Finite element modeling and experimental verification of lightweight steel floor vibration

3.1. Rigid link

Unlike concrete slab floor structure, light-weight steel floor structure is not an integrated system of structural elements (beams, joists, and flooring materials). Therefore, beam elements are used for beam and joist while plate elements are used for flooring material in finite element analysis modeling. In this case, it is necessary to constrain different elements meeting at one node appropriately to ensure the same behavior. Especially, it is important to secure the identical vertical behaviors of members because the floor vibration generates vertically. Furthermore, counter measures are necessary to resolve the phase difference because the central line positions of those elements are different.

In this study, rigid links are used to solve the problem of difference in the centroid of beam, joist, and flooring material and to guarantee the same behavior of members that meet at one node. A rigid link is a rigid bar element that provides a stiff connection between nodes in finite element model. This is accomplished by subordinating the degree of freedom of one or more nodes to the degree of freedom of a certain node; the certain node is called a master node, and the subordinated ones are called slave nodes. Types of rigid link are: method of limiting displacement and rotation by restraining relative motion as if the master nodes and slave nodes are connected by a 3-dimensional rigid body; a method to restrain relative motion as if master nodes and slave nodes are connected by a planar rigid body; and a method of restraining only displacement or rotation.

The flooring material and joist are fixed by screws in a light-weight floor structure, which is not considered to restrain rotational motion fully. Meanwhile, connection between the joist and beams are generally performed by welding, which prevents the rotation effectively. Considering these two different connection details, the rigid link that restrains rotation and displacement simultaneously is used for the convenience of modeling in this study. Fig. 2 shows the schematic of rigid links between floor material and joist, while Fig. 3 illustrates rigid links that connect edge beam, mid-beam, joist, and flooring material that have phase differences. In Fig. 3, M represents a master node and S means a slave node. Note that two rigid links must be used when two different slave nodes (S1 and S2) are connected to one master node (M1) as indicated in dotted box of Fig. 3. If two master nodes with respective slave nodes are assigned at the same location, an error will be produced from the overlap of the nodes.

Fig. 2Schematic of rigid link between flooring material and joist

Fig. 3Rigid rinks connecting edge beam, mid-beam, joist, and flooring material with phase differences

3.2. Support restraints

The natural vibration frequency of a floor is calculated by the following equation [9, 10]:

where, $L$ is beam length, $E$ is the modulus of elasticity of the beam, $I_x$ is moment of inertia, $m$ is the mass per unit length, and $\alpha$ is the variable depending on support restraint. $\alpha$ is 9.869 for simply supported beams while 22.37 for fixed beams.

It is not appropriate to apply the above equation directly to actual floors because the support restraint of the beam-column joints is in-between of fixed and simply supported support. Similarly, it is very difficult to model the actual support conditions of beam-column connection in light-weight floor structures because members are connected by welding, bolt, and/or screws. In this study, two different support restraints, all fixed restraint and mixture of fixed and released restraints are investigated in the finite element analysis in order to represent the actual floors as summarized in Table 1.

When all rotational and displacement degrees of freedoms of entire supports are restrained, it is termed as “All Fixed”; and when all rotational degrees of freedoms are freed but the displacement degrees of freedom are freed or restrained depending on the support locations, it is termed as “Fixed+Roller”. In the Fixed+Roller restraints, Support 1 in Fig. 4 is restrained in all three translational directions, and Support 2 is restrained in the vertical direction ($z$-direction) and in the out-of plane direction ($y$-direction). The diagonally opposite support, Support 4 in Fig 4, is restrained in the vertical direction and in the $x$-direction. Support 3 is restrained in the vertical direction ($z$-direction) only.

Fig. 4Support restraints

4.1. Full-scale test floor specimens

Four full-scale floor specimens were fabricated for the verification of the finite element analysis model presented in this study. The size is same for all specimens as 3.3 m×3.3 m with the joist span of 400 mm. H-200×100×5.5×8 was used as the corner beam and the corner beams were bolt-connected to columns. Two of the specimens were built without center beam while the other two had a mid-beam of H-148×100×6×9 to investigate the stiffness increase effect. In addition, each specimen had two different type of joist: standard cold-formed C section or rectangular pipe section with two different moments of inertia. The standard cold-formed C-shape joist and the rectangular pipe joist were selected to have almost same moment of inertia as presented in Table 2. Fig. 5 shows the dimensions of the specimens.

The specimens were named as S (square pipes) or C (cold-formed C-shape) according to the joist shapes, and X (do not have) or H (have) depending on the existence of the hot-rolled steel beam at the center. Fig. 6 shows full-scale specimens used for the test. All specimens are finished with 900 mm wide cement particle boards.

Fig. 5Dimensions of specimens

a) Specimen without a mid-beam

b) Specimen with a mid-beam

Fig. 6Test specimens

a) SX

b) CX

c) SH

d) CX

4.2. Finite element modeling of full-scale test floor specimens

Midas/Gen [11] was used in this paper for the floor vibration analysis of light-weight steel floor structure. The mesh size for flooring material was set 200 mm×200 mm to obtain more accurate mode shape and natural vibration frequencies. Fig. 7(a) shows the final mesh configuration of the analysis model, and rigid links applied to all nodes are shown in Fig. 7(b). The rigid lines in Fig. 7(b) represent the location of joist, and the small number 1 at each node indicates the degree of restraints of each rigid link. Fig. 8 shows the finalized 3-dimensional finite element model for specimens SX and CH. The finite element models of specimens CX and SH are almost identical to specimens SH and CH, respectively, and thus omitted here.

Fig. 7Meshing and application of rigid links

a) Meshing

b) Application of rigid links

Fig. 8Finite element model of specimens

a) SX

b) CH

4.3. Results of finite element analysis

Fig. 9 shows the first mode shape of specimens SX and SH obtained from the finite element analysis. The natural vibration frequencies in Fig. 9 are for the Fixed+Roller support restraint, which are quite different from the result of All Fixed support restraint despite the similar mode shape.

Fig. 9First mode shape and natural frequencies

a) SX (natural frequency: 23.71 Hz)

b) CH (natural frequency: 25.36 Hz)

Table 3 summarizes the first mode natural frequencies of specimens calculated from the finite element analysis. The first mode natural frequencies under All Fixed support restraints appeared to be higher than those under Fixed+Roller support restraints for all specimens. It can be noted that the first mode natural frequencies under All Fixed support restraints are about 1.33 times greater than those under Fixed+Roller support restraints when there exist no mid-beam regardless of joist shapes; and 1.18 times with the existence of a mid-beam.

The increase in stiffness and the corresponding increase in natural frequency owing to a mid-beam were shown as expected under Fixed+Roller support restraints, but the opposite effect appeared under All Fixed support restraint. The reason seems that the mid-beam does not increase the stiffness but only adds the mass due to the confining effect of rotation and displacement at supports under All Fixed support restraint. Regardless of support restraints, the first mode natural frequency does not show much difference depending on joist shapes if the moment of inertia is similar. It can be noticed that the moment of inertia of the joist affects more on the floor vibration than the shape of the joist in the finite element analysis.

4.4. Experimental results

A human impact loading test on full-scale floor specimens was carried out to verify the validity of the finite element analysis model. The experiment was performed by applying the impact load by means of jumping of an adult person while acceleration data were measured at the center of floor (Fig. 10(a)). For the acceleration measurement, a servo-type accelerator ES-U2 of Kinemerics was used and a data-logger NetPod 4003 was used for data collection. Fig. 10(b) shows the measured acceleration time history of CX specimen.

Fig. 10Human impact loading test

a) Test on specimen CX

b) Measured acceleration time hisoty

Fig. 11 shows the power spectral density (PSD) of measured acceleration data to find the first mode natural frequency from the human impact loading test. The resulting first mode natural frequencies of four specimens are summarized in Table 4. It can be seen from Table 4 that the increase of stiffness owing to the mid-beam does not occur regardless of joist shape. That is, the measured first mode natural frequencies of specimens SH and CH with a mid-beam were lower than those of the specimens SX and CX. It can be concluded that the mass increase effect was greater that the stiffness increase effect if the mid-beam of hot rolled steel section is used. This finding is just as similar to finite element analysis result under All Fixed support restraints.

Unlike the finite element analysis result, however, the first mode natural frequencies of specimens SX and SH with square pipes joist ware measured higher than those of specimens CX and CH with cold-formed C-shape joist. It can be concluded that closed-section joist exhibits better floor vibration performance than open-section joist even though they have similar moment of inertia.

4.5. Comparison of analysis result and measured result

Table 5 compares the first mode natural frequencies between from the finite element analysis and from the experiment. It can be seen that the finite element analysis result under Fixed+Roller support restraint shows errors of 0.16 %-9.10 % compared to the experiment result, while All Fixed condition shows errors of 18.41 %-25.45 %. From this result, it can be conclude that the bolt connection for beam-column joint used in light-weight steel floor structures needs to be modeled with Fixed+Roller support restraint in order to represent the behavior of actual floor.

Fig. 11PSD of measured acceleration

a) SX

b) CX

c) SH

d) CX