Abstract: In this study, the impact of boundary condition in dynamic behaviour of a straight beam under action of moving force is investigated. Furthermore, the effectiveness of various parameters such as beam geometry and material type on dynamic magnification factor is studied. This study shows that maximum dynamic magnification factor only depends on boundary condition. This value is obtained at a critical to moving load velocity ratio that is independent of material and beam geometry and only depends on boundary condition. The Maximum dynamic magnification factor for composite and isotropic material in each boundary condition is slightly different.
INTRODUCTION
Transport engineering structures such as bridges are subjected to loads that vary in both time and space (moving forces), in the form of vehicular traffic, which cause them to vibrate. A moving vehicle on a bridge causes deflections and stresses that are generally greater than those caused by the same vehicular loads applied statically. This is due to dynamic interaction between the bridge and the vehicle.
Given a particular beam and the magnitude of the traveling force, the following characteristics may be of interest: maximum response of the beam for a given velocity of the force, maximum response over all possible velocities and the velocity at which it is developed, dependence of forced response and the amplitude of free vibration on velocity and the like.
There is not much work done on the study of impact of the material, beam dimensions and boundary conditions on dynamic behaviour of straight beams under the action of moving loads.
Fryba (1999) reported, analytical solutions for simple problems of simply supported and continuous beams with uniform cross-section.
Pesterev et al. (2003), Gbadeyan and Oni (1995), Renard and Taazount (2002) and Ichikawa et al. (2000) performed dynamic analysis of beams due with uniform cross-section to moving masses by means of various analytical or numerical methods. Zheng et al. (1998) studied dynamic response of a non-uniform continuous Euler-Bernoulli beam with Hamilton’s principle. Zhu and Law (2001) used generalized orthogonal function approach to determine the responses of a bridge deck under the action of moving loads. Kadivar and Mohebpour (1997, 1998) investigated forced vibration of composite laminated beams under action of moving loads. He only considered a simply supported isotropic and composite beam. Abu-Hilal and Mohen (2000) studied vibrational behavior of isotropic single span beams with general boundary conditions. Although Abu-Hilal studied boundary condition effects on dynamic behaviour, but he did not investigated these effects on dynamic magnification factor.
In these studies, simply supported is the major feature of boundary conditions that is considered and the effect of other boundary conditions is not investigated.
The objective of this study is to demonstrate the effect of boundary conditions in dynamic magnification factor. Dynamic magnification factor is defined as the ratio of dynamic deflection to static deflection of a beam. This investigation is performed in various boundary conditions with different cross-sections and material properties.
FINITE ELEMENTS MODEL (FEM)
Dynamic behavior of a straight beam subjected to a vertical constant force P traversing the beam with a constant velocity C (Fig. 1 ) is studied in three-dimensional element by use of finite element software Msc/Nastran.
In isotropic materials such as steel, the beam model is discritized with 20 node solid elements, while for composite material the 20 node continuum composite element is applied (Anonymous, 2004). Each node has three degrees of freedom. Various mesh density in each beam is used in each beam to achieve a good mesh density and therefore improving the accuracy of results. Densifying is iteratively repeated until the difference between consecutive results becomes less than 0.5%. The number of nodes in each model is shown in Table 1.
Finite element models are consisted of 245722 nodes for isotropic beam with cross-section C to 345682 nodes for isotropic beam with cross section A. Various cross-sections are shown in Fig. 2. For transient analysis the Newmark (Anonymous, 2004) method is used to carry out integration with respect to time where, all parameters for tn are known, tn+1 can be calculated using the following equation of motion
(1) |
Where, M is the mass matrix, B the damping matrix, K
the stiffness matrix,
The estimates of qn+1,
(2) |
Where, Δt is the time step and γ and β are constants. The
(3) |
The analyses stages are illustrated in the flow chart are shown in Fig. 2.
Validation of Finite Element Model
For this purpose, a uniform simply supported steel beam with below
specifications analysis with FE and analytical method (Saranjam et
al., 2006) and the results are compared.
| Fig. 1: | Simply supported beam subjected to a moving load |
| Fig. 2: | A Flowchart of Analysis Scheme |
| Table 1: | No. of nodes and elements in various beams |
| Table 2: | Dynamic magnification factor for moving load on simply supported steel beam |
| Table 3: | Material properties of aluminum |
The beam has length L = 100 cm, width b = 0.5 cm, thickness α = 0.5 cm (Fig. 1 ), modulus of elasticity E = 2.1x1011Pa, mass density ρ = 7800 kg m-3, Poisson ratio υ= 0.3, moment of inertia I = 0.00520833 cm4 and fundamental period Tf = 0.085005 s (Meirovitch, 1967).
All degrees of freedom in Z direction are constrained in order to calculate the modes which are in X-Y plane and contribute to dynamic response of the beam in Y direction. The beam is simply supported and can be analyzed employing the following conditions (Fryba, 1999):
| • | Initial conditions: |
(4) |
| • | Boundary conditions: |
(5) |
Where,
| W (x,t) | = Beam deflection in at point x and time t. |
| M (x,t) | = The bending moment at point x and time t. |
To validate the finite elements model, the results of dynamic response of a
simply supported isotropic beam subjected to a P = 10.0KN (Fig.
1 ) moving force is compared with analytical results.
The error e is defined as
(6) |
Where, D mf exact and D mf FEM are dynamic magnification factors calculated with exact formula and finite elements method, respectively,
The dynamic magnification factor for a simply supported beam is calculated by dividing dynamic deflection by static deflection of beam center.
Table 2 shows calculated dynamic magnification factors, for involved beam. The quantity of T is the time required for the moving load to travel the beam span. Closeness of the results of the agreement between the finite elements method and theoretical solution is acceptable (Table 2).
Parametric Study
In previous part the finite elements model is validated for a uniform simply
supported steel beam subjected to a single moving force with a constant velocity,
thus for evaluating boundary conditions effect on dynamic magnification factor,
the finite elements method can be used.
In this part four boundary conditions, Pinned-fixed, Fixed-pinned, Fixed-fixed and Pinned-pinned is investigated.
| Fig. 3: | Cross-sections Dimensions (cm) |
Dynamic magnification factor for each boundary condition is calculated in the location of maximum static deflection. For a simply supported beam, this location is in the mid-point of the beam span but for other boundary condition, this location is changed.
For investigation of beam geometry (moment of inertia), four cross-sections are considered. The beam length is similar to simply supported beam that is described earlier (Fig. 3).
In each boundary condition, steel, aluminum and composite material is used. Cross-section for composite beam is the section A in Fig. 3. Mechanical property of aluminum is shown (Table 3)
The composite material is consisted of 20 layers with equal thickness of 1 mm and angle sequence (0/90)10s. Property of each lamina shown in (Table 4).
The principle directions 1 and 2 are in X and Y directions, respectively (Fig. 1 ).
RESULTS AND DISCUSSION
As mentioned in previous section, dynamic magnification
factor is calculated by dividing the maximum dynamic deflection by maximum
static deflection of beam. In simply supported beam, the maximum dynamic
and static deflections are in the mid-point of the beam span, but in other
boundary conditions, this location may be changed. In Table
5 the location and magnitude of maximum static deflection is shown
for involved boundary conditions.
| Fig. 4: | Comparison of relative displacement in various boundary conditions |
| Table 4: | Material properties of Lamina |
| Table 5: | Location of maximum static deflection for various boundary conditions (Abu-hilal, 2000) |
Relative displacement of the points that are specified in this table is compared in Fig. 4. Relative displacement is defined as dynamic deflection divided by maximum static deflection that is specified in Table 5. As shown in this figure, for each boundary condition the location of maximum deflection is the location of static deflection. Thus, dynamic and static deflection for involved boundary condition are calculated on the points that are specified in Table 5.
As it can be seen from Fig. 5-7
the dynamic magnification factor in various beams with different materials,
boundary conditions and cross-sections (moment of inertia) that are subjected
to a moving constant load can be divided into two regions: undercritical
and overcritical. In the undercritical region, the dynamic magnification
factor increases with the increasing velocity of the moving load, but
in the overcritical region dynamic magnification factor decreases when
the moving load velocity increases. The quantity of Cf in these
figures is the velocity of which the time for traveling span is equal
to the fundamental period of the beam.
| Fig. 5: | Dynamic magnification factor in steel beam with different cross-sections and boundary conditions |
| Fig. 6: | Dynamic magnification factor in aluminum beam with different cross-sections and boundary conditions |
According to these figures in various beams with different boundary conditions, material types and cross-sections, maximum dynamic magnification factors in Pinned-pinned and Pinned-fixed conditions are the same and higher than the others. The Fixed-fixed condition has a lower maximum dynamic magnification factor.
The important results of this investigation are that maximum dynamic magnification factors for Pinned-pinned and Fixed-pinned conditions in various materials and cross-sections are developed at Cc/Cf = 1.25 but for Fixed-fixed and Pinned-fixed conditions this ratio will be 1 (Cc is the critical velocity in which the maximum dynamic magnification factor is developed). Thus in uniform straight beams Cc/Cf ratio is independent of material type and beam geometry and only depends on boundary conditions.
As shown in Fig. 5-7, in spite of the fact that, total weight of composite and aluminum beams are about 5 and 3 times less than the total weight of steel beam, respectively; maximum dynamic magnification factor is approximately close to the value of that the same boundary conditions. Thus as in Cc/ Cf ratio, maximum dynamic magnification factor for a uniform straight beam only depends on boundary conditions.
In Fig. 8-10, the variation of dynamic deflection of the beam (Wd) divided by the static deflection (Ws) is drawn with respect to the position of the moving load for steel and composite beams. Note that, the curves obtained from aluminum beams are similar to that of steel beams thus for avoidance of chaos, behaviours of steel beam only, shown in Fig. 8.
The velocity of moving load is the critical velocity for each boundary condition.
It can be seen that maximum dynamic deflections of steel and aluminum beams
are 1.73 times greater than static deflection and belong to Pinned-fixed and
Pinned-pinned conditions (Fig. 8-10). This
maximum is developed when the moving load passes from the location of maximum
static deflection and has a delay with respect to the position of the moving
load.
| Fig. 7: | Dynamic magnification factor in composite beam with different boundary conditions |
| Fig. 8: | Comparison between dynamic and static deflection of steel beam with cross-section A in critical velocities |
| Fig. 9: | Comparison of dynamic and static deflection of steel beam with cross-section D in critical velocities |
| Fig. 10: | Comparison of dynamic deflection of steel beam for various boundary conditions in critical velocities |
| Fig. 11: | Comparison of dynamic deflection of composite beam in critical velocities for various boundary conditions |
In other boundary conditions, this delay increases and the location of maximum dynamic deflection shift to the right of the beam so that in Fixed-pinned condition maximum dynamic magnification factor approximately reaches the third quarter of the beam span. The higher delay time belongs to Fixed-pinned condition.
According to Fig. 11 these behaviours are also seen in a composite beam. In a composite beam, the Fixed-pinned condition has maximum delay time and maximum dynamic magnification factor is about 1.74 that is the same as in to steel and aluminum beams.
CONCLUSION
Formulations and comparative finite elements results have been presented for a conforming finite elements method that may be used to model a beam under action of several moving loads and this comparison shows that finite elements method has a good accuracy for moving load problem.
In a straight uniform beam, maximum points of dynamic and static deflections are in the same position.
Dynamic magnification factor (Wd/Ws) of a beam can be separated into undercritical and overcritical regions.
Pinned-fixed and Pinned-pinned conditions have maximum dynamic magnification factors and maximum delay time is belong to Fixed-fixed condition in various materials and beam geometries.
Important conclusion of this investigation may be described as that the ratio of Cc/Cf is independent of material types and beam dimensions. This ratio is only depending on boundary conditions.
Another important conclusion is that maximum dynamic magnification factors for a uniform straight beam are only dependent on boundary conditions. These values are approximately are the same for both isotropic and composite material in a boundary conditions (about 1.73).