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Sandwich beams are better strength to weight ratios and rigidity in the comparison of individual beams, because composed of hard and stiff faces and core with light weight (Gdoutos and Daniel, 2008). Such beams can resist on more loading rather than the element of sandwich beams separately. This structures are used in high-rise buildings, skyscraper.
In the state that vertical loads affected on layers of sandwich beams independently, the slabs hinder as individual element and the relative shear slip take place between the layers. Such structures are considered as a composite beam without composite action.
Shear connection that happened in interface of layers, determine the behavior of sandwich structures. Beam manner with full composite action or partially composite action as a result of rigid or flexible shear connection respectively. If the beams withstand vertical loads unity that means shear connector can be propagate between layer interface for maintaining shear slip depends on that shear connector (Viest et al., 1997).
The flexural rigidity of the beam with partial composite action depends on slip strain furthermore sectional and materials parameters; therefore the relationship between moment and curvature of the composite beam is nonlinear (Gue-Qiang and Jin-Jun, 2009).
Finite element Method is useful for gaining deflection of beams involving complex geometries, combined loading and material properties, in which the analytical solutions are not available (Budynas1999), so for composite sandwich structures this approach are used for calculating deflection. Account must be taken of the effect of relative shear slip, in finite element analysis of sandwich beams with partial composite action. For reaching this assumption, at the two ends of such beams two autonomous axial degree of freedom are considered, but inconsistency of degree of freedom take place in finite element analysis. (faella and et al., 2001). To keep away from this problem, the elastic stiffness equation of composite beam element, with respect to relative slip, have to be derived (Gue-Qiang and Jin-Jun, 2009). Because of, the nonlinearity of relationship between moment and curvature of sandwich beam with partial composite action, which happened as a result of existence of sleep strain in flexural rigidity of such beams, the elastic stiffness equation cannot drive by using the equilibrium of internal and external moments of beams directly. For solving this problem, according to Newmark and at el. (1951) the elastic stiffness matrix is derived found on elastic interaction theory through the solution of the governing differential equilibrium equation of the composite beams.
The elastic stiffness matrix for composite beams that composed of two layers has gained in previous researches. The majority of those researches especially consisted on the composite that consist of concrete slabs laid on the steel beams. In this research, this method will extended for sandwich beams with three layers with different materials and sectional for top and bottom layers. Lastly, the gaining equation will simplify for symmetric composite sandwich beam with partial composite action as a special case.
After that, the deflection of sandwich beams can gain by finite element analysis considering the Timoshenko theory (Reddy, 200), or in the simple state of loading and supporting, the analytical solution based on Timoshenko Theory can use (Wang, 1995). But in both of the above approach, the flexural rigidity and shearing rigidity have to be considered. At the end of this paper, these equations are derived and explained.
2 The Composite Action Effects on Elastic Stiffness of Composite Sandwich Beams
2.1 Beams with Partial Composite Action
The strain diagram is given in Figure 2.1 . Denote and as the distances from the neutral axes of any layer components to their top surfaces, respectively. Note that , , , and .
Fig 2.1 Composite beam with partial composite action: partial composite action section, strain distribution along sectional height, and internal forces
A strain difference can be seen along the sandwich layers interface, which is defined as slip strain or . In partial composite action restrained slip occurs. The strain diagram is given in figure 4.1. The slip strain at the two top and bottom layers (faces) with the layers located in the middle (core) can then be expressed as
The compression in the top and bottom slabs and the tension in the core of the beam are given by
The equilibrium of N and T, i.e. N=T, results in
and can be expressed with and :
Combining equations 1 and 6 yields:
Substituting equation 8 back into equation 6:
From equations 1 or 8, can be expressed as:
Substituting equation 9 back into yields:
Substituting 9 back into 3, yields
As shown in the figure 4.1
By considering that:
From equations 11 and 12 is:
From the equation 5, can be expressed as:
From equation 7, or if equations 4 and 5 are equal, yields:
Combining equations 2 and 7 yields:
Substituting 14 back into 7, denote
Substituting equation 15 back into 14 yields:
Substituting 15 back into 5 yields:
From the figure 4.1, can be expressed:
By considering that:
From equations 17 and 18 can be expressed:
The equilibrium of internal and external moments with considering the figure 4.1 gives:
With respect to the static, can be expressed
By approximating and with respect to Radius Zhyrasyvn , can be expressed
And and is given by
Obviously, the relationship between moment and curvature of the composite beam with partial composite action is no longer linear. In addition to sectional and material parameters, the bending stiffness of the partially composite beam depends also on the slip strain. In next Section, the elastic stiffness equation of the partially composite beam, based on Newmark partial interaction theory, will be derived.
2.2 Elastic Stiffness Equation of Composite Beam Element
2.2.1 Basic Assumptions
The following assumptions are employed in this section:
(1) Both faces and core layers are in elastic state.
(2) The shear stud is also in elastic state, and the shearââ‚¬"slip relationship for single shear stud is
where K is the shear stiffness of a stud.
(3) The composite action is smeared uniformly on the face-core interface, although the actual shear studs providing composite action are discretely distributed.
(4) The plane section of the faces slabs and the core beam remains plane independently, which indicates that the strains are linearly distributed along layers section heights, respectively.
(5) Lift-up of shear studs, namely pull-out of shear studs form the face slab, is prevented. The deflection of the core layer of the beam and the faces slab at the same position along the length is identical, or the layer components of the composite beam are subjected to the same curvature in deformation.
2.2.2 Differential equilibrium equation of partially composite beam:
The strains of the layers components at the interface can be expressed with internal forces as
Consider a differential unit of the top and bottom flange (see figure 2.2), and the force equilibrium of the unit in horizontal is
FIG 2.2 Horizontal balance of the top and bottom layers
For top layer
For bottom layer
The shear density transferred by single shear stud on the interface is:
Combining equations 23, 24 and 25 lead to
The slip strain at the interface of layers of the sandwich beams can then be expressed as
Equalling equations 22 to equation 27 result in
By considering the assumptions, the moment curvature relationships are
And it leads to
By equation, one has
Substituting equations 30, 31, and 33 back into equation 28 leads to the following fourth order differential equilibrium equation of the partially composite beam between the interface of top and middle layer
Substituting equations 30, 31, and 34 back into equation 29 leads to the following fourth order differential equilibrium equation of the partially composite beam between the interface of bottom and middle layer:
With adding equations 35 and 36, yields
From equation 32, can be expressed
Two differentiate from the above equation, yields
Substituting equations 38b-43 into equation 37, leads to
Where is the shear modulus of the interface of composite beam.
and are parameters that relevant to the material properties and section dimensions, and are defined as
2.2.3 Stiffness equation of composite beam element:
The typical forces and deformations of the beam element are as in figure 2.3
Fig 2.3 the typical forces and deformation of the beam element
The moment at an arbitrary location distance x away from end 1 can be expressed with the end moment and the end shear
The force balance also determines
Substituting equations 45-50 into equation 44, yield
The solution of the fourth order differential equation 51 is
Where and are integration constants.
Integrating equation 52 twice results in the deflection of the composite beam element with slip as
Where and are also integration constants.
Consider the following boundary conditions with considering the figure 2.3
With use above boundary conditions into equation 53 with considering the crammer rules yields four simultaneous algebra equations as following:
In most cases, the middle layer are connected to columns fixedly, and when the anchor-hold of negative reinforcement bars in top and bottom slabs has good performance, it is reasonable to assume that the slip between the middle layer and 2 up and bottom slabs at the ends of composite beams is negligible, namely
Substituting equation 61 into equation 26 leads to
Differentiate from equation 38.a, one has
Substituting equations 31 and 48 into above equation, yields
Three differentiate from equation 53, leads to
Substitute equation back into 63, result in
Substituting equation 64 into equation 62.a, yields
Substituting equation 64 into equation 62.b, yields
From equation 60 and 65.a, with considering the following assumptions, , , and calculated:
Substitute equation 60.b into 65.a, yields
Assume that is equal to coefficient of in equation 67
From equation 67 with considering to 68, one has
Now, the equation has to summarized as following
Then rewrite equation with considering the following assumptions
So, one has
Substitute assumption 66 in equation 65.b, lead to
From equations 71.a and 71.b, yields
Substitute 71.c into back 65.b, yields
Extract the coefficient of from equation 72, yields:
Substituting equation 73 into equation 72
Substituting equation 74 into equation 72, yields
From equation and 75 with considering to crammer rules, one has
Denominator of equations 76 and 77 are
Now, have to solve equation 78
Numerator of equation 76 is
With considering equations 69.a, 69.c, 74.a, and 74.c yields
After solving equation 81 by using equation 82 with considering equations 66.b, 66.c, 66.d, and 66.e, the coefficient of equation 81, can be written as
Coefficient of :
Coefficient of :
Coefficient of :
Coefficient of :
Substituting equation 83 into equation 81, yields
From equation 76, one has
Equation 84, is the solution of equation 76, with considering the following formula
That - in equation 85 have gained by equation 83, and have gained by equation 80.
If the equation 77 has been considered
At first, and with considering equations 69.b, 69.c, 74.b, and 74.c, will be calculated. then has been gained
Substituting equation 86 into equation 77 and considering equations 66 yields
From equations 80 and 87, one has
From equation 50, one has
Combining equations 88, 89, 84, and 85, lead to
The matrix expression of equations 84, 88, and 90 is
Equation 91 or 93 is the elastic stiffness equation for the sandwich composite beam element with partial composite action for three nonsymmetrical layers and is the corresponding elastic stiffness matrix of the element.