Complex core system 1: Analytical multilayer modeling
5.3 New analytical models
New analytical bending and shear stiffness models have been developed for a multilayer sandwich beam used to predict its deflection, axial and shear stresses in the face sheets, cores and laminate layers. The models are based on the single-core classical sandwich theory.
5.3.1 New bending stiffness model
The bending stiffness, D, of a single-core sandwich beam, as shown in Fig. 5.4a, is the summation of the individual rigidities of the face sheets and the core, obtained about the neutral axis of the entire sandwich beam [20]. This model can be extended, in a first step, to multilayer sandwich beams with two cores as shown in Fig. 5.4b, by splitting the rigidity of the single core into two core layers as follows:
1 1 1 1 2 2 2 2
ml f f c c c c f f
D E I E I E I E I (1)
where subscripts ml, f and c denote multilayer sandwich beam, face sheet and core respectively, and Ef1,If1,Ef2, If2,Ec1, Ic1, Ec2,Ic2 are the Young’s moduli and moments of inertia of the top face sheet, bottom face sheet, and core layers 1 and 2 about the neutral axis of the entire multilayer sandwich beam respectively.
Equation (1) can then be generalized for n layers between the top and bottom face sheets, which could either be cores or laminates as:
Chapter 5: Complex core system 1-analytical multilayer modeling
where i is the top or bottom face sheet, subscript c/l denotes any core/laminate layer in between the top and bottom face sheets and k is the kth core/laminatelayer.
d2
Figure 5.4: Schematic for a) single-core sandwich model and b) multilayer sandwich beam
5.3.2 New shear stiffness model
The shear stiffness, S, of a sandwich beam with a single core is simply a product of the core shear modulus, Gc, and the cross sectional area, Ac, of the core [21]. This model can be extended again to a multilayer sandwich beam with two cores in a first step using the energy method. Figure 5.5 shows a deformed multilayer sandwich beam element subjected to an arbitrary shear force, Vy.
Figure 5.5: Deformed multilayer sandwich element illustrating shear deformation
Structural performance of complex core systems for FRP-balsa composite sandwich bridge decks
88 The potential energy, Up, of the load, which causes a vertical shear deflection, ws, is:
1
The strain energy in the multilayer sandwich element due to the shearing of core layers 1 and 2 is the summation of the individual strain energies in each core as follows:
strains in core layers 1 and 2 respectively. Substituting shear stress, τxy= Vy/bd, where b is the beam width (assumed as being 1.0 m), d is the distance between the face sheet axis, and shear strain, γxy= τxy/Gxy, into Eq. 4, and integrating over the entire core thickness gives: layers 1 and 2 respectively, and hc is the total core thickness of the multilayer sandwich beam.The energy balance equation Up=Us, results in:
2 1
Taking into account that the derivative of the vertical shear deflection with respect to the beam direction denotes the average shear strain, dws/dx=γxy, the shear stiffness of a multilayer sandwich beam with two core layers can be expressed for a beam width, b, as:
1 2
Equation (7) can then be extended for the general case of n core/laminate layers to:
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Chapter 5: Complex core system 1-analytical multilayer modeling
89 5.3.3 Deflections, axial and shear stresses in multilayer sandwich beam
The total beam deflection, wt, of a sandwich beam is, according to the classical sandwich theory [21], the sum of the bending deflection, wb,and shear deflection, ws,as follows:
t b s
w w w (9)
The bending deflection is analytically obtained based on a shape function, magnified by a coefficient A, which is the maximum bending deflection at mid-span, as follows:
4 3
where L is the beam span and A depends on the type of applied load, such as, A=qL4/24Dml
for a uniformly distributed load, q, and A=23PL3/648Dml for a four-point bending load configuration with loads, P, applied at the third points of the span. The shear deflection is related to the bending deflection according to [21] as follows:
3
For the case of the four-point bending load configuration described above, the shear deflection results as:
The axial stresses in the face sheets and core of a single-core sandwich beam, according to the classical sandwich theory, which assumes a plane strain distribution through the beam thickness, are obtained from:
, stress location with reference to the neutral axis (see section A-A of Fig. 5.4a). Again assuming plane strain distribution, these equations can be extended for a multilayer sandwich beam and the axial stresses in the top or bottom face sheets and in the kth of n core/laminate layers result in:
The axial stresses at any depth, y, therefore vary linearly in each individual core/laminate layer depending on the Young’s modulus of the core/laminate layers. The axial stress is zero at the neutral axis of the entire multilayer system.
Structural performance of complex core systems for FRP-balsa composite sandwich bridge decks
90 The out-of-plane shear stresses in a single-core sandwich beam exhibit a parabolic distribution in the core, which decreases to zero in the top and bottom face sheets according to the classical sandwich theory. The magnitude of the shear stress at any beam section depends on the shear force at that section and the bending stiffness of the sandwich beam. The variation of the shear stress through the thickness depends on the first moments of area of the face sheets (top and bottom), Sf, and the core, Sc, about the neutral axis of the entire section. part of the cross section above y in section A-A of Fig. 5.4a. For a single core, this product is expressed as:
where d1 is the distance between the neutral axis of the entire sandwich beam and the top face sheet axis and hf is the face sheet thickness (see Fig. 5.4a). The shear stress model for the single core can be modified for a sandwich beam with two cores by considering two separate shear stress distributions, τc1 and τc2, in the core layers 1 and 2, which depend on their individual Young’s moduli and first moments of area, represented by the shaded areas in section A-A of Fig. 5.4b. Extending to the general case, the shear stress in the kth of n core/laminate layers results in:
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Chapter 5: Complex core system 1-analytical multilayer modeling
91