Microsoft Word - ETASR_V13_N3_pp10843-10848 Engineering, Technology & Applied Science Research Vol. 13, No. 3, 2023, 10843-10848 10843 www.etasr.com Kumar & Kumar: Free Vibration Analysis of Steel-Concrete Pervious Beams Free Vibration Analysis of Steel-Concrete Pervious Beams Prashant Kumar Department of Civil Engineering, National Institute of Technology Patna, India pk0895300@gmail.com (corresponding author) Ajay Kumar Department of Civil Engineering, National Institute of Technology Delhi, India sajaydce@gmail.com Received: 4 April 2023 | Revised: 22 April 2023 | Accepted: 24 April 2023 Licensed under a CC-BY 4.0 license | Copyright (c) by the authors | DOI: https://doi.org/10.48084/etasr.5913 ABSTRACT This study investigated the free vibration analysis of steel-concrete porous beams with partial or complete shear interface using a finite-element model based on the cubic order beam theory. The present model assumes uniform porosity distribution along the beam thickness. It is assumed that the axial displacement will vary cubically along the thickness of the layer. The cubic order beam theory is implemented using a continuous C 0 finite element containing three nodes and each node has eight degrees of freedom. Shear locking is eliminated in the present model by the numerical integration of the stiffness matrix. Comparing the present model with the published literature, it is found that the present model is robust in predicting the free vibration of the steel-concrete porous beam. Keywords-steel-concrete porous beams; finite element method; free vibration; porosity; partial shear interface I. INTRODUCTION New construction materials are constantly developed for various innovative engineering structures which must withstand static and dynamic loads. More and more companies are adopting cost-effective materials, while enhancing the safety and efficiency of engineering projects. Many engineering practices use composite beams, such as steel- concrete beams, in buildings and bridges, wood-concrete floors, linked shear walls, etc. [1, 2]. In structural design, it is necessary to focus on the materials and the strength of the connectors. The structure of composite beams is affected by the interlayer slip under loading. The impact of partial contact on the structural behavior was investigated in [3]. The Euler– Bernoulli beam theory was used in [4] to investigate the free vibration characteristics of steel-concrete composite continuous beams. Various experimental and analytical studies have been conducted to analyze composite beams [5]. Applying the eigenfunction expansion method along with the quasistatic decomposition method, an analytical solution is presented in [6]. Authors in [7, 8] worked on the vibrational behavior of nanocomposite beams. Under the effect of a moving load, the beam is reinforced by random straight Single-Walled Carbon NanoTubes (SWCNTs). The accurate results of Higher-order Beam Theory (HBT) intrigued researchers, so TBT has been replaced with HBT in order to obtain more accurate solutions. Authors in [9] built on Reddy's approach to an HSDT for multi-layered anisotropic composite laminates having complete shear interaction. C 0 finite element models were presented for evaluating composite and sandwich beams and the penalty function approach was used to find a C 0 continuous finite element formulation [10, 11]. Analysis of 3 different porosity variations in the thickness direction and their impact on the vibrational properties of the beam are presented in [12]. Free and forced lateral vibration analysis of beams made from Functionally Graded Materials (FGMs) using the Finite Element Method (FEM) is presented in [13]. The state-space technique was used in [14] to expand the static analysis to dynamic analysis without the use of axial force. From the literature review, it can be seen that there are a few studies regarding the free and forced vibration of a porous steel-concrete beam with a partial shear interface. Therefore, the present study uses the cubic order equation for axial displacement. A linear finite element model is developed for the dynamic analysis of a porous composite beam. The proposed composite beam has a partial or complete shear interaction. The homogeneous porosity distribution along the beam thickness is used to parametrically calculate the material properties. Different interfacial stiffness values are used to calculate the fundamental frequency. The present model is effective for analyzing the free vibrations of porous beams made of steel and concrete under various boundary conditions and moving loads. Several new findings are presented in this paper, making it useful for the subsequent analysis of the free vibration of porous composite beams in the future. Engineering, Technology & Applied Science Research Vol. 13, No. 3, 2023, 10843-10848 10844 www.etasr.com Kumar & Kumar: Free Vibration Analysis of Steel-Concrete Pervious Beams II. FORMULATION A. Porosity Distribution This study evaluates how steel and concrete porosity influence the free vibration of composite beams. Due to the uniform distribution of porosity, the variation in Young’s modulus, shear modulus, and mass density of the concrete and the steel layer is calculated by:           max max max 1 1 1 i i i i i mi i E y E G y G y e               (1) where Ei(y), Gi(y), ρi(y) are the Young’s modulus, shear modulus, and mass density of the concrete (upper) layer and steel (lower) layer of the beam along the transverse direction (where i = c-concrete, s-steel). m i n m i n m a x m a x 1 1i E G e E G     (2) 1 1mi ie e   (3) where the range of the porosity coefficient for mass density (em), porosity coefficient of concrete (e0), and porosity coefficient of steel (e1) are 0 < ��� < 1 , and 0 < �� < 1 . ei = emi = 0 means that porosity is zero at maximum elastic modulus and greater porosity gives lower elastic modulus. 2 2 2 1 1 1i ie             (4) B. Mathematical Formulation High-order beam theory is used to investigate the free vibration analysis of a composite beam having a shear interface as shown in Figure 1. The displacement field that was selected is unique. Transverse shear stresses must cease on the beam surfaces and remain nonzero elsewhere for the requirements to be satisfied, which determines the shape. The beam's axial displacement is considered as the thickness's cubic function. The axial displacement equation for the upper layer is expressed in (5). h c /2 h c /2 h s/ 2 h s/ 2 yc ys Fig. 1. Porous steel-concrete composite beam with shear flexible interface.    2 31 Ti i i c io i i iu y y y u       (5) where ��� = ���, �� , �� is the axial displacement along the top layer's reference axis passing through its centroid, �� = ���� is the bending rotation, and �� = ���� and �� = ���� are high- order terms. The transverse displacement is assumed to be the same for both layers and can be represented as: ( , , ) ( , , ) ( ) c c s s W x y z W x y z W x W   (6) The partial shear interaction between two layers of the composite beam is modelled by taking distributed shear springs at their interface. The interfacial stiffness and the shear slip at the interface are used to figure out the shear stress at the interface. Interfacial slip (s) is calculated in (7) with u’c and u’s being the axial displacement of the upper and lower layer at the interface. ' ' ( ) c s s u u  (7) Axial displacement equations, such as (5), are higher-order equations that are concerned with the warping of transverse sections, but do not describe the commonly used displacement parameters adopted in beam theories. So, higher order terms are eliminated by using shear stress-free conditions at the extreme surfaces of the composite beam. The shear stress at any point in the upper layer is calculated by:      2 1 2 3 1 c c c c c c c c T c c c G u W y y y x W x                           (8) where γc is the shear strain and Gc the shear modulus of porous concrete layer. 2 3 2 3 2 3 2 3 2 3 2 2 3 2 2 3 2 3 12 16 1 , 5 5 12 16 , 5 5 4 12 , 5 5 2 4 5 5 12 16 1 , 5 5 c c c c c c c c c c c c s s s s y y A h h y y B h h y y C y h h y y D h h y y A h h B                                                                                     2 3 2 3 2 3 2 2 3 2 12 16 , 5 5 4 12 , 5 5 2 4 , and 5 5 s s s s s s s s y y h h y y C y h h dWy y D h h dx                                                (9) Shear stress-free conditions are applied at the top and bottom surface of the beam to find the displacement equation. Engineering, Technology & Applied Science Research Vol. 13, No. 3, 2023, 10843-10848 10845 www.etasr.com Kumar & Kumar: Free Vibration Analysis of Steel-Concrete Pervious Beams     ' T i i i i i io ii u A B c D u u      (10) The normal stress and normal strain at any point of the beam are calculated by:     j jj D     (11)       j j j jj jj j u x H u W y x                             (12) where σj is the normal stress, τj the shear stress, Ej the modulus of elasticity, Gj the shear modulus, εj the normal strain, and γj the shear strain of the j th layer, j=c represents the upper layer and j= s represents the lower layer of two-material composite beams. �� � is a function of yj and ���� is the function of xj which is coordinated along the axial direction. The expressions of �� � and ���� are:   0 0 0 0 0 0 0 0 0 1 j j j j j j j jj j j j j A B C D dA dB dC dDH dy dy dy dy           (13)   ' 'jo j j jo jj T j ddu d d W u dx dx dx dx x u u               (14) The strain energy as a function of stress and strain can be found by (11)-(12).                     1 2 1 2 T T c s c s T T c sc s c s U dAdx D D dx                       (15) where       T c cc cc HD D H dA    ,       T ss sss D D H dAH       Numerical integration is used to evaluate the cross-section rigidity matrix �� � and �� �. Stored strain energy is calculated by (7) and distributed shear springs stiffness (ks):   2 ' '' 21 1 2 2 s s c s U k s dx k dxu u    (16) C. Finite Element Formulation First of all, the 3-nodded iso-parametric C o element is selected. This theory assumes 8 nodal degrees of freedom to solve the present problem using a one-dimensional finite element approximation written as:    ' ' T coj cj cj j j soj s sf u u W u u   (17) The unknown nodal displacement vector ��� on the middle surface of a typical element is given by:       3 1 , .i i i d x y d   (18) where ψi is the shape function. The generalized strain vector as a function of the nodal displacement vector {d} can be found by:      3 1 i ij i P d    (19) where [Pi] is the interpolation function differential operator matrix. The strain vectors are calculated by (13), (16), and (18) as a function of the stiffness matrix lK   .     1 2 T l U d K d    (20) where:             Ts Tl c c s scP P K D P D P dx      (21) Similarly, the interfacial stiffness and stiffness due to the penalty function approach are given below:  ' ' 'T sK P k P dx           (22)  Tp p ppK P k P dx           (23) Now we can integrate (21), (22), and (23) and combine them to find out the element stiffness matrix [K].   'l pK K KK              (24) An element's mass matrix and its geometric stiffness matrix can be found in the same way as the element stiffness matrix is calculated above. Equations (16)-(18) are used to find out the displacement component vector at a point in the beam layer:       j j j u f F f F X d W                (25) where jF   is a matrix of order 2×8 which contains the coefficient of displacement component expressed in (25) and [X] is a shape function matrix of order 8×24. Now, the consistent mass matrix of 3-nodded elements is developed by using (25) and is expressed as:        .T Tj j jX F F dzM X dx dy        (26) Free vibration analysis is conducted to find out the fundamental natural frequency from the given equation:    2 0g gK M     (27) where ω is the vibration frequency and   the eigenvector. Engineering, Technology & Applied Science Research Vol. 13, No. 3, 2023, 10843-10848 10846 www.etasr.com Kumar & Kumar: Free Vibration Analysis of Steel-Concrete Pervious Beams III. RESULTS AND DISCUSSION A finite element technique based on the cubic order beam theory was used to examine steel-concrete beam features to assess the effectiveness of the recommended model. The findings are produced by implementing the proposed model in FORTRAN code. A. Comparison Study For the validation of the results, a two-layered simple support composite beam having a T-cross section is considered. The four different Boundary Conditions (BCs) SS, SC, CC, and CF (SS = Simply Supported, C = Clamped, F = Free) and the partial interaction of the composite beams were taken into consideration. The results used for validation are taken [14], in which the state-step approach is used to tackle this issue. In this example, the cross-sectional size and material characteristics of a two-layered composite beam are stated as: span length (L) of beam equal to 4m, the upper and lower elements are 0.05×0.30m 2 and 0.15×0.05m 2 , respectively, the modulus of elasticity of the upper (E1) and lower elements (E2) are 12Gpa and 8Gpa, respectively, the modulus of rigidity of the upper (G1) and lower element (G2) are 5GPa and 3GPa, respectively, and the mass densities of the upper and lower elements are taken as ρ1 = 2400Kg/m 3 and ρ2 = 500Kg/m 3 , respectively, using partial interfacial spring stiffness ks = 50MPa. The free vibration frequencies (rad/s) for various BCs are listed in Table I. Fifty elements throughout the length of the beam are used to determine the present findings. The natural frequency of a Simply Supported (SS) beam determined in [14] and using the state-space technique is contrasted with the present results. The error percentages are also listed in parentheses. In the present simulation, the results are quite accurate and in close agreement with those in [14] with an accuracy of 99.76%. TABLE I. VALIDATION OF THE FUNDAMENTAL FREQUENCY OF A COMPOSITE BEAM (E0=E1=0). S.No. BC Fundamental frequency (rad/s) Present [14] 1. CF 25.06 25.12 (0.24) 2. SS 64.56 64.85 (0.45) 3. SC 88.78 89.56 (0.88) 4. CC 116.66 118.50 (1.58) Note: Parameters in parentheses represent percentage errors. B. Steel-Concrete Porous Beam In the present study, steel-concrete two-material composite beams with different porosity, BCs, and different interfacial shear stiffness values are taken for analysis. The beam's cross- section (Figure 2) is made up of a rectangular slab with a thickness and width of 0.15m and 2.25m respectively, an I- shaped steel joist with a flange dimension of 0.1780×013m, and a web dimension of 0.380×0078m. The material properties of the two layers are as follows: beam span L = 15m, the modulus of elasticity EC and the modulus of rigidity GC of the upper (concrete) element are 13.55GPa and 6.775GPa, respectively, and the modulus of elasticity ES and the modulus of rigidity GS of the lower (steel joist) element are 200GPa and 100GPa, respectively. The maximum densities of the concrete element and the steel layer are taken as ρC = 2396.45Kg/m 3 and ρS = 7948.89Kg/m 3 , respectively, and the uniform porosity distribution of the concrete and steel element is taken as 0, 0.05, 0.10, 0.20, 0.30, and 0.4. In the case of partial interaction, it is assumed that the interfacial shear spring stiffness ks is 100MPa. A moving point load of magnitude 100KN is applied. The point load is traveling from the left support to the right support at a constant speed v0 of 16.67m/s. bc tc bf tw tf hw Fig. 2. Cross section of the steel-concrete porous flanged composite beam. C. Effect of Boundary Conditions, Porosity, and Interlayer Spring Stiffness on Fundamental Frequency In both cases, the fundamental frequency of a SS steel- concrete porous beam with partial or complete shear interface varies with the porosity as shown in Figure 3. (a) (b) Fig. 3. Variations of fundamental frequency as a function of (a) porosity of the steel element and (b) porosity of concrete and steel elements. Engineering, Technology & Applied Science Research Vol. 13, No. 3, 2023, 10843-10848 10847 www.etasr.com Kumar & Kumar: Free Vibration Analysis of Steel-Concrete Pervious Beams It decreases with an increase in the porosity of the concrete slab and steel joist. The fundamental frequency of the steel- concrete porous beam is greater with a complete shear interface than with a partial shear interface. Table II describes the effect of the porosity of steel (e1=0, 0.05, 0.1, 0.2, 0.3, and 0.4) and BCs on the fundamental frequency of complete and partial interface steel-concrete porous beams without considering the damping effect. It is maximum for the CC (clamped, clamped) and minimum for the CF (clamped, free) BC. Table III shows the effect of steel and concrete layer porosity variations on the fundamental frequency (e0=e1=0, 0.05, 0.1, 0.2, 0.3, and 0.4). The different boundary conditions are considered. Table IV analyzes the impact of the interfacial shear stiffness (ks) on the fundamental frequency. Porosity variation is taken into account in this scenario. Fundamental frequencies increase as interfacial shear stiffness increases, as can be easily seen. The fundamental frequency of a steel-concrete porous simply supported beam is expected to change with its porosity. TABLE II. FUNDAMENTAL FREQUENCY OF POROUS BEAM FOR DIFFERENT POROSITY (e1) AND BOUNDARY CONDITIONS Porosity (e1) Fundamental frequency(rad/s) Partial composite Full composite CF SS SC CC CF SS SC CC 0.00 5.9217 15.5493 22.0437 29.6486 6.2464 17.4810 27.0829 38.8882 0.05 5.8527 15.3843 21.8207 29.3481 6.1676 17.2605 26.7407 38.3962 0.10 5.7815 15.2140 21.5914 29.0400 6.0863 17.0332 26.3879 37.8891 0.20 5.6318 14.8553 21.1112 28.3985 5.9155 16.5555 25.6468 36.8240 0.30 5.4703 14.4680 20.5966 27.7164 5.7320 16.0420 24.8503 35.6798 0.40 5.2944 14.0451 20.0388 26.9833 5.5327 15.4845 23.9859 34.4384 TABLE III. FUNDAMENTAL FREQUENCY OF POROUS BEAM FOR eo, e1, AND BOUNDARY CONDITIONS Porosity Fundamental frequency(rad/s) Partial composite Full composite eo e1 CF SS SC CC CF SS SC CC 0.00 0.00 5.9217 15.5493 22.0437 29.6486 6.2464 17.4810 27.0829 38.8882 0.05 0.05 5.8354 15.3453 21.7723 29.2861 6.1470 17.2027 26.6518 38.2691 0.10 0.10 5.7489 15.1407 21.5010 28.9246 6.0473 16.9239 26.2198 37.6488 0.20 0.20 5.5744 14.7276 20.9562 28.2019 5.8467 16.3625 25.3501 36.4000 0.30 0.30 5.3964 14.3058 20.4039 27.4742 5.6429 15.7920 24.4661 35.1307 0.40 0.40 5.2130 13.8700 19.8374 26.7337 5.4335 15.2062 23.5586 33.8276 TABLE IV. FUNDAMENTAL FREQUENCY OF POROUS SS COMPOSITE BEAM FOR DIFFERENT POROSITY eo, e1, AND INTERFACIAL STIFFNESS (ks) Porosity Fundamental frequency(rad/s) eo e1 ks=10 -2 ks=10 -1 ks=10 0 ks=10 ks=10 2 ks=500 ks=10 3 ks=10 4 0.00 0.00 10.6498 10.6660 10.8244 12.0652 15.5493 16.9769 17.2189 17.4538 0.05 0.05 10.4803 10.4968 10.6577 11.9090 15.3453 16.7213 16.9527 17.1768 0.10 0.10 10.3105 10.3273 10.4909 11.7535 15.1407 16.4649 16.6858 16.8993 0.20 0.20 09.9686 09.9861 10.1557 11.4437 14.7276 15.9478 16.1479 16.3404 0.30 0.30 09.6212 09.6395 09.8164 11.1336 14.3058 15.4207 15.6003 15.7722 0.40 0.40 09.2644 09.2837 09.4695 10.8208 14.8700 14.8777 15.0371 15.1888 IV. CONCLUSIONS In this paper, one-dimensional finite element methods based on cubic order beam theory are used to analyze the free vibration of pervious steel-concrete beams. The distribution of linear shear springs is used to simulate partial shear interaction between layers. The cubic order beam theory is implemented using a continuous C 0 finite element containing 3 nodes. This work presents a novel analysis of the effect of porosity on the fundamental frequency of layered composite beams. The main findings of this study are:  As the porosity of the steel-concrete composite beam increases from 0 to 0.40, the fundamental frequency decreases by 10% in partial interaction and approximately 13.0% in the complete composite.  The presented finite element formulation shows the relationship between the fundamental frequency and various parameters such as the interfacial shear stiffness and end supports.  In CF support, the fundamental frequency is minimum, whereas it is maximum in CC support.  It is greater for a complete shear interface than for a partial shear interface. An increase in interfacial shear stiffness from 10 -2 to 10 4 leads to an increase in fundamental frequency by more than 63%.  The proposed model is more accurate in predicting the free vibration of composite steel-concrete porous beams than the existing models based on EBT and TBT. ACKNOWLEDGMENT The authors would like to thank the National Institute of Technology Patna (India) for its financial support. Engineering, Technology & Applied Science Research Vol. 13, No. 3, 2023, 10843-10848 10848 www.etasr.com Kumar & Kumar: Free Vibration Analysis of Steel-Concrete Pervious Beams REFERENCES [1] D. Mohammed and S. R. Al-Zaidee, "Deflection Reliability Analysis for Composite Steel Bridges," Engineering, Technology & Applied Science Research, vol. 12, no. 5, pp. 9155–9159, Oct. 2022, https://doi.org/ 10.48084/etasr.5146. [2] K. Swaminathan, D. T. Naveenkumar, A. M. Zenkour, and E. 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