PSI - Issue 41

Christos F. Markides et al. / Procedia Structural Integrity 41 (2022) 351–360 Christos F. Markides et al. / Structural Integrity Procedia 00 (2019) 000 – 000

358

8

50

0

P frame =36.5kN

0.00 0.01 0.02 0.03 0.04 0.05

-50

 x   

P j

P frame

-200 Stresses along y -axis [MPa] -150 -100

y

F j

 r y  

Platen Stamp

Analytical Numerical

– L

L

t =1cm

t

-250

R =5cm

y -axis [m]

(b)

FBD

Ο

x

45

 r x  

Stamp Platen

0

0.00 0.01 0.02 0.03 0.04 0.05

-90 Stresses along x -axis [MPa] -45

 y   

P frame

Analytical Numerical

-135

x -axis [m]

(a)

(c)

Fig. 10. Stress distributions by the analytical and numerical analyses: (a) the loci considered, (b) along y -axis, and (c) along x -axis.

4.2 Displacements Assuming the same geometry and material for the FBD, its deformed configuration is plotted in Fig.11 for several load levels, in juxtaposition to the respective configuration of the unstressed state (dotted black line). Especially for the load level of P frame =36 kN the predictions of the numerical analysis are also plotted in the same figure. The agreement between analytical and numerical results is evident, justifying once again the efficiency of the theoretical approach. The other two examples, for exaggeratedly high load levels (for P frame =82 kN and 125 kN) are given just for the deformation tendency to be clearly seen, showing the fulfillment of the demanded conditions regarding the flat edge of the FBD, namely: (a) to remain flat after deformation, and (b) not to shrink inwards (due to the friction imposed) pro hibiting FBD‟s material from penetrating the stamp (theoretically speaking, the stamp must also have the necessary height that is required to prevent any contact or even penetration of the platen from the deformed FBD in the vertical sense). 5. Conclusions and discussion In this study a first step in the direction of curing some drawbacks of previously introduced analytic solutions for the distribution of stresses and displacements in an FBD was described. Attention was paid to properly reconsider the boundary conditions prevailing along the FBD-platen interface. In this context, an analytic solution was presented for an FBD compressed by two rigid stamps (loading platens) assuming that their length is equal to that of the flat edges of the FBD. Closed form expressions for both the stresses and the displacements were obtained. Comparing the analytic results with the respective numerical ones, it was indicated that the mathematical solution proposed here approaches quite efficiently the actual problem.