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Publication Number: FHWA-HRT-05-063
Date: May 2007
Evaluation of LS-DYNA Concrete Material Model 159
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Chapter 5. Size Effect
Numerous articles in the literature, particularly those by Bazant, suggest that larger structures are softer than smaller structures.(9) This observation is schematically demonstrated in Figure 43. In general, a larger structure will be more brittle and have lower strength than a smaller, scaled-down structure. The larger structure will experience more damage than the smaller structure. The size effect is examined computationally by analyzing reinforced beams in four-point bending.
Figure 43. Schematic of the size effect, as suggested by Bazant and Planas.(9)
The effect of size on computed response was analyzed for two geometrically similar beams: one at full scale and the other at ⅓-scale. The full-scale beam is 4,572 mm (180 inches) long, with a cross section 457.2 mm (18 inches) deep and 342.9 mm (13.5 inches) wide. Two #3 longitudinal rebars are modeled on the tension side using beam elements with nodes common to the concrete hex elements. The rebars are located 228.6 mm (9 inches) apart and 276.2 mm (10.9 inches) deep (from the compression face of the beam). The concrete and rebar dimensions are reduced correspondingly in the ⅓-scale beam.
The beams are loaded in four-point bending, using nodal velocities to apply the load. The nodal velocity is 3,810 mm/sec (150 inches/sec) in the full-scale beam and 1,270 mm/sec (50 inches/sec) in the ⅓-scale beam. These velocities are intended to be slow enough to produce quasi-static conditions in the beam (no wave propagation effects). Pinned supports are modeled using nodal constraints. No contact surfaces are modeled. The concrete material was modeled with rate effects properties turned off. The steel reinforcement is modeled with elastic material properties.
The mesh of the concrete beam is shown in Figure 44. The full-scale and ⅓-scale beams are geometrically similar and are meshed with the same number of elements. The elements differ in length by a factor of 3. Only one mesh is shown, because the full-scale and ⅓-scale mesh refinements are identical.
Figure 44. Refinement of the concrete beam mesh used in the size effect analyses.
Damage fringes and stress versus relative displacement curves, calculated with regulation of the softening formulation, are shown in Figures 45 and 46. The larger beam experiences substantially more damage, and softens with more brittleness, than the smaller beam. This result is consistent with the size effect. The nominal stress is the vertical reaction force divided by the beam cross-sectional area. The relative deflection is the midpoint beam deflection (u) divided by the beam depth (D).
Figure 45. The damage simulated in the full-scale beam is more severe than that simulated in the ⅓-scale beam, which is consistent with the size effect.
Figure 46. The stress-deflection behavior of the full-scale beam is softer than that of the ⅓-scale beam, which is consistent with the size effect.
It is interesting that the size effect is calculated when the softening response is regulated, but not when it is unregulated. Without regulation, the full and ⅓-scale beams exhibit the same amount of damage, as shown in Figure 47. In addition, the stress versus relative deflection behaviors are nearly the same. This similarity is because the strain distributions in the full and ⅓-scale beams are identical in the elastic regime and quite similar in the damage regime. The unregulated softening formulation keeps the softening stress versus strain consistent, regardless of size. Hence, damage is nearly equivalent because strain is nearly equivalent. But equivalent damage is not consistent with the size effect.
On the other hand, the regulated softening formulation keeps the softening stress versus displacement (crack opening) consistent. Displacements in the full-scale beam are 3 times as large as those in the ⅓-scale beam. Hence, damage is more pronounced in the full-scale beam (as shown in Figure 48).
Figure 47. Without regulation of the softening formulation, the damage simulated in the full-scale and ⅓-scale beams is nearly the same, which is inconsistent with the size effect.
Figure 48. Without regulation of the softening formulation, the stress versus relative deflection curves of the full and ⅓-scale beams are nearly identical, which is inconsistent with the size effect.