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FINITE ELEMENT ANALYSIS OF MEMBRANE ACTION IN STEEL DECK PLATES AND FIBROUS CONCRETE SLABS

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Introduction Tests on steel plate bridge decks have shown that, if a load on the deck is increased beyond the usual wheel load limits, or if the ratio of the deflection to the span of a ribs is relatively large, internal axial stresses begin to appear in the loaded ribs, in addition to the purely flexural stresses. An additional tension, due to membrane action of the deck plate, occurs in the directly loaded rib, and has to be balanced by an equally large compression in the adjoining ribs [8]. As the loads and the corresponding deflection increase, a complete redistribution of the stresses takes place in the system, and the membrane stresses almost entirely replace the flexural stresses which are predominant under working loads. Thus with sufficiently large deflections, a steel plate deck behaves in a manner radically different from that predicted by the usual flexural theory which disregards the effect of the deformations of a system on its stresses. Most importantly, its strength has been found to be many times greater than predicted by the ordinary flexural theory.

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The Aim of the Research The main aim of this research is to use finite element approach in analyzing steel fiber concrete slabs and steel deck plate allowing for membrane action. Then, study the effect of number of parameters on the structural behavior of the above structures.

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ANALYSIS OF STEEL DECK PLATES ALLOWING FOR MEMBRANE ACTION Comparison of small- and large-deflection theories using an approximate method Eq.1 An alternate form of Eq.1, obtained by taking =0.3, is: Eq.2 Fig.1: Maximum Deflections and Stresses Plates Having Large Deflections [78] (a) Clamped edge(b) Simply supported h h Eh 4 The variation of load deflection for a uniformly loaded circular plate with clamped edge, w max /h and p o a 4 /Eh 4 are plotted in Fig.3. It is observed that the linear theory which, neglects the membrane action, is satisfactory for w max

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Von-Karman assumptions The Von-Karmen assumptions for large deformation of plates and shells should take the following forms when applying shells and plates [22] : 1.The magnitude of the deflection (w) is of the order of the thickness (h). 2.The thickness is much less than the length (L); (this hypothesis is not restrictive) since otherwise the displacements are not large. 3.The slope is small everywhere 4. The tangent displacements (u, v) are small, only nonlinear term, which depend on have to be retained in the strain-displacement relations. 5.All strain components are small. General equations for large deflections of plates The general differential equation for a thin plate, subjected to combined lateral (p) and in-plane (N x, N y, and N xy ) forces [78] : Eq.3 For a thin plate element, the x and y equilibria of direct forces are expressed by: Eq.4 Eq.5 The resultant strain components may be expressed as:

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The in-plane forces can be expressed as [86] : Eq.6a Eq.6b Eq.6c The non-linear stiffness matrix: K= K+K Eq.7 Where: Eq.8 Eq.9 B:strain displacement matrix D: elasticity matrix v: volume

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[G]: is a matrix with two rows and number of columns equal to the total number of element nodal variables. The first row contains the contribution of each nodal variable to the local derivative corresponding shape function derivatives( ) and the second row contains similar contributions for ( ) andEq.10 Material Nonlinearity The material nonlinearity deals (for steel deck plates) with elasto-plastic behavior and the anisotropic affect in the yielding behavior. Flow theory of plasticity The flow theory of plasticity [44,45] is employed as the nonlinear material model. For anisotropic material to be considered in this work is a generalization of the Huber-Mises law and can be written in general form as: F( )=f( )-Y( ) in which f( ) is some function of the deviatoric stress invariant and the yield level Y( ) can be a function of a hardening parameter, . Tangent stiffness matrix The tangential stiffness matrix of the material can be expressed as: Eq.11 Eq.12

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where D ep : is the elasto-plastic rigidity matrix and equal: Eq.13 a: is the flow vector and H`: is the hardening parameter Computer program (ANSYS software) ANSYS software is a general purpose FE-program for static, dynamic as well as multiphysics analysis and includes a number of shell elements with corner nodes only and with corner and mid- side nodes. From the available element library in ANSYS, SHELL93 element is used in this work.

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Case studies 1.Steel deck plate The results of a full scale loading test [8] of a 10 mm thick deck plate supported on ribs spaced 300 mm o.c. is analyzed by ANSYS software, Fig.2. It is represented and modeled as shown in Figs.3 and 4. The load deflection curves for both small displacement analysis and large displacement analysis (allowing for membrane action) are explained in Fig.5. Conservative solution was obtained in small displacement analysis but good agreement with experimental tests was gotten in case of large displacement analysis. The deflected shape at ultimate load is listed in Fig.6. a a=300 mm aa Loadin g Point A Rigid steel frame 16 0 m m P/ 2 Point m 10 m m Fig.2: Steel Deck Plate Fig.3: Representation of Steel Deck Plate Poi nt A Fig.4: FE-Model of Steel Deck Plate Fig.5: Load Deflection Relationships of Steel Deck Plate central deflection (mm) Fig.6: Deflected Shape of Steel Deck Plate

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2. Steel deck plate with open ribs A steel deck plate with open ribs under a concentrated wheel load, which was conducted at the Technological University in Darmstadt [8] (Germany), is analyzed using both large and small displacement analysis. The 5-span continuous test panel was a half-scale model of the deck plating intended for Save River Bridge in Belgrade, Fig.7. Also conservatives solution and good agreement compared with experimental test was shown in both small and large deformation analysis respectively, Fig10. P Load P Point A Rubber bed 4.5 mm deck plate 150 mm PL 250 mm x 4.5 mm PL 70 mm x 4.5 mm 3750 mm 300 mm P 1480 mm Ribs 70 mm x 4.5 mm Fig.7: Steel Deck Plate with Open Ribs Fig.9: FE-Model of Steel Deck with Open Ribs Plate in ANSYS Fig.8: Representation of Steel Deck Plate with Open Ribs

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Fig.10: Load Deflection Relationships of Steel Deck Plate with Open Ribs central deflection (mm) Fig.11: Deflected Shape of Steel Deck Plate with Open Ribs

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3. Steel deck plate with closed ribs A steel deck plate with closed (torsionally rigid) longitudinal stiffening ribs was tested at the Technological University in Stuttgart. The dimensions of test panel with trapezoidal ribs, which may be regarded as a half scale model of an actual bridge penal, are illustrated in Fig. 12. the same previous notes for large and small displacements analysis can be observed in Fig.15. Piont A 4000 mm Edge beamRigid support P P P Edge beam 150 mm 3 mm longitudinal ribs 5 mm deck plate 85 mm 3450 mm 1800 mm Fig.12: Steel Deck Plate with Closed Ribs Fig. 13: Representation of Steel Deck Plate with Closed Ribs Fig.14: FE-Model of Steel Deck with Closed Ribs

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Fig.15: Load Deflection Relationships of Steel Deck Plate with Closed Ribs central deflection (mm) Fig.16: Deflected Shape of Steel Deck Plate with Closed Ribs

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Computer Program To analyze fibrous concrete slabs, a program which is coded by Hinton and Owen [40] and called "conshell", is used. The program was modified by Al-Shather [11] to take into account the presence of steel fiber in concrete slab. This is done by introducing tension stiffening technique. In the present study the membrane action is introduced, and this done by using geometrical nonlinearity, and allowing for high strain limit m during the running of the program. ANALYSIS OF FIBROUS CONCRETE SLABS ALLOWING FOR MEMBRANE ACTION

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Case Studies To verify the computer program which is called "conshell program", number of fibrous concrete slabs, which were tested by Mohammed [61], are analyzed. The material parameters and are assumed to obtain good agreement with the experimental tests. These specimens are also analyzed by ANSYS software. The so-called "SOLID65 element", which is favorable with the analysis of reinforced concrete structures, is used. Geometrical and material nonlinearities are adopted. The tested slabs [6] had dimensions of 1.0 m x 0.7 m and 25 mm thick with compressive strength as shown in table 1. Table 1: Compressive Strength of Fibrous Concrete The steel fiber aspect ratio (l f /d f ) was 20. For slabs with reinforcement, they were reinforced with smooth wires of 2 mm diameter, with yield strength of MPa. These steel wire reinforcement provided at shorter span were mm 2 /m ( x =0.33%) and mm 2 /m ( ' x =0.49%) at positive and negative moment, respectively. Conversely at longer direction an amount of mm 2 /m ( y =0.17%) and mm 2 /m ( ' y =0.23%) at positive and negative moment, respectively. Compressive strength f' cf (MPa)Volume fraction V f % 28.60no fiber

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1.Fibrous concrete slabs axially unrestrained at all edges a. Without reinforcement Fig.18: Load Deflection Relationships of Fibrous Concrete Slab Axially Unrestrained with V f =0.5% uniformly distributed load (kN/m 2 ) Fig.19: Load Deflection Relationships of Fibrous Concrete Slab Axially Unrestrained with V f =1.0% uniformly distributed load (kN/m 2 ) Fig. 20: Load Deflection Relationships of Fibrous Reinforced Concrete Slab Axially Unrestrained with V f =0.5% uniformly distributed load (kN/m 2 ) b. With reinforcement

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2. Fibrous concrete slabs axially restrained at all edges a. Without reinforcement Fig.21: Load Deflection Relationships of Plain Concrete Slab Axially Restrained uniformly distributed load (kN/m 2 ) Fig.22: Load Deflection Relationships of Fibrous Concrete slab Axially Restrained with V f =0.5% uniformly distributed load (kN/m 2 ) Fig.23: Load Deflection Relationships of Fibrous Concrete Slab Axially Restrained with V f =1.0% uniformly distributed load (kN/m 2 )

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b. With reinforcement Fig.24: Load Deflection Relationships of Reinforced Concrete Slab Axially Restrained uniformly distributed load (kN/m 2 ) Fig.25: Load Deflection Relationships of Fibrous Reinforced Concrete Slab Axially Restrained with V f =0.5% uniformly distributed load (kN/m 2 ) Fig.26: Load Deflection Relationships of Fibrous Reinforced Concrete Slab Axially Restrained with V f =1.0% uniformly distributed load (kN/m 2 ) Fig.27: Load Deflection Relationships of Fibrous Reinforced Concrete Slab Axially Restrained with V f =1.5% uniformly distributed load (kN/m 2 )

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3. Fibrous concrete slabs axially restrained at two longer edges a. Without reinforcement Fig.28: Load Deflection Relationships of Fibrous Concrete Slab Axially Restrained at Two Longer Edges with V f =0.5% uniformly distributed load (kN/m 2 ) Fig.29: Load Deflection Relationships of Fibrous Concrete Slab Axially Restrained at Two Longer Edges with V f =1.0% uniformly distributed load (kN/m 2 ) b. With reinforcement Fig.30: Load Deflection Relationships of Fibrous Reinforced Concrete Slab Axially Restrained at Two Longer Edges with V f =0.5% uniformly distributed load (kN/m 2 )

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THE EFFECT OF SOME PARAMETERS ON THE BEHAVIOR OF STEEL DECK PLATES The effect of a number of parameters on the behavior of steel deck plates are studied. ANSYS software is used to perform analyses of the considered steel deck plate, using SHELL93 element to model the considered problems. Large deformation analysis is adopted to take into account the membrane action as well as material nonlinearity. Four types of the structures are studied in this work. These are: steel rectangular plate, steel deck plate with open rips running in one direction, steel deck plate with open ribs supported by transverse floor I - beams, and steel deck plate with closed ribs running in one direction. Two types of loading conditions are used, they are central point load and uniformly distributed load above the entire structure. Dimensions of deck plate 2000 mm 1500 mm Symmetry Dimensions of deck plate with open ribs Symmetry 2250 mm Symmetry PL 70mm x 5 mm as ribs spaced 150mm o.c. Dimensions of deck plate with open ribs and stiffened by floor beams Symmetry 2625 mm 2250 mm Symmetry PL 70 mm x 5 mm as ribs spaced 150 mm o.c. PL 70 mm x 5 mm. PL 250 mm x 5 mm. Dimensions of deck plate with closed ribs Symmetry 2250 mm Symmetry PL 5 mm 150 mm 63.1 mm 100 mm Deck plate 5 mm thick Fig. 31: Types of Steel Deck Plates

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1.Effect of Changing Thickness of Rectangular or Top Plate of Steel Deck Plate a. Steel rectangular plate b. Steel deck plate with one-way open ribs Fig.32: Load Deflection Relationships of Rectangular Plate with Varying Plate Thickness under Central Point Load Fig.33: Load Deflection Relationships of Rectangular Plate with Varying Plate Thickness under Uniformly Distributed Load Fig.34: Load Deflection Relationships of Steel Deck Plate with One-Way Open Ribs and Varying Thickness of the Top Plate under Central Point Load central deflection (mm) Fig.35: Load Deflection Relationships of Steel Deck Plate with One-Way Open Ribs and Varying Thickness of the Top Plate under Uniformly Distributed Load central deflection (mm)

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c. Steel deck plate with open ribs and stiffened by transverse floor I-beams d. Steel deck plate with closed ribs Fig.36: Load Deflection Relationships of Steel Deck plate with One-Way Open Ribs Stiffened by Floor I-Beams and Varying Thickness of the Top Plate under Central Point Load Fig.37: Load Deflection Relationships of Steel Deck Plate with One-Way Open Ribs Stiffened by Floor I-Beams and Varying Thickness of the Top Plate under Uniformly Distributed Load Fig.38: Load Deflection Relationships of Steel Deck Plate with Closed Ribs and Varying Thickness of the Top Plate under Central Point Load central deflection (mm) Fig.39: Load Deflection Relationships of Steel Deck Plate with Closed Ribs and Varying Thickness of the Top Plate under Uniformly Distributed Load central deflection (mm)

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2. Effect of Changing the Length of Thickening Edge Plate a. Steel rectangular plate b. Steel deck plate with one-way open ribs Fig.40: Load Deflection Relationships of Steel Rectangular Plate with Varying Lengths of Thickening Plate under Central Point Load Fig.41: Load Deflection Relationships of Steel Rectangular Plate with Varying Lengths of Thickening Plate under Uniformly Distributed Load Fig.42: Load Deflection Relationships of Steel Deck Plate with One Way-Open Ribs and Varying Lengths of Thickening Plate under Central Point Load Fig.43: Load Deflection Relationships of Steel Deck Plate with One Way-Open Ribs and Varying Lengths of Thickening Plate under Uniformly Distributed Load

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c. Steel deck plate with open ribs and stiffened by transverse floor I-beams d. Steel deck plate with closed ribs Fig.44: Load Deflection Relationships of Steel Deck Plate with One Way-Open Ribs Stiffened by Floor I- Beams and Varying Lengths of Thickening Plate under Central Point Load Fig.45: Load Deflection Relationships of Steel Deck Plate with One Way-Open Ribs Stiffened by Floor I- Beams and Varying Lengths of Thickening Plate under Uniformly Distributed Load Fig.46: Load Deflection Relationships of Steel Deck Plate with Closed Ribs and Varying Lengths of Thickening Plate under Central Point Load Fig.47: Load Deflection Relationships of Steel Deck Plate with Closed Ribs and Varying Lengths of Thickening Plate under Uniformly Distributed Load

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3. Effect of Changing the Aspect Ratio of the Steel Rectangular Plate 4. Effect of Changing the Spacing between Ribs with Keeping the Same Total Weight a. Steel deck plate with one-way open ribs Fig.48: Load Deflection Relationships of Steel Rectangular Plate with Varying Aspect Ratios under Central Point Load Fig.49: Load Deflection Relationships of Steel Rectangular Plate with Varying Aspect Ratios under Uniformly Distributed Load Fig.50: Load Deflection Relationship of Steel Deck Plate with One-Way Open Ribs, Changing the Spacing between Ribs and Keeping the Total Weight under Central Point Load Fig.51: Load Deflection Relationship of Steel Deck Plate with One-Way Open Ribs, Changing the Spacing between Ribs and Keeping the Total Weight under Uniformly Distributed Load

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b. Steel deck plate with open ribs and stiffened by transverse floor I-beams c. Steel deck plate with closed ribs Fig.45: Load Deflection Relationship of Steel Deck Plate with One-Way Open Ribs Stiffened by Floor I- Beams, Changing the Spacing between Ribs and Keeping the Total Weight under Central Point Load Fig.46: Load Deflection Relationship of Steel Deck Plate with One-Way Open Ribs Stiffened by Floor I- Beams, Changing the Spacing between Ribs and Keeping the Total Weight under Uniformly Distributed Load Fig.47: Load Deflection Relationship of Steel Deck Plate with Closed ribs, Changing the Spacing between Ribs and Keeping the Total Weight under Central Point Load central deflection (mm) Fig.48: Load Deflection Relationship of Steel Deck Plate with Closed ribs, Changing the Spacing between Ribs and Keeping the Total Weight under Uniformly Distributed Load central deflection (mm)

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5. Effect of Changing the Spacing between Ribs a. Steel deck plate with one-way open ribs b. Steel deck plate with open ribs and stiffened by transverse floor I-beams Fig.52: Load Deflection Relationship of Steel Deck Plate with One-Way Open Ribs and Changing the Spacing between Ribs under Central Point Load central deflection (mm) Fig.53: Load Deflection Relationship of Steel Deck Plate with One-Way Open Ribs and Changing the Spacing between Ribs under Uniformly Distributed Load central deflection (mm) Fig.54: Load Deflection Relationship of Steel Deck Plate with One-Way Open Ribs Stiffened by Floor I- Beams and Changing the Spacing between Ribs under Central Point Load Fig.55: Load Deflection Relationship of Steel Deck Plate with One-Way Open Ribs Stiffened by Floor I- Beams and Changing the Spacing between Ribs under Uniformly Distributed Load

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c. Steel deck plate with closed ribs Fig.56: Load Deflection Relationship of Steel Deck Plate with Closed Ribs and Changing the Spacing between Ribs under Central Point Load Fig.57: Load Deflection Relationship of Steel Deck Plate with Closed Ribs and Changing the Spacing between Ribs under Uniformly Distributed Load 6. Effect of Changing the Support Conditions Fig.58: Load Deflection Relationships of Steel Rectangular Plate with Different Edge Conditions under Central Point Load Fig.59: Load Deflection Relationships of Steel Rectangular Plate with Different Edge Conditions under Uniformly Distributed Load a. Steel rectangular plate

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Fig.60: Load Deflection Relationships of Steel Deck Plate with One-Way Open Ribs and Different Edge Conditions under Central Point Load Fig.61: Load Deflection Relationships of Steel Deck Plate with One-Way Open Ribs and Different Edge Conditions under Uniformly Distributed Load c. Steel deck plate with open ribs and stiffened by transverse floor I-beams Fig.62: Load Deflection Relationships of Steel Deck Plate with One-Way Open Ribs Stiffened by Floor I- Beams and Different Edge Conditions under Central Point Load Fig.63: Load Deflection Relationships of Steel Deck Plate with One-Way Open Ribs Stiffened by Floor I- Beams and Different Edge Conditions under Uniformly Distributed Load b. Steel deck plate with one-way open ribs

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d. Steel deck plate with closed ribs Fig.64: Load Deflection Relationships of Steel Deck Plate with Closed Ribs and Different Edge Conditions under Central Point Load central deflection (mm) Fig.65: Load Deflection Relationships of Steel Deck Plate with Closed Ribs and Different Edge Conditions under Uniformly Distributed Load central deflection (mm) 7. Effect of Changing of the Material Properties a. Steel rectangular plate Fig.66: Load Deflection Relationships of Steel Rectangular Plate with Different Types of Steel Property under Central Point Load Fig.67: Load Deflection Relationships of Steel Rectangular Plate with Different Types of Steel Property under Uniformly Distributed Load

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Fig.68: Load Deflection Relationships of Steel Deck Plate with One-Way Open Ribs and Different Types of Steel Property under Central Point Load Fig.69: Load Deflection Relationships of Steel Deck Plate with One-Way Open Ribs and Different Types of Steel Property under Uniformly Distributed Load c. Steel deck plate with open ribs and stiffened by transverse floor I-beams Fig.70: Load Deflection Relationships of Steel Deck Plate with One-Way Open Ribs Stiffened by Floor I- Beams and Different Types of Steel Property under Central Point Load Fig.71: Load Deflection Relationships of Steel Deck Plate with One-Way Open Ribs Stiffened by Floor I- Beams and Different Types of Steel Property under Uniformly Distributed Load b. Steel deck plate with one-way open ribs

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d. Steel deck plate with closed ribs Fig.72: Load Deflection Relationships of Steel Deck Plate with Closed Ribs and Different Types of Steel Property under Central Point Load Fig.73: Load Deflection Relationships of Steel Deck Plate with Closed Ribs and Different Types of Steel Property under Uniformly Distributed Load 8. The Effect of Initial Imperfection on the Steel Rectangular Plate Fig.74: Load Deflection Relationships of Steel Rectangular Plate with Initial Imperfection under Central Point Load central deflection (mm) Fig.75: Load Deflection Relationships of Steel Rectangular Plate with Initial Imperfection under Uniformly Distributed Load central deflection (mm)

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SENSITIVITY ANALYSIS OF SOME PARAMETERS AFFECTING FIBROUS CONCRETE SLABS The dimensions of the slab are 4.5 m by 4.5 m and 150 mm thick. The steel fiber volume fraction (V f ) is 1.0 percent with aspect ratio (l f /d f ) of 30. For fibrous reinforced concrete slabs, it is assumed that top and bottom reinforcements in two directions are used of the following values: the reinforcements in x-direction are mm 2 /m ( x =0.33%) and mm 2 /m ( ' x =0.49%) at positive and negative moment, respectively. Conversely at y- direction an amount of mm 2 /m ( y =0.17%) and mm 2 /m ( ' y =0.23%) at positive and negative moment, respectively. The compressive strength is assumed equal to MPa and the other material properties are calculated by using the equations listed previously.

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1.The Effect of Changing Thickness of Fibrous Concrete Slabs c. Restrained fibrous reinforced concrete slab b. Restrained fibrous concrete slaba. Unrestrained fibrous concrete slab Fig.76: Load Deflection Relationship of Unrestrained Fibrous Concrete Slab with Varying Thickness under Central Point Load central deflection (mm) Fig.77: Load Deflection Relationship of Unrestrained Fibrous Concrete Slab with Varying Thickness under Uniformly Distributed Load central deflection (mm) Fig.78: Load Deflection Relationship of Restrained Fibrous Concrete Slab with Varying Thickness under Central Point Load Fig.79: Load Deflection Relationship of Restrained Fibrous Concrete Slab with Varying Thickness under Uniformly Distributed Load Fig.80: Load Deflection Relationship of Restrained Fibrous Reinforced Concrete Slab with Varying Thickness under Central Point Load central deflection (mm ) Fig.81: Load Deflection Relationship of Restrained Fibrous Reinforced Concrete Slab with Varying Thickness under Uniformly Distributed Load central deflection (mm)

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2. The Effect of Changing Length of the Edge Thickening Slabs c. Restrained fibrous reinforced concrete slab b. Restrained fibrous concrete slaba. Unrestrained fibrous concrete slab Fig.82: Load Deflection Relationship of Unrestrained Fibrous Concrete Slab with Varying Length of Edges Thickening Slab under Central Point Load central deflection (mm) Fig.83: Load Deflection Relationship of Unrestrained Fibrous Concrete Slab with Varying Length of Edges Thickening Slab under Uniformly Distributed Load central deflection (mm) Fig.84: Load Deflection Relationship of Restrained Fibrous Concrete Slab with Varying Length of Edges Thickening Slab under Central Point Load Fig.85: Load Deflection Relationship of Restrained Fibrous Concrete Slab with Varying Length of Edges Thickening Slab under Uniformly Distributed Load Fig. 86: Load Deflection Relationship of Restrained Fibrous Reinforced Concrete Slab with Varying Length of Edges Thickening Slab under Central Point Load central deflection (mm) Fig.87: Load Deflection Relationship of Restrained Fibrous Reinforced Concrete Slab with Varying Length of Edges Thickening Slab under Uniformly Distributed Load central deflection (mm)

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3. Effect of Changing Edge Support Conditions b. Fibrous reinforced concrete slaba. fibrous concrete slab Fig.88: Load Deflection Relationship of Fibrous Concrete Slab with Varying Edge Support Conditions under Central Point Load Fig.89: Load Deflection Relationship of Fibrous Concrete Slab with Varying Edge Support Conditions under Uniformly Distributed Load Fig.90: Load Deflection Relationship of Fibrous Concrete Reinforced Slab with Varying Edge Support Conditions under Central Point Load central deflection (mm) Fig.91: Load Deflection Relationship of Fibrous Concrete Reinforced Slab with Varying Edge Support Conditions under Uniformly Distributed Load central deflection (mm)

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4. Effect of Initial Imperfection c. Restrained fibrous reinforced concrete slab b. Restrained fibrous concrete slaba. Unrestrained fibrous concrete slab Fig.92: Load Deflection Relationship of Unrestrained Fibrous Concrete Slab with Varying Initial Imperfection under Central Point Load central deflection (mm) Fig.93: Load Deflection Relationship of Unrestrained Fibrous Concrete Slab with Varying Initial Imperfection under Uniformly Distributed Load central deflection (mm) Fig. 94: Load Deflection Relationship of Restrained Fibrous Concrete Slab with Varying Initial Imperfection under Central Point Load Fig.95: Load Deflection Relationship of Restrained Fibrous Concrete Slab with Varying Initial Imperfection under Uniformly Distributed Load Fig.96: Load Deflection Relationship of Restrained Fibrous Reinforced Concrete Slab with Varying Initial Imperfection under Central Point Load central deflection (mm) Fig.97: Load Deflection Relationship of Restrained Fibrous Reinforced Concrete Slab with Varying Initial Imperfection under Uniformly Distributed Load central deflection (mm)

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5. Effect of Changing Strength of Concrete c. Restrained fibrous reinforced concrete slab b. Restrained fibrous concrete slaba. Unrestrained fibrous concrete slab Fig.98: Load Deflection Relationship of Unrestrained Fibrous Concrete Slab with Varying Ultimate Strength under Central Point Load central deflection (mm) Fig.99: Load Deflection Relationship of Unrestrained Fibrous Concrete Slab with Varying Ultimate Strength under Uniformly Distributed Load central deflection (mm) Fig.100: Load Deflection Relationship of Restrained Fibrous Concrete Slab with Varying Ultimate Strength under Central Point Load Fig.101: Load Deflection Relationship of Restrained Fibrous Concrete Slab with Varying Ultimate Strength under Uniformly Distributed Load Fig.102: Load Deflection Relationship of Restrained Reinforced Fibrous Concrete Slab with Varying Ultimate Strength under Central Point Load central deflection (mm) Fig.103: Load Deflection Relationship of Restrained Fibrous Reinforced Concrete Slab with Varying Ultimate Strength under Uniformly Distributed Load central deflection (mm)

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6. Effect of Changing Aspect Ratio of the Steel Fibers c. Restrained fibrous reinforced concrete slab b. Restrained fibrous concrete slaba. Unrestrained fibrous concrete slab Fig.104: Load Deflection Relationship of Unrestrained Fibrous Concrete Slab with Varying Aspect Ratio under Central Point Load central deflection (mm) Fig.105: Load Deflection Relationship of Unrestrained Fibrous Concrete Slab with Varying Aspect Ratio under Uniformly Distributed Load central deflection (mm) Fig.106: Load Deflection Relationship of Restrained Fibrous Concrete Slab with Varying Aspect Ratio under Central Point Load Fig.107: Load Deflection Relationship of Restrained Fibrous Concrete Slab with Varying Aspect Ratio under Uniformly Distributed Load Fig.108: Load Deflection Relationship of Restrained Fibrous Reinforced Concrete Slab with Varying Aspect Ratio under Central Point Load central deflection (mm) Fig.109: Load Deflection Relationship of Restrained Fibrous Reinforced Concrete Slab with Varying Aspect Ratio under Uniformly Distributed Load central deflection (mm)

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7. The Effect of Changing Reinforcement Steel Ratio in Restrained Fibrous Reinforced Concrete Slab b. Changing top steel ratioa. Changing bottom steel ratio Fig.110: Load Deflection Relationship of Restrained Fibrous Reinforced Concrete Slab with Varying Bottom Steel Ratio under Central Point Load Fig.111: Load Deflection Relationship of Restrained Fibrous Reinforced Concrete Slab with Varying Bottom Steel Ratio under Uniformly Distributed Load Fig.112: Load Deflection Relationship of Restrained Fibrous Reinforced Concrete Slab with Varying Top Steel Ratio under Central Point Load Fig.113: Load Deflection Relationship of Restrained Fibrous Reinforced Concrete Slab with Varying Top Steel Ratio under Uniformly Distributed Load

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Conclusions 1.The results obtained using ANSYS software, in the analysis of steel deck plates adopting large deformation analysis (allowing for membrane action), are in agreement with the experimental tests. 7.Restraining the top plate of the steel deck plate and releasing the ribs give a structure having softening behavior at the beginning of loading but as the deformation become large the stiffening behavior appears, since the inducing of the membrane action. A reverse behavior could be gotten when releasing the top plate and restraining ribs against horizontal movement. 2.While very stiff solutions are obtained, when using ANSYS software in the analysis of fibrous reinforced concrete slabs, if compared with the experimental tests. 3.Conshell program gives very stiff solutions when Leonard's material parameters are used. But agreement with the experimental tests is obtained when the assumed material parameters are adopted. 4.For fibrous concrete slabs without reinforcement, an equivalent smeared layer for steel fibers shows good modification in the analysis results compared with other analysis results. 5.Small deformation analysis (membrane action is not allowing) could not give full load deflection curve. But it fails in a very small load compared with the experimental tests. So, conservative solution is obtained with small displacement analysis. 6.The stiffness of steel deck plate is affected by increasing the spacing between ribs for the same total weight. At the beginning of loading, the stiffness increases significantly. This is due to increasing the moment of inertia of the structure which makes the bending action be dominant. But as the deflection becomes large, the membrane action will be the dominant. So the changes in the stiffness of the steel deck plate has restricted values with increasing the spacing between ribs and keeping the same total weight.

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9.In fibrous concrete slab, as the initial imperfection increases the strength of unrestrained fibrous concrete slab and restrained fibrous reinforced concrete slab increase. This is due to increasing the inclination of the induced membrane forces which increases the vertical components of them. But, when the initial imperfection increases the strength of restrained fibrous concrete slab reduces. This may attributed to formation more than one region having high tensile stresses which causes cracks in concrete slabs and reduce their strength. 8.When the initial imperfection increases the stiffness of the steel deck plate increases. This is attributed to increasing the inclination of the induced membrane forces which increase the resisting of these forces to the applied loads.

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THANK YOU THANK YOU

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FINITE ELEMENT ANALYSIS OF MEMBRANE ACTION IN STEEL DECK PLATES AND FIBROUS CONCRETE SLABS FINITE ELEMENT ANALYSIS OF MEMBRANE ACTION IN STEEL DECK PLATES AND FIBROUS CONCRETE SLABS

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