PSI - Issue 72

Victor Rizov / Procedia Structural Integrity 72 (2025) 120–127

121

the functionally graded materials is the relatively low resistance to longitudinal fracture. This refers especially to functionally graded materials which are produced layer by layer (in fact, this is one of the widely used technologies at present (Mahamood and Akinlabi (2017))). Although significant advance has been made in studying longitudinal fracture (or delamination) (Dowling (2007), Hutchinson and Suo (1991), Rizov (2017), Rizov (2017)), there are issues that are relatively less investigated and understood. For instance, such issue is the longitudinal fracture in functionally graded frames under dynamic loading. The current article explores one aspect of this issue, namely, longitudinal fracture under periodic dynamic loading. The reason for the current theoretical exploration is that periodic dynamic loads are very common in the engineering. One of the basic aims of the article is to find out to what degree the longitudinal fracture may be influenced by the supporting conditions of the frames under periodic dynamic loads. The exploration described in the current article uses a theoretical model that treats the frame as deformable body with inhomogeneity across thickness. The frame explored represents a single bay frame that is acted upon by a periodical dynamic load in the middle of the horizontal bar. The right vertical member of the frame is split by a longitudinal crack. The maximum and minimum SERR induced by the periodic dynamic load is derived for three types of supporting conditions of the frame to see their influence. Besides, basic geometrical parameters of the frame and the longitudinal crack are varied to explore how they affect the SERR for the different supporting conditions. The combined effects of the variation of the frame geometrical parameters and the supporting conditions on the maximum SERR under periodic dynamic loading are illustrated by several diagrams. 2. Single bay frames with three types of supporting conditions Figure 1 illustrates the frames explored in the current article. Three different support conditions are considered to explore their influence on the longitudinal fracture of frames. The frame in Fig. 1a is supported by three links (one horizontal link in point, D 2 , and two vertical links in points, D 1 and D 6 ). The frame in Fig. 1b is supported by two links (one vertical link in point, D 1 , and one horizontal link in point, D 2 ) and one pinned support in point, D 6 . The third type of support conditions is illustrated in Fig. 1c. The frame in Fig. 1c is supported by vertical and horizontal links in points, D 1 and D 2 , respectively. Besides, a fixed support is used in point, D 6 , as reported in Fig. 1c. The part, D 4 D 5 , of the frame vertical member, D 4 D 6 , is split by a longitudinal crack (Fig. 1). A periodic load, F 2 , acts in point, D 3, on the frame horizontal member. F 2 f is the maximum magnitude of F 2 . The first step in the analysis of the frame in Fig. 1a is to obtain the static vertical displacement, δ , of point, D 3 , due to the weight, F 1 , of the motor that induces the periodic load. By employing the integrals of Maxwell-Mohr (Mladenov et al. (2012)), δ in this case is = ∑ ∫ 1 , (1) where κ i is the curvature, M i is the bending moment due to the virtual loading. Equations (2) and (3) are used for determining i  and the neutral axis coordinate, z mi .

h

2 

(2)

N b dz  

,

1

i

i

2 h



h

2 

1 1 M b z dz i i  

(3)

,

2 h



where b is the width, h is the thickness.