concrete opening of 20 cm; second from the three samples tested under the same conditions)

Figure 9
Figure 9 Failure modes: (A) diagonal buckling; (B) lower flange buckling A B

Figure 10
Figure 10 Static scheme of the tested lattice joists Source: Authors Another reason is related to the fixity of the lower bars to the concrete base of the joist.The lattice trusses I. S. STORCH | J. G. S. DOBELIN | L. C. BATALHA | A. L. SARTORTI Figure 12:  x is the distance from the center of gravity position (of the transformed cross section) to the bottom face of the section; the theoretical stiffness of the lattice truss steel section.If P a is equal to the limit deflection lim a , Equation 26 is obtained.

Figure 13
Figure 13Center of gravity position for the lattice truss steel elements Source: Authors

Table 1
Lattice reinforcement characteristics The base cross section has a width of 12 cm, a height of 3 cm and a length of 240 cm.For the 72 joists, 15 mixtures in a concrete mixer were necessary, regarding to the concrete mixer capacity.

Table 2
Results for the negative bending tests (Part 1)

Table 2
Results for the negative bending tests (Part 2) pp -self weight; PD -test equipment weight; F limit -equivalent load for a 4 mm displacement (ℓ/500); F failure -buckling load for any of the lattice joist elements or welded node rupture.

Table 3
Values of l e,real for lattice joists with lower flange buckling failure α and β are the truss angles, respectively defined by where ci E is the tangent modulus of elasticity of the concrete; α i In this article, the elasticity modulus values presented in Table1were reduced in 30% in order to correlate the dynamic and static modulus of elasticity.

Table 5 pres
ents the results of the deflection values.It can be seen that the real

Table 5
Results of deflection values