Stress recovery of laminated non-prismatic beams under layerwise traction and body forces

Abstract

Emerging manufacturing technologies, including 3D printing and additive layer manufacturing, offer scope for making slender heterogeneous structures with complex geometry. Modern applications include tapered sandwich beams employed in the aeronautical industry, wind turbine blades and concrete beams used in construction. It is noteworthy that state-of-the-art closed form solutions for stresses are often excessively simple to be representative of real laminated tapered beams. For example, centroidal variation with respect to the neutral axis is neglected, and the transverse direct stress component is disregarded. Also, non-classical terms arise due to interactions between stiffness and external load distributions. Another drawback is that the external load is assumed to react uniformly through the cross-section in classical beam formulations, which is an inaccurate assumption for slender structures loaded on only a sub-section of the entire cross-section. To address these limitations, a simple and efficient yet accurate analytical stress recovery method is presented for laminated non-prismatic beams with arbitrary cross-sectional shapes under layerwise body forces and traction loads. Moreover, closed-form solutions are deduced for rectangular cross-sections. The proposed method invokes Cauchy stress equilibrium followed by implementing appropriate interfacial boundary conditions. The main novelties comprise the 2D transverse stress field recovery considering centroidal variation with respect to the neutral axis, application of layerwise external loads, and consideration of effects where stiffness and external load distributions differ. A state of plane stress under small linear-elastic strains is assumed, for cases where beam thickness taper is restricted to ${15}^{\circ }$ . The model is validated by comparison with finite element analysis and relevant analytical formulations.

Keywords: Laminated beam; Layerwise load; Non-prismatic beam; Stress recovery; Tapered beam; Traction.

Fig. 1

a Side perspective. b Generic cross-section

Fig. 2

Laminated non-prismatic beam in equilibrium

Fig. 3

Interface equilibrium in: a x-direction, b z-direction

Fig. 4

Case (1) boundary conditions

Fig. 5

Case (1) typical mesh

Fig. 6

${\sigma }_{\mathit{xx}}$ for case (1): a $x=0.25L$, b $x=0.90L$

Fig. 7

${\tau }_{\mathit{xz}}$ for case (1): a $x=0.25L$, b $x=0.90L$

Fig. 8

${\sigma }_{\mathit{zz}}$ for case (1): a $x=0.25L$, b $x=0.90L$

Fig. 9

Case (2) boundary conditions

Fig. 10

Case (2) typical mesh

Fig. 11

${\sigma }_{\mathit{xx}}$ for case (2): a $x=0.25L$, b $x=0.60L$

Fig. 12

${\tau }_{\mathit{xz}}$ for case (2): a $x=0.25L$, b $x=0.60L$

Fig. 13

${\sigma }_{\mathit{zz}}$ for case (2): a $x=0.25L$, b $x=0.60L$

Fig. 14

Case (3) typical mesh

Fig. 15

Case (3.1) boundary conditions

Fig. 16

${\sigma }_{\mathit{xx}}$ for case (3.1): a $x=0.25L$, b $x=0.75L$

Fig. 17

${\tau }_{\mathit{xz}}$ for case (3.1): a $x=0.25L$, b $x=0.75L$

Fig. 18

${\sigma }_{\mathit{zz}}$ for case (3.1): a $x=0.25L$, b $x=0.75L$

Fig. 19

Case (3.2) boundary conditions

Fig. 20

${\sigma }_{\mathit{xx}}$ for case (3.2): a $x=0.25L$, b $x=0.75L$

Fig. 21

${\tau }_{\mathit{xz}}$ for case (3.2): a $x=0.25L$, b $x=0.75L$

Fig. 22

${\sigma }_{\mathit{zz}}$ for case (3.2): a $x=0.25L$, b $x=0.75L$

Fig. 23

Rectangular cross-section