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Research Papers

Application of the Shakedown Theory to Brittle-Matrix Fiber-Reinforced Cracked Composite Beams Under Combined Traction and Flexure

[+] Author and Article Information
Andrea Spagnoli

e-mai: spagnoli@unipr.it

Lorenzo Montanari

Department of Civil-Environmental
Engineering and Architecture,
University of Parma,
Parco Area delle Scienze 181/A,
Parma 43124, Italy

Manuscript received April 3, 2013; final manuscript received August 29, 2013; accepted manuscript posted September 3, 2013; published online October 16, 2013. Assoc. Editor: Daining Fang.

J. Appl. Mech 81(3), 031012 (Oct 16, 2013) (8 pages) Paper No: JAM-13-1147; doi: 10.1115/1.4025313 History: Received April 03, 2013; Revised August 29, 2013; Accepted September 03, 2013

The cracking behavior of a composite beam with multiple reinforcing fibers under periodic traction-flexure is analyzed through a fracture mechanics-based model, where the edge-cracked beam section is exposed to external loads and crack bridging reactions due to the fibers. Assuming a rigid-perfectly plastic bridging law for the fibers and a linear-elastic law for the matrix, the statically indeterminate bridging forces are obtained from compatibility conditions. Under general load paths, shakedown conditions are explored by making use of the Melan's theorem, here reformulated for the discrete problem under consideration, where crack opening displacement at the fiber level plays the role of plastic strain in the counterpart problem of an elastic-plastic solid. The limit of shakedown is determined through an optimization procedure based on a linear programming technique.

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Figures

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Fig. 1

(a) Schematic of the model, (b) crack profile for fibers in their rigid stage, (c) crack profile for some fibers in their plastic stage

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Fig. 2

(a) Time histories of axial force and bending moment in the case of Mmax/Nmax = 0.25 and cross load path; (b) time histories of axial force and bending moment in the case of Mmax/Nmax = 0.25 and elliptic load path; (c) corresponding M-N load paths

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Fig. 3

Strength distribution of fibers along the height of the beam: (a) constant distribution; (b) linear distribution; (c) symmetrical and linear distribution

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Fig. 4

Bree-like diagram showing the elastic domain and the shakedown domain for the (a), (b), and (c) fiber distributions with ξ = 0.1

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Fig. 5

Bending moment versus rotation and axial force versus rotation for (a) cross load path and (b) elliptic load path in the case of Mmax/Nmax = 0.25 and symmetrical and linear fiber distribution

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Fig. 6

Bridging force versus crack-opening displacement in the case of Mmax/Nmax = 0.25: (a) μ = μSD and cross load path; (b) μ = 1.05μSD and cross load path; (c) μ = μSD and elliptic load path; (d) μ = 1.05μSD and elliptic load path. The symmetrical and linear fiber distribution is considered.

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Fig. 7

Shakedown domain for different crack depths and fiber strength distributions: (a) constant distribution; (b) linear distribution; (c) symmetrical and linear distribution

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Fig. 8

Shakedown domain for intermediate crack depth (ξ = 0.3) and different distributions of fiber strength

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